The composition of compost variants used in mushroom cultures
\\n\\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 179 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 252 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
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He was a visiting Professor in Georgia Institute of Technology in 2002 and 2005. He has published more than 225 papers in scientific journals. He received IEICE Electronics Society Award in 2004, IEICE Achievement Award in 2013, and IEEJ Outstanding Achievement Award in 2014. He served as Vice President of JSAP in 2014-2015. He is JSAP Fellow, IEEJ Fellow, and IEICE Fellow.",institutionString:"Tokyo Institute of Technology",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"0",totalChapterViews:"0",totalEditedBooks:"0",institution:null},coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"158",title:"Metals and Nonmetals",slug:"metals-and-nonmetals"}],chapters:[{id:"63477",title:"Evaluation Methods of Mechanical Properties of Micro-Sized Specimens",slug:"evaluation-methods-of-mechanical-properties-of-micro-sized-specimens",totalDownloads:586,totalCrossrefCites:0,authors:[null]},{id:"64415",title:"Morphology Controlled Synthesis of the Nanostructured Gold by Electrodeposition Techniques",slug:"morphology-controlled-synthesis-of-the-nanostructured-gold-by-electrodeposition-techniques",totalDownloads:559,totalCrossrefCites:1,authors:[null]},{id:"64406",title:"Cu Wiring Fabrication by Supercritical Fluid Deposition for MEMS Devices",slug:"cu-wiring-fabrication-by-supercritical-fluid-deposition-for-mems-devices",totalDownloads:525,totalCrossrefCites:0,authors:[null]},{id:"63479",title:"Pulse-Current Electrodeposition of Gold",slug:"pulse-current-electrodeposition-of-gold",totalDownloads:532,totalCrossrefCites:0,authors:[null]},{id:"64201",title:"Electrodeposition of Gold Alloys and the Mechanical Properties",slug:"electrodeposition-of-gold-alloys-and-the-mechanical-properties",totalDownloads:555,totalCrossrefCites:0,authors:[null]},{id:"64607",title:"Hard Pure-Gold and Gold-CNT Composite Plating Using Electrodeposition Technique with Environmentally Friendly Sulfite Bath",slug:"hard-pure-gold-and-gold-cnt-composite-plating-using-electrodeposition-technique-with-environmentally",totalDownloads:589,totalCrossrefCites:0,authors:[null]},{id:"64769",title:"Electrodeposition of High-Functional Metal Oxide on Noble Metal for MEMS Devices",slug:"electrodeposition-of-high-functional-metal-oxide-on-noble-metal-for-mems-devices",totalDownloads:482,totalCrossrefCites:0,authors:[null]},{id:"64336",title:"Multi-Physics Simulation Platform and Multi-Layer Metal Technology for CMOS-MEMS Accelerometer with Gold Proof Mass",slug:"multi-physics-simulation-platform-and-multi-layer-metal-technology-for-cmos-mems-accelerometer-with-",totalDownloads:680,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"252211",firstName:"Sara",lastName:"Debeuc",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/252211/images/7239_n.png",email:"sara.d@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"6426",title:"Titanium Dioxide",subtitle:"Material for a Sustainable Environment",isOpenForSubmission:!1,hash:"5626c0fe0b53330717e73094946cfd86",slug:"titanium-dioxide-material-for-a-sustainable-environment",bookSignature:"Dongfang Yang",coverURL:"https://cdn.intechopen.com/books/images_new/6426.jpg",editedByType:"Edited by",editors:[{id:"177814",title:"Dr.",name:"Dongfang",surname:"Yang",slug:"dongfang-yang",fullName:"Dongfang Yang"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"6282",title:"Noble and Precious Metals",subtitle:"Properties, Nanoscale Effects and Applications",isOpenForSubmission:!1,hash:"e4c28d6be4fd7b5f5b787d4dabbf721b",slug:"noble-and-precious-metals-properties-nanoscale-effects-and-applications",bookSignature:"Mohindar Singh Seehra and Alan D. Bristow",coverURL:"https://cdn.intechopen.com/books/images_new/6282.jpg",editedByType:"Edited by",editors:[{id:"48086",title:"Prof.",name:"Mohindar",surname:"Seehra",slug:"mohindar-seehra",fullName:"Mohindar Seehra"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7213",title:"Shape-Memory Materials",subtitle:null,isOpenForSubmission:!1,hash:"4e3e756cd4f8a8617dffdc36f8dce7c7",slug:"shape-memory-materials",bookSignature:"Alicia Esther Ares",coverURL:"https://cdn.intechopen.com/books/images_new/7213.jpg",editedByType:"Edited by",editors:[{id:"91095",title:"Dr.",name:"Alicia Esther",surname:"Ares",slug:"alicia-esther-ares",fullName:"Alicia Esther Ares"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"6529",title:"Bismuth",subtitle:"Advanced Applications and Defects Characterization",isOpenForSubmission:!1,hash:"55ed997d678e9c18382af23ab873ba85",slug:"bismuth-advanced-applications-and-defects-characterization",bookSignature:"Ying Zhou, Fan Dong and Shengming Jin",coverURL:"https://cdn.intechopen.com/books/images_new/6529.jpg",editedByType:"Edited by",editors:[{id:"176372",title:"Prof.",name:"Ying",surname:"Zhou",slug:"ying-zhou",fullName:"Ying Zhou"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"5825",title:"Superalloys for Industry Applications",subtitle:null,isOpenForSubmission:!1,hash:"4cbaaafeb4958d641b74988e33229020",slug:"superalloys-for-industry-applications",bookSignature:"Sinem Cevik",coverURL:"https://cdn.intechopen.com/books/images_new/5825.jpg",editedByType:"Edited by",editors:[{id:"117212",title:"MSc.",name:"Sinem",surname:"Cevik",slug:"sinem-cevik",fullName:"Sinem Cevik"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"8787",title:"Bismuth",subtitle:"Fundamentals and Optoelectronic Applications",isOpenForSubmission:!1,hash:"7751170d0b538f61d14a27a56e6567a5",slug:"bismuth-fundamentals-and-optoelectronic-applications",bookSignature:"Yanhua Luo, Jianxiang Wen and Jianzhong Zhang",coverURL:"https://cdn.intechopen.com/books/images_new/8787.jpg",editedByType:"Edited by",editors:[{id:"226148",title:"Dr.",name:"Yanhua",surname:"Luo",slug:"yanhua-luo",fullName:"Yanhua Luo"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9949",title:"Lead Chemistry",subtitle:null,isOpenForSubmission:!1,hash:"b2f999b9583c748f957f612227976570",slug:"lead-chemistry",bookSignature:"Pipat Chooto",coverURL:"https://cdn.intechopen.com/books/images_new/9949.jpg",editedByType:"Edited by",editors:[{id:"197984",title:"Ph.D.",name:"Pipat",surname:"Chooto",slug:"pipat-chooto",fullName:"Pipat Chooto"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7787",title:"Rare Earth Elements and Their Minerals",subtitle:null,isOpenForSubmission:!1,hash:"7ba4060b0830f7a68f00557da8ed8a39",slug:"rare-earth-elements-and-their-minerals",bookSignature:"Michael Aide and Takahito Nakajima",coverURL:"https://cdn.intechopen.com/books/images_new/7787.jpg",editedByType:"Edited by",editors:[{id:"185895",title:"Dr.",name:"Michael",surname:"Aide",slug:"michael-aide",fullName:"Michael Aide"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7775",title:"Metallic Glasses",subtitle:null,isOpenForSubmission:!1,hash:"665fb007e1e410d119fc09d709c41cc3",slug:"metallic-glasses",bookSignature:"Dragica Minić and Milica Vasić",coverURL:"https://cdn.intechopen.com/books/images_new/7775.jpg",editedByType:"Edited by",editors:[{id:"30470",title:"Prof.",name:"Dragica",surname:"Minić",slug:"dragica-minic",fullName:"Dragica Minić"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1591",title:"Infrared Spectroscopy",subtitle:"Materials Science, Engineering and Technology",isOpenForSubmission:!1,hash:"99b4b7b71a8caeb693ed762b40b017f4",slug:"infrared-spectroscopy-materials-science-engineering-and-technology",bookSignature:"Theophile Theophanides",coverURL:"https://cdn.intechopen.com/books/images_new/1591.jpg",editedByType:"Edited by",editors:[{id:"37194",title:"Dr.",name:"Theophanides",surname:"Theophile",slug:"theophanides-theophile",fullName:"Theophanides Theophile"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"42489",title:"Environmental Biotechnology for Bioconversion of Agricultural and Forestry Wastes into Nutritive Biomass",doi:"10.5772/55204",slug:"environmental-biotechnology-for-bioconversion-of-agricultural-and-forestry-wastes-into-nutritive-bio",body:'The cellulose is the most widely distributed skeletal polysaccharide and represents about 50% of the cell wall material of plants. Beside hemicellulose and lignin, cellulose is a major component of agricultural wastes and municipal residues. The cellulose and hemicellulose comprise the major part of all green plants and this is the main reason of using such terms as “cellulosic wastes” or simply “cellulosics” for those materials which are produced especially as agricultural crop residues, fruit and vegetable wastes from industrial processing, and other solid wastes from canned food and drinks industries.
The cellulose biodegradation using fungal cells is essentially based on the complex interaction between biotic factors, such as the morphogenesis and physiology of fungi, as the cellulose composition and its complexness with hemicellulose and lignin (Andrews & Fonta, 1988; Carlile & Watkinson, 1996).
An efficient method to convert cellulose materials, in order to produce unconventional high-calorie foods or feeds, is the direct conversion by cellulolytic microorganisms. Theoretically, any microorganism that can grow as pure culture on cellulose substrata, used as carbon and energy sources, should be considered a potential organism for “single-cell protein” (SCP) or “protein rich feed” (PRF) producing.
The submerged cultivation of mushroom mycelia is a promising method which can be used in novel biotechnological processes for obtaining pharmaceutical substances of anticancer, antiviral, immuno-modulating, and anti-sclerotic action from fungal biomass and cultural liquids and also for the production of liquid spawn (Breene, 1990).
The researches that were carried out to get nutritive supplements from the biomass of Ganoderma lucidum species (Reishi) have shown that the nutritive value of its mycelia is owned to the huge protein content, carbohydrates and mineral salts. Lentinula edodes species (Shiitake) is a good source of proteins, carbohydrates (especially polysaccharides) and mineral elements with beneficial effects on human nutrition (Wasser & Weis, 1994; Mizuno et al., 1995).
It is well known the anti-tumor activity of polysaccharide fractions extracted from mycelia of Pleurotus ostreatus, known on its popular name as Oyster Mushroom (Mizuno et al., 1995; Hobbs, 1996).
The main purpose of this research work consists in the application of biotechnology for continuous cultivation of edible and medicinal mushrooms by submerged fermentation in agro-food industry which has a couple of effects by solving the ecological problems generated by the accumulation of plant wastes in agro-food industry through biological means to valorise them without pollutant effects as well as getting fungal biomass with high nutritive value which can be used to prepare functional food (Carlile & Watkinson, 1996; Moser, 1994).
The continuous cultivation of medicinal mushrooms was applied using the submerged fermentation of natural wastes of agro-food industry, such as different sorts of grain by-products as well as winery wastes that provided a fast growth as well as high biomass productivity of the investigated strains (Petre & Teodorescu, 2012; Petre & Teodorescu, 2011).
Ganoderma lucidum (Curt. Fr.) P. Karst, Lentinula edodes (Berkeley) Pegler and Pleurotus ostreatus (Jacquin ex Fries) Kummer were used as pure strains. The stock cultures were maintained on malt-extract agar (MEA) slants, incubated at 25°C for 5-7 d and then stored at 4°C. The seed cultures were grown in 250-ml flasks containing 100 ml of MEA medium (20% malt extract, 2% yeast extract, 20% agar-agar) at 23°C on rotary shaker incubator at 100 rev.min-1 for 7 d (Petre & Petre, 2008; Petre et al., 2007).
The fungal cultures were grown by inoculating 100 ml of culture medium using 3-5% (v/v) of the seed culture and then cultivated at 23-25°C in rotary shake flasks of 250 ml. The experiments were conducted under the following conditions: temperature, 25°C; agitation speed, 120 rev. min -1; initial pH, 4.5–5.5.
After 10–12 d of incubation the fungal cultures were ready to be inoculated aseptically into the glass vessel of a laboratory-scale bioreactor (Fig. 1).
For fungal growing inside the culture vessel of this bioreactor, certain special culture media were prepared by using liquid nutritive broth, having the following composition: 15% cellulose powder, 5% wheat bran, 3% malt extract, 0.5% yeast extract, 0.5% peptone, 0.3% powder of natural argillaceous materials. After the steam sterilization at 121oC, 1.1 atm., for 15 min. this nutritive broth was transferred aseptically inside the culture vessel of the laboratory scale bioreactor shown in figure 1.
Laboratoy-scale bioreactor for submerged cultivation of edible and medicinal mushrooms
The culture medium was aseptically inoculated with activated spores belonging to G. lucidum, L. edodes and P. ostreatus species. After inoculation into the bioreactor vessel, a slow constant flow of nutritive liquid broth was maintained inside the nutritive culture medium by recycling it and adding from time to time a fresh new one.
The submerged fermentation was set up at the following parameters: constant temperature, 23°C; agitation speed, 80-100 rev. min-1; pH level, 5.7–6.0 units; dissolved oxygen tension within the range of 30-70%. After a period of submerged fermentation lasting up to 120 h, small fungal pellets were developed inside the broth (Petre & Teodorescu, 2010; Petre & Teodorescu, 2009).
The experimental model of biotechnological installation, represented by the laboratory scale bioreactor shown in figure 1, was designed to be used in submerged cultivation of the mentioned mushroom species that were grown on substrata made of wastes resulted from the industrial processing of cereals and grapes (Table 1).
\n\t\t\t\tVariants of culture substrata\n\t\t\t | \n\t\t\t\n\t\t\t\tComposition\n\t\t\t | \n\t\t
S1 | \n\t\t\tMixture of winery wastes and wheat bran 2.5% | \n\t\t
S2 | \n\t\t\tMixture of winery wastes and barley bran 2.5% | \n\t\t
S3 | \n\t\t\tMixture of winery wastes and rye bran 2.5% | \n\t\t
Control | \n\t\t\tPure cellulose | \n\t\t
The composition of compost variants used in mushroom cultures
The whole process of mushroom mycelia growing lasts for a single cycle between 5-7 days in case of L. edodes and between 3 to 5 days for G. lucidum and P. ostreatus. All experiments regarding the fermentation process were carried out by inoculating the growing medium volume (15 L) with secondary mycelium inside the culture vessel of the laboratory-scale bioreactor (see Fig. 1).
The strains of these fungal species were characterized by morphological stability, manifested by its ability to maintain the phenotypic and taxonomic identity. Observations on morphological and physiological characters of these two tested species of fungi were made after each culture cycle, highlighting the following aspects:
sphere-shaped structure of fungal pellets, sometimes elongated, irregular, with various sizes (from 7 to 12 mm in diameter), reddish-brown colour of G. lucidum specific culture (Fig. 2a);
Fungal pellets of G. lucidum, b. Fungal pellets of L. Edodes,. c. Fungal pellets of P. ostreatus
globular structures of fungal pellets, irregular with diameters of 5 up to 10 mm or mycelia congestion, which have developed specific hyphae of L. edodes (Fig. 2b);
round-shaped pellets with diameter measuring between 5 and 15 mm, having a white-cream colour and showing compact structures of P. ostreatus mycelia (Fig. 2c).
The experiments were carried out in three repetitions. Samples for analysis were collected at the end of the fermentation process, when pellets formed specific shapes and characteristic sizes. For this purpose, fungal biomass was washed repeatedly with double distilled water in a sieve with 2 mm diameter eye, to remove the remained bran in each culture medium (Petre at al., 2005a).
Biochemical analyses of fungal biomass samples obtained by submerged cultivation of edible and medicinal mushrooms were carried out separately for the solid fraction and extract fluid remaining after the separation of fungal biomass by pressing and filtering. Also, the most obvious sensory characteristics (color, odor, consistency) were evaluated and presented at this stage of biosynthesis taking into consideration that they are very important in the prospective view of fungal biomass using as raw matarials for nutraceuticals producing. In each experimental variant the amount of fresh biomass mycelia was analyzed.
Percentage amount of dry biomass was determined by dehydration at 70° C, until constant weight. The total protein content was investigated by using the biuret method, whose principle is similar to the Lowry method, being recommended for the protein content ranging from 0.5 to 20 mg/100 mg sample (Bae et al., 2000; Lamar et al., 1992).
The principle method is based on the reaction that takes place between copper salts and compounds with two or more peptides in the composition in alkali, which results in a red-purple complex, whose absorbance is read in a spectrophotometer in the visible domain (λ 550 nm). In addition, this method requires only one sample incubation period (20 min) eliminating the interference with various chemical agents (ammonium salts, for example).
In table 2 are presented the amounts of fresh and dry biomass as well as the protein contents for each fungal species and variants of culture media.
According to registered data, using a mixture of wheat bran 2.5% and winery wastes the growth of G. lucidum biomass was stimulated, while the barley bran led to increased growth of L. edodes mycelium and G. lucidum as well.
In contrast, the dry matter content was significantly higher when using barley bran 2.5% mixed with winery wastes for both species used. Protein accumulation was more intense when using barley bran compared with those of wheat bran and rye bran, at both mushroom species.
The sugar content of dried mushroom pellets collected after the biotechnological experiments was determined by using Dubois method. The mushroom extracts were prepared by immersion of dried pellets inside a solution of NaOH pH 9, in the ratio 1:5. All dispersed solutions containing the dried pellets were maintained 24 h at the precise temperature of 25 oC, in full darkness, with continuous homogenization to avoid the oxidation reactions.
\n\t\t\t\tMushroom species\n\t\t\t | \n\t\t\t\n\t\t\t\tCulture variants\n\t\t\t | \n\t\t\tFresh biomass(g) | \n\t\t\tDry biomass(%) | \n\t\t\tTotal proteins(g % d.w.) | \n\t\t
\n\t\t\t\tG. lucidum\n\t\t\t | \n\t\t\tI | \n\t\t\t25.94 | \n\t\t\t9.03 | \n\t\t\t0.67 | \n\t\t
\n\t\t\t\tG. lucidum\n\t\t\t | \n\t\t\tII | \n\t\t\t22.45 | \n\t\t\t10.70 | \n\t\t\t0.55 | \n\t\t
\n\t\t\t\tG. lucidum\n\t\t\t | \n\t\t\tIII | \n\t\t\t23.47 | \n\t\t\t9.95 | \n\t\t\t0.73 | \n\t\t
\n\t\t\t | Control | \n\t\t\t5.9 | \n\t\t\t0.7 | \n\t\t\t0.3 | \n\t\t
\n\t\t\t\tL. edodes\n\t\t\t | \n\t\t\tI | \n\t\t\t20.30 | \n\t\t\t5.23 | \n\t\t\t0.55 | \n\t\t
\n\t\t\t\tL. edodes\n\t\t\t | \n\t\t\tII | \n\t\t\t23.55 | \n\t\t\t6.10 | \n\t\t\t0.53 | \n\t\t
\n\t\t\t\tL. edodes\n\t\t\t | \n\t\t\tIII | \n\t\t\t22.27 | \n\t\t\t4.53 | \n\t\t\t0.73 | \n\t\t
\n\t\t\t | Control | \n\t\t\t4.5 | \n\t\t\t0.5 | \n\t\t\t0.2 | \n\t\t
\n\t\t\t\tP. ostreatus\n\t\t\t | \n\t\t\tI | \n\t\t\t21.50 | \n\t\t\t5.73 | \n\t\t\t0.63 | \n\t\t
\n\t\t\t\tP. ostreatus\n\t\t\t | \n\t\t\tII | \n\t\t\t23.95 | \n\t\t\t7.45 | \n\t\t\t0.55 | \n\t\t
\n\t\t\t\tP. ostreatus\n\t\t\t | \n\t\t\tIII | \n\t\t\t23.25 | \n\t\t\t4.79 | \n\t\t\t0.75 | \n\t\t
\n\t\t\t | Control | \n\t\t\t4.7 | \n\t\t\t0.5 | \n\t\t\t0.3 | \n\t\t
Fresh and dry biomass and protein content of G. lucidum, L. edodes and P. ostreatus mycelia grown by submerged fermentation
After the removal of solid residues by filtration the samples were analyzed by the previous mention method (Wasser & Weis, 1994).
The nitrogen content of mushroom pellets was analyzed by Kjeldahl method. All the registered results are related to the dry weight of mushroom pellets that were collected at the end of each biotechnological culture cycle (Table 3).
Comparing all the registered data, it could be noticed that the correlation between the dry weight of mushroom pellets and their sugar and nitrogen contents is kept at a balanced ratio for each tested mushroom species.
From these mushroom species that were tested in biotechnological experiments G. lucidum (variant III) showed the best values concerning the sugar and total nitrogen content. On the very next places, L. edodes (variant I) and G. lucidum (variant II) could be mentioned from these points of view.
The registered results concerning the sugar and total nitrogen contents have higher values than those obtained by other researchers (Bae et al., 2000; Jones, 1995; Moo-Young, 1993). The nitrogen content in fungal biomass is a key factor for assessing its nutraceutical potential, but the assessing of differential protein nitrogen compounds requires additional investigations.
\n\t\t\t\tMushroom species\n\t\t\t | \n\t\t\t\n\t\t\t\tCulture variants\n\t\t\t | \n\t\t\tMushroom pelletsd. w. (%) | \n\t\t\t\n\t\t\t\tSugar content of dried pellets (mg/ml)\n\t\t\t | \n\t\t\t\n\t\t\t\tKjeldahl nitrogen of dried pellets (%)\n\t\t\t | \n\t\t
\n\t\t\t\tG. lucidum\n\t\t\t | \n\t\t\tI | \n\t\t\t17.64 | \n\t\t\t4.93 | \n\t\t\t5.15 | \n\t\t
\n\t\t\t\tG. lucidum\n\t\t\t | \n\t\t\tII | \n\t\t\t14.51 | \n\t\t\t3.70 | \n\t\t\t5.35 | \n\t\t
\n\t\t\t\tG. lucidum\n\t\t\t | \n\t\t\tIII | \n\t\t\t20.16 | \n\t\t\t5.23 | \n\t\t\t6.28 | \n\t\t
\n\t\t\t | Control | \n\t\t\t0.7 | \n\t\t\t0.45 | \n\t\t\t0.30 | \n\t\t
\n\t\t\t\tL. edodes\n\t\t\t | \n\t\t\tI | \n\t\t\t19.67 | \n\t\t\t4.35 | \n\t\t\t6.34 | \n\t\t
\n\t\t\t\tL. edodes\n\t\t\t | \n\t\t\tII | \n\t\t\t17,43 | \n\t\t\t3.40 | \n\t\t\t5.03 | \n\t\t
\n\t\t\t\tL. edodes\n\t\t\t | \n\t\t\tIII | \n\t\t\t15.55 | \n\t\t\t4.75 | \n\t\t\t6.05 | \n\t\t
\n\t\t\t | Control | \n\t\t\t0.5 | \n\t\t\t0.45 | \n\t\t\t0.35 | \n\t\t
\n\t\t\t\tP. ostreatus\n\t\t\t | \n\t\t\tI | \n\t\t\t19.70 | \n\t\t\t5.15 | \n\t\t\t6.43 | \n\t\t
\n\t\t\t\tP. ostreatus\n\t\t\t | \n\t\t\tII | \n\t\t\t14.93 | \n\t\t\t4.93 | \n\t\t\t6.25 | \n\t\t
\n\t\t\t\tP. ostreatus\n\t\t\t | \n\t\t\tIII | \n\t\t\t15.63 | \n\t\t\t5.10 | \n\t\t\t5.83 | \n\t\t
\n\t\t\t | Control | \n\t\t\t0.55 | \n\t\t\t0.50 | \n\t\t\t0.35 | \n\t\t
The sugar and total nitrogen contents of dried mushroom pellets
The agricultural works as well as the industrial activities related to apple and grape processing have generally been matched by a huge formation of wide range of cellulosic wastes that cause environmental pollution effects if they are allowed to accumulate in the environment or much worse they are burned on the soil (Petre, 2009; Verstrate & Top, 1992).
The solid substrate fermentation of plant wastes from agro-food industry is one of the challenging and technically demanding biotechnology that is known so far (Petre & Petre, 2008; Carlile & Watkinson, 1996).
The major group of fungi which are able to degrade lignocellulose is represented by the edible mushrooms of Basidiomycetes Class. Taking into consideration that most of the edible mushrooms species requires a specific micro-environment including complex nutrients, the influence of physical and chemical factors upon fungal biomass production and mushroom fruit bodies formation were studied by testing new biotechnological procedures (Petre & Petre, 2008; Moser, 1994; Beguin & Aubert, 1994; Chahal & Hachey, 1990).
The main aim of research was to find out the best biotechnology of recycling the apple and winery wastes by using them as a growing source for edible mushrooms and, last but not least, to protect the environment (Petre et al., 2008; Smith, 1998; Raaska, 1990).
Two fungal species of Basidiomycetes group, namely Lentinula edodes (Berkeley) Pegler (folk name: Shiitake) as well as Pleurotus ostreatus (Jacquin ex Fries) Kummer (folk name: Oyster Mushroom) were used as pure mushroom cultures isolated from the natural environment and now being preserved in the local collection of the University of Pitesti.
The stock cultures were maintained on malt-extract agar (MEA) slants (20% malt extract, 2% yeast extract, 20% agar-agar). Slants were incubated at 25°C for 120-168 h and stored at 4°C. The pure mushroom cultures were expanded by growing in 250-ml flasks containing 100 ml of liquid malt-extract medium at 23°C on rotary shaker incubators at 110 rev. min-1 for 72-120 h. To prepare the inoculum for the spawn cultures of L. edodes and P. ostreatus the pure mushroom cultures were inoculated into 100 ml of liquid malt-yeast extract culture medium with 3-5% (v/v) and then maintained at 23-25°C in 250 ml rotary shake flasks.
After 10–12 d of incubation the fungal cultures were inoculated aseptically into glass vessels containing sterilized liquid culture media in order to produce the spawn necessary for the inoculation of 10 kg plastic bags filled with compost made of winery and apple wastes.
These compost variants were mixed with other needed natural ingredients in order to improve the enzymatic activity of mushroom mycelia and convert the cellulose content of winery and apple wastes into protein biomass. The best compositions of five compost variants are presented in Table 4.
\n\t\t\t\tCompost variants\n\t\t\t | \n\t\t\t\n\t\t\t\tCompost composition\n\t\t\t | \n\t\t
S1 | \n\t\t\tWinery and apple wastes (1:1) | \n\t\t
S2 | \n\t\t\tWinery wastes + wheat bran (9:1) | \n\t\t
S3 | \n\t\t\tWinery wastes and rye bran (9:1) | \n\t\t
S4 | \n\t\t\tApple wastes and wheat bran (9:1) | \n\t\t
S5 | \n\t\t\tApple wastes + rye bran (9:1) | \n\t\t
Control | \n\t\t\tPoplar, beech and birch sawdust (1:1:1) | \n\t\t
The composition of five compost variants used in mushroom culture cycles
In this way, the whole bags filled with compost were steam sterilized at 121oC, 1.1 atm., for 30 min. In the next stage, all the sterilized bags were inoculated with liquid mycelia, and then, all inoculated bags were transferred into the growing chambers for incubation. After 10-15 d, on the surface of sterilized plastic bags filled with compost, the first buttons of mushroom fruit bodies emerged. For a period of 20-30 d there were harvested between 1.5–3.5 kg of mushroom fruit bodies per 10 kg compost of one bag (Petre et al., 2012; Oei, 2003; Stamets, 1993; Wainwright, 1992; Ropars et al., 1992).
To increase the specific processes of winery and apple wastes bioconversion into protein of fungal biomass, there were performed experiments to grow the mushroom species of P. ostreatus and L. edodes on the previous mentioned variants of culture substrata (see Table 1).
During the mushroom growing cycles the specific rates of cellulose biodegradation were determined using the direct method of biomass weighing the results being expressed as percentage of dry weight (d.w.) before and after their cultivation (Stamets, 1993; Wainwright, 1992).
In order to determine the evolution of the total nitrogen content in the fungal biomass there were collected samples at precise time intervals of 50 h and they were analyzed by using Kjeldahl method. The registered results concerning the evolution of total nitrogen content in P. ostreatus biomass are presented in figure 3 and the data regarding L. edodes biomass could be seen in figure 4.
The evolution of total nitrogen content in P. ostreatus biomass
The evolution of total nitrogen content in L. edodes biomass
During the whole period of fruit body formation, the culture parameters were set up and maintained at the following levels, depending on each mushroom species:
air temperature, 15–17oC;
the air flow volume, 5–6m3/h;
air flow speed, 0.2–0.3 m/s;
the relative moisture content, 80–85%;
light intensity, 500–1,000 luces for 8–10 h/d.
According to the registered results of the performed experiments the optimal laboratory-scale biotechnology for edible mushroom cultivation on composts made of marc of grapes and apples was established (Fig. 5).
As it is shown in figure 5, two technological flows were carried out simultaneously until the first common stages of the inoculation of composts with liquid mushroom spawn followed by the mushroom fruit body formation.
The whole period of mushroom growing from the inoculation to the fruit body formation lasted between 30–60 d, depending on each fungal species used in experiments.
Scheme of laboratory-scale biotechnology for edible mushroom producing on winery and apple wastes
The registered data revealed that by applying such biotechnology, the winery and apple wastes can be recycled as useful raw materials for mushroom compost preparation in order to get significant mushroom production.
In this respect, the final fruit body production of these two mushroom species was registered as being between 20–28 kg relative to 100 kg of composts made of apple and winery wastes.
The most part of wastes produced all over the world arise from industrial, agricultural and domestic activities. These wastes represent the final stage of the technical and economical life of products (Verstraete & Top 1992).
As a matter of fact, the forestry works as well as the industrial activities related to forest management and wood processing have generally been matched by a huge formation of wide range of waste products (Beguin & Aubert 1994, Wainwright 1992).
Many of these lignocellulosic wastes cause serious environmental pollution effects, if they are allowed to accumulate in the forests or much worse to be burned for uncontrolled domestic purposes. So far, the basis of most studies on lignocellulose-degrading fungi has been economic rather than ecological, with emphasize on the applied aspects of lignin and cellulose decomposition, including biodegradation and bioconversion (Carlile & Watkinson 1996).
In this respect, the main aim of this work was focused on finding out the best way to convert the wood wastes into useful food supplements, such as mushroom fruit bodies, by using them as growing sources for the edible and medicinal mushrooms (Smith, 1998).
According to the main purpose of this work, three fungal species from Basidiomycetes, namely Ganoderma lucidum (Curt.:Fr.) P. Karst, Lentinus edodes (Berkeley) Pegler and Pleurotus ostreatus (Jacquin ex Fries) Kummer were used as pure mushroom cultures during all experiments. The stock mushroom cultures were maintained by cultivating on malt-extract agar (MEA) slants. After that, they were incubated at 25° C for 5-7 d and then stored at 4° C. These pure mushroom cultures were grown in 250-ml flasks containing 100 ml of MEA medium (20% malt extract, 2% yeast extract) at 23°C on rotary shaker incubators at 110 rev min -1 for 5-7 d.
The pure mushroom cultures for experiments were prepared by inoculating 100 ml of culture medium with 3-5% (v/v) of the seed culture and then cultivated at 23-25°C in rotary shake flasks of 250 ml. The experiments were conducted under the following conditions:
temperature, 25°C;
agitation speed, 90-120 rev min-1;
initial pH, 4.5–5.5.
The seed culture was transferred to the fungal culture medium and cultivated for 7–12 d (Petre et al., 2005a; Glazebrook et al., 1992).
The experiments were performed by growing all the previous mentioned fungal species in special culture rooms, where all the culture parameters were kept at optimal levels in order to get the highest production of fruit bodies. The effects of culture compost composition (carbon, nitrogen and mineral sources) as well as other physical and chemical factors (such as: temperature, inoculum size and volume and incubation time) on mycelial net formation and especially, on fruit body induction were investigated (Petre & Petre, 2008).
All the culture composts for mushroom growing were inoculated using liquid inoculum with the age of 5–7 days and the volume size ranging between 3-7% (v/w). During the period of time of 18–20 d after this inoculation, all the fungal cultures had developed a significant biomass on the culture substrata made of wood wastes, such as: white poplar and beech wood sawdusts. These woody wastes were used as main ingredients to prepare natural composts for mushroom growing. The optimal temperatures for incubation and mycelia growth were maintained between 23–25°C. The whole period of mushroom growing from the inoculation to the fruit body formation lasted between 30–60 days, depending on each fungal species used in experiments (Petre & Teodorescu, 2010).
The lignocellulosic materials were mechanical pre-treated to breakdown the lignin and cellulose structures in order to induce their susceptibility to the enzyme actions during the mushroom growing. All these pre-treated lignocellulosic wastes were disinfected by steam sterilization at 120o C for 60 min (Petre et al., 2005b; Leahy & Colwell 1990).
The final composition of culture composts was improved by adding the following ingredients: 15-20% grain seeds (wheat, rye, rice) in the ratio 2:1:1, 0.7–0.9% CaCO3, 0.3–0.5% NH4H2PO4, each kind of culture medium composition depending on the fungal species used to be grown. As control samples for each variant of culture composts used for the experimental growing of all these fungal species were used wood logs of white poplar and beech that were kept in water three days before the experiments and after that they were steam sterilized to be disinfected.
3000 g of white poplar sawdust and 1500 g of beech sawdust were mixed with cleaned and ground rye grain, 640 g of CaCO3, 50 g of NH4H2PO4 and 3550 ml of water, in order to obtain the growth substratum for mushroom spawn. The ingredients of such smal compost were mixed and then they were sterilized at 121° C, for 20 min. and allowed to cool until the mixture temperature decreased below 35° C. The spawn mixture was inoculated with 100-200 ml of liquid fungal inoculums and mixed for 10 min. to ensure complete homogeneity. Sterile polyethylene bags, containing microporus filtration strips, were filled with the smal composts and incubated at 25° C, until the spawn fully colonized the whole composts. At this point the spawn may be used to inoculate the mushroom growing substrate or alternatively it may be stored for up to 6 months at 4° C before use (Chahal & Hachey, 1990).
All the culture composts were inoculated using inoculum with the age of 5–7 d and the volume size ranging between 3-7% (v/w). The optimal temperatures for incubation and mycelia growth were maintained between 23–25°C. The whole period of mushroom growing from the inoculation to the fruit body formation lasted between 30–50 days.
The experiments were carried out inside such in vitro growing rooms, where the main culture parameters (temperature, humidity, aeration) were kept at optimal levels to get the highest production of mushroom fruit bodies (Moser, 1994).
In order to find a suitable carbon source for the mycelia growth and consequently for fungal biomass synthesis, the pure cultures of P. ostreatus (Oyster Mushroom), as well as L. edodes (Shiitake) and G. lucidum (Reishi) were cultivated in different nutritive culture media containing various carbon sources, and each carbon source was added to the basal medium at a concentration level of 1.5% (w/v) for 7-12 d (Raaska, 1990).
To investigate the effect of nitrogen sources on mycelia growth and fungal biomass production, the pure cultures of these two fungal species were cultivated in media containing various nitrogen sources, where each nitrogen source was added to the basal medium at a concentration level of 10 g/l. At the same time, malt extract was one of the better nitrogen sources for a high mycelia growth. Peptone, tryptone and yeast extract are also known as efficient nitrogen sources for fungal biomass production by using the pure cultures of such fungal species (Chang & Hayes, 1978). In comparison with organic nitrogen sources, inorganic nitrogen sources gave rise to relatively lower mycelia growth and fungal biomass production (Bae et al., 2000).
The influence of mineral sources on fungal biomass production was examined at a standard concentration level of 5 mg. In order to study the effects of initial pH correlated with the incubation temperature upon fruit body formation, G. lucidum, P. ostreatus and L. edodes were cultivated on substrates made of wood wastes of white poplar and beech at different initial pH values (4.5–6.0). The experiments were carried out for 6 days at 25°C with the initial pH 5.5. Similar observations were made by Stamets (1993), during the experiments. K2HPO4 could improve the productivity through its buffering action, being favourable for mycelia growth. The experiments were carried out between 30-60 days at 25°C.
The effects of carbon, nitrogen and mineral sources as well as other physical and chemical factors on mycelial net formation and especially, on fruit body induction were investigated by adding them to the main composts made of white poplar and beech sawdusts in the ratio 2:1. For the experimental growing of all these fungal species white poplar and beech logs were used as control samples.
When the cells were grown in the maltose medium, the fungal biomass production was the highest among the tested variants. Data presented in the following table are the means ± S.D. of triple determinations (Table 5).
Carbon source(g/l) | \n\t\t\tFresh Fungal Biomass Weight(g/l) | \n\t\t\t\n\t\t\t\tFinal pH\n\t\t\t | \n\t\t||||
\n\t\t\t | \n\t\t\t\tG. lucidum\n\t\t\t | \n\t\t\t\n\t\t\t\tL. edodes\n\t\t\t | \n\t\t\t\n\t\t\t\tP. ostreatus\n\t\t\t | \n\t\t\t\n\t\t\t\tG. l\n\t\t\t | \n\t\t\t\n\t\t\t\tL. e\n\t\t\t | \n\t\t\t\n\t\t\t\tP. o\n\t\t\t | \n\t\t
Glucose | \n\t\t\t27±0.10 | \n\t\t\t41±0.05 | \n\t\t\t43±0.03 | \n\t\t\t5.5 | \n\t\t\t5.3 | \n\t\t\t5.1 | \n\t\t
Maltose | \n\t\t\t27±0.14 | \n\t\t\t45±0.12 | \n\t\t\t49±0.05 | \n\t\t\t5.8 | \n\t\t\t5.4 | \n\t\t\t5.3 | \n\t\t
Sucrose | \n\t\t\t25±0.23 | \n\t\t\t35±0.03 | \n\t\t\t37±0.09 | \n\t\t\t5.1 | \n\t\t\t5.1 | \n\t\t\t5.7 | \n\t\t
Xylose | \n\t\t\t26±0.07 | \n\t\t\t38±0.07 | \n\t\t\t35±0.07 | \n\t\t\t5.3 | \n\t\t\t5.5 | \n\t\t\t5.9 | \n\t\t
The effect of carbon sources upon the mycelia growth of pure mushroom cultures on white poplar and beech composts
What is very important to be noticed is that the maltose has a significant effect upon the increasing of mycelia growth and fungal biomass synthesis. The experiments were carried out for 12 days at 25 °C with the initial pH 5.5 (Petre, 2002).
Among five nitrogen sources examined, rice bran was the most efficient for mycelia growth and fungal biomass production. The experiments were carried out for 12 days at 25 °C with the initial pH 5.5 (Table 6).
Nitrogen sources(1%, w/v) | \n\t\t\tFresh Fungal Biomass Weight(g/l) | \n\t\t\t\n\t\t\t\tFinal pH\n\t\t\t | \n\t\t||||
\n\t\t\t | \n\t\t\t\tG. lucidum\n\t\t\t | \n\t\t\t\n\t\t\t\tL. edodes\n\t\t\t | \n\t\t\t\n\t\t\t\tP. ostreatus\n\t\t\t | \n\t\t\t\n\t\t\t\tG. l\n\t\t\t | \n\t\t\t\n\t\t\t\tL. e\n\t\t\t | \n\t\t\t\n\t\t\t\tP. o\n\t\t\t | \n\t\t
Rice bran | \n\t\t\t37±0.21 | \n\t\t\t57±0.05 | \n\t\t\t73±0.23 | \n\t\t\t5.5 | \n\t\t\t5.5 | \n\t\t\t5.1 | \n\t\t
Malt extract | \n\t\t\t36±0.12 | \n\t\t\t55±0.03 | \n\t\t\t69±0.20 | \n\t\t\t5.3 | \n\t\t\t5.2 | \n\t\t\t5.7 | \n\t\t
Peptone | \n\t\t\t35±0.03 | \n\t\t\t41±0.12 | \n\t\t\t57±0.15 | \n\t\t\t4.6 | \n\t\t\t4.9 | \n\t\t\t5.3 | \n\t\t
Tryptone | \n\t\t\t36±0.15 | \n\t\t\t38±0.07 | \n\t\t\t55±0.17 | \n\t\t\t5.1 | \n\t\t\t5.3 | \n\t\t\t5.9 | \n\t\t
Yeast extract | \n\t\t\t37±0.20 | \n\t\t\t30±0.01 | \n\t\t\t61±0.14 | \n\t\t\t4.3. | \n\t\t\t5.1 | \n\t\t\t5.1 | \n\t\t
The effect of nitrogen sources upon the mycelia growth of pure mushroom cultures on white poplar and beech composts
Data presented in table 6 are the means ± S.D. of triple determinations.
Among the various mineral sources examined, K2HPO4 yielded good mycelia growth as well as fungal biomass production and for this reason it was recognized as a favourable mineral source (Table 7). Data presented in table 7 are the means ± S.D. of triple determinations
Mineral Sources(5 mg) | \n\t\t\tFresh Fungal Biomass Weight(g/l) | \n\t\t\t\n\t\t\t\tFinal pH\n\t\t\t | \n\t\t||||
\n\t\t\t | \n\t\t\t\tG. lucidum\n\t\t\t | \n\t\t\t\n\t\t\t\tL. edodes\n\t\t\t | \n\t\t\t\n\t\t\t\tP. ostreatus\n\t\t\t | \n\t\t\t\n\t\t\t\tG. l\n\t\t\t | \n\t\t\t\n\t\t\t\tL. e\n\t\t\t | \n\t\t\t\n\t\t\t\tP. o\n\t\t\t | \n\t\t
KH2PO4\n\t\t\t | \n\t\t\t37±0.15 | \n\t\t\t45±0.07 | \n\t\t\t53±0.12 | \n\t\t\t5.5 | \n\t\t\t5.3 | \n\t\t\t5.9 | \n\t\t
K2HPO4\n\t\t\t | \n\t\t\t45±0.07 | \n\t\t\t57±0.05 | \n\t\t\t59±0.07 | \n\t\t\t5.1 | \n\t\t\t5.1 | \n\t\t\t5.7 | \n\t\t
MgSO4· 5H2O | \n\t\t\t35±0.25 | \n\t\t\t55±0.09 | \n\t\t\t63±0.28 | \n\t\t\t5.6 | \n\t\t\t5.4 | \n\t\t\t6.1 | \n\t\t
The effect of mineral source upon mycelia growth of pure mushroom cultures on white poplar and beech composts
The optimal pH and temperature levels for fungal fruit body production were 5.0–5.5 and 21–23°C (Table 8).
To find the optimal incubation temperature for mycelia growth, these fungal species were cultivated at different temperatures ranging from 20-25°C, and, finally, the optimum level of temperature was found at 23°C, being correlated with the appropriate pH level 5.5, at it is shown in Table 8. All data presented in the previous table are the means ± S.D. of triple determinations
Initial pH(pH units) | \n\t\t\tInitialtemperature (to) | \n\t\t\tFinal Weight of Fresh Mushroom Fruit Bodies(g / kg substratum) | \n\t\t||
\n\t\t\t | \n\t\t\t | \n\t\t\t\tG. lucidum\n\t\t\t | \n\t\t\t\n\t\t\t\tL. edode\n\t\t\t | \n\t\t\t\n\t\t\t\tP. ostreatus\n\t\t\t | \n\t\t
4.5 | \n\t\t\t18 | \n\t\t\t175±0.23 | \n\t\t\t191±0.10 | \n\t\t\t180±0.02 | \n\t\t
5.0 | \n\t\t\t21 | \n\t\t\t193±0.15 | \n\t\t\t203±0.05 | \n\t\t\t297±0.14 | \n\t\t
5.5 | \n\t\t\t23 | \n\t\t\t198±0.10 | \n\t\t\t195±0.15 | \n\t\t\t351±0.23 | \n\t\t
6.0 | \n\t\t\t26 | \n\t\t\t181±0.12 | \n\t\t\t179±0.12 | \n\t\t\t280±0.03 | \n\t\t
6.5 | \n\t\t\t29 | \n\t\t\t173±0.09 | \n\t\t\t105±0.23 | \n\t\t\t257±0.15 | \n\t\t
The effects of initial pH and temperature upon mushroom fruit body formation on white poplar and beech composts
Amongst several fungal physiological properties, the age and volume of mycelia inoculum may play an important role in fungal hyphae development as well as in fruit body formation (Petre & Teodorescu, 2012).
To examine the effect of inoculum age and inoculum volume, mushroom species G. lucidum, P. ostreatus and L. edodes were grown on substrates made of vineyard wastes during different time periods between 30 and 60 days, varying the inoculum volume (5 - 7 v/w).
All the experiments were carried out at 25°C and initial pH 5.5. As it is shown in Tables 9 and 10, the inoculum age of 120 h as well as an inoculum volume of 6.0 (v/w) have beneficial effects on the fungal biomass production.
Inoculum age(h) | \n\t\t\tFinal Weight of Fresh Mushroom Fruit Bodies(g /kg substratum) | \n\t\t||
\n\t\t\t | \n\t\t\t\tG. lucidum\n\t\t\t | \n\t\t\t\n\t\t\t\tL. edodes\n\t\t\t | \n\t\t\t\n\t\t\t\tP. ostreatus\n\t\t\t | \n\t\t
264 | \n\t\t\t123±0.14 | \n\t\t\t128±0.05 | \n\t\t\t135±0.23 | \n\t\t
240 | \n\t\t\t141±0.10 | \n\t\t\t150±0.28 | \n\t\t\t157±0.17 | \n\t\t
216 | \n\t\t\t154±0.12 | \n\t\t\t195±0.90 | \n\t\t\t193±0.15 | \n\t\t
192 | \n\t\t\t155±0.23 | \n\t\t\t221±0.25 | \n\t\t\t215±0.05 | \n\t\t
168 | \n\t\t\t169±0.37 | \n\t\t\t235±0.78 | \n\t\t\t241±0.07 | \n\t\t
144 | \n\t\t\t210±0.20 | \n\t\t\t248±0.03 | \n\t\t\t259±0.12 | \n\t\t
120 | \n\t\t\t230±0.15 | \n\t\t\t253±0.05 | \n\t\t\t264±0.21 | \n\t\t
96 | \n\t\t\t215±0.09 | \n\t\t\t230±0.15 | \n\t\t\t253±0.10 | \n\t\t
72 | \n\t\t\t183±0.05 | \n\t\t\t205±0.23 | \n\t\t\t210±0.05 | \n\t\t
The effect of inoculum age upon mushroom fruit body formation on white poplar and beech composts
Inoculum Volume(v/w) | \n\t\t\tFinal Weight of Fresh Mushroom Fruit Bodies(g /kg substratum) | \n\t\t||
\n\t\t\t | \n\t\t\t\tG. lucidum\n\t\t\t | \n\t\t\t\n\t\t\t\tL. edodes\n\t\t\t | \n\t\t\t\n\t\t\t\tP. ostreatus\n\t\t\t | \n\t\t
7.0 | \n\t\t\t234±0.12 | \n\t\t\t215±0.20 | \n\t\t\t220±0.05 | \n\t\t
6.5 | \n\t\t\t245±0.15 | \n\t\t\t248±0.23 | \n\t\t\t251±0.20 | \n\t\t
6.0 | \n\t\t\t253±0.1 | \n\t\t\t257±0.07 | \n\t\t\t280±0.15 | \n\t\t
5.5 | \n\t\t\t243±0.12 | \n\t\t\t235±0.03 | \n\t\t\t247±0.07 | \n\t\t
5.0 | \n\t\t\t255±0.23 | \n\t\t\t215±0.15 | \n\t\t\t235±0.03 | \n\t\t
The effect of inoculum volume upon mushroom fruit body formation on white poplar and beech composts
From all these fungal species tested, P. ostreatus was registered as the fastest mushroom (25–30 days), then L. edodes (35–45 days) and eventually, G. lucidum as the longest mushroom culture (40–50 days).
The registered data revealed that the white poplar and beech wood wastes have to be used as substrates for mushroom growing only after some mechanical pre-treatments (such as grinding) that could breakdown the whole lignocellulose structure in order to be more susceptible to the fungal enzyme action (Chahal, 1994).
Due to their high content of carbohydrates and nitrogen, the variants of culture composts supplemented with wheat grains at the ratio 1:10 and rice grains at the ratio 1:5 as well as a water content of 60% were optimal for the fruit body production of P. ostreatus and, respectively, L. edodes. The mushroom culture of G. lucidum does not need such supplements (Ropars et al., 1992; Lamar et al., 1992).
So far, lignocellulose biodegradation made by mushroom species of Ganoderma genus had been little studied, mostly because of their slow growth, difficulty in culturing as well as little apparent biotechnological potential. Only, Stamets (1993) reported a few experimental data concerning the cultivation of such fungal species in natural sites and he noticed its slowly growing.
In spite of these facts, some strains of G. lucidum were grown in our experiments on culture substrates made of wood wastes of white poplar and beech mixed with rye grains at the ratio 1:7 and a water content of 50%.
Higher ratio of rye grains might lead to an increase of total dry weight of fruit body, but also could induce the formation of antler branches and smaller fruit bodies than those of the control samples.
The final fruit body mushroom production ranged between 15 and 20 kg relative to 100 kg of compost made of wood, depending on the specific strains of those tested mushroom species.
The cereal by-products and winery wastes used as substrata for growing the fungal species G. lucidum,\n\t\t\t\t\t\tL. edodes and P. ostreatus by controlled submerged fermentation showed optimal effects on the mycelia development in order to get high nutritive biomass.
The dry matter content of fungal biomass produced by submerged fermentation of barley bran was higher for both tested species.
The protein accumulation is more intense when using barley bran compared with those of wheat and rye, at both fungal species.
G. lucidum (variant III) registered the best values of sugar and total nitrogen contents, being followed by L. edodes (variant I)
The winery and apple wastes can be recycled as useful raw materials for mushroom compost preparation in order to get significant mushroom fruit body production and protect the natural environment surrounding apple juice factories as well as wine making industrial plants.
By applying the biotechnology of recycling the grape and apple wastes can be produced between 20–28 kg of mushroom fruit bodies relative to 100 kg of composts made of winery and apple wastes.
From all these fungal species tested in experiments, P. ostreatus was registered as the fastest mushroom culture (25–30 days), then L. edodes (35–45 days) and finally, G. lucidum as the longest mushroom culture (40–50 days).
The registered data revealed that when the cells were grown in the maltose medium, the fungal biomass production was the highest among the tested variants.
From five nitrogen sources examined, rice bran was the most efficient for mycelia growth and fungal biomass production
Among the various mineral sources examined, K2HPO4 yielded good mycelia growth as well as fungal biomass production and for this reason it was as a favourable mineral source.
The inoculum age of 120 h as well as an inoculum volume of 6.0 (v/w) have beneficial effects on the fungal biomass production and the optimal pH and temperature levels for fungal fruit body production were 5.0–5.5 and 21–23° C.
The final fruit body mushroom production ranged between 15 and 20 kg relative to 100 kg compost made of wood, depending on the specific strains of those tested mushroom species.
The authors express their highest respect and deepest gratefulness for the professional competence and outstanding scientific contribution which were proven by Dr. Paul Adrian during so many research works.
Well testing is a valuable and economical formation evaluation tool used in the hydrocarbon industry. It has been supported by mathematical modeling, computing, and the precision of measurement devices. The data acquired during a well test are used for reservoir characterization and description. However, the biggest drawback is that the system dealt with is neither designed nor seen by well test interpreters, and the only way to make contact with the reservoir is through the well by making indirect measurements.
Four methods are used for well test interpretation: (1) The oldest one is the conventional straight‐line method which consists of plotting pressure or the reciprocal rate—if dealing with transient rate analysis—in the y‐axis against a function of time in the x‐axis. This time function depends upon the governing equation for a given flow. For instance, radial flow uses the logarithm of time and linear flow uses the square root of time. The slope and intercept of such plot are used to find reservoir parameters. The main disadvantage of this method is the lack of confirmation and the difficulty to define a given flow regime. The method is widely used nowadays. (2) Type‐curve matching uses predefined dimensionless pressure and dimensionless time curves (some also use dimensionless pressure derivative), which are used as master guides to be matched with well pressure data to obtain a reference point for reservoir parameter determination. This method is basically a trial‐and‐error procedure which becomes into its biggest disadvantage. The method is practically unused. (3) Simulation of reservoir conditions and automatic adjustment to well test data by non‐linear regression analysis is the method widely used by petroleum engineers. This method is also being widely disused since engineers trust the whole task to the computer. They even perform inverse modeling trying to fit the data to any reservoir model without taking care of the actual conditions. However, the biggest weakness of this method lies on the none uniqueness of the solution. Depending on the input starting values, the results may be different. (4) The newest method known as Tiab’s direct synthesis (TDS) [1, 2] is the most powerful and practical one as will be demonstrated throughout the book. It employs characteristic points and features found on the pressure and pressure derivative versus time log‐log plot to be used into direct analytic equations for reservoir parameters’ calculation. It is even used, without using the original name, by all the commercial software. One of them calls it “Specialized lines.” Because of its practicality, accuracy and application is the main object of this book. Conventional analysis method will be also included for comparison purposes.
The TDS technique can be easily implemented for all kinds of conventional or unconventional systems. It can be easily applied on cases for which the other methods fail or are difficult to be applied. It is strongly based on the pressure derivative curve. The method works by sector or regions found on the test. This means once a given flow regime is identified, a straight line is drawn throughout it, and then, any arbitrary point on this line and the intersection with other lines as well are used into the appropriate equations for the calculation of reservoir parameters.
The book contains the application and detailed examples of the TDS technique to the most common or fundamental reservoir/fluid scenarios. It is divided into seven chapters that are recommended to be read in the other they appear, especially for academic purposes in senior undergraduate level or master degree level. Chapter 1 contains the governing equation and the superposition principle. Chapter 2 is the longest one since it includes drawdown for infinite and finite cases, elongated system, multi‐rate testing, and spherical/hemispherical flow. All the interpretation methods are studied in this chapter which covers about 45% of the book. Chapter 3 deals with pressure buildup testing and average reservoir pressure determination. Distance to barriers and interference testing are, respectively, treated in Chapters 4 and 5. Since the author is convinced that all reservoirs are naturally fractured, Chapter 6 covers this part which is also extended in hydraulically fractured wells in Chapter 7. In this last chapter, the most common flow regime shown in fractured wells: bilinear, linear, and elliptical are discussed with detailed for parameter characterization. The idea is to present a book on TDS technique as practical and short as possible; then, horizontal well testing is excluded here because of its complexity and extension, but the most outstanding and practical publications are named here.
My book entitled “Recent Advances in Practical Applied Well Test Analysis,” published in 2015, was written for people having some familiarity with the TDS technique, so that, it can be read in any order. This is not the case of the present textbook. It is recommended to be read in order from Chapter 1 and take especial care in Chapter 2 since many equations and concepts will be applied in the remaining chapters. TDS technique applies indifferently to both pressure drawdown and pressure buildup tests.
Finally, this book is an upgraded and updated version of a former one published in Spanish. Most of the type curves have been removed since they have never been used by the author on actual well test interpretations. However, the first motivation to publish this book is the author’s belief that TDS technique is the panacea for well test interpretation. TDS technique is such an easy and practical methodology that his creator, Dr. Djebbar Tiab, when day said to me “I still don’t believe TDS works!” But, it really does. Well, once things have been created, they look easy.
Pressure test fundamentals come from the application of Newton’s law, especially the third one: Principle of action‐reaction, since it comes from a perturbation on a well, as illustrated in Figure 1.1.
Diagram of the mathematical representation of a pressure test.
A well can be produced under any of two given scenarios: (a) by keeping a constant flow rate and recording the well‐flowing pressure or (b) by keeping a constant well‐flowing pressure and measuring the flow rate. The first case is known as pressure transient analysis, PTA, and the second one is better known as rate transient analysis, RTA, which both are commonly run in very low permeable formations such as shales.
Basically, the objectives of the analysis of the pressure tests are:
Reservoir evaluation and description: well delivery, properties, reservoir size, permeability by thickness (useful for spacing and stimulation), initial pressure (energy and forecast), and determination of aquifer existence.
Reservoir management.
There are several types of tests with their particular applications. DST and pressure buildup tests are mainly used in primary production and exploration. Multiple tests are most often used during secondary recovery projects, and multilayer and vertical permeability tests are used in producing/injectors wells. Drawdown, interference, and pulse tests are used at all stages of production. Multi‐rate, injection, interference, and pulse tests are used in primary and secondary stages [3, 4, 5, 6, 7].
Pressure test analysis has a variety of applications over the life of a reservoir. DST and pressure buildup tests run in single wells are mainly used during primary production and exploration, while multiple tests are used more often during secondary recovery projects. Multilayer and vertical permeability tests are also run in producing/injectors wells. Drawdown, buildup, interference, and pulse tests are used at all stages of production. Multi‐rate, injection, interference, and pulse testing are used in the primary and secondary stages. Petroleum engineers should take into account the state of the art of interpreting pressure tests, data acquisition tools, interpretation methods, and other factors that affect the quality of the results obtained from pressure test analysis.
Once the data have been obtained from the well and reviewed, the pressure test analysis comprises two steps: (1) To establish the reservoir model and the identification of the different flow regimes encountered during the test and (2) the parameter estimation. To achieve this goal, several plots are employed; among them, we have log‐log plot of pressure and pressure derivative versus testing time (diagnostic tool), semilog graph of pressure versus time, Cartesian graph of the same parameters, etc. Pressure derivative will be dealt later in this chapter.
The interpretation of pressure tests is the primary method for determining average permeability, skin factor, average reservoir pressure, fracture length and fracture conductivity, and reservoir heterogeneity. In addition, it is the only fastest and cheapest method to estimate time‐dependent variables such as skin factor and permeability in stress‐sensitive reservoirs.
In general, pressure test analysis is an excellent tool to describe and define the model of a reservoir. Flow regimes are a direct function of the characteristics of the well/reservoir system, that is, a simple fracture that intercepts the well can be identified by detection of a linear flow. However, whenever there is linear flow, it does not necessarily imply the presence of a fracture. The infinite‐acing behavior occurs after the end of wellbore storage and before the influence of the limits of the deposit. Since the boundaries do not affect the data during this period, the pressure behavior is identical to the behavior of an infinite reservoir. The radial flow can be recognized by an apparent stabilization of the value of the derivative.
Well tests can be classified in several ways depending upon the view point. Some classifications consider whether or not the well produces or is shut‐in. Other engineers focus on the number of flow rates. The two main pressure tests are (a) pressure drawdown and (b) buildup. While the first one involves only one flow rate, the second one involves two flow rates, one of which is zero. Then, a pressure buildup test can be considered as a multi‐rate test.
Drawdown pressure test (see Figure 1.2): It is also referred as a flow test. After the well has been shut‐in for a long enough time to achieve stabilization, the well is placed in production, at a constant rate, while recording the bottom pressure against time. Its main disadvantage is that it is difficult to maintain the constant flow rate.
Schematic representation of pressure drawdown and pressure buildup tests.
Pressure buildup test (see Figure 1.2): In this test, the well is shut‐in while recording the static bottom‐hole pressure as a function of time. This test allows obtaining the average pressure of the reservoir. Although since 2010, average reservoir pressures can be determined from drawdown tests. Its main disadvantage is economic since the shut‐in entails the loss of production.
Injection test (see Figure 1.3): Since it considers fluid flow, it is a test similar to the pressure drawdown test, but instead of producing fluids, fluids, usually water, are injected.
Injection pressure test (left) and falloff test (right).
Falloff test (see Figure 1.3): This test considers a pressure drawdown immediately after the injection period finishes. Since the well is shut‐in, falloff tests are identical to pressure buildup tests.
Interference and/or multiple tests: They involve more than one well and its purpose is to define connectivity and find directional permeabilities. A well perturbation is observed in another well.
Drill stem test (DST): This test is used during or immediately after well drilling and consists of short and continuous shut‐off or flow tests. Its purpose is to establish the potential of the well, although the estimated skin factor is not very representative because well cleaning can occur during the first productive stage of the well (Figure 1.4).
Well test classification based on the number of flow rates.
Short tests: There are some very short tests mainly run in offshore wells. They are not treated in this book. Some of them are slug tests, general close chamber tests (CCTs), surge tests, shoot and pool tests, FasTest, and impulse tests.
As stated before, in a pressure drawdown test, the well is set to a constant flow rate. This condition is, sometimes, difficult to be fulfilled; then, multi‐rate tests have to be employed. According to [8], multi‐rate tests fit into four categories: (a) uncontrolled variable rate [9, 10], series of constant rates [11, 12], pressure buildup testing, and constant bottom‐hole pressure with a continuous changing flow rate [13]. This last technique has been recently named as rate transient analysis (RTA) which is included in PTA, but its study is not treated in this book.
At the beginning of production, the pressure in the vicinity of the well falls abruptly and the fluids near the well expand and move toward the area of lower pressure. Such movement is retarded by friction against the walls of the well and the inertia and viscosity of the fluid itself. As the fluid moves, an imbalance of pressure is created, which induces the surrounding fluids to move toward the well. The process continues until the pressure drop created by the production dissipates throughout the reservoir. The physical process that takes place in the reservoir can be described by the diffusivity equation whose deduction is shown below [5]:
According to the volume element given in Figure 1.5,
Radial volume element.
The right‐hand side part of Eq. (1.1) corresponds to the mass accumulated in the volume element. Darcy’s law for radial flow:
The cross‐sectional area available for flow is provided by cylindrical geometry, 2πrh. Additionally, flow rate must be multiplied by density, ρ, to obtain mass flow. With these premises, Eq. (1.2) becomes:
Replacing Eq. (1.3) into (1.1) yields:
If the control volume remains constant with time, then, Eq. (1.4) can be rearranged as:
Rearranging further the above expression:
The left‐hand side of Eq (1.6) corresponds to the definition of the derivative; then, it can be rewritten as:
The definition of compressibility has been widely used;
By the same token, the pore volume compressibility is given by:
The integration of Eq. (1.8) will lead to obtain:
The right‐hand side part of Eq. (1.7) can be expanded as:
Using the definitions given by Eqs. (1.9) and (1.10) into Eq. (1.11) leads to:
Considering that the total compressibility, ct, is the result of the fluid compressibility, c, plus the pore volume compressibility, cf, it yields:
The gradient term can be expanded as:
Combination of Eqs. (1.14) and (1.13) results in:
Taking derivative to Eq. (1.10) with respect to both time and radial distance and replacing these results into Eq. (1.15) yield:
After simplification and considering permeability and viscosity to be constant, we obtain:
The hydraulic diffusivity constant is well known as
Then, the final form of the diffusivity equation in oilfield units is obtained by combination of Eqs. (1.17) and (1.18):
In expanded form:
The final form of the diffusivity equation strongly depends upon the flow geometry. For cylindrical, [11, 14], spherical [14], and elliptical coordinates [15], the diffusivity equation is given, respectively,
Here, ξ is a space coordinate and represents a family of confocal ellipses. The focal length of these ellipses is 2a. The space coordinate, η, represents a family of confocal hyperbolas that represent the streamlines for elliptical flow. These two coordinates are normal to each other.
Isotropic, horizontal, homogeneous porous medium, permeability, and constant porosity
A single fluid saturates the porous medium
Constant viscosity, incompressible, or slightly compressible fluid
The well completely penetrates the formation. Negligible gravitational forces
The density of the fluid is governed by an equation of state (EOS). For the case of slightly compressible fluid, Eq. (1.8) is used as the EOS.
Similar to the analysis of gas well tests as will be seen later, multiphase tests can be interpreted using the method of pressure approximation (Perrine method), [6, 7, 16], which is based on phase mobility:
The total compressibility is defined by [17, 18]:
For practical purposes, Eq. (1.25) can be expressed as:
As commented before Eq. (1.19) is limited to a single fluid. However, it can be extended to multiphase flow using the concept expressed by Eq. (1.24):
Perrine method assumes negligible pressure and saturation gradients. Martin [19] showed that (a) the method loses accuracy as the gas saturation increases, (b) the estimation of the mobility is good, and (c) the mobility calculations are sensitive to the saturation gradients. Better estimates are obtained when the saturation distribution is uniform and (d) underestimates the effective permeability of the phase and overestimates the damage factor.
It is well known that gas compressibility, gas viscosity, and gas density are highly dependent pressure parameters; then, the liquid diffusivity equation may fail to observe pressure gas behavior. Therefore, there exist three forms for a better linearization of the diffusivity equation to better represent gas flow: (a) the pseudopressure approximation [20], (b) the P2 approximation, and (c) linear approximation. The first one is valid for any pressure range; the second one is valid for reservoir pressures between 2000 and 4000 psia, and the third one is for pressures above 4000 psia [20].
Starting from the equation of continuity and the equation of Darcy:
The state equation for slightly compressible liquids does not model gas flow; therefore, the law of real gases is used [21, 22]:
Combining the above three equations:
Since M, R, and T are constants and assuming that the permeability is constant, the above equation reduces to:
Applying the differentiation chain rule to the right‐hand side part of Eq. (1.32) leads to:
Expanding and rearranging,
Using the definition of compressibility for gas flow:
Using Eqs. (1.9) and (1.35) into Eq. (1.34),
If
The above is a nonlinear partial differential equation and cannot be solved directly. In general, three limiting assumptions are considered for its solution, namely: (a) P/μz is constant; (b) μct is constant; and (c) the pseudopressure transformation, [20], for an actual gas.
Assuming the term P/μz remains constant with respect to the pressure, Eq. (1.17) is obtained.
Eq. (1.37) can be written in terms of squared pressure, P2, starting from the fact that, [3, 4, 5, 6, 7, 9, 17, 21, 22]:
Assuming the term μz remains constant with respect to the pressure, and of course, the radius, then the above equation can be written as:
This expression is similar to Eq. (1.37), but the dependent variable is P2. Therefore, its solution is similar to Eq. (1.17), except that it is given in terms of P2. This equation also requires that μct remain constant.
The diffusivity equation in terms of P2 can be applied at low pressures, and Eq. (1.17) can be applied at high pressures without incurring errors. Therefore, a solution is required that applies to all ranges. Ref. [20] introduced a more rigorous linearization method called pseudopressure that allows the general diffusivity equation to be solved without limiting assumptions that restrict certain properties of gases to remain constant with pressure [3, 4, 5, 6, 7, 9, 17, 20, 21, 22]:
Taking the derivative with respect to both time and radius and replacing the respective results in Eq. (1.37), we obtain:
After simplification,
Expanding the above equation and expressing it in oilfield units:
The solution to the above expression is similar to the solution of Eq. (1.17), except that it is now given in terms of m(P) which can be determined by numerical integration if the PVT properties are known at each pressure level.
For a more effective linearization of Eq. (1.45), [23] introduced pseudotime, ta, since the product μgct in Eq. (1.45) is not constant:
With this criterion, the diffusivity equation for gases is:
The incomplete linearization of the above expression leads to somewhat longer semilog slopes compared to those obtained for liquids. Sometimes it is recommended to use normalized variables in order to retain the units of time and pressure, [6]. The normalized pseudovariables are:
The line‐source solution: The line‐source solution assumes that the wellbore radius approaches zero. Furthermore, the solution considers a reservoir of infinite extent and the well produces as a constant flow rate. Ref. [4] presents the solution of the source line using the Boltzmann transform, the Laplace transform, and Bessel functions. The following is the combinations of independent variables method, which is based on the dimensional analysis of Buckingham’s theorem [24]. This takes a function f = f(x, y, z, t), it must be transformed into a group or function containing fewer variables, f = f(s1,s2…). A group of variables whose general form is proposed as [24]:
The diffusivity equation is:
where f is a dimensionless term given by:
Eq. (1.51) is subjected to the following initial and boundary conditions:
Multiplying the Eq. (1.51) by ∂s/∂s:
Exchanging terms:
The new derivatives are obtained from Eq. (1.50):
Replacing the above derivatives into Eq. (1.56) and rearranging:
Solving from rb from Eq. (1.50) and replacing this result into Eq. (1.6). After rearranging, it yields:
Comparing the term enclosed in square brackets with Eq. (1.50) shows that b = 2, c = −1, then
From Eq. (1.61) follows r2t‒1 = s/a, then
The term enclosed in square brackets is a constant that is assumed equal to 1 for convenience. Since c/(b2a) = 1, then a = −1/4. Therefore, the above expression leads to:
Writing as an ordinary differential equation:
The differential equation is now ordinary, and only two conditions are required to solve it. Applying a similar mathematical treatment to both the initial and boundary conditions to convert them into function of s. Regarding Eq. (1.62) and referring to the initial condition, Eq. (1.53), when the time is set to zero; then, then s function tends to infinite:
Darcy’s law is used to convert the internal boundary condition. Eq. (1.54) multiplied by ∂s/∂s gives:
Replacing Eqs. (1.57) in the above equation; then, replacing Eq. (1.62) into the result, and after simplification, we obtain
Since b = 2, then,
For the external boundary condition, Eq. (1.55), consider the case of Eq. (1.62) when r → ∞ then:
Then, the new differential equation, Eq. (1.65) is subject to new conditions given by Eqs. (1.66), (1.69), and (1.70). Define now,
Applying this definition into the ordinary differential expression given by Eq. (1.65), it results:
Integration of the above expression leads to:
Rearranging the result and comparing to Eq. (1.71) and applying the boundary condition given by Eq. (1.69):
Solving for df and integrating,
Eq. (1.75) cannot be analytically integrated (solved by power series). Simplifying the solution:
When s = 0, es = 0, then c1 = ½ and Eq. (1.76) becomes:
Applying the external boundary condition, Eq. (1.69), when s → ∞, f = 0, therefore, Eq. (1.77) leads,
Replacing c1 and c2 into Eq. (1.76) yields:
This can be further simplified to:
The integral given in Eq. (1.80) is well known as the exponential integral, Ei(−s). If the f variable is changed by pressure terms:
In dimensionless form,
The above equation is a very good approximation of the analytical solution when it is satisfied (Mueller and Witherspoon [2, 9, 18, 19, 25, 26]) that rD ≥ 20 or tD/rD2 ≥ 0.5, see Figure 1.6. If tD/rD2≥ 5, an error is less than 2%, and if tD/rD2 ≥ 25, the error is less than 5%. Figure 1.7 is represented by the following adjustment which has a correlation coefficient, R2 of 0.999998. This plot can be easily rebuilt using the algorithm provided in Figure 1.8. The fitted equation was achieved with the data generated from simulation.
Dimensionless pressure for different values of the dimensionless radius, taken from [9, 25].
Dimensionless well pressure behavior for a well without skin and storage effects in an infinite reservoir, taken from [9, 25].
BASIC code function to calculate Ei function, taken from [29].
being x = log(tD/rD2) > −1.13.
The exponential function can be evaluated by the following formula, [27], for x ≤ 25:
Figure 1.8 shows a listing of a program code in Basic, which can be easily added as a function in Microsoft Excel to calculate the exponential function. Figure 1.9 and Table 1.1, 1.2, 1.3, and 1.4 present solutions of the exponential function.
Values of the exponential integral for 1 ≤ x ≤ 10 (left) and 0.0001 ≤ x ≤ 1 (right).
a | b | c | d | e | f |
---|---|---|---|---|---|
—0.0906765673563653 | 0.5133959845491270 | —0.0243644307428167 | —0.0000014346860800 | —0.4865489789766050 | — |
0.7480202919199570 | 1.3629598993866700 | —0.5960091961168400 | 0.0275653486990893 | —0.7768782064908800 | −0.0010740336145794 |
Constants for Eqs. (1.85) and (1.86).
x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
0.000 | 8.63322 | 7.94018 | 7.53481 | 7.24723 | 7.02419 | 6.84197 | 6.68791 | 6.55448 | 6.43680 | |
0.001 | 6.33154 | 6.23633 | 6.14942 | 6.06948 | 5.99547 | 5.92657 | 5.86214 | 5.80161 | 5.74455 | 5.69058 |
0.002 | 5.63939 | 5.59070 | 5.54428 | 5.49993 | 5.45747 | 5.41675 | 5.37763 | 5.33999 | 5.30372 | 5.26873 |
0.003 | 5.23493 | 5.20224 | 5.17059 | 5.13991 | 5.11016 | 5.08127 | 5.05320 | 5.02590 | 4.99934 | 4.97346 |
0.004 | 4.94824 | 4.92365 | 4.89965 | 4.87622 | 4.85333 | 4.83096 | 4.80908 | 4.78767 | 4.76672 | 4.74620 |
0.005 | 4.72610 | 4.70639 | 4.68707 | 4.66813 | 4.64953 | 4.63128 | 4.61337 | 4.59577 | 4.57847 | 4.56148 |
0.006 | 4.54477 | 4.52834 | 4.51218 | 4.49628 | 4.48063 | 4.46523 | 4.45006 | 4.43512 | 4.42041 | 4.40591 |
0.007 | 4.39162 | 4.37753 | 4.36365 | 4.34995 | 4.33645 | 4.32312 | 4.30998 | 4.29700 | 4.28420 | 4.27156 |
0.008 | 4.25908 | 4.24676 | 4.23459 | 4.22257 | 4.21069 | 4.19896 | 4.18736 | 4.17590 | 4.16457 | 4.15337 |
0.009 | 4.14229 | 4.13134 | 4.12052 | 4.10980 | 4.09921 | 4.08873 | 4.07835 | 4.06809 | 4.05793 | 4.04788 |
0.01 | 4.03793 | 3.94361 | 3.85760 | 3.77855 | 3.70543 | 3.63743 | 3.57389 | 3.51425 | 3.45809 | 3.40501 |
0.02 | 3.35471 | 3.30691 | 3.26138 | 3.21791 | 3.17634 | 3.13651 | 3.09828 | 3.06152 | 3.02614 | 2.99203 |
0.03 | 2.95912 | 2.92731 | 2.89655 | 2.86676 | 2.83789 | 2.80989 | 2.78270 | 2.75628 | 2.73060 | 2.70560 |
0.04 | 2.68126 | 2.65755 | 2.63443 | 2.61188 | 2.58987 | 2.56838 | 2.54737 | 2.52685 | 2.50677 | 2.48713 |
0.05 | 2.46790 | 2.44907 | 2.43063 | 2.41255 | 2.39484 | 2.37746 | 2.36041 | 2.34369 | 2.32727 | 2.31114 |
0.06 | 2.29531 | 2.27975 | 2.26446 | 2.24943 | 2.23465 | 2.22011 | 2.20581 | 2.19174 | 2.17789 | 2.16426 |
0.07 | 2.15084 | 2.13762 | 2.12460 | 2.11177 | 2.09913 | 2.08667 | 2.07439 | 2.06228 | 2.05034 | 2.03856 |
0.08 | 2.02694 | 2.01548 | 2.00417 | 1.99301 | 1.98199 | 1.97112 | 1.96038 | 1.94978 | 1.93930 | 1.92896 |
0.09 | 1.91874 | 1.90865 | 1.89868 | 1.88882 | 1.87908 | 1.86945 | 1.85994 | 1.85053 | 1.84122 | 1.83202 |
0.10 | 1.82292 | 1.81393 | 1.80502 | 1.79622 | 1.78751 | 1.77889 | 1.77036 | 1.76192 | 1.75356 | 1.74529 |
0.11 | 1.73711 | 1.72900 | 1.72098 | 1.71304 | 1.70517 | 1.69738 | 1.68967 | 1.68203 | 1.67446 | 1.66697 |
0.12 | 1.65954 | 1.65219 | 1.64490 | 1.63767 | 1.63052 | 1.62343 | 1.61640 | 1.60943 | 1.60253 | 1.59568 |
0.13 | 1.58890 | 1.58217 | 1.57551 | 1.56890 | 1.56234 | 1.55584 | 1.54940 | 1.54301 | 1.53667 | 1.53038 |
0.14 | 1.52415 | 1.51796 | 1.51183 | 1.50574 | 1.49970 | 1.49371 | 1.48777 | 1.48188 | 1.47603 | 1.47022 |
0.15 | 1.46446 | 1.45875 | 1.45307 | 1.44744 | 1.44186 | 1.43631 | 1.43080 | 1.42534 | 1.41992 | 1.41453 |
0.16 | 1.40919 | 1.40388 | 1.39861 | 1.39338 | 1.38819 | 1.38303 | 1.37791 | 1.37282 | 1.36778 | 1.36276 |
0.17 | 1.35778 | 1.35284 | 1.34792 | 1.34304 | 1.33820 | 1.33339 | 1.32860 | 1.32386 | 1.31914 | 1.31445 |
0.18 | 1.30980 | 1.30517 | 1.30058 | 1.29601 | 1.29147 | 1.28697 | 1.28249 | 1.27804 | 1.27362 | 1.26922 |
0.19 | 1.26486 | 1.26052 | 1.25621 | 1.25192 | 1.24766 | 1.24343 | 1.23922 | 1.23504 | 1.23089 | 1.22676 |
0.2 | 1.22265 | 1.21857 | 1.21451 | 1.21048 | 1.20647 | 1.20248 | 1.19852 | 1.19458 | 1.19067 | 1.18677 |
Values of the exponential integral for 0.0001 ≤ x ≤ 0.209.
x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
4 | 37.7940000 | 37.7927530 | 33.4888052 | 29.6876209 | 26.3291192 | 23.3601005 | 20.7340078 | 18.4100584 | 16.3524950 | 14.5299393 |
5 | 11.4839049 | 11.4829557 | 10.2130008 | 9.0862158 | 8.0860830 | 7.1980442 | 6.4092603 | 5.7084015 | 5.0854647 | 4.5316127 |
6 | 3.6017735 | 3.6008245 | 3.2108703 | 2.8637634 | 2.5547143 | 2.2794796 | 2.0342987 | 1.8158374 | 1.6211385 | 1.4475779 |
7 | 1.1557663 | 1.1548173 | 1.0317127 | 0.9218812 | 0.8238725 | 0.7363972 | 0.6583089 | 0.5885877 | 0.5263261 | 0.4707165 |
8 | 0.3776052 | 0.3766562 | 0.3369951 | 0.3015486 | 0.2698641 | 0.2415382 | 0.2162112 | 0.1935625 | 0.1733060 | 0.1551866 |
9 | 0.1254226 | 0.1244735 | 0.1114954 | 0.0998807 | 0.0894849 | 0.0801790 | 0.0718477 | 0.0643883 | 0.0577086 | 0.0517267 |
10 | 0.0425187 | 0.0415697 | 0.0372704 | 0.0334186 | 0.0299673 | 0.0268747 | 0.0241031 | 0.0216191 | 0.0193925 | 0.0173966 |
11 | 0.0149520 | 0.0140030 | 0.0125645 | 0.0112746 | 0.0101178 | 0.0090804 | 0.0081498 | 0.0073151 | 0.0065663 | 0.0058946 |
12 | 0.0057001 | 0.0047511 | 0.0042658 | 0.0038303 | 0.0034395 | 0.0030888 | 0.0027739 | 0.0024913 | 0.0022377 | 0.0020099 |
13 | 0.0025709 | 0.0016219 | 0.0014570 | 0.0013090 | 0.0011761 | 0.0010567 | 0.0009495 | 0.0008532 | 0.0007667 | 0.0006890 |
14 | 0.0015056 | 0.0005566 | 0.0005002 | 0.0004496 | 0.0004042 | 0.0003633 | 0.0003266 | 0.0002936 | 0.0002640 | 0.0002373 |
15 | 0.0011409 | 0.00019186 | 0.00017251 | 0.00015513 | 0.00013950 | 0.00012545 | 0.00011282 | 0.00010146 | 9.1257E−05 | 8.2079E−05 |
16 | 0.0010155 | 6.6405E−09 | 5.9732E−09 | 5.3732E−09 | 4.8336E−09 | 4.3483E−09 | 3.9119E−09 | 3.5194E−09 | 3.1664E−09 | 2.8489E−09 |
17 | 0.0009725 | 2.3064E−09 | 2.0754E−09 | 1.8675E−09 | 1.6805E−09 | 1.5123E−09 | 1.3609E−09 | 1.2248E−09 | 1.1022E−09 | 9.9202E−10 |
18 | 0.0009563 | 8.0361E−10 | 7.2331E−10 | 6.5105E−10 | 5.8603E−10 | 5.2752E−10 | 4.7486E−10 | 4.2747E−10 | 3.8482E−10 | 3.4643E−10 |
19 | 0.0009511 | 2.8078E−10 | 2.5279E−10 | 2.2760E−10 | 2.0492E−10 | 1.8451E−10 | 1.6613E−10 | 1.4959E−10 | 1.3470E−10 | 1.2129E−10 |
20 | 0.0009526 | 9.8355E−11 | 8.8572E−11 | 7.9764E−11 | 7.1833E−11 | 6.4692E−11 | 5.8263E−11 | 5.2473E−11 | 4.7260E−11 | 4.2566E−11 |
21 | 0.0009248 | 3.4532E−11 | 3.1104E−11 | 2.8017E−11 | 2.5237E−11 | 2.2733E−11 | 2.0478E−11 | 1.8447E−11 | 1.6617E−11 | 1.4970E−11 |
22 | 0.0009183 | 1.2149E−11 | 1.0945E−11 | 9.8610E−12 | 8.8842E−12 | 8.0043E−12 | 7.2117E−12 | 6.4976E−12 | 5.8544E−12 | 5.2750E−12 |
23 | 0.0009464 | 4.2827E−12 | 3.8590E−12 | 3.4773E−12 | 3.1334E−12 | 2.8236E−12 | 2.5444E−12 | 2.2929E−12 | 2.0663E−2 | 1.8621E−12 |
24 | 0.0009316 | 1.5123E−12 | 1.3629E−12 | 1.2283E−12 | 1.1070E−12 | 9.9772E−13 | 8.9922E−13 | 8.1046E−13 | 7.3048E−13 | 6.5839E−13 |
25 | 0.0000779 | 5.3489E−13 | 4.8213E−13 | 4.3458E−13 | 3.9172E−13 | 3.5310E−13 | 3.1829E−13 | 2.8692E−13 | 2.5864E−13 | 2.3315E−13 |
Values of the exponential integral, Ei(−x) × 10−4, for 4 ≤ x ≤ 25.9.
x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
0.20 | 1.222651 | 1.182902 | 1.145380 | 1.109883 | 1.076236 | 1.044283 | 1.013889 | 0.984933 | 0.957308 | 0.930918 |
0.30 | 0.905677 | 0.881506 | 0.858335 | 0.836101 | 0.814746 | 0.794216 | 0.774462 | 0.755442 | 0.737112 | 0.719437 |
0.40 | 0.702380 | 0.685910 | 0.669997 | 0.654614 | 0.639733 | 0.625331 | 0.611387 | 0.597878 | 0.584784 | 0.572089 |
0.50 | 0.559774 | 0.547822 | 0.536220 | 0.524952 | 0.514004 | 0.503364 | 0.493020 | 0.482960 | 0.473174 | 0.463650 |
0.60 | 0.454380 | 0.445353 | 0.436562 | 0.427997 | 0.419652 | 0.411517 | 0.403586 | 0.395853 | 0.388309 | 0.380950 |
0.70 | 0.373769 | 0.366760 | 0.359918 | 0.353237 | 0.346713 | 0.340341 | 0.334115 | 0.328032 | 0.322088 | 0.316277 |
0.80 | 0.310597 | 0.305043 | 0.299611 | 0.294299 | 0.289103 | 0.284019 | 0.279045 | 0.274177 | 0.269413 | 0.264750 |
0.90 | 0.260184 | 0.255714 | 0.251337 | 0.247050 | 0.242851 | 0.238738 | 0.234708 | 0.230760 | 0.226891 | 0.223100 |
1.00 | 0.2193840 | 0.2157417 | 0.2121712 | 0.2086707 | 0.2052384 | 0.2018729 | 0.1985724 | 0.1953355 | 0.1921606 | 0.1890462 |
1.10 | 0.1859910 | 0.1829936 | 0.1800526 | 0.1771667 | 0.1743347 | 0.1715554 | 0.1688276 | 0.1661501 | 0.1635218 | 0.1609417 |
1.20 | 0.1584085 | 0.1559214 | 0.1534793 | 0.1510813 | 0.1487263 | 0.1464135 | 0.1441419 | 0.1419107 | 0.1397191 | 0.1375661 |
1.30 | 0.1354511 | 0.1333731 | 0.1313314 | 0.1293253 | 0.1273541 | 0.1254169 | 0.1235132 | 0.1216423 | 0.1198034 | 0.1179960 |
1.40 | 0.1162194 | 0.1144730 | 0.1127562 | 0.1110684 | 0.1094090 | 0.1077775 | 0.1061734 | 0.1045960 | 0.1030450 | 0.1015197 |
1.50 | 0.1000197 | 0.0985445 | 0.0970936 | 0.0956665 | 0.0942629 | 0.0928822 | 0.0915241 | 0.0901880 | 0.0888737 | 0.0875806 |
1.60 | 0.0863084 | 0.0850568 | 0.0838252 | 0.0826134 | 0.0814211 | 0.0802477 | 0.0790931 | 0.0779568 | 0.0768385 | 0.0757379 |
1.70 | 0.0746547 | 0.0735886 | 0.0725392 | 0.0715063 | 0.0704896 | 0.0694888 | 0.0685035 | 0.0675336 | 0.0665788 | 0.0656387 |
1.80 | 0.0647132 | 0.0638020 | 0.0629048 | 0.0620214 | 0.0611516 | 0.0602951 | 0.0594516 | 0.0586211 | 0.0578032 | 0.0569977 |
1.90 | 0.0562045 | 0.0554232 | 0.0546538 | 0.0538960 | 0.0531496 | 0.0524145 | 0.0516904 | 0.0509771 | 0.0502745 | 0.0495824 |
2.00 | 0.0489006 | 0.0482290 | 0.0475673 | 0.0469155 | 0.0462733 | 0.0456407 | 0.0450173 | 0.0444032 | 0.0437981 | 0.0432019 |
2.10 | 0.0426144 | 0.0420356 | 0.0414652 | 0.0409032 | 0.0403493 | 0.0398036 | 0.0392657 | 0.0387357 | 0.0382133 | 0.0376986 |
2.20 | 0.0371912 | 0.0366912 | 0.0361984 | 0.0357127 | 0.0352340 | 0.0347622 | 0.0342971 | 0.0338387 | 0.0333868 | 0.0329414 |
2.30 | 0.0325024 | 0.0320696 | 0.0316429 | 0.0312223 | 0.0308077 | 0.0303990 | 0.0299961 | 0.0295988 | 0.0292072 | 0.0288210 |
2.40 | 0.0284404 | 0.0280650 | 0.0276950 | 0.0273301 | 0.0269704 | 0.0266157 | 0.0262659 | 0.0259210 | 0.0255810 | 0.0252457 |
2.50 | 0.0249150 | 0.0245890 | 0.0242674 | 0.0239504 | 0.0236377 | 0.0233294 | 0.0230253 | 0.0227254 | 0.0224296 | 0.0221380 |
2.60 | 0.0218503 | 0.0215666 | 0.0212868 | 0.0210109 | 0.0207387 | 0.0204702 | 0.0202054 | 0.0199443 | 0.0196867 | 0.0194326 |
2.70 | 0.0191820 | 0.0189348 | 0.0186909 | 0.0184504 | 0.0182131 | 0.0179790 | 0.0177481 | 0.0175204 | 0.0172957 | 0.0170740 |
2.80 | 0.0168554 | 0.0166397 | 0.0164269 | 0.0162169 | 0.0160098 | 0.0158055 | 0.0156039 | 0.0154050 | 0.0152087 | 0.0150151 |
2.90 | 0.0148241 | 0.0146356 | 0.0144497 | 0.0142662 | 0.0140852 | 0.0139066 | 0.0137303 | 0.0135564 | 0.0133849 | 0.0132155 |
3.00 | 0.0130485 | 0.0128836 | 0.0127209 | 0.0125604 | 0.0124020 | 0.0122457 | 0.0120915 | 0.0119392 | 0.0117890 | 0.0116408 |
3.10 | 0.0114945 | 0.0113502 | 0.0112077 | 0.0110671 | 0.0109283 | 0.0107914 | 0.0106562 | 0.0105229 | 0.0103912 | 0.0102613 |
3.20 | 0.0101331 | 0.0100065 | 0.0098816 | 0.0097584 | 0.0096367 | 0.0095166 | 0.0093981 | 0.0092811 | 0.0091656 | 0.0090516 |
3.30 | 0.0089391 | 0.0088281 | 0.0087185 | 0.0086103 | 0.0085035 | 0.0083981 | 0.0082940 | 0.0081913 | 0.0080899 | 0.0079899 |
3.40 | 0.0078911 | 0.0077935 | 0.0076973 | 0.0076022 | 0.0075084 | 0.0074158 | 0.0073244 | 0.0072341 | 0.0071450 | 0.0070571 |
3.50 | 0.0069702 | 0.0068845 | 0.0067999 | 0.0067163 | 0.0066338 | 0.0065524 | 0.0064720 | 0.0063926 | 0.0063143 | 0.0062369 |
3.60 | 0.0061605 | 0.0060851 | 0.0060106 | 0.0059371 | 0.0058645 | 0.0057929 | 0.0057221 | 0.0056523 | 0.0055833 | 0.0055152 |
3.70 | 0.0054479 | 0.0053815 | 0.0053160 | 0.0052512 | 0.0051873 | 0.0051242 | 0.0050619 | 0.0050003 | 0.0049396 | 0.0048796 |
3.80 | 0.0048203 | 0.0047618 | 0.0047041 | 0.0046470 | 0.0045907 | 0.0045351 | 0.0044802 | 0.0044259 | 0.0043724 | 0.0043195 |
3.90 | 0.0042672 | 0.0042157 | 0.0041647 | 0.0041144 | 0.0040648 | 0.0040157 | 0.0039673 | 0.0039194 | 0.0038722 | 0.0038255 |
4.00 | 0.0037794 | 0.0037339 | 0.0036890 | 0.0036446 | 0.0036008 | 0.0035575 | 0.0035148 | 0.0034725 | 0.0034308 | 0.0033896 |
Values of the exponential integral for 0.1 ≤ x ≤ 4.09.
Dimensional parameters do not provide a physical view of the parameter being measured but rather a general or universal description of these parameters. For example, a real time of 24 hours corresponds to a dimensionless time of approximately 300 hours in very low permeability formations or more than 107 in very permeable formations [3, 9, 21, 25, 28].
A set number of Ei values for 0.0001 ≤ x ≤ 25 with the aid of the algorithm given in Figure 1.8. Then, a fitting of these data was performed to obtain the polynomials given by Eqs. (1.85) and (1.90). The first one has a R2 of 1, and the second one has a R2 of 0.999999999 which implies accuracy up to the fifth digit can be obtained.
Adapted from [29] and generated with the Ei function code given in Figure 1.8.
Define dimensionless radius, dimensionless time, and dimensionless pressure as:
Adapted from [29] and generated with the Ei function code given in Figure 1.8.
For pressure drawdown tests, ΔP = Pi − Pwf. For pressure buildup tests, ΔP = Pws − Pwf (Δt = 0).
This means that the steady‐state physical pressure drop for radial flow is equal to the dimensionless pressure multiplied by a scalable factor, which in this case depends on the flow and the properties of the reservoir, [3, 4, 5, 6, 7, 9, 21, 26, 30]. The same concept applies to transient flow and to more complex situations, but in this case, the dimensionless pressure is different. For example, for transient flow, the dimensionless pressure is always a function of dimensionless time.
Taking derivative to Eqs. (1.87) and (1.88),
Replacing the above derivatives into Eq. (1.20),
Adapted from [5] and generated with the Ei function code given in Figure 1.8.
Definition of to requires assuming
Replacing this definition into Eq. (1.88) and solving for the dimensionless time (oilfield units),
Replacing Eq. (1.93) in Eq. (1.92) leads, after simplification, to:
The dimensionless pressure is also affected by the system geometry, other well systems, storage coefficient, anisotropic characteristics of the reservoir, fractures, radial discontinuities, double porosity, among others. In general, the pressure at any point in a single well system that produces the constant rate, q, is given by [25]:
Taking twice derivative to Eq. (1.87), excluding the conversion factor, will provide:
Replacing Eqs. (1.97) and (1.98) in Eq. (1.95) and simplifying leads to:
If the characteristic length is the area, instead of wellbore radius, Eq. (1.92) can be expressed as:
Example 1.1
A square shaped reservoir produces 300 BPD through a well located in the center of one of its quadrants. See Figure 1.10. Estimate the pressure in the well after 1 month of production. Other relevant data:
Geometry of the reservoir for example 1.1.
Pi = 3225 psia, h = 42 ft
ko = 1 darcy, ϕ = 25%
μo = 25 cp, ct = 6.1 × 10−6/psia
Bo = 1.32 bbl/BF, rw = 6 in
A = 150 Acres, q = 300 BPD
Solution
Assuming the system behaves infinitely, it means, during 1 month of production the transient wave has not yet reached the reservoir boundaries, the problem can be solved by estimating the Ei function. Replacing Eqs. (1.82) and (1.92) into the argument of Eq. (1.82), it results:
Using Eq. (1.101) with the above given reservoir and well data:
This x value allows finding Ei(−x) = 17.6163 using the function provided in Figure 1.8. From the application of Eq. (82), PD = 8.808. This dimensionless pressure is meaningless for practical purposes. Converting to oilfield units by means of Eq. (1.87), the well‐flowing pressure value after 1 month of production is given as:
Pwf = 2931.84 psia.
How it can be now if the example was correctly done? A good approximation consists of considering a small pressure drop; let us say ± 0.002 psia (smallest value that can be read from current pressure recorders) at the closest reservoir boundary. Use Eq. (1.87) to convert from psia to dimensionless pressure:
Eq. (1.82) allows finding Ei(−x) = 0.00012. This value can be used to determine an x value from Table 1.2. However, a trial‐and‐error procedure with the function given in Figure 1.8 was performed to find an x value of 6.97. Then, the time at which this value takes place at the nearest reservoir boundary is found from Eq. (1.101). The nearest boundary is obtained from one‐fourth of the reservoir size area (3.7 Ac or 1663500 ft2). Then, for a square geometry system (the system may also be approached to a circle):
The radial distance from the well to the nearest boundary corresponds to one half of the square side, the r = 639.04 ft. Solving for time from Eq. (1.101);
This means that after 2 h and 7 min of flow, the wave has reached the nearest reservoir boundary; therefore, the infinite‐acting period no longer exists for this reservoir, then, a pseudosteady‐state solution ought to be applied (Figures 1.11–1.14). To do so, Eq. (1.98) is employed for the whole reservoir area:
With this tDA value of 0.76, the normal procedure is to estimate the dimensionless pressure for a given reservoir‐well position configuration, which can be found in Figures C.13 through C.16 in [25] for which data were originally presented in [31]. These plots provide the pressure behavior for a well inside a rectangular/square no-flow system, without storage wellbore and skin factor; A0.5/rw = 2000 can also be found in [3, 9, 26]. This procedure is avoided in this textbook. Instead new set of data was generated and adjusted to the following polynomial fitting in which constants are reported in Table 1.5:
Using Eq. (1.102) will result:
Constants for Eq. (1.102).
PD = 12.05597.
The well‐flowing pressure is estimated with Eq. (1.87); thus,
Pwf = 2823.75 psia.
A straight‐line behavior can be observed in mostly the whole range on the right‐hand plot of Ei versus x plot given in Figure 1.9. Then, it was concluded, [3, 4, 5, 6, 7, 9, 11, 19, 21, 26, 30], when x < 0.0025, the more complex mathematical representation of Eq. (1.82) can be replaced by a straight line function, given by:
this leads to,
Replacing this new definition into Eq. (1.82) will result in:
At the well rD = 1, after rearranging,
The above indicates that the well pressure behavior obeys a semi‐logarithmic behavior of pressure versus time.
Example 1.2
A well and infinite reservoir has the following characteristics:
q = 2000 STB/D, μ = 0.72 cp, ct = 1.5 × 10−5 psia−1
ϕ = 23%, Pi = 3000 psia, h = 150 ft
B = 1.475 bbl/STB, k = 10 md, rw = 0.5 ft
Estimate the well‐flowing pressure at radii of 0.5, 1, 5, 10, 20, 50, 70, 100, 200, 500, 1000, 2000, 2500, 3000, and 4000 feet after 1 month of production. Plot the results.
Solution
For the wellbore radius, find x with Eq. (1.101);
Using the function given in Figure 1.9 or Eq. (1.103), a value of Ei(−x) of 15.7421 is found. Then, Eq. (1.82) indicates that PD = 7.871. Use of Eq. (1.87) allows estimating both pressure drop and well‐flowing pressure:
The remaining results are summarized in Table 1.6 and plotted in Figure 1.11. From this, it can be inferred that the highest pressure drop takes place in the near‐wellbore region which mathematically agrees with the continuity equation stating that when the area is reduced, the velocity has to be increased so the flow rate can be constant. The higher the fluid velocity, the higher the pressure drops.
r, ft | x | Ei(−x) | P, psia | Pwf, psia |
---|---|---|---|---|
0.5 | 8.18E−08 | 15.7421 | 1537.74 | 1462.26 |
1 | 3.27E−07 | 14.3558 | 1435.15 | 1564.85 |
5 | 8.18E−06 | 11.137 | 1113.36 | 1886.64 |
10 | 3.27E−04 | 9.75 | 974.78 | 2025.22 |
20 | 1.31E−04 | 8.365 | 836.2 | 2163.8 |
50 | 8.18E−04 | 6.533 | 653.07 | 2346.93 |
70 | 1.60E−03 | 5.86 | 585.87 | 2414.13 |
100 | 3.27E−03 | 5.149 | 514.72 | 2485.28 |
200 | 1.31E−02 | 3.772 | 377.11 | 2622.89 |
500 | 8.17E−02 | 2.007 | 200.616 | 2799.384 |
1000 | 3.27E−01 | 0.8425 | 84.225 | 2915.775 |
2000 | 1.31E+00 | 0.1337 | 13.368 | 2986.632 |
2500 | 2.04E+00 | 0.046 | 4.6 | 2995.4 |
3000 | 2.94E+00 | 0.014 | 1.401 | 2998.599 |
4000 | 5.23E+00 | 0.0009 | 0.087 | 2999.913 |
Summarized results for example 1.2.
Pressure versus distance plot for example 1.2.
Example 1.3
Re‐work example 1.2 to estimate the sand‐face pressure at time values starting from 0.01 to 1000 h. Show the results in both Cartesian and semilog plots. What does this suggest?
Solution
Find x with Eq. (1.101);
A value of Ei(−x) of 6.385 is found with Eq. (1.103). Then, Eq. (1.82) gives a PD value of 3.192 and Eq. (1.87) leads to calculate a well‐flowing pressure of;
The remaining well‐flowing pressure values against time are given in Table 1.7 and plotted in Figure 1.12. The semilog behavior goes in the upper part of the plot (solid line), and the Cartesian plot corresponds to the lower dashed line. The semilog line behaves linearly while the Cartesian curve does not. This situation perfectly agrees with Eq. (1.106), which ensures that the behavior of pressure drop versus time obeys a semilog trend. In other word, in a transient radial system, pressure drops is a linear function of the logarithm of time.
t, h | x | Ei(−x) | PD | Pwf, Psia | t, h | x | Ei(−x) | PD | Pwf, psia |
---|---|---|---|---|---|---|---|---|---|
0.01 | 9.480E−04 | 6.385 | 3.192 | 2361.71 | 6 | 1.580E−06 | 12.781 | 6.390 | 1722.30 |
0.02 | 4.740E−04 | 7.078 | 3.539 | 2292.46 | 7 | 1.354E−06 | 12.935 | 6.468 | 1706.89 |
0.03 | 3.160E−04 | 7.483 | 3.741 | 2251.94 | 8 | 1.185E−06 | 13.069 | 6.534 | 1693.54 |
0.04 | 2.370E−04 | 7.770 | 3.885 | 2223.19 | 9 | 1.053E−06 | 13.186 | 6.593 | 1681.77 |
0.05 | 1.896E−04 | 7.994 | 3.997 | 2200.89 | 10 | 9.480E−07 | 13.292 | 6.646 | 1671.23 |
0.06 | 1.580E−04 | 8.176 | 4.088 | 2182.66 | 20 | 4.740E−07 | 13.985 | 6.992 | 1601.94 |
0.07 | 1.354E−04 | 8.330 | 4.165 | 2167.25 | 30 | 3.160E−07 | 14.390 | 7.195 | 1561.41 |
0.08 | 1.185E−04 | 8.464 | 4.232 | 2153.91 | 40 | 2.370E−07 | 14.678 | 7.339 | 1532.65 |
0.09 | 1.053E−04 | 8.581 | 4.291 | 2142.13 | 50 | 1.896E−07 | 14.901 | 7.451 | 1510.34 |
0.1 | 9.480E−05 | 8.687 | 4.343 | 2131.60 | 60 | 1.580E−07 | 15.083 | 7.542 | 1492.11 |
0.2 | 4.740E−05 | 9.380 | 4.690 | 2062.31 | 70 | 1.354E−07 | 15.238 | 7.619 | 1476.70 |
0.3 | 3.160E−05 | 9.785 | 4.893 | 2021.78 | 80 | 1.185E−07 | 15.371 | 7.686 | 1463.35 |
0.4 | 2.370E−05 | 10.073 | 5.036 | 1993.02 | 90 | 1.053E−07 | 15.489 | 7.744 | 1451.58 |
0.5 | 1.896E−05 | 10.296 | 5.148 | 1970.71 | 100 | 9.480E−08 | 15.594 | 7.797 | 1441.05 |
0.6 | 1.580E−05 | 10.478 | 5.239 | 1952.49 | 200 | 4.740E−08 | 16.287 | 8.144 | 1371.75 |
0.7 | 1.354E−05 | 10.632 | 5.316 | 1937.08 | 300 | 3.160E−08 | 16.693 | 8.346 | 1331.22 |
0.8 | 1.185E−05 | 10.766 | 5.383 | 1923.73 | 400 | 2.370E−08 | 16.981 | 8.490 | 1302.46 |
0.9 | 1.053E−05 | 10.884 | 5.442 | 1911.95 | 500 | 1.896E−08 | 17.204 | 8.602 | 1280.15 |
1 | 9.480E−06 | 10.989 | 5.495 | 1901.42 | 600 | 1.580E−08 | 17.386 | 8.693 | 1261.92 |
2 | 4.740E−06 | 11.682 | 5.841 | 1832.13 | 700 | 1.354E−08 | 17.540 | 8.770 | 1246.51 |
3 | 3.160E−06 | 12.088 | 6.044 | 1791.59 | 800 | 1.185E−08 | 17.674 | 8.837 | 1233.17 |
4 | 2.370E−06 | 12.375 | 6.188 | 1762.84 | 900 | 1.053E−08 | 17.792 | 8.896 | 1221.39 |
5 | 1.896E−06 | 12.599 | 6.299 | 1740.53 | 1000 | 9.480E−09 | 17.897 | 8.948 | 1210.86 |
Summarized results for example 1.3.
Pressure versus time plot for example 1.3.
Once the dimensionless parameters are plugged in Eq. (1.82), this yields:
At point N, Figure 1.13, the pressure can be calculated by Eq. (1.107). At the wellbore rD = r/rw = 1, then, r = rw and P(r,t) = Pwf. Note that application of the line‐source solution requires the reservoir to possess an infinite extent, [3, 9, 18, 21, 25, 26].
Pressure distribution in the reservoir.
There are several ways to quantify damage or stimulation in an operating well (producer or injector). These conditions are schematically represented in Figure 1.14. The most popular method is to represent a well condition by a steady‐state pressure drop occurring at the wellbore, in addition to the transient pressure drop normally occurring in the reservoir. This additional pressure drop is called “skin pressure drop” and takes place in an infinitesimally thin zone: “damage zone,” [4, 5, 9, 11, 19, 30]. It can be caused by several factors:
Skin factor influence.
Invasion of drilling fluids
Partial well penetration
Partial completion
Blocking of perforations
Organic/inorganic precipitation
Inadequate drilling density or limited drilling
Bacterial growth
Dispersion of clays
Presence of cake and cement
Presence of high gas saturation around the well
Skin factor is a dimensionless parameter; then, it has to be added to the dimensionless pressure in Eq. (1.87), so that:
From the above expression can be easily obtained:
Therefore, the skin factor pressure drop is given by:
Assuming steady state near the wellbore and the damage area has a finite radius, rs, with an altered permeability, ks, the pressure drop due to the damage is expressed as the pressure difference between the virgin zone and the altered zone, that is to say:
Rearranging;
Comparing Eqs (1.112) and (1.107), the following can be concluded:
rs and ks are not easy to be obtained.
Equation (1.82) and (1.106) can be respectively written as:
Replacing the dimensionless quantities given by Eqs. (1.87) and (1.95) in Eq. (1.115) will result:
Taking natural logarithm to 0.0002637 and adding its result to 0.80908 results in:
Multiplying and dividing by the natural logarithm of 10 and solving for the well‐flowing pressure:
Thus, a straight line is expected to develop from a semilog plot of pressure against the time, as seen on the upper curve of Figure 1.12.
In closed systems, the radial flow is followed by a transition period. This in turn is followed by the pseudosteady, semi‐stable, or quasi‐stable state, which is a transient flow regime where the pressures change over time, dP/dt, is constant at all points of the reservoir:
Eq. (1.99) is now subjected to the following initial and boundary conditions:
The pseudosteady‐state period takes place at late times (t > 948ϕμctre2/k), so that as time tends to infinity, summation tends to zero, then:
At the well, rD = 1 and as reD >>>> 1, the above expression is reduced to:
This can be approximated to:
Invoking Eq. (1.98) for a circular reservoir area,
It follows that;
The final solution to the pseudosteady‐state diffusivity equation is obtained from using the definition given by Eq. (1.128) in Eq. (1.129):
The derivative with respect to time of the above equation in dimensional form allows obtaining the pore volume:
An important feature of this period is that the rate of change of pressure with respect to time is a constant, that is, dPD/dtDA = 2π.
When the reservoir pressure does not change over time at any point, the flow is said to be stable. In other words, the right side of Eq. (1.99) is zero, [3]:
Similar to the pseudosteady‐state case, steady state takes place at late times. Now, its initial, external, and internal boundary conditions are given by:
The solution to the steady‐state diffusivity equation is [3]:
As time tends to infinity, the summation tends to infinity, then:
In dimensional terms, the above expression is reduced to Darcy’s equation. The dimensionless pressure function for linear flow is given by:
Steady state can occur in reservoirs only when the reservoir is fully recharged by an aquifer or when injection and production are balanced. However, a reservoir with a very active aquifer will not always act under steady‐state conditions. First, there has to be a period of unsteady state, which will be followed by the steady state once the pressure drop has reached the reservoir boundaries. Extraction of fluids from a pressurized reservoir with compressible fluids causes a pressure disturbance which travels throughout the reservoir. Although such disturbance is expected to travel at the speed of sound, it is rapidly attenuated so that for a given duration of production time, there is a distance, the drainage radius, beyond which no substantial changes in pressure will be observed. As more fluid is withdrawn (or injected), the disturbance moves further into the reservoir with continuous pressure decline at all points that have experienced pressure decline. Once a reservoir boundary is found, the pressure on the boundary continues to decline but at a faster rate than when the boundary was not detected. On the other hand, if the pressure transient reaches an open boundary (water influx), the pressure remains constant at some point; the pressure closest to the well will decline more slowly than if a closed boundary were found. Flow changes or the addition of new wells cause additional pressure drops that affect both the pressure decline and the pressure distribution. Each well will establish its own drainage area that supplies fluid. When a flow boundary is found, the pressure gradient—not the pressure level—tends to stabilize after sufficiently long production time. For the closed boundary case, the pressure reaches the pseudosteady state with a constant pressure gradient and general pressure drop everywhere, which is linear over time. For constant‐pressure boundaries, steady state is obtained; both the pressure and its gradient remain constant over time.
Pressure derivative has been one of the most valuable tools ever introduced to the pressure transient analysis field. In fact, [32] affirms that pressure derivative and deconvolution have been the best elements added for well test interpretation. However, here it is affirmed that besides these two “blessings,” TDS technique, [1, 2], is the best and practical well test interpretation method in which application will be very devoted along this textbook. Actually, in the following chapters, TDS is extended for long, homogeneous reservoirs, [33], interference testing [34], drainage area determination in constant‐pressure reservoirs, [35], and recent applications on fractured vertical wells, [36], among others. More complex scenarios, for instance finite‐conductivity faults, [37], are treated extensively in [38].
Attempts to introduce the pressure derivative are not really new. Some of them try to even apply the derivative concept to material balance. Just to name a few of them, [39] in 1961, tried to approach the rate of pressure change with time for detection of reservoir boundaries. Later, in 1965, [40] presented drawdown curves of well pressure change with time for wells near intersecting faults (36 and 90°). These applications, however, use numerical estimations of the pressure rate change on the field data regardless of two aspects: (1) an understanding of the theoretical situation behind a given system and (2) noise in the pressure data.
Between 1975 and 1976, Tiab’s contributions on the pressure derivative were remarkable. Actually, he is the father of the pressure derivative concept as used nowadays. Refs. [41, 42] include detailed derivation and application of the pressure derivative function. These results are further summarized on [41, 42, 43, 44, 45]. Ref. [46] applied Tiab’s finding to provide a type‐curve matching technique using the natural logarithm pressure derivative.
It was required to obtain the pressure derivative from a continuous function, instead of attempting to work on discrete data in order to understand the pressure derivative behavior in an infinite system. Then, Tiab decided to apply the Leibnitz’s rule of derivation of an integral to the Ei function.
Applying Leibnitz’s rule to the Ei function in Eq. (1.81) to differentiate with respect to tD (see Appendix B in [42]),
Taking the derivative Δu/ΔtD and replacing u by rD2/4tD,
After simplification,
From inspection of Eq. (1.81) results:
In oilfield units,
At the well, rD = 1, then, Eq. (1.142) becomes:
For tD > 250, e−1/4tD = 1; then, Eq. (1.144) reduces to
The derivative of equation (1.145) is better known as the Cartesian derivative. The natural logarithmic derivative is obtained from:
Later on, [46] use the natural logarithmic derivative to develop a type‐curve matching technique.
Appendix C in [42] also provides the derivation of the second pressure derivative:
Conversion of Eq. (1.145) to natural logarithmic derivative requires multiplying both sides of it by tD; then, it results:
Eq. (1.148) suggests that a log‐log plot of dimensionless pressure derivative against dimensionless time provides a straight line with zero slope and intercept of ½. Taking logarithm to both sides of Eq. (1.145) leads to:
The above expression corresponds of a straight line with negative unit slope. In dimensional form:
Taking logarithm to both sides of the above expression:
As shown in Figure 1.15, Eq. (1.151) corresponds to a straight line with negative unit slope and intercept of:
Log‐log plot of Pwf′ against t.
Eq. (1.152) is applied to find permeability from the intersect plot of the Cartesian pressure derivative versus time plot. This type of plot is also useful to detect the presence of a linear boundary (fault) since the negative unit slope line displaces when the fault is felt as depicted in Figure 1.16.
Fault identification by means of a log‐log plot of PD′ vs. tD.
The noise that occurs in a pressure test is due to such factors as (1) turbulence, (2) tool movements, (3) temperature variations, (4) opening and closing wells in the field, and (5) gravitational effects of the sun and moon on the tides (near the great lakes the noise is about 0.15 psia and offshore up to 1 psia).
The estimation of the pressure derivative with respect to time to actual data, of course, must be performed numerically since data recorded from wells are always discrete. During the derivative calculation, the noise is increased by the rate of change that the derivative imposes, so it is necessary to soften the derivative or to use smoothing techniques. The low resolution of the tool and the log‐log paper also increase or exaggerate the noise. Therefore, calculating the derivative of pressure requires some care because the process of data differentiation can amplify any noise that may be present. Numerical differentiation using adjacent points will produce a very noisy derivative, [8, 47, 48].
Ref. [8] conducted a comparative study of several algorithms for estimation of the pressure derivative. They obtained synthetic pressure derivatives for seven different reservoir and well configuration scenarios and, then, estimated the pressure derivative using several comparative methods. They found that the Spline algorithm (not presented here) is the best procedure to derive pressure versus time data since it produces minimal average errors. It is the only algorithm of polynomial character that to be continuous can be smoothed during any derivation process and the form of the curve obtained is in agreement with the worked model. The Horne and Bourdet algorithms when the smoothing window is of either 0.2 or 0.4 are good options for derivation processes. Ref. [8] also found the best procedure for data analysis of pressure against time is to differentiate and then smooth the data.
By itself, the central finite difference formula fails to provide good derivative computation. Instead, some modifications are introduced by [18, 20, 46], respectively:
Horne equation [32]:
When the data are distributed in a geometrical progression (with the time difference from one point to the next much larger as the test passes), then the noise in the derivative can be reduced using a numerical differentiation with respect to the logarithm of time. The best method to reduce noise is to use data that is separated by at least 0.2 logarithmic cycles, rather than points that are immediately adjacent. This procedure is recognized as smoothing and is best explained in Figure 1.17.
Equation of Bourdet et al. [46]:
Smoothing diagram.
Let X is the natural logarithm of the time function.
This differentiation algorithm reproduces the test type curve over the entire time interval. It uses a point before and a point after the point of interest, i, to calculate the corresponding derivative and places its weighted mean for the objective point. Smoothing can also be applied.
This principle is not new. It was first introduced to the petroleum literature by van Everdingen and Hurst in 1949, [49]. However, its application is too important and many field engineers fail or neglect to use it. Superposition is too useful for systems having one well producing at variable rate or the case when more than one well produces at different flow rates.
As quoted from [25], the superposition principle is defined by:
“Adding solutions to the linear differential equation will result in a new solution of that differential equation but for different boundary conditions,” which mathematically translates to:
where ψ is the general solution and ψ1 f1, ψ2 f2 and ψ3 f3… are the particular solutions.
If the wells produce at a constant flow rate, the pressure drop at point N, Figure 1.18, will be [3, 9, 19, 21, 25]:
Pressure at the point N.
If reservoir and fluid properties are considered constant, then, Eq. (1.87) can be applied to the above expression, so that:
The dimensionless radii are defined by:
Extended to n number of wells:
If point N is an active well, its contribution to the total pressure drop plus the skin factor pressure drop, Eq. (1.108), must be included in Eq. (1.159), then,
Notice that in Eqs. (1.159) and (1.160), changes of pressures or dimensionless pressures are added. If the point of interest is a well in operation, the damage factor should be added to the dimensionless pressure of that well only.
Sometimes there are changes in flow rate when a well produces as referred in Figures 1.19 and 1.22. Then, the superposition concept must be applied. To do this, [25], a single well is visualized as if there were two wells at the same point, one with a production rate of q1 during a time period from t = 0 to t and another imaginary well with a production rate of q2 − q1 for a time frame between t1 and t − t1. The total rate after time t1 is q1+ (q2 − q1) = q2. The change in well pressure due to the rate change [19, 25] is,
Time superposition.
where tD2 = (t−t1)D. If there are more variations in flow rate,
Example 1.4
This example is taken [25]. The below data and the schematic given in Figure 1.20 correspond to two wells in production:
Flow rate changes for example 1.4.
k = 76 md, ϕ = 20 %, B = 1.08 bbl/STB
Pi = 2200 psia, μ = 1 cp, ct = 10 × 10−6/psia
h = 20 ft
Calculate the pressure in (a) well 1 after 7 h of production and (b) in well 2 after 11 h of production. Assume infinite behavior.
Solution
Part (a):
ΔP(7 hr)= ΔP caused by production from well 1 to well 1 + ΔP caused by production from well 2 to well 1. Mathematically,
Using Eq. (1.101) for the well,
Since x <<<< 0.0025, it implies the use of Eq. (1.82) with Eq. (1.103); then,
In well 2, x = 0.03564 from Eq. (1.101). Interpolating this value in Table 1.2, Ei(−x) = 2.7924; then, PD ≅ 1.4. Estimating ΔP in well 1 will result:
Pwf @ well1 = 2200−113.7 = 2086.4 psia (notice that skin factor was only applied to well 1)
Part (b);
At 11 h, it is desired to estimate the pressure in well 2. Two flow rates should be considered for in each well. Then, the use of Eq. (1.162) will provide:
Using Eq. (1.101), the four respective values of x are: x =0.02268, 0.2494, 2.268 × 10−6, and 8.316 × 10−6. Estimation of Ei requires the use of Table 1.2 for the first two values and use of Eq. (1.103) for the last two values. The four values of Ei(−x) are: 0.0227, 0.811, 12.42, and 11.12. Therefore, the respective values of PD are 1.605, 0.405, 6.209, and 5.56. The total pressure drop is found with Eq. (1.161) as follows:
Pwf @ well2 = 2200 − 87.75 = 2112.25 psia
The method of images applies to deal with either no‐flow or constant‐pressure boundaries. If a well operates at a constant flow rate at a distance, d, from an impermeable barrier (fault), the systems acts as if there were two wells separated 2d from each other [3, 25]. For no‐flow boundaries, the image well corresponds to the same operating well. For constant‐pressure boundary, the resulting image corresponds to an opposite operating well. In other words, if the well is a producer near a fault, the image well corresponds to an injector well. These two situations are sketched in Figure 1.21. For the no‐flow boundary, upper system in Figure 1.21, the dimensionless pressure can be expressed as:
Well near a linear barrier.
For the constant‐pressure boundary, lower part in Figure 1.21, the dimensionless pressure can be expressed as:
The negative sign in Eq. (1.165) is because of dealing with an imaginary injector well.
For the case of two intersecting faults, the total number of wells depends on the value of the angle formed by the two faults, thus:
The image method is limited to one well per quadrant. If this situation fails to be fulfilled, then, the method cannot be applied. In the system of Figure 1.22, an angle of 90° is formed from the intersecting faults. According to Eq. (1.166), nwells = 360/90 = 4 wells, as shown there. The ratio of the distances from the well to each fault is given by:
Well between two intersecting faults.
The practical way to apply space superposition for generating the well system resulting from two intersecting faults consist of extending the length of the faults and setting as many divisions as suggested by Equation (1.166); that is, for example, 1.5, Figure 1.23 left, six well spaces are obtained. Then, draw a circle with center at the fault intersection and radius at well position. This guarantees that the total length corresponds to the double length value from the well to the fault. Draw from the well a line to be perpendicular to the nearest fault and keep drawing the line until the circle line has been reached. See Figure 1.24 left. Set the well. A sealing fault provides the same type of well as the source well, that is, a producing well generates another producing well to the other side of the fault. A constant‐pressure boundary provides the opposite well type of the source well, that is, a producing well generates an injector well on the other side of the line. Draw a new line from the just drawn imaginary well normal to the fault and keep drawing the line until the line circle is reached. See Figure 1.24 right. Repeat the procedure until the complete well set system has been drawn.
Location of well A and resulting well number system for example 1.5.
Generating the well system for two intersecting faults.
For more than six well spaces generated, that is angles greater than 60°, as the case of example 1.5, when a fault intersects a constant‐pressure boundary injector and producer imaginary wells ought to be generated. What type of line should be drawn? A solid line representing a sealing fault, or a dash line, representing a constant‐pressure boundary? The answer is any of both. The lines should be drawn alternatively and as long as the system closes correctly, superposition works well.
Example 1.5
Well A in Figure 1.23 has produced a constant rate of 380 BPD. It is desired (a) to estimate the well‐flowing pressure after one week of production. The properties of the reservoir, well and fluid are given as follows:
Pi = 2500 psia, B = 1.3 bbl/STB, μ = 0.87 cp
h = 40 ft, ct = 15×10−6/psia, ϕ = 18 %
rw = 6 in, k = 220 md, s = −5
(b) What would be the well‐flowing pressure after a week of production if the well were in an infinite reservoir?
Solution
Part (a)
The pressure drop in well A is affected by its own pressure drop and pressure drop caused by its well images. The distance from well A to its imaginary wells is shown in Figure 1.23 (right‐hand side). The total pressure drop for well A is:
By symmetry, the above expression becomes:
Using Eq. (1.101) for the well:
Since x <<<< 0.0025, Eq. (1.163) applies:
Estimation for the image wells are given below. In all cases, x > 0.0025, then, Table 1.2 is used to find Ei and the resulting below divided by 2 for the estimation of PD,
Then, the pressure drop in A will be:
Pwf @ well A = 2500 − 76.3 = 2423.7 psia
Part (b)
If the well were located inside an infinite reservoir, the pressure drop would not include imaginary wells, then:
The well‐flowing pressure would be (2500 − 25.3) = 2474.4 psia. It was observed that the no‐flow boundaries contribute with 66.4% of total pressure drop in well A.
A | area, ft2 or Ac |
Bg | gas volume factor, ft3/STB |
Bo | oil volume factor, bbl/STB |
Bw | oil volume factor, bbl/STB |
bx | distance from closer lateral boundary to well along the x‐direction, ft |
by | distance from closer lateral boundary to well along the y‐direction, ft |
c | compressibility, 1/psia |
cf | pore volume compressibility, 1/psia |
ct | total or system compressibility, 1/psia |
d | distance from a well to a fault, ft |
f | a given function |
h | formation thickness, ft |
k | permeability, md |
ks | permeability in the damage zone, md |
krf | phase relative permeability, f = oil, water or gas |
L | reservoir length, ft |
m | slope |
m(P) | pseudopressure function, psia2/cp |
M | gas molecular weight, lb/lbmol |
P | pressure |
dP/dr | pressure gradient, psia/ft |
PD′ | dimensionless pressure derivative |
PD″ | dimensionless second pressure derivative |
PD | dimensionless pressure |
Pi | initial reservoir pressure, psia |
Pwf | well flowing pressure, psia |
q | flow rate, bbl/D. For gas reservoirs the units are Mscf/D |
Rs | gas dissolved in crude oil, SCF/STB |
Rsw | gas dissolved in crude water, SCF/STB |
rD | dimensionless radius |
rDe | dimensionless drainage radius = re/rw |
r | radial distance, radius, ft |
re | drainage radius, ft |
rs | radius of the damage zone, ft |
rw | well radius, ft |
Sf | fluid saturation, f = oil, gas or water |
s | skin factor |
T | reservoir temperature, ºR |
t | time, h |
ta | pseudotime, psia h/cp |
to | dummy time variable |
ur | radial flow velocity, ft/h |
tD | dimensionless time based on well radius |
tDA | dimensionless time based on reservoir area |
tD*PD′ | logarithmic pressure derivative |
V | volume, ft3 |
z | vertical direction of the cylindrical coordinate, real gas constant |
Δ | change, drop |
Δt | shut‐in time, h |
ϕ | porosity, fraction. Spherical coordinate |
λ | phase mobility, md/cp |
η | hydraulic diffusivity constant, md‐cp/psia |
ρ | density, lbm/ft3 |
θ | cylindrical coordinate |
μ | viscosity, cp |
ζ | time function |
1 hr | reading at time of 1 h |
D | dimensionless |
DA | dimensionless with respect to area |
f | formation |
g | gas |
i | initial conditions |
o | oil, based condition |
w | well, water |
p | pore |
As can be seen in Figure 1.4, well pressure test analysis (PTA) considers this as the most basic and simple test, which does not mean that it is not important. In these tests, bottom‐hole well‐flowing pressure, Pwf, is continuously recorded keeping the flow constant. These tests are also referred as flow tests. Similar to an injection test, these tests require either production/injection from/into the well.
These tests are performed with the objective of (a) obtaining pore volume of the reservoir and (b) determining heterogeneities (in the drainage area). In fact, what is obtained is (a) transmissibility and (b) porous volume by total compressibility. In fact, a recent study by Agarwal [1] allows using drawdown tests to estimate the average permeability in the well drainage area. To run a pressure decline test, the following steps are generally followed:
The well is shut‐in for a long enough time to achieve stabilization throughout the reservoir, if this is not achieved, multirate testing is probably required;
The recording pressure tool is lowered to a level immediately above the perforations. This is to reduce Joule‐Thompson effects. It is important to have at least two pressure sensors for data quality control purposes;
The well opens in production at constant flow and in the meantime the well‐flowing pressure is continuously recorded.
Ideally, the well is closed until the static reservoir pressure. The duration of a drawdown test may last for a few hours or several days, depending upon the test objectives and reservoir characteristics. There are extensive pressure drawdown tests or reservoir limit tests (RLT) that run to delimit the reservoir or estimate the well drainage volume. Other objectives are the determination of: well‐drainage area permeability, skin factor, wellbore storage coefficient (WBS), porosity, reservoir geometry, and size of an adjacent aquifer.
It is the continuous flow of the formation to the well after the well has been shut‐in for stabilization. It is also called after‐flow, postproduction, postinjection, loading, or unloading (for flow tests). The flow occurs by the expansion of fluids in the wellbore. In pressure buildup tests, after‐flow occurs. Figure 2.1 illustrates the above [2].
Traditional pressure tests had to be long enough to cope with both wellbore storage and skin effects so that a straight line could be obtained indicating the radial flow behavior. Even this approach has disadvantages since more than one apparent line can appear and analysts have problems deciding which line to use. In addition, the scale of the graph may show certain pressure responses as straight lines when in fact they are curves. To overcome these issues, analysts developed the method the type‐curve matching method.
There is flow in the wellbore face after shutting‐in the well in surface. Wellbore storage affects the behavior of the pressure transient at early times. Mathematically, the storage coefficient is defined as the total volume of well fluids per unit change in bottom‐hole pressure, or as the capacity of the well to discharge or load fluids per unit change in background pressure:
As commented by Earlougher [2], wellbore storage causes the flow rate at the face of the well to change more slowly than the surface flow rate. Figure 2.2 schematizes the relation qsf/q when the surface rate is changed from 0 to q, when C = 0, qsf/q = 1, while for C > 0, the relation qsf/q gradually changes from 0 to 1. The greater the value of C, the greater the transition is. As the storage effects become less severe, the formation begins to influence more and more the bottom‐hole pressure until the infinite behavior is fully developed. Pressure data that are influenced by wellbore storage can be used for interpretation purposes since fluids unload or load has certain dependence on reservoir transmissibility; however, this analysis is risky and tedious. TDS technique, presented later in this chapter, can provide a better solution to this problem.
Typically, the flow rate is surface‐controlled (unless there is a bottom shut‐in tool), the fluids in the well do not allow an immediate transmission of the disturbance from the subsurface to the surface, resulting in uneven surface and wellbore face flow [2, 3, 4, 5, 6, 7]. Wellbore storage can change during a pressure test in both injector and producer wells. Various circumstances cause changes in storage, such as phase redistribution and increase or decrease in storage associated with pressure tests in injector wells. In injector wells, once the well is closed, the surface pressure is high but could decrease to atmospheric pressure and go to vacuum if the static pressure is lower than the hydrostatic pressure. This causes an increase in storage (up to 100 times) of an incompressible system to one in a system where the liquid level drops [2]. The inverse situation occurs in injector wells with a high level of increase of liquid storage level and in producing wells with a high gas‐oil ratio or by redissolution of the free gas. Both for increase or decrease of storage, the second storage coefficient determines the beginning of the semilogarithmic straight line.
When the relationship between ΔV and ΔP does not change during the test, the wellbore storage coefficient is constant and can be estimated from completion data [2, 3, 4].
where Vu is the wellbore volume/unit length, bbl/ft, r is the density of the fluid in the wellbore, lbm/ft3, and C is the wellbore storage coefficient, bbl/psia.
Effects of wellbore storage on buildup and drawdown tests, taken from [2].
Effect of storage on the flow rate at the face of the well, C3>C2>C1, taken from [2].
For injector wells or wells completely filled with fluids:
where Cwb is the wellbore fluid compressibility = 1/Pwb, Vwb is the total wellbore volume, and Vu can be estimated with internal casing, IDcsg, and external tubing, ODtbg, diameters.
When opening a well, see Figure 2.3, the oil production will be given by the fluid that is stored in the well, qsf = 0. As time goes by, qsf tends to q and storage is neglected and the amount of liquid in the wellbore will be constant. The net accumulation volume will be (assuming constant B) [3, 5]:
Schematic representation of wellbore storage, taken from [3].
The flow rate is given by:
The rate of volume change depends upon the difference between the subsurface and surface rates:
Since (assuming g/gc = 1):
Taking the derivative to Eq. (2.8),
Combining Eqs. (2.7) and (2.9) will result:
Define
Assuming constant, Pt, replacing the definition given by Eq. (2.11) and solving for the wellbore face flow rate, qsf, leads to:
Taking derivative to Eqs. (1.89) and (1.94) with respect to time and taking the ratio of these will yield:
Combining Eqs. (2.12) and (2.13);
Defining the dimensionless wellbore storage coefficient;
Rewriting Eq. (2.14);
The main advantage of using downhole shut‐in devices is the minimization of wellbore storage effects and after‐flow duration.
Rhagavan [5] presents the solution for the radial flow diffusivity equation considering wellbore storage and skin effects in both Laplace and real domains, respectively:
where f(x) = 1−CD(s) x2, and K0, K1, J0, J1,Y0, and Y1 are Bessel functions.
There exist four methods for well test interpretation as follows: (a) conventional straight‐line, (b) type‐curve matching, (c) regression analysis, and (d) modern method: TDS technique. Although they were named chronologically, from oldest to most recent, they will be presented in another way:
This is the most widely used method. It consists of automatically matching the pressure versus time data to a given analytical solution (normally) of a specific reservoir model. The automatic procedure uses nonlinear regression analysis by taking the difference between a given matching point and the objective point from the analytical solution.
This method has been also widely misused. Engineers try to match the data with any reservoir model without considering the reservoir physics. The natural problem arid=sing with this method is the none‐uniqueness of the solution. This means that for a given problem, the results are different if the starting simulation values change. This can be avoided if the starting values for the simulation values are obtained from other techniques, such as TDS technique or conventional analysis, and then, the range of variation for a given variable is reduced. This technique will not be longer discussed here since this book focused on analytical and handy interpretation techniques.
As seen before, this technique was the second one to appear. Actually, it came as a solution to the difficulty of identity flow regimes in conventional straight‐line plots. However, as observed later, the technique is basically a trial‐and‐error procedure. This makes the technique tedious and risky to properly obtain reservoir parameters.
The oldest type‐curve method was introduced by Ramey [2, 8, 9]. If CD = 0 in Eq. (2.16), then, qsf = q. Therefore;
By integration between 0 and a given PwD and from dimensionless time zero to tD, and taking logarithm to both terms, it yields:
Suffix w is used to emphasize that the pressure drop takes place at the wellbore bottom‐hole. This will be dropped for practical purposes. It is clearly observed in Eq. (2.19) that the slope is one. Then in any opportunity that is plotted PD vs. tD and a straight line with a unitary slope is observed at early times, is a good indication that storage exists. Substituting the dimensionless quantities given by Eqs. (1.89), (1.94) and (2.15) in Eq. (2.20), we have:
Eq. (2.21) serves to determine the storage coefficient from data from a pressure decline test using a log‐log plot of ΔP versus time. Any point N is taken from the unit‐slope straight line portion. The value of C obtained using Eq. (2.21) must match the value obtained from Eq. (2.5). Otherwise, there may be an indication that the liquid level is going down or rising inside the well. The reasons most commonly attributed to this phenomenon are high gas‐oil ratios, highly stimulated wells, exhaust gaskets or spaces in the well connections caused by formation collapse or poor cementation and wells used for viscous fluid injection. In conclusion, the properties of Ramey\'s type curves allow (a) a unitary slope to be identified which indicates wellbore storage and (b) the fading of wellbore storage effects.
It can also be seen that each curve deviates from the unitary slope and forms a transition period lasting approximately 1.5 logarithmic cycles. This applies only to constant wellbore storage, otherwise, refer to [10]. If every ½ cycle is equal to (100.5 = 3.1622), it means that three half cycles (3.16223 = 31.62) represent approximately a value of 30. That is to say that a line that deviates at 2 min requires 1 h forming the transient state or radial flow regime. In other words, the test is masked for 1 h by wellbore storage effects [2, 5, 11]. It is also observed that a group of curves that present damage are mixed at approximately a dimensionless time,
After which time, the test is free of wellbore storage effects [2, 5, 6]. Along with TDS technique [10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73] which will be discussed later in this chapter, type‐curve matching is the only manual procedure that can be applied in short tests where radial flow has not been developed (semilog line). However, type‐curve matching is risky because it is a trial‐and‐error technique, but can provide approximate results even when conventional methods fail. One millimeter shifting can cause pressure differences of up to 200 psia. The procedure is as follows [2, 9]:
Type curve of dimensionless pressure against dimensionless time for a well in an infinite reservoir (wellbore storage and skin), taken from [2, 9].
Prepare a plot of DP vs. t on logarithmic paper using the same scale as the master curve given in Figure 2.4. This is recognized as the field data plot, fdp.
Place the fdp on the master curve so that the axes are parallel.
Find the best match with one of the curves in Figure 2.4.
Choose a suitable match point and read the corresponding coordinates DPM, tM, PDM, tDM, and CDM. The two first parameters are read from the fdp. The remaining from the type‐curve (Figure 2.4).
Estimate permeability, porosity, and wellbore storage coefficient, respectively:
The results from the Ramey’s type curve must be verified with some other type curve. For instance, Earlougher and Kersch [8], formulated another type curve, Figure 2.5, which result should agree with those using Ramey method. The procedure for this method [8] is outlined as follows:
Plot ΔP/t vs. t (fdp) on logarithmic paper using the same scale as the master curve given in Figure 2.5. Match the plotted curve, fdp, with the appropriate curve of Figure 2.5. Choose any convenient point and read from the master graph (CDe2s)M, (ΔP/t 24C/qB)M and (kh/µ t/C)M. Read from the fdp: (ΔP/t)M and tM.
Earlougher and Kersch type‐curve for a well in infinite reservoir with wellbore storage and skin, taken from [2, 8].
Find wellbore storage coefficient, formation permeability, and skin factor using, respectively, the below expressions:
Another important type curve that is supposed to provide a better match was presented by Bourdet et al. [73], Figure 2.6. This includes both pressure and pressure derivative curves. The variables to be matched are ΔPM, (t*ΔP′)M, (PD)M, [(tD/CD)PD′]M, tM, (tD/CD)M, and (CDe2s)M. The equations use after the matching are [73]:
Bourdet et al. [73] pressure and pressure‐derivative versus time‐type curve.
The conventional method implies plotting either pressure or pressure drop against a given time function. The intercept and slope of such plot is used for reservoir and well parameters estimation. When the fluid initiates its path from the farthest reservoir point until the well head, several states and flow regimes are observed depending on the system geometry. For instance, if the reservoir has an elongated shape, probably linear flow will be observed. Linear flow obeys a pressure dependency on the square‐root of time, or, if the fluid experiences radial flow regime, the relation between pressure and time observes a semilog behavior, or, either inside the well or the limitation of the reservoir boundaries imply a pseudosteady‐state condition, then, pressure is a linear function of time.
The time function depends on the system geometry and could be any of the kinds described by Eq. (2.34).
Normally, the pressure or pressure drop are plotted in Cartesian coordinates, except certain few cases as for the Muskat method, see Chapter 3, which requires a potential plot, meaning, logarithm scale of pressure drop in the y‐axis and Cartesian scale for time in the x‐direction.
It is commonly referred as the “semilog method” since the radial flow is the most important regime found on a pressure test. Then, a semilogarithm plot is customary used in well test analysis.
Starting by including the skin factor in Eq. (1.106);
Replacing the dimensionless terms given by Eqs. (1.89) and (1.94) into Eq. (2.35) and dividing both terms by ln 10 will lead to:
Solving for the well‐flowing pressure;
Behavior of the well‐flowing pressure observed in a semilog graph, taken from [68].
Eq. (2.37) suggests a straight‐line behavior which is represented in the central region of Figure 2.7. The other two regions are affected by wellbore storage and skin effects, at early times and boundary effects at late times. Reservoir transmissivity, mobility, or permeability can be determined from the slope;
The intercept of Eq. (2.34) is used for the determination of the mechanical skin factor. For practical purposes, the well‐flowing pressure at time of 1 h, P1hr, is read from the straight‐line portion of the semilog behavior, normally extrapolated as sketched in Figure 2.7, so solving for skin factor, s, from Eq. (2.34) results:
Since the slope possesses a negative signed, so does the P1hr− Pi term. Therefore, the first fractional in the above equation is always positive unless the well is highly stimulated.
According to Eq. (2.39), the contribution to the pressure drop caused by the mechanical skin factor is included to the last term: 0.8686s multiplied by the slope. Then:
Eq. (2.40) is similar to Eq. (1.110) and works for either pressure drawdown or pressure buildup tests.
Eq. (1.110) is useful to find either skin factor, s, formation damaged permeability, ks, or the damaged or affected skin zone radius, rs. However, since the skin zone covers an infinitely thin area and the pressure wave travels at high speed, it is difficult to detect transmissivity changes, then, rs and ks are difficult to be measured.
Eqs. (1.110) and (2.37) imply the skin factor along flow rate just increases or decreases the well pressure drop. However, this occurs because the well radius behaves as if its radius was modified by the value of the skin factor. Brons and Miller [74] defined the apparent or effective wellbore radius, rwa, to be used in Eqs. (1.89), (1.94), and (1.100)
Example 2.1
A well with a radius of 0.25 ft was detected to have a skin factor of 2. A skin factor of −2 was obtained after a stimulation procedure. Find the apparent radii and the percentage of change in the radius due to the stimulation. What conclusion can be drawn?
Solution
Application of Eq. (2.42) for the damaged‐well case gives:
Application of Eq. (2.42) for the damaged‐well case gives:
It can be observed that 1.847 × 100/0.034 ≅ 5460%, meaning that the stimulation helps the well to increase its radius 55 times. It can be concluded from the example that for positive skin factor values, the effective wellbore radius decreases (rwa<rw) and for negative skin factor values, the effective wellbore radius increases (rwa>rw).
The starting time of the semilog straight line defined by Ramey [9] in Eq. (2.22) allows determining mathematically where the radial flow starts, i.e., the moment wellbore storage effects no longer affect the test. Replacing into Eq. (2.22) the dimensionless parameters given by Eqs. (1.94) and (2.15) results [2]:
The application of Eq. (2.40) is twofolded. (1) It can be used for test design purposes. The duration of a pressure drawdown test should be last 10 times the value of tSSL, so a significant portion of the radial flow regime can be observed and analyzed and (2) finding the semilog slope can be somehow confusing. Once the semilog line is drawn and permeability, skin factor, and wellbore storage are calculated, then, Eq. (2.40) can be used to find the starting point of the radial flow regime. Radial flow is correctly found if the tSSL value agrees with the one chosen in the plot. This last situation is avoid if the pressure and pressure derivative plot is available since radial flow is observed once the pressure derivative curve gets flat as seen in Figure 2.6.
The declination stabilization time (time required to reach the boundaries and develop the pseudosteady‐state period) during the test can be from the maximum time at which the maximum pressure drops (not shown here) take place. This is:
From which;
For square or circular geometries, tDA = 0.1 from Table 2.1. Replacing this value in Eq. (1.100) and solving for time leads to:
from
For any producing time, tp, the radius of investigation—not bigger than re—can be found.
The point reached by the disturbance does not imply fluid movement occurs there. The drainage radius is about 90% that value, then
Skin factor is a dimensionless quantity. This does not necessarily reflect the degree of either damage or stimulation of a well. Then, more practical measurement parameters ought to be used. One of this is the flow efficiency, FE, which implies what percentage of the total pressure drawdown is due to skin factor. The flow efficiency is defined as the ratio between the actual productivity index, J, and the ideal productivity index. The productivity index involves money since it is defined as the amount of pressure drop needed to produce a barrel of fluid per day. In other words, it is the energy required to produce one BPD. Mathematically;
FE < 1 is an indication that well damage exists, otherwise there is stimulation. The productivity index can be increased by:
Increasing the permeability in the zone near the well—hydraulic fracturing;
Reduce viscosity—steam injection, dissolvent, or in situ combustion;
Damage removing—acidification;
Increase well penetration;
Reduce volumetric factor—choosing correct surface separators.
Other parameters to quantify well damage are [68]:
Damage ratio, DR
Damage ratios less than the unity indicate stimulation.
Damage factor, DF
Negative values of damage factors indicate stimulation. The damage factor can also be estimated from [68]:
Eq. (2.54) applies to circular‐shaped reservoir.
Productivity ratio, PR
Annual loss income, FD$L (USD$)
where OP is oil price.
Example 2.2
What will be the annual loss of a well that produces 500 BFD, which has a damage factor of 8, drains an area of 120 acres and has a radius of 6 inches? Assume circular reservoir area and a price of oil crude of USD $ 55/barrel.
Characteristics found in the Cartesian graph, taken from [68].
Solution
120 acres = 5,227,200 ft2. If the area is circular, then: r = 1290 ft. Find the damage factor from Eq. (2.54);
Find the yearly loss income using Eq. (2.56)
This indicates that the well requires immediate stimulation.
It is a drawdown test run long enough to reach the reservoir boundaries. Normal pressure drawdown tests, during either radial flow or transient period test, are used to estimate formation permeability and artificial well conditions (C and s), while an RLT test—introduced by [76]—deals with boundaries and is employed to determine well drainage area or well drainage pore volume. In a Cartesian graph for a closed boundary system, Figure 2.8, three zones are distinguished [8, 68]: (i) skin and wellbore storage dominated zone, (ii) transient zone (radial flow), and (iii) pseudosteady‐state zone. As indicated by Eq. (1.129), the pressure drop is a linear function of time. Eq. (1.129) is given for circular reservoir geometry. For any geometry, the late time pseudosteady‐state solution involves the Dietz shape factor, [75], to extent the use of Eq. (1.129) for other reservoir geometries, as described in Table 2.1. Under this condition, Eq. (1.129) becomes [77]:
Replacing in the above expression the dimensionless quantities given by Eqs. (1.89) and (1.94), it results:
From the slope, m*, and intercept, PINT, of Eq. (2.58), the reservoir pore volume and Dietz shape factor [74] can be obtained from either:
Once the value of CA is obtained from Eq. (2.60), the reservoir geometry can be obtained from Table 2.1 by using the closest tabulated value (“exact for tDA”) and confront with the time to develop pseudosteady‐state regime, (tDA)pss which is found from:
tpss can be read from the Cartesian plot. However, this reading is inexact; therefore, it is recommended to plot the Cartesian pressure derivative and to find the exact point at which this becomes flat.
TDS technique is the latest methodology for well test interpretation. Its basis started in 1989 [70]. TDS’ creator was Tiab [71], who provided analytical and practical solutions for reservoir characterization using characteristic points or features—called by him “fingerprints”—read from a log‐log plot of pressure and pressure derivative [15], versus time. Since the introduction of TDS in 1995, several scenarios, reservoir geometries, fluid types, well configurations, and operation conditions. For instance, extension of TDS technique to elongated systems can be found in [13, 14, 16–19, 23, 24, 28, 30, 31]. Some applications of conventional analysis in long reservoirs are given in [20, 29, 38, 54]. For vertical and horizontal gas wells with and without use of pseudotime, refer to [22, 36, 39]. Special cases of horizontal wells are found in [12, 47]. For transient rate analysis, refer to [27, 35, 49]. Applications on heavy oil (non‐Newtonian fluids) can be found in [32, 34, 41, 42, 45, 52, 56, 62, 64]. For cases on shales reservoirs, refer to [49, 51, 56, 78]. Well test analysis by the TDS technique on secondary and tertiary oil recovery is presented by [25, 33, 60, 79]. For multirate testing in horizontal and vertical wells, refer, respectively, to [65, 67]. References [43, 46] are given for conductive faults. For deviated and partially penetrated wells, refer to [37, 64], respectively. TDS technique extended to multiphase flow was presented by [26]. Wedged and T‐shaped reservoirs can be found in [48] and coalbed‐methane reservoirs with bottom water drive are given in [53]. TDS technique is excellent for interpreting pressure test in hydraulically fractured vertical wells since unseen flow regimes can be generated [50, 69, 80]. The first publications on horizontal wells in naturally fractured and anisotropic media are given in [81, 82]. The threshold pressure gradient is dealt by [57, 72]. For vertical wells in double porosity and double permeability formations, refer, respectively, to [41, 83]. A book published by Escobar [56] presents the most recent topics covered by the TDS technique, and a more comprehensive state‐of‐the‐art on TDS technique is given by [58]. This book revolves around this methodology; therefore, practically, the whole content of [71]—pioneer paper of TDS technique—will be brought here:
The starting point is the definition of the dimensionless pressure derivative from Eq. (1.89);
By looking at Eqs. (2.17) and (2.18), we can conclude the difficulty of using hand mathematical operations with them. Instead of using these general solutions, Tiab [71] obtained partial solutions to the differential equation for each flow regime or time period. For instance, during early pseudosteady‐state, the governing equation reduces to:
Combination of Eqs. (1.94) and (2.15) results in:
Replacing Eq. (1.89) in the above expression yields;
Solving for C;
The pressure derivative curve also has a straight line of unitary slope at early times. The equation of this line is obtained by taking the derivative of Eq. (2.63) with respect to the natural logarithm of tD/CD. So:
Where the derivative of the dimensionless pressure is:
Rearranging;
Converting to dimensional form, the left‐hand side of Eq. (2.67) by using the definitions given by Eqs. (2.64) and (2.68):
Multiplying and dividing by 0.8935;
Recalling Eq. (2.15), the above becomes:
Since the unit slope is one, then CD = 1, thus;
From looking at Figure 2.6, both pressure and pressure derivative curves display a unitary slope at early times. Replacing Eqs. (2.64), (2.73) in (2.67) and solving for C will result:
As seen in Figure 2.6, the infinitely acting radial flow portion of the pressure derivative is a horizontal straight line with intercept of 1/2. The governing equation is:
Combining the above equation with Eq. (2.73) results the best expression to estimate reservoir permeability:
Subscript r stands for radial flow line. A customary use of TDS, as established by Tiab [71], is to provide suffices to identify the different flow regimes. For instance, pss stands for pseudosteady state, i stands for either initial or intercept, etc. In terms of pressure, the equation of this line is:
It is recommended to draw a horizontal line throughout the radial flow regime and choose one convenient value of (t*ΔP\')r falling on such line.
Tiab [71] also obtained the start time of the infinite line of action of the pressure derivative is:
Replacing Eqs. (1.92) and (2.15) in the above equation will yield:
A better form of Eq. (2.78) was given by [84];
Setting a = 0.05 in the above equation and solving for C:
tDsr is calculated with Eq. (1.94) letting t = tsr.
The point of intersection, i, between the early time unit‐slope line defined by Eqs. (2.63) and (2.67) and the late‐time infinite‐acting line of the pressure derivative, defined by Eq. (2.75), is given by:
where i stands for intersection. After replacing the definitions given by Eqs. (1.94), (2.15), and (2.72) will, respectively, provide:
For the unit‐slope line, the pressure curve is the same as for the pressure derivative curve. Then, at the intersection point:
Tiab [71] correlated for CDe2s > 100 permeability, wellbore storage coefficient, and skin factor with the coordinates of the maximum point—suffix x—displayed once the “hump” observed once wellbore storage effects start diminishing. These correlations are given as follows:
Replacing Eqs. (2.64) and (2.73) into Eq. (2.87) leads to:
Either formation permeability or wellbore storage coefficient can be determined using the coordinates of the peak, tx and (t*ΔP′)x. Solving for both of these parameters from Eq. (2.90) results:
The constants in Eqs. (2.91) and (2.92) are slightly different as those in [58]. These new unpublished versions were performed by TDS’ creator.
Eq. (2.91) is so helpful to find reservoir permeability in short test when radial flow is absent which is very common in fall‐off tests. Once permeability is found from Eq. (2.91), solved for (t*ΔP′)r from Eq. (2.76) and plot on a horizontal line throughout this value. Then, compare with the actual derivative plot and use engineering criterion to determine if the permeability value is acceptable. This means, if the straight line is either lower or higher than expected. Otherwise, new coordinates of the peak ought to be read for repeating the calculations since the hump should look some flat.
Substitution of Eqs. (2.64) and (2.73) in Eqs. (2.88) and (2.89) allows obtaining two new respective correlations for the determination of the mechanical skin factor:
Sometimes, the reading of the peak coordinates may be wrong due to the flat appearance of it. Then, it should be a good practice to estimate the skin factor using both Eqs. (2.93) and (2.94). These values should match each other.
Divide Eq. (2.87) by Eq. (2.75); then, in the result replace Eqs. (2.64) and (2.73) and solve for both permeability and wellbore storage:
This last expression is useful to find wellbore storage coefficient when the early unitary slope line is absent.
TDS technique has a great particularity: for a given flow regime, the skin factor equation can be easily derived from dividing the dimensionless pressure equation by the dimensionless derivative equation of such flow regime. Then, the division of Eq. (2.77) by Eq. (2.75) leads to the below expression once the dimensionless parameters given by Eqs. (1.89), (1.94), and (2.73) are replaced in the resulting quotient. Solving for s from the final replacement leads to:
being tr any convenient time during the infinite‐acting radial flow regime throughout which a horizontal straight line should have been drawn. Read the ΔPr corresponding to tr. Comparison between Eqs. (2.38) and (2.76) allows concluding:
which avoids the need of using the semilog plot if the skin pressure drop is needed to be estimated by Eq. (2.40), otherwise, Eq. (2.40) becomes:
For the determination of well‐drainage area, Tiab [69] expressed Eq. (2.75) as:
Also, Tiab [69] differentiated the dimensionless pressure with respect to dimensionless time in Eq. (2.57), so:
Then, Tiab [69] based on the fact that two given flow regime governing equations can be intersected each other, regardless the physical meaning of such intersection, and solving for any given parameter, intercepted Eqs. (2.100) with (2.101), then, replaced in the resulting expression the dimensionless quantities given by Eqs. (2.92), (2.97), and (2.62) and solved for the area given in ft2:
Furthermore, Chacon et al. [85] replaced the dimensionless time given by Eq. (1.100) and the dimensionless pressure derivative of Eq. (2.62) into Eq. (2.102) and also solved for the area in ft2:
The above expression uses any convenient point, tpss and (t*ΔP’)pss, during the late time pseudosteady‐state period. Because of noisy pressure derivative data, the readings of several arbitrary points may provide, even close, different area values. Therefore, it is convenient to use an average value. To do so, it is recommended to draw the best late‐time unit‐slope line passing through the higher number of pressure derivative points and extrapolate the line at the time of 1 h and read the pressure derivative value, (t*ΔP\')pss1. Under these circumstances, Eq. (2.103) becomes:
Eqs. (2.102) through (2.104) apply only to closed‐boundary reservoirs of any geometrical shape. For constant‐pressure reservoirs, the works by Escobar et al. [28, 54] for TDS technique (summary given in Table 2.2) and for conventional analysis are used for well‐drainage area determination in circular, square, and elongated systems.
TDS technique has certain step‐by‐step procedures which not necessarily are to be followed since the interpreter is welcome to explore and use TDS as desired. Then, they are not provided here but can be checked in [69, 71].
Example 2.3
Taken from [68]. The pressure and pressure derivative data given in Table 2.3 corresponds to a drawdown test of a well. Well, fluid, and reservoir data are given below:
rw = 0.267 ft | q = 250 BPD | μ = 1.2 cp | |
ct = 26.4 × 10−5psi−1 | h = 16 pies | ϕ = 18% | B = 1.229 bbl/BF |
Find permeability, skin factor, drainage area, and flow efficiency by conventional analysis. Find permeability, skin factor, and three values of drainage area using TDS technique:
Solution
Conventional analysis. Figure 2.9 and 2.10 present the semilog and Cartesian plots, respectively, to be used in conventional analysis. From Figure 2.9, the semilog slope, m, is of 18 psia/cycle and P1hr = 2308 psia. Permeability and skin factor are calculated using Eqs. 2.38 and 2.39, respectively, thus:
Find the pressure loss due to skin factor with Eq. (2.40);
Since the average reservoir pressure is not reported, then, the initial pressure value is taken instead. Eq. (2.51) allows estimating the flow efficiency.
Summary of equations, taken from [28].
t, h | Pwf, psia | DP, psia | t*DP′, psia/h | t, h | Pwf, psia | DP, psia | t*DP′, psia/h |
---|---|---|---|---|---|---|---|
0.00 | 2733 | 0 | 5 | 2312 | 421 | 65.42 | |
0.10 | 2703 | 30 | 31.05 | 7 | 2293 | 440 | 35.32 |
0.20 | 2672 | 61 | 58.95 | 9.6 | 2291 | 442 | 5.86 |
0.30 | 2644 | 89 | 84.14 | 12 | 2290 | 443 | 5.85 |
0.40 | 2616 | 117 | 106.30 | 16.8 | 2287 | 446 | 7.63 |
0.65 | 2553 | 180 | 129.70 | 33.6 | 2282 | 451 | 7.99 |
1.00 | 2500 | 233 | 135.15 | 50 | 2279 | 454 | 7.94 |
1.50 | 2440 | 293 | 151.90 | 72 | 2276 | 457 | 10.50 |
2.00 | 2398 | 335 | 127.26 | 85 | 2274 | 459 | 12.18 |
3.00 | 2353 | 380 | 102.10 | 100 | 2272 | 461 | 13.36 |
4.00 | 2329 | 404 | 81.44 |
Pressure and pressure derivative versus time data for example 2.3.
Semilog plot for example 2.3.
Cartesian plot for example 2.3.
Pressure and pressure derivative plot for example 2.3.
From the Cartesian plot, Figure 2.10, is read the following data:
m* = −0.13 psia/h | PINT = 2285 psia | tpss ≈ 50 h |
Use Eq. (2.59) to find well drainage area:
Find the Dietz shape factor with Eq. (2.60);
As observed in Table 2.1, there exist three possible well drainage area geometry values (hexagon, circle, and square) close to the above value. To discriminate which one should be the appropriate system geometry find the dimensionless time in which pseudosteady‐state period starts by using Eq. (2.61):
TDS technique. The following are the characteristic points read from Figure 2.11:
(t*ΔP\')r = 7.7 psia | tr = 33.6 h | ΔPr = 451 psia |
tpss = 85 h | (t*ΔP\')pss = 12.18 psia | trpi = 58 h |
(t*ΔP\')pss1 = 0.14 psia |
Find permeability and skin factor with Eqs. (2.76) and (2.97), respectively:
Determine the well drainage area with Eqs. (2.102) and (2.103), thus;
Even, more parameters can be reestimated with TDS technique for verification purposes but it will not be performed for saving‐space reasons. However, the reader is invited to read the coordinates of the peak and the intersection point of the wellbore storage and radial flow lines. Then, estimate formation permeability with Eqs. (2.84), (2.85), and (2.91). Also, find the wellbore storage coefficient using Eqs. (2.74), (2.81), (2.92), and (2.96) and skin factor with Eqs. (2.93) and (2.94).
Example 2.4
Taken from [68] with the data from the previous example, Example 2.3, determine tSSL and find if the well fluid level is increasing or decreasing in the annulus if the well has a drill pipe with 2 in external diameter inside a liner with 5 in of inner diameter including joint gaskets. The density of the wellbore fluid is 42.5 lbm/ft3.
Solution
From Figure 2.11, a point is chosen on the early unit‐slope line. This point has coordinates: DP = 59 psia and t = 0.2 h. Wellbore storage coefficient is found with Eq. (2.21):
Solving for annulus capacity from Eq. (2.5);
The theoretical capacity is found with Eq. (2.45), so:
This leads to the conclusion that the annular liquid is falling.
According to Perrine [86], the single fluid flow may be applied to the multiple fluid flow systems when the gas does not dominate the pressure tests, it means liquid production is much more relevant than gas flow. Under this condition, the diffusivity equation, Eq. (1.27), will result and the total fluid mobility is determined by Eq. (1.24). We also mentioned in Chapter 1 that Martin [63] provided some tips for a better use of Perrine method. Actually, Perrine method works very well in liquid systems.
The semilog equations for drawdown and build tests are, respectively, given below:
The flow rate is estimated by:
Eq. (2.107) is recommended when oil flow dominates the test. It is removed from the denominator, otherwise. It advised to use consistent units in Eq. (2.107) meaning that the gas flow rate must be in Mscf/D and the gas volume factor bbl/SCF.
Once the semilog slope has been estimated, the total mobility, the phase effective permeabilities, and the mechanical skin factor are found from:
The best way of interpreting multiphasic flow tests in by using biphasic and/or triphasic pseudofunctions. Normally, well test software uses empirical relationships to estimate relative permeability data. The accuracy of the following expression is sensitive to the relative permeability data:
The expressions used along this textbook for reservoir characterization may apply for both single fluid and multiple fluid production tests. Single mobility has to be changed by total fluid mobility and individual flow rate ought to be replaced by the total fluid rate. Just to cite a few of them, Eqs. (2.66), (2.76), (2.85), (2.91), (2.92), and (2.97) become:
Also, the effective liquid permeabilities are found using the individual viscosity, rate, and volume factor. Then, Eq. (2.76) applied to oil and water will yield:
However, from a multiple fluid test, it is a challenge to find the reservoir absolute permeability. Several methods have been presented. For instance, Al‐Khalifah et al. [87] presented a sophisticated method applied to either drawdown or multiple rate tests. Their method even includes the estimating of the saturation change respect to pressure. However, we presented the method by Kamal and Pan [88] which applies well for liquid fluid. Relative permeabilities must be known for its application. Once effective permeabilities are found, let us say from Eqs. (2.119) and (2.120), estimate the permeability ratio ko/kw and find the water saturation from the relative permeability curves as schematically depicted in Figure 2.12 (left). Then, using the estimated water saturation value, enter Figure 2.12 (right) and read a value from a relative permeability curve. Use the most dominant flow curve. The dominant phase is assumed to be oil for the example in Figure 2.12. Since both phase effective permeability and phase relative permeability are known, the absolute permeability is found from the definition of relative permeability:
Determination of absolute permeability as outlined by Kamal and Pan [88].
Further recommendations for handling multiphase flow tests are presented by Al‐Khalifah et al. [89] and are also reported by Stanislav and Kabir [7].
When a well penetrates a small part of the formation thickness, hemispherical flow takes place. See Figure 2.13 top. When the well is cased above the producer range and only a small part of the casing is perforated, spherical flow occurs in the region near the face of the well. See Figure 2.13 bottom. As the transient moves further into depth of the formation, the flow becomes radial, but if the test is short, the flow will be spherical. Both types of flow are characterized by a slope of −1/2 in the log‐log plot of pressure derivative versus time [90, 91]. Theoretically, before either hemispherical or spherical flow takes place, there exists a radial flow regime occurring by fluids withdrawn from the formation thickness that is close in height to the completion interval. This represents the transmissibility of the perforated interval. Actually, this flow regime is unpractical to be seen mainly because of wellbore storage effects. We will see further in this chapter that there are especial conditions for hemispherical/spherical flow to be observed which occur later that the completion‐interval‐limited radial flow regime. Both hemispherical and spherical flow vanished when the top and bottom boundaries have been fully reached by the transient wave; the true radial flow is developed throughout the full reservoir thickness.
Ideal flow regimes in partial penetration (top) and partial completion (bottom) systems, after [66].
The apparent skin factor, sa, obtained from pressure transient analysis is a combination of several “pseudoskin” factors such as [91]:
where s is the true damage factor caused by damage to the well portion, sp is the pseudoskin factor due to the restricted flow entry, sq is the pseudoskin factor resulting from a well deviation angle, and scp is the pseudoskin due to a change in permeability near the face of the well. sp can be estimated from [92]:
hp = length of perforated or open interval. The equations of dimensionless thickness, hD, for hemispherical and spherical flow, respectively:
where kh is the horizontal permeability, kz = kv is the vertical permeability. The contribution of the pseudoskin of an inclined well is given by Cinco et al. [92]:
According to Cinco et al. [92], the above equation is valid for 0° ≤ q ≤ 75°, h/rw > 40, and tD > 100. Note that Eq. (2.127) could provide a negative value. This is because the deviation at the face of the well increases the flow area or presents reservoir pseudothickness. The pseudoskin responding for permeability changes near wellbore is given by [93]:
Example 2.4
Taken from [91]. A directional well which has an angle to the vertical of 24.1° has a skin factor s = −0.8. The thickness of the formation is 100 ft, the radius of the wellbore is 0.3 ft, and the horizontal to vertical permeability ratio is 5. Which portion of the damaged corresponds to the deviation of the well?
Solution
The deviation angle affected by the anisotropy is estimated with Eq. (2.126);
The pseudoskin factor caused by well deviation is found from Eq. (2.127):
From Eq. (2.122);
Therefore, 66.1 % of the skin factor is due to the well deviation.
The diffusivity equation for spherical flow assuming constant porosity, compressibility, and mobility is given by Abbott et al. [90]:
where ksp is the spherical permeability which is defined as the geometrical mean of the vertical and horizontal permeabilities:
The physical system is illustrated in Figure 2.14, right. This region is called a “spherical sink.” rsw is given by:
The spherical flow equations for pressure drawdown and pressure buildup when the flow time is much longer than the shut‐in time were presented by [94]:
Cylindrical, hemispherical, and spherical sinks, after [66].
The spherical pressure buildup equation when the flow time is shorter than the shut‐in time:
Then, from a Cartesian plot of Pwf as a function of t −1/2, for drawdown, or Pws as a function of either [tp−1/2 + Δt−1/2−(tp + Δt)−1/2] or [Δt−1/2−(tp + Δt)−1/2] for buildup, we obtain a line which slope, m, and intercept, I, can be used to estimate tridimensional permeability and geometrical (spherical) skin factor.
Once the spherical permeability is known, we solve for the vertical permeability from Eq. (2.130), and then, estimate the value of skin effects due to partial penetration [94]:
where b = hp/h. hD can be estimated from Eq. (2.125), and G is found from [94]:
The model for hemispheric flow is very similar to that of spherical flow [94]. The difference is that a boundary condition considers half sphere. Figure 2.14 (left) outlines the geometry of such system. The drawdown and pressure equations are given below [94]:
As for the spherical case, from a Cartesian plot of Pwf as a function of t−1/2, for drawdown, or Pws as a function of either [tp−1/2 + Δt−1/2−(tp + Δt)−1/2] or [Δt−1/2− (tp + Δt)−1/2] for buildup, we obtain a line which slope, m, and intercept, I, can be used to estimate spherical permeability and geometrical (spherical) skin factor.
Moncada et al. [66] presented the expressions for interpreting both pressure drawdown or buildup tests in either gas or oil reservoirs using the TDS methodology. Spherical permeability is estimated by reading the pressure derivative at any arbitrary time during which spherical flow can calculate spherical permeability and the spherical skin factor also uses the pressure reading at the same chosen time:
The total skin, st, is defined as the sum of all skin effects at the well surroundings:
If the radial flow were seen, the horizontal permeability can be estimated from:
The suffix r1 implies the first radial flow.
Moncada et al. [66] observed that the value of the derivative for the late radial flow in spherical geometry is equivalent to 0.0066 instead of 0.5 as of the radial system. In addition, the slope line −½ corresponding to the spherical flow and the late radial flow line of the curve of the dimensionless pressure derivative in spherical symmetry intersect in:
Replacing the dimensionless time results:
In the above equation, suffix i denotes the intersection between the spherical flow and the late radial flow. If the radial flow is not observed, this time can give an initial point to draw the horizontal line corresponding to the radial flow regime, from which horizontal permeability is determined. This point can also be used to verify spherical permeability, ksp. Another equation defining the mentioned dimensionless time can be found from the intersection of the slope line −½ (spherical flow) with the radial line of late radial flow but in radial symmetry, knowing that:
Replacing the dimensionless time will give:
Combining Eqs. (2.149) and (2.151), an expression to find the spherical wellbore radius, rsw:
Here the same considerations are presented in Section 2.4.3. Using a pressure and a pressure derivative value reading at any time during hemispherical flow allows finding hemispherical permeability and partial penetration skin [66],
Moncada et al. [66] also found that the derivative in spherical geometry of the late radial flow corresponds to 0.0033 instead of 0.5 as of the radial system. This time the line of radial flow and hemispheric flow, in hemispherical symmetry, intersect in:
From where,
As for the spherical case, there exists an expression to define the intersection time of the −½ slope line of the hemispherical flow regime pressure derivative and the late radial flow line pressure derivative but, now, in radial symmetry:
This point of intersection in radial symmetry gives the following equation:
Skin factors are estimated in a manner similar to Section 2.4.3.
It is important to identify the range of WBS values, which can influence the interpretation of the spherical and hemispheric flow regime. Figure 2.15 is a plot of PD vs. tD providing an idea of the storage effect. As can be seen, the pressure response for several CD values can be distinguished when storage is low (<10), whereas for larger CD values, the response is almost identical. For CD < 10, the slope of −½ that characterizes both spherical and hemispherical flow is well distinguished. For values of 10 < CD < 100, the slope of −½ is more difficult to identify. For values of CD > 100, the spherical flow regime has been practically masked by storage, which makes it impossible to apply the technique presented above to estimate the vertical permeability. Then, to ensure there is no CD masking, it should be less than 10 [66].
The length of the completed interval or the length of the partial penetration, hp, plays an important role in defining the spherical/hemispherical flow. The presence of spherical or hemispheric flow is characterized by a slope of −½. This characteristic slope of −½ is absent when the penetration ratio, b = hp/h, is greater than 20% [66], as shown in Figure 2.16.
Example 2.5
Abbott et al. [90] presented pressure‐time data for a pressure drawdown test. Well no. 20 is partially completed in a massive carbonate reservoir. The well was shut‐in for stabilization and then flowed to 5200 BOPD for 8.5 h. The pressure data are given in Table 2.4 and reservoir and fluid properties are given below:
h = 302 ft | rw = 0.246 f | Pi = 2298 psia |
hp = 20 ft | q = 5200 BPD | B = 1.7 bbl/STB |
φ = 0.2 | μ = 0.21 cp | ct = 34.2 × 10−6 psia−1 |
Pressure derivative spherical source solution for a single well in an infinite system including WBS and no skin, after [66].
Pressure derivative behavior for a single well in an infinite reservoir with different partial penetration lengths (CD = 0, s = 0), after [66].
Solution by conventional analysis
Using the slope value of −122 psia/cycle read from the semilog plot of Figure 2.17, the reservoir permeability is calculated with Eq. (2.38);
The mechanical skin factor is determined with Eq. (2.39) once the intercept of 2252 psia is read from Figure 2.17.
t, h | t−0.5, h−0.5 | Pwf, psia | ΔP, psia | t*ΔP′, psia |
---|---|---|---|---|
0.0 | 2266 | 0 | ||
0.5 | 1.414 | 2255 | 11 | 11.5 |
1.0 | 1.000 | 2243 | 23 | 24.5 |
1.6 | 0.791 | 2228 | 38 | 40.0 |
2.0 | 0.707 | 2218 | 48 | 45.0 |
2.5 | 0.632 | 2208 | 58 | 52.5 |
3.0 | 0.577 | 2197 | 69 | 69.0 |
3.5 | 0.535 | 2185 | 81 | 66.5 |
4.0 | 0.500 | 2178 | 88 | 60.0 |
4.5 | 0.471 | 2170 | 96 | 56.3 |
5.5 | 0.426 | 2161 | 105 | 46.8 |
6.0 | 0.408 | 2157 | 109 | 48.0 |
6.5 | 0.392 | 2153 | 113 | 52.0 |
7.0 | 0.378 | 2149 | 117 | 49.0 |
7.5 | 0.365 | 2146 | 120 | 52.5 |
8.0 | 0.354 | 2142 | 124 | 48.0 |
8.5 | 0.343 | 2140 | 126 |
Pressure and pressure derivative versus time data for example 2.5.
Figure 2.18 contains a Cartesian graph of Pwf as a function of t−1/2. From there, the observed slope is m = 250 psia (h−1/2) and intercept, I = 2060 psia, spherical permeability, and spherical skin factors are calculated using Eqs. (2.239) and (2.240), respectively:
Vertical permeability and spherical wellbore radius are found with Eq. (2.130) and (2.131), respectively,
With the value of the vertical permeability, it is possible to estimate the skin factor caused by partial penetration with Eqs. (2.125), (2.138), and (2.137), thus:
Semilog plot for well no. 20.
Cartesian spherical plot for well no. 20.
Pressure and pressure derivative versus time log‐log plot for well no. 20.
Solution by TDS technique
The following data points were read from Figure 2.19.
tN = 1 h | ΔP = 23 psia | |
(t*ΔP′)sp = 56.25 psia | ΔPs = 96 psia | tsp = 4.5 h |
(t*ΔP′)r2 = 52.5 psia | ΔPr2 = 96 psia | tr2 = 7.5 h |
Wellbore storage coefficient is found from Eq. (2.66)
From the spherical flow pressure derivative line, m = −1/2, the spherical permeability and mechanical spherical skin factor are, respectively, estimated by Eqs. (2.144) and (2.145);
The horizontal permeability and mechanical skin are found during the late radial flow using Eqs. (2.76) and (2.97), respectively;
Vertical permeability is determined from Eq. (2.130);
Table 2.5 presents the comparison of the results obtained by the conventional method and TDS technique.
Parameter | Conventional | TDS |
---|---|---|
ksp, md | 7.01 | 8.05 |
ssp | −0.86 | −0.93 |
8.19 | 8.26 | |
sr | −5.03 | −5.53 |
kv, md | 7.10 | 7.65 |
Comparison of results.
So far, the considerations revolve around a single flow test, meaning the production rate is kept constant for the application of the solution of the diffusivity equation. However, there are cases in which the flow rate changes; in such cases, the use of the solution to the diffusivity equation requires the application of the time superposition principle already studied in Section 1.14.2. Some reasons for the use of multirate testing are outlined as follows:
It is often impractical to keep a constant rate for a long time to perform a complete pressure drawdown test.
When the well was not shut‐in long enough to reach the static pressure before the pressure drawdown test started. It implies superposition effects.
When, it is not economically feasible shutting‐in a well to run a pressure buildup test.
Whether the production rates are constant or not during the flow periods, there are mainly four types of multirate tests:
Uncontrolled variable flow rate;
Series of constant flow rates;
Variable flow rate while keeping constant bottom‐hole pressure, Pwf. This test is common in gas wells producing very tight formations and more recently applied on testing of shale formations;
Pressure buildup (fall‐off) tests.
Actually, a holistic classification of transient well testing is given in Figure 1.4. It starts with PTA which is known in the oil argot as pressure transient analysis. As seen in the figure, it is divided in single well tests, normally known as drawdown (flow) tests for production cases or injection tests for injection fluid projects. Our field of interest focuses on more than one rate operation (multirate testing) which includes all the four types just above described. It is worth to mention types 3 and 4. Type three is also known as rate transient analysis (RTA) which has been dealt with in a full chapter by this book\'s author in reference [56]. As far as case 4 is concerned, pressure buildup testing is the most basic multirate test ever existed since it comprises two flow rates: (1) one time period at a given q value different than zero and (2) another time period with a zero flow rate. This is because when a well is shut‐in, the flow stops at surface by the formation keeps still providing fluid to the well due to inertia.
Considering the sketch of Figure 2.20, application of the superposition principle [2, 3, 4, 6, 7, 11, 27, 44, 56, 60, 65, 67, 95, 96, 97] leads to:
Rearranging;
Next step is to replace PD by an appropriate diffusivity equation solution which depends upon the flow regime dealt with. Figure 2.21 presents the most typical superposition functions applied to individual flow regimes. The normal case is to use radial flow, top function in Figure 2.21. However, Escobar et al. [44] presented the inconvenience of not applying the appropriate superposition function for a given flow regime. They found, for instance, that if the radial superposition is used, instead of the linear, for characterization of an infinite‐conductivity hydraulic fracture, the estimated half‐fracture length would be almost three times longer than the actual one.
Coming back to Eq. (2.161), the assumed superposition function to be used is the radial one; then, this equation becomes:
Schematic representation of a multirate test (typ. 1).
Flow regime superposition functions.
Since it is uneasy to find natural log paper in the stationary shops, then, dividing for the natural log of 10 is recommended to express Eq. (2.162) in decadic log; then,
Simplifying;
Let;
Solving for skin factor from Eq. (2.165);
Plugging Eqs. (2.165), (2.166), and (2.168) into Eq. (2.164) will lead to:
which indicates that a Cartesian plot of ΔP/qn against the superposition time, Xn, provides a straight line which slope, m\', and intercept, m\'b’ allows finding reservoir permeability and skin factor using Eqs. (2.166) and (2.167), respectively. However, it is customary for radial flow well interpretation to employ a semilog plot instead of a Cartesian plot. This issue is easily solved by taking the antilogarithm to the superposition function resulting into the equivalent time, teq. Under this situation, Eq. (2.169) becomes:
And the equivalent time is then defined by,
For a two‐rate case, Russell [96] developed the governing well‐flowing pressure equation, as follows:
Therefore, the slope, m\'1, and intercept, PINT, of a Cartesian plot of Pwf versus log[(t1+Δt)/Δt] + (q2/q1)log(Δt) allows finding permeability and skin factor from the following relationships:
In general, the lag time, tlag, transition occurred during the rate change, is shorter when there is a rate reduction than a rate increment, i.e., if q2 < q1, then the tlag will be short and if q2 > q1, then the tlag will be longer due to wellbore storage effects.
The pressure drop across the damage zone is:
And;
P* is known as “false pressure” and is often used to estimate the average reservoir pressure which is treated in Chapter 3.
The mathematical details of the derivation of the equations are presented in detail by Perrine [86]. Application of TDS technique requires estimating the following parameters:
And equivalent time, teq, estimation is achieved using Eq. (2.172). Mongi and Tiab [67] suggested for moderate flow rate variation, to use real time rather than equivalent time with excellent results. In contrast, sudden changes in the flow rate provide unacceptable results. However, it is recommended here to always use equivalent time as will be demonstrated in the following exercise where using equivalent time the pressure derivative provides a better description. Mongi and Tiab [67] also recommended that test data be recorded at equal intervals of time to obtain smoother derivatives. However, it is not a practical suggestion since derivative plot is given in log coordinates. TDS is also applicable to two‐rate tests and there is also a TDS technique where there is a constant flow rate proceeded by a variable flow rate. For variable injection tests, refer to [60].
With the equivalent time, Eq. (2.172) determines the pressure derivative, teq*(DP/q)\', and plot the derivative in a similar fashion as in Section 2.2.4; wellbore storage coefficient can be obtained by taking any point on the early‐time unit‐slope line by:
Permeability and mechanical skin factor are estimated from:
Once again, rigorous time instead of equivalent time can be used in Eqs. (2.182) and (2.183); however, a glance to Figure 2.23 and 2.24 tells us not to do so.
Example 2.6
Earlougher and Kersch [8] presented an example to estimate permeability using a Cartesian plot of flowing pressure, Pwf, versus superposition time, Xn, and demonstrated the tedious application of Eq. (2.168). A slope of 0.227 psia/(BPD/cycle) was estimated which was used in Eq. (2.166) to allow finding a permeability value of 13.6. We determined an intercept of 0.5532 psia/(BPD/cycle) which led us to find a skin factor of −3.87 with Eq. (2.167).
Use semilog conventional analysis and TDS technique to find reservoir permeability and skin factor, as well. Pressure and rate data are given in Table 2.6 along another parameters estimated here. Reservoir, fluid, and well parameters are given below:
Pi = 2906 psia | B = 1.27 bbl/STB | µ = 0.6 cp | |
h = 40 ft | rw = 0.29 ft | φ = 11.2% | ct = 2.4 × 10−61/psia |
Figure 2.22 is a semilog graph of [Pi−Pwf(t)]/qn versus t and teq. The purpose of this graph is to compare between the rigorous analysis using equivalent time, teq, and analysis using the real time of flow, t. Note that during the first cycle, the graphs of t and teq are practically the same. By regression for the real‐time case gave a slope m\' = 0.2411 psia/BPD/cycle and intercept ΔP/q(1hr) = 0.553 psia/BPD/cycle. Permeability and skin factors are calculated with the Eqs. (2.166) and (2.167), respectively:
The straight line with teq has a slope m’ = 0.2296 psia/BPD/cycle, and intercept ΔP/q(1hr) = 0.5532 psia/BPD/cycle. Then, permeability and skin factor estimated by Eqs. (2.166) and (2.167) are 13.49 md and −3.87, respectively.
Solution by TDS technique
The derivative of normalized pressure is also reported in Table 2.6. Figure 2.23 illustrates a log‐log plot of ΔPq versus teq and (t*ΔP\'q) and (teq*ΔP\'q) versus t and teq. Both derivatives were estimated with a smooth value of 0.5. During the first cycle, the two sets of data have roughly the same trend; also the flow regimes are quite different. Also, the equivalent normalized pressure derivative suggests a faulted system and possibly the pseudosteady‐state period has been reached. This last situation is unseen in the normalized pressure derivative. From this graph, the following values are read:
(t*ΔP\'q)r = 0.097 psia/BPD/cycle | (ΔPq)r = 0.693 psia/BPD/cycle | (teq)r = 4.208 h |
Permeability and skin factor are estimated, respectively, using Eqs. (2.182) and (2.183):
The comparison of the results obtained by the different methods is summarized in Table 2.7. The permeability absolute deviation with respect to arithmetic mean is less than 5% using actual time. Note that all results agree well. Even though, when Earlougher and Kersch [8] written, pressure derivative function was still in diapers; then, it was not possible to differentiate the second straight‐line which for Earlougher and Kersch [8] corresponded to pseudosteady‐state period instead of a fault as clearly seen in Figure 2.23. Also, the absolute deviation of the flow rate (referred to the first value) is less than 10% during radial flow regime. However, when using real time, the radial flow regime is different; then, the recommendation is to always use equivalent time.
n | t, h | q, BPD | Pwf, psia | ΔP, psia | ΔP /q, psia/BPD | Xn | teq, h | t*(ΔP/q)\', psia/BPD | teq*(ΔP/q)\', psia/BPD |
---|---|---|---|---|---|---|---|---|---|
0 | 2906 | ||||||||
1 | 1 | 1580 | 2023 | 883 | 0.559 | 0.000 | 1.000 | 0.559 | 0.261 |
1 | 1.5 | 1580 | 1968 | 938 | 0.594 | 0.176 | 1.500 | 0.594 | 0.131 |
1 | 1.89 | 1580 | 1941 | 965 | 0.611 | 0.276 | 1.890 | 0.611 | 0.102 |
1 | 2.4 | 1580 | |||||||
2 | 3 | 1490 | 1892 | 1014 | 0.681 | 0.519 | 3.306 | 0.681 | 0.099 |
2 | 3.45 | 1490 | 1882 | 1024 | 0.687 | 0.569 | 3.707 | 0.687 | 0.103 |
2 | 3.98 | 1490 | 1873 | 1033 | 0.693 | 0.624 | 4.208 | 0.693 | 0.099 |
2 | 4.5 | 1490 | 1867 | 1039 | 0.697 | 0.673 | 4.712 | 0.697 | 0.095 |
2 | 4.8 | 1490 | |||||||
3 | 5.5 | 1440 | 1853 | 1053 | 0.731 | 0.787 | 6.124 | 0.731 | 0.104 |
3 | 6.05 | 1440 | 1843 | 1063 | 0.738 | 0.819 | 6.596 | 0.738 | 0.111 |
3 | 6.55 | 1440 | 1834 | 1072 | 0.744 | 0.849 | 7.056 | 0.744 | 0.120 |
3 | 7 | 1440 | 1830 | 1076 | 0.747 | 0.874 | 7.481 | 0.747 | 0.128 |
3 | 7.2 | 1440 | |||||||
4 | 7.5 | 1370 | 1827 | 1079 | 0.788 | 0.974 | 9.412 | 0.788 | 0.148 |
4 | 8.95 | 1370 | 1821 | 1085 | 0.792 | 1.009 | 10.212 | 0.792 | 0.154 |
4 | 9.6 | 1370 | |||||||
5 | 10 | 1300 | 1815 | 1091 | 0.839 | 1.124 | 13.311 | 0.839 | 0.192 |
5 | 12 | 1300 | 1797 | 1109 | 0.853 | 1.153 | 14.239 | 0.853 | 0.188 |
6 | 14.4 | 1260 | |||||||
7 | 15 | 1190 | 1775 | 1131 | 0.950 | 1.337 | 21.746 | 0.950 | 0.205 |
7 | 18 | 1190 | 1771 | 1135 | 0.954 | 1.355 | 22.662 | 0.954 | 0.206 |
7 | 19.2 | 1190 | |||||||
8 | 20 | 1160 | 1772 | 1134 | 0.978 | 1.423 | 26.457 | 0.978 | 0.208 |
8 | 21.6 | 1160 | |||||||
9 | 24 | 1137 | 1756 | 1150 | 1.011 | 1.485 | 30.553 | 1.011 | 0.208 |
10 | 28.8 | 1106 | |||||||
11 | 30 | 1080 | 1751 | 1155 | 1.069 | 1.607 | 40.426 | 1.069 | 0.248 |
11 | 33.6 | 1080 | |||||||
12 | 36 | 1000 | |||||||
13 | 36.2 | 983 | 1756 | 1150 | 1.170 | 1.788 | 61.414 | 1.170 | 0.447 |
13 | 48 | 983 | 1743 | 1163 | 1.183 | 1.799 | 63.020 | 1.183 | 0.463 |
Slider [11, 98, 99] suggested a methodology to analyze pressure tests when there are no constant conditions prior to the test. Figure 2.24 schematizes a well with the shutting‐in pressure declining (solid line) before the flow test started at a time t1. The dotted line represents future extrapolation without the effect of other wells in the reservoir. The production starts at t1 and the pressure behaves as shown by the solid line [11].
Semilog of normalized pressure versus actual and equivalent time for example 2.6.
Normalized pressure and pressure derivative versus time and equivalent time log‐log plot for example 2.6.
The procedure suggested by Slider [11, 99] to correctly analyze such tests is presented below:
Extrapolate the shutting‐in pressure correctly (dotted line in Figure 2.24).
Estimate the difference between the observed well‐flowing pressure and the extrapolated pressure, ΔPΔt.
Graph ΔPΔt vs. Log Δt. This should give a straight line which slope and intercept can be used for estimation of permeability and skin factor using Eqs. (2.38) and (2.39), respectively. For this particular case, Eq. (2.39) is rewritten as:
However, this analysis could be modified as follows [8, 11, 21, 98, 99]. Consider a shut‐in developed with other wells in operation. There is a pressure decline in the shut‐in well resulting from the production of the other wells (superposition). After the test, well has been put into production at time t1, its pressure will be:
According to Figure 2.24, ΔPwo(t) is the pressure drop referred to Pi caused by other wells in the reservoir and measured at a time t = t1 + Δt. ΔPwo(t) can be estimated by superposing by:
Eq. (2.186) assumes that all wells start to produce at t = 0. This is not always true. Including wells that start at different times require a more complex superposition. If the other wells in the reservoir operate under pseudosteady‐state conditions, as is usually the case, Eq. (2.152) becomes:
The slope, m*, is negative when ΔPwo(t) vs. t is plotted. Instead, it is positive, if Pw vs. t is plotted. m* is estimated before the test well is opened in production at the pressure decline rate:
Behavior of a declination test in a depleted well, after [11, 21].
If pressure data is available before the test, m* can be easily estimated. Also, it can be estimated by an equation resulting from replacing Eq. (2.57) in (2.186):
The reservoir volume is given in ft3. Combining Eq. (1.106) with rD = 1, (1.94), (2.185), and (2.187), results:
Eq. (2.190) indicates that a graph of Pwf−m*Δt vs. log Δt gives a straight line of slope m and intercept ΔP1hr at Δt = 1 h. The permeability can be found from Eq. (2.38). The skin is estimated from an arrangement of Eq. (2.39):
TDS technique for developed reservoirs was extended by Escobar and Montealegre [21]. Escobar and Montealegre [21] showed that the technique could be applied taking the derivative to the pressure in a rigorous way, that is to say, without considering the effect of the production of other wells. As it will be seen in the example 2.7, this is not recommended since the derivative is not correctly defined and, therefore, the results could include deviations above 10%. In this case, it is advisable to correct or extrapolate the pressure by means of Eq. (2.192) and, then, take the extrapolated pressure derivative and apply the normal equations of the TDS technique given in Section 2.2.4. Needless to say that any TDS technique equation can also be used once the pressure derivative has been properly estimated with the extrapolated pressure:
Example 2.7
Escobar and Montealegre [21] presented a simulated pressure test of a square‐shaped reservoir with an area of 2295.7 acres having a testing well 1 in the center and another well 2 at 1956 ft north of well 1. Well 2 produced at a rate of 500 BPD during 14000 h. After 4000 h of flow, well 1 was opened at a flow rate of 320 BPD to run a pressure drawdown test which data are reported in Table 2.8 and Figure 2.26. The data used for the simulation were:
Methodology | k, md | s |
---|---|---|
Superposition time, Cartesian plot | 13.6 | −3.87 |
Equivalent time, semilog plot | 13.49 | −3.87 |
Actual time, semilog plot | 12.84 | −3.98 |
TDS | 13.86 | −3.794 |
Average | 13.45 | −3.88 |
Comparison of estimated results of example 2.6.
t, h | Pwf, psia | t, h | Pwf, psia | t, h | Pwf, psia |
---|---|---|---|---|---|
0 | 5000 | 4000.00 | 4278.93 | 7091.28 | 2007.41 |
4.51 | 5000.0001 | 4000.10 | 4134.44 | 7511.28 | 1899.99 |
10.10 | 4999.98 | 4000.20 | 4015.56 | 7931.28 | 1792.61 |
56.79 | 4991.08 | 4000.40 | 3830.82 | 8351.28 | 1685.19 |
100.98 | 4970.97 | 4000.64 | 3676.32 | 8771.28 | 1577.72 |
201.48 | 4926.98 | 4001.13 | 3478.40 | 9191.28 | 1470.25 |
319.33 | 4887.16 | 4001.80 | 3345.40 | 9611.28 | 1362.85 |
402.02 | 4864.59 | 4005.06 | 3166.11 | 10031.28 | 1255.45 |
506.11 | 4840.13 | 4017.96 | 3039.90 | 10451.28 | 1148.04 |
637.15 | 4813.27 | 4090.00 | 2891.65 | 10871.28 | 1040.57 |
802.13 | 4782.99 | 4201.48 | 2807.85 | 11291.28 | 933.06 |
1009.82 | 4747.74 | 4402.02 | 2720.00 | 11711.28 | 825.63 |
1271.28 | 4705.41 | 4637.15 | 2644.70 | 12131.28 | 718.26 |
1551.28 | 4661.10 | 5009.82 | 2542.16 | 12551.28 | 610.85 |
2111.28 | 4573.46 | 5411.28 | 2437.61 | 12971.28 | 503.40 |
2671.28 | 4486.12 | 5831.28 | 2329.70 | 13391.28 | 395.95 |
3091.28 | 4420.63 | 6251.28 | 2222.26 | 13811.28 | 288.51 |
3511.28 | 4355.13 | 6671.28 | 2114.85 | 14000.00 | 240.23 |
Pressure data of a developed reservoir in example 2.7, after [21].
rw = 0.3 pie | μ = 3 cp | ct = 3 × 10−6 psia−1 | h = 30 pies |
ϕ = 10% | B = 1.2 bbl/BF | k = 33.33 md | s = 0 |
Interpret this test using conventional and TDS techniques considering and without considering the presence of well 2.
Solution by conventional analysis
A pressure change is observed in well 1 up to a time of 4000 h, after which it is put into production for the declination test, as shown in Figure 2.25. Figure 2.26 presents a plot of Pwf vs. log Δt obtained with the information in Table 2.9. Hence, the slope and intercept are, respectively, −230 psia/cycle and 3330.9 psia. Permeability and skin factor are, respectively, estimated from Eqs. (2.38) and (2.191):
Δt, h | Pwf, psia | ΔPwf, psia | t*ΔPwf′, psia | Pext, psia | ΔPext, psia | t*ΔPext′, psia |
---|---|---|---|---|---|---|
0.00 | 4278.93 | 0.00 | 0.00 | 4278.93 | 0.00 | 0.00 |
0.01 | 4263.17 | 15.76 | 16.17 | 4263.17 | 15.76 | 16.17 |
0.02 | 4247.80 | 31.13 | 31.50 | 4247.80 | 31.13 | 31.50 |
0.03 | 4232.77 | 46.16 | 46.15 | 4232.77 | 46.16 | 46.14 |
0.05 | 4203.65 | 75.28 | 73.43 | 4203.66 | 75.27 | 73.42 |
0.06 | 4189.53 | 89.40 | 86.24 | 4189.54 | 89.39 | 86.23 |
0.08 | 4162.11 | 116.83 | 110.14 | 4162.12 | 116.81 | 110.13 |
0.113 | 4118.69 | 160.24 | 145.70 | 4118.71 | 160.22 | 145.69 |
0.160 | 4062.04 | 216.89 | 187.89 | 4062.06 | 216.87 | 187.86 |
0.226 | 3989.51 | 289.42 | 234.79 | 3989.55 | 289.38 | 234.75 |
0.319 | 3899.81 | 379.12 | 281.46 | 3899.86 | 379.07 | 281.41 |
0.451 | 3793.92 | 485.01 | 319.72 | 3793.99 | 484.94 | 319.65 |
0.637 | 3676.32 | 602.61 | 339.60 | 3676.42 | 602.51 | 339.50 |
0.900 | 3555.47 | 723.47 | 333.33 | 3555.61 | 723.33 | 333.19 |
1.271 | 3442.11 | 836.82 | 300.48 | 3442.31 | 836.63 | 300.27 |
1.796 | 3345.40 | 933.53 | 250.08 | 3345.68 | 933.25 | 249.79 |
2.537 | 3269.25 | 1009.68 | 196.69 | 3269.65 | 1009.28 | 196.28 |
3.583 | 3211.35 | 1067.58 | 152.72 | 3211.91 | 1067.02 | 152.14 |
5.061 | 3166.11 | 1112.82 | 123.22 | 3166.90 | 1112.03 | 122.39 |
7.149 | 3128.10 | 1150.84 | 106.39 | 3129.21 | 1149.72 | 105.23 |
10.098 | 3093.70 | 1185.23 | 97.75 | 3095.28 | 1183.65 | 96.11 |
16.005 | 3050.46 | 1228.47 | 92.62 | 3052.96 | 1225.97 | 90.01 |
22.61 | 3018.94 | 1259.99 | 91.11 | 3022.47 | 1256.46 | 87.43 |
31.93 | 2987.69 | 1291.25 | 90.86 | 2992.67 | 1286.26 | 85.67 |
45.11 | 2956.32 | 1322.61 | 91.76 | 2963.37 | 1315.57 | 84.42 |
63.72 | 2924.46 | 1354.47 | 93.89 | 2934.41 | 1344.52 | 83.53 |
90.00 | 2891.65 | 1387.28 | 97.54 | 2905.70 | 1373.23 | 82.89 |
127.13 | 2857.33 | 1421.60 | 103.15 | 2877.17 | 1401.76 | 82.47 |
179.57 | 2820.74 | 1458.19 | 111.39 | 2848.77 | 1430.16 | 82.17 |
253.65 | 2780.81 | 1498.12 | 123.24 | 2820.41 | 1458.52 | 81.97 |
358.30 | 2736.24 | 1542.70 | 140.71 | 2792.17 | 1486.76 | 82.41 |
506.11 | 2684.75 | 1594.18 | 167.76 | 2763.76 | 1515.17 | 85.41 |
714.90 | 2622.34 | 1656.59 | 211.07 | 2733.94 | 1544.99 | 94.76 |
1009.82 | 2542.16 | 1736.77 | 279.69 | 2699.80 | 1579.13 | 115.40 |
1411.28 | 2437.61 | 1841.32 | 380.45 | 2657.93 | 1621.01 | 150.84 |
1831.28 | 2329.70 | 1949.23 | 489.81 | 2615.59 | 1663.35 | 191.86 |
2251.28 | 2222.26 | 2056.67 | 600.72 | 2573.71 | 1705.22 | 234.44 |
2811.28 | 2079.04 | 2199.90 | 749.48 | 2517.91 | 1761.03 | 292.10 |
3371.28 | 1935.78 | 2343.15 | 898.68 | 2462.07 | 1816.86 | 350.18 |
4071.28 | 1756.82 | 2522.11 | 1085.26 | 2392.38 | 1886.55 | 422.88 |
4771.28 | 1577.72 | 2701.21 | 1271.94 | 2322.57 | 1956.36 | 495.67 |
5611.28 | 1362.85 | 2916.08 | 1495.85 | 2238.83 | 2040.10 | 582.91 |
6451.28 | 1148.04 | 3130.89 | 1719.92 | 2155.15 | 2123.78 | 670.32 |
6591.28 | 1112.23 | 3166.71 | 1757.27 | 2141.19 | 2137.74 | 684.89 |
7711.28 | 825.63 | 3453.31 | 2055.97 | 2029.43 | 2249.50 | 801.38 |
8831.28 | 539.22 | 3739.71 | 2354.68 | 1917.88 | 2361.06 | 917.87 |
9951.28 | 252.69 | 4026.24 | 2653.47 | 1806.19 | 2472.74 | 1034.44 |
10000.00 | 240.23 | 4038.70 | 2666.47 | 1801.33 | 2477.60 | 1039.51 |
Data of Pwf, Pext = Pwf−m*Δt, t*ΔPwf′, t*ΔPext′ for example 2.7, after [21].
Cartesian plot of pressure versus time data simulated for well 1, after [21].
Table 2.9 also reports the data of Pwf−m*Δt. Figure 2.26 presents, in addition, the plot of Pwf−m*Δt vs. log Δt. Now, the slope and intercept are, respectively, 193.9 psia/cycle and 3285.9 psia. A permeability of 32.2 md is found from Eq. (2.38) and a skin factor of −0.28 is estimated from Eq. (2.191).
Semilog plot for example 2.7, after [21].
Log‐log plot of pressures and pressure derivatives versus time for example 2.7, after [21].
From the derivative plot, Figure 2.27, we can observe that the pseudosteady‐state period has been perfectly developed; as a consequence, we can obtain the Cartesian slopes performing a linear regression with the last 10 pressure points, namely: m* (Pwf vs. Δt) = −0.256 psia/h and m* (Pext vs. Δt) = −0.0992 psia/h. Eq. (2.59) allows obtaining the well drainage area of well 2:
Solution by TDS technique
Application of TDS, the pressure derivative is initially taken to the well‐flowing pressure data, see Table 2.9. Then, the derivative is taken to the corrected pressure, Pwf−m*Δt. Both pressure derivatives are reported in Figure 2.27. For the uncorrected pressure, the following information was read from Figure 2.27:
tr = 35.826 h | (t*ΔP′)r = 90.4 psia | ΔPr = 1301.7 psia |
Permeability and skin factor are calculated with Eqs. (2.76) and (2.97);
Then, for the corrected pressure case, the following data were read from Figure 2.27;
tr = 319.3321 h | (t*ΔP′)r = 82.1177 psia | ΔPr = 1477.3508 psia |
With these data, Eq. (2.76) provided a permeability value of 33.07 md and Eq. (2.97) allows estimating a skin factor of −0.087. Eq. (2.102) is used to find the well drainage area using trpi = 376.6049 h (uncorrected pressure) and trpi = 800.5503 h (corrected pressure) read from Figure 2.28, then,
Method | k, md | Abs. error, % | s | Abs. error, % |
---|---|---|---|---|
Simulation | 33.33 | 0 | ||
Semilog with Pwf | 27.15 | 18.54 | −1.35 | 135 |
Semilog with Pext | 32.2 | 3.39 | −0.29 | 29 |
TDS with Pwf | 30 | 9.99 | −0.74 | 74 |
TDS with Pext | 33.07 | 0.78 | −0.087 | 8.7 |
Permeability and skin factor results for example 2.7, after [21].
Figure 2.28 provides a comparison of the derivative of the flowing bottom pressure ignoring the effect of well 2 and the pressure derivative including the effect of well 2. It is noted there that the radial flow zone is shorter and less defined. On the other hand, the pseudosteady‐state zone appears first when the effect of the adjacent well is not included, indicating that the well drainage area, and therefore, the reserves present therein will be substantially underestimated. Table 2.10 shows all the permeability and skin factor values obtained for this example with their respective absolute errors with reference to the input simulation value. TDS when corrected pressure is taken gives the best results.
These deposits can be approximated to the geometry described by Figure 2.28. They mainly result from fluvial depositions (deltaic), commonly called channels, terrace faulting, and carbonate reefs. The possible flow regimes when the well is completely off‐center are presented in Figure 2.28b when the parallel reservoir boundaries are no‐flow type (closed). Once radial flow vanishes, two linear flows take place at both sides of the reservoir. This flow regime is normally known as linear flow regime, see Figure 2.27b; actually, it consists of two linear flow regimes forming a 180° angle between each other. Therefore, Escobar et al. [19] named it dual‐linear flow. Once the shorter reservoir boundary has been reached by the transient wave, only a unique linear flow is kept and lasts until the other boundary is reached. This unique flow is referred as single‐linear flow by Escobar et al. [19]. However, since linear flow is taken on one side of the reservoir, it is also known as hemilinear flow regime. This is the only linear flow taken place in the system depicted in Figure 2.28c.
Both linear flows are characterized by a slope of 0.5 in the pressure derivative curve. Figure 2.29 sketches the pressure derivative behavior of the mentioned systems.
The governing pressure and pressure derivative equations for the single‐linear and dual‐linear flow regimes are, respectively, given below [13, 16, 17, 18, 19, 20, 23, 24, 28, 29, 31, 35, 38, 55, 56]:
Being sL is the geometrical skin factor caused by converging from either radial to linear flow regime (well located at one end of reservoir sides, Figure 2.28c or from dual‐linear to linear flow (well off‐center well). sDL is the geometrical skin factor caused by converging from either radial to linear flow regime. The dimensionless parameters are defined by Escobar et al. [19] as:
Reservoir geometry and description of flow regimes. (a) Reservoir approximated geometry, (b) Dual linear flow, (c) Single linear or hemilinear flow.
The dimensionless distances are given by:
Variables bx and by correspond to the nearest distances from the well to the reservoir boundaries in the directions x and y, respectively. See Figure 2.28a. Replacing Eqs. (1.94), (2.62) and (2.197) in Eq. (2.194) and solving for the root product of permeability by the reservoir width, YE, will yield:
Since, TDS equations apply to either drawdown or buildup tests; then, when either t or Δt = 1 h, Eq. (2.200) becomes:
Dimensionless well pressure derivative versus time behavior for a rectangular reservoir with the well located off‐center, after [19].
The root product of permeability by the reservoir width can be also calculated from the dual‐linear flow, DL. This can be performed by replacing also Eqs. (1.94), (2.62), and (2.197) into the dimensionless pressure derivative equation into Eq. (2.196) leading to:
Again at either t or Δt = 1 h, the above equation becomes:
For long production times, the pseudosteady‐state period is reached. Both pressure and pressure derivative are joined into a unit‐slope line, we obtain a straight line. The governing pressure derivative equation at this time is given by Eq. (2.101). For the systems dealt with in this section, Eq. (2.102) which uses the point of intersection radial‐pseudosteady state, Eq. (2.103) and (2.104) also apply. The straight line given by Eq. (2.101) also intersects the lines given by Eqs. (2.96) and (2.98); then, reservoir area can be found from such intersection times, thus, [13, 19]:
Likewise, the intersection times of the line of infinite radial behavior of the pressure derivative (horizontal straight line) with the hemilinear and dual‐linear flow regimes lead to obtain reservoir width from: