Pressure, pseudopressure, time, and pseudotime data for example 1.
\\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!
Note: Edited in March 2021
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 191 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 261 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!
Note: Edited in March 2021
\n'}],latestNews:[{slug:"intechopen-partners-with-ehs-for-digital-advertising-representation-20210416",title:"IntechOpen Partners with EHS for Digital Advertising Representation"},{slug:"intechopen-signs-new-contract-with-cepiec-china-for-distribution-of-open-access-books-20210319",title:"IntechOpen Signs New Contract with CEPIEC, China for Distribution of Open Access Books"},{slug:"150-million-downloads-and-counting-20210316",title:"150 Million Downloads and Counting"},{slug:"intechopen-secures-indefinite-content-preservation-with-clockss-20210309",title:"IntechOpen Secures Indefinite Content Preservation with CLOCKSS"},{slug:"intechopen-expands-to-all-global-amazon-channels-with-full-catalog-of-books-20210308",title:"IntechOpen Expands to All Global Amazon Channels with Full Catalog of Books"},{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"}]},book:{item:{type:"book",id:"1494",leadTitle:null,fullTitle:"Current Frontiers in Cryobiology",title:"Current Frontiers in Cryobiology",subtitle:null,reviewType:"peer-reviewed",abstract:'Almost a decade has passed since the last textbook on the science of cryobiology, Life in the Frozen State, was published. Recently, there have been some serious tectonic shifts in cryobiology which were perhaps not seen on the surface but will have a profound effect on both the future of cryobiology and the development of new cryopreservation methods. We feel that it is time to revise the previous paradigms and dogmas, discuss the conceptually new cryobiological ideas, and introduce the recently emerged practical protocols for cryopreservation. The present books, "Current Frontiers in Cryobiology" and "Current Frontiers in Cryopreservation" will serve the purpose. This is a global effort by scientists from 27 countries from all continents and we hope it will be interesting to a wide audience.',isbn:null,printIsbn:"978-953-51-0191-8",pdfIsbn:"978-953-51-4331-4",doi:"10.5772/1962",price:159,priceEur:175,priceUsd:205,slug:"current-frontiers-in-cryobiology",numberOfPages:594,isOpenForSubmission:!1,isInWos:1,hash:"14595adb0882149d0e3c7a5701990ca3",bookSignature:"Igor I. 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In general, gas viscosity is about a 100 times lower than the least viscous crude oil. It is important, however, to try to provide the same mathematical treatment to oil and gas hydrocarbons, so interpretation methodologies can easily be applied in a more practical way. Then, the gas flow equation is normally linearized to allow the liquid diffusivity solution to satisfy the gas behavior when analyzing transient test data of gas reservoirs. Depending on the values of reservoir pressure, viscosity, and gas compressibility factor, the gas flow behavior can be treated as a function of either pressure to the second power or linear pressure with a region which does not correspond to any of these and it is better represented by a synthetic function call pseudopressure. Pseudopressure is a function that integrates pressure, density, and compressibility factor. The gas system’s total compressibility highly depends on gas compressibility which for ideal gases changes inversely with the pressure. Then, another artificial function referred as pseudotime is included to further understand the transient behavior of gas flow in porous media. For instance, when wellbore storage conditions are insignificant, drawdown tests are best analyzed using the pseudopressure function. On the other hand, buildup pressure tests require linearization of both pseudotime and pseudopressure.
This chapter will be devoted to provide both fundamental of gas flow in porous media as well as interpretation of pressure and rate data in gas reservoirs. The use of the oil flow equations and interpretation techniques is carefully extended for gas flow so that reservoir permeability, skin factor, and reservoir area can be easily estimated from a gas pressure or gas rate test by using conventional analysis and characteristic points found on the pressure derivative plot (
The chapter will include both interpretation techniques
It is convenient to mention some other important aspects concerning gas well testing which have appeared recently. The first case is the transient rate analysis in hydraulically fractured wells which was presented by [19] for both oil and gas wells. The traditional model for elliptical flow included the reservoir area as a variable. Handling the interpretation using
Practical exercises will provide in the chapter provide a better understanding and applicability of the interpretation techniques.
The purpose of this chapter is two folded: (1) to present the governing equation for gas flow used in well test interpretation and (2) to use both conventional and
Transient pressure analysis is performed measuring the bottom-hole pressure while the flow rate is kept constant.
The gas diffusivity equation in oil-field units is given by:
Which can be modified to respond for three-phase flow (oil, water, and gas):
where, the total compressibility,
As can be inferred from Eq. (3), the total compressibility varies significantly when dealing with monophasic gas flow since the gas compressibility varies along with the pressure. Agarwal [1] introduced the pseudotime function to alleviate such problem. This function accounts for the time dependence of gas viscosity and total system compressibility:
Pseudotime is better defined as a function of pressure as a new function given in hr psi/cp:
Notice that
As expressed by Eq. (1), viscosity and gas compressibility factor are strong functions of pressure; then, to account for gas flow behavior, Al-Hussainy et al. [2] introduced the pseudopressure function which basically includes the variation of gas viscosity and compressibility into a single function which is given by:
After replacing Eqs. (6) and (7) into Eq. (1), it yields:
Contrary to liquid well testing, rapid gas flow has a strong influence on well testing, [32]. As the flow rate increases, so does the skin factor, then:
Eq. (9) shows that the apparent skin factor is a function of the mechanical skin factor—which is assumed to be constant during the test— and the product of the flow rate with the turbulence factor or non-Darcy term. This implies that two flow test ought to be run at different flow rates to find mechanical skin factor and the turbulence factor from:
Solving the simultaneous equations:
where, the skin factors 1 and 2 are estimated from each pressure test. However, there is a need of estimating the turbulence factor by empirical correlations for buildup cases or when a single test exists. Then, the non-Darcy flow coefficient is defined by [26]:
The above equation is also applied to partially completed or partially penetrated wells.
Parameter
The consideration on the skin factor effect on gas testing was recognized by Fligelman et al. [25] who provided correction charts to account for apparent skin factor values.
The solution to the transient diffusivity equation, Eq. (8), is given by:
The dimensionless parameters used in this chapter are given below. The rigorous dimensionless time is:
Including the pseudotime function,
Notice that the viscosity-compressibility product is not seen in Eq. (16) since they are included in the pseudotime function. However, if we multiply and, then, divide by (
The dimensionless pseudopressure and pseudopressure derivatives are:
And the dimensionless wellbore storage coefficient is given by:
The dimensionless radii are given:
For practical purposes, Eq. (16) will end up in a semilog behavior of pseudopressure drops against time. After replacing the respective dimensionless quantities into the mentioned straight-line semilog expression, it is obtained [4]:
The above equations are applied during transient or radial flow regime. They are used to find reservoir transmissibility and apparent skin factor from the slope and intercept, respectively, of a semilog plot of well-flowing pressure versus time. After applying the superposition principle, the above equations for the buildup case are converted into:
From a semilog plot of pseudopressure versus time (or pseudotime), its slope allows calculating the reservoir permeability and the intercept is used to find the pseudoskin factor, respectively:
Notice that for the pseudotime case, (
The governing dimensionless pressure equation during pseudosteady-state period is given by [28]:
By replacing the dimensionless quantities, changing the log base, the above equation leads to:
A Cartesian plot of
Such deliverability tests as backpressure, isochronal, modified isochronal, and flow after flow are conducted for the purpose of determining the flow exponent
Tiab [33] proposed a revolutionary technique which is very useful to interpret pressure tests using characteristics points found on the pressure and pressure derivative versus time log-log plot. He obtained practical analytical solutions for the determination of reservoir parameters.
From a log-log plot of pseudopressure and pseudopressure derivative against pseudotime, Figure 1, several main characteristics are outlined:
Log-log plot of pseudopressure and pseudopressure derivative versus pseudotime. After Ref. [
1. The early unit-slope line originated by wellbore storage is described by the following equation:
Replacing the dimensionless parameters in Eq. (36), a new equation to estimate the wellbore storage coefficient is obtained:
2. The intersection of the early unit-slope line with the radial horizontal straight line gives:
From this, an equation to estimate either permeability or wellbore storage is obtained once the dimensionless parameters are replaced.
As presented by Tiab [33], the governing equation for the well pressure behavior during radial flow reformulated by Escobar et al. [7] in terms of pseudofunctions is expressed by:
3. According to Ref. [28], another form of Eq. (35) is obtained when wellbore storage and skin factor are included:
From the above equation, the derivative of pseudopressure with respect to the natural log of
From Eq. (21), the dimensionless pseudopressure derivative with respect to the natural log of log
Combination of Eqs. (41) and (42) will result into an equation to estimate permeability:
3. Dividing Eq. (40) by Eq. (41), replacing the dimensionless quantities and, then, solving for the pseudoskin factor will yield:
Finally, the pressure derivative during the pseudosteady-state flow regime of closed systems is governed by:
The intersection point of the above straight line and the radial flow regime straight line is:
After substituting the dimensionless pseudotime function into Eq. (46), a new equation for the well drainage area is presented:
Further applications of gas well test can be found in the literature. Escobar et al. [12] introduced the mathematical expressions for interpretation of pressure tests using the pseudopressure and pseudopressure derivative as a function of pseudotime for hydraulically fractured wells and naturally fractured (heterogeneous) formations. Fligelman [30] presented an interpretation methodology using
Chaudhry [4] presented a reservoir limit test for a gas reservoir (example 5-2 of Ref. [4]). However, once the pressure derivative was taken to the test data, no late pseudosteady state regime was observed. Then, the input data given below were used to simulate a pressure test given in Table 1.
0 | 3965 | 1.2713 | 3677.2527 |
0.001 | 3960.629 | 1.6005 | 3670.3779 |
0.002 | 3956.4313 | 2.0148 | 3663.602 |
0.003 | 3952.3774 | 2.5365 | 3656.8388 |
0.004 | 3948.4516 | 3.1933 | 3650.1477 |
0.005 | 3944.6431 | 4.0202 | 3643.5144 |
0.006 | 3940.9438 | 5.0107 | 3637.2126 |
0.007 | 3937.3469 | 6.0107 | 3632.0323 |
0.008 | 3933.8466 | 7.0107 | 3627.6664 |
0.009 | 3930.4382 | 8.0107 | 3623.8936 |
0.0113 | 3922.8277 | 9.0107 | 3620.5718 |
0.0143 | 3913.8402 | 10.0107 | 3617.6047 |
0.018 | 3903.2573 | 12.0107 | 3612.4479 |
0.0226 | 3891.1325 | 21.0107 | 3596.5482 |
0.0285 | 3877.5006 | 30.0107 | 3586.4447 |
0.0358 | 3862.4816 | 39.0107 | 3579.0231 |
0.0451 | 3846.2054 | 48.0107 | 3573.1536 |
0.0568 | 3829.2592 | 57.0107 | 3568.2967 |
0.0715 | 3812.1329 | 66.0107 | 3564.1453 |
0.09 | 3795.2335 | 75.0107 | 3560.4805 |
0.1133 | 3779.2686 | 84.0107 | 3557.2312 |
0.1271 | 3771.7594 | 93.0107 | 3554.3136 |
0.16 | 3757.7512 | 102.0107 | 3551.6672 |
0.2015 | 3745.1803 | 179.5107 | 3535.5164 |
0.2537 | 3734.0438 | 269.5107 | 3523.5847 |
0.3193 | 3724.1258 | 359.5107 | 3514.1663 |
0.402 | 3715.1658 | 449.5107 | 3505.545 |
0.5061 | 3706.8112 | 539.5107 | 3497.2804 |
0.6372 | 3698.9669 | 629.5107 | 3489.1576 |
0.8021 | 3691.489 | 719.5107 | 3481.0959 |
1.0098 | 3684.2741 | 809.5107 | 3473.056 |
Δ | Δ | ||||
---|---|---|---|---|---|
0 | 0 | 0 | 45802.414 | 1870739.384 | 17533667.77 |
37.7745 | 270218.7615 | 265698.0159 | 57569.7648 | 1838960.003 | 17954428.54 |
75.5167 | 518758.6126 | 520866.1041 | 72365.7851 | 1815682.598 | 18369258.28 |
113.2278 | 751325.0471 | 767304.1906 | 90970.2467 | 1799003.726 | 18783428.06 |
150.9089 | 968476.5902 | 1005966.456 | 114363.6923 | 1786548.918 | 19193297.17 |
188.561 | 1172014.904 | 1237504.102 | 143779.3051 | 1774358.756 | 19599754.75 |
226.185 | 1362245.214 | 1462410.249 | 178978.2667 | 1766967.699 | 19986008.75 |
263.7817 | 1544967.139 | 1681099.321 | 214477.474 | 1764620.952 | 20303606.64 |
301.352 | 1713052.797 | 1893921.256 | 249946.7071 | 1752899.594 | 20571329.92 |
338.8966 | 1876882.193 | 2101169.292 | 285390.4587 | 1717299.587 | 20802733.84 |
426.2923 | 2225474.894 | 2563944.683 | 320812.0523 | 1692510.182 | 21006509.26 |
536.1368 | 2605002.302 | 3110507.871 | 356214.0473 | 1681851.999 | 21188553.27 |
674.1563 | 3004035.218 | 3754179.95 | 426967.5286 | 1725356.98 | 21505009.16 |
847.5255 | 3404366.431 | 4491744.317 | 744772.1706 | 1759676.341 | 22480949.29 |
1065.235 | 3781729.348 | 5321159.089 | 1061858.072 | 1799660.974 | 23101515.46 |
1338.5511 | 4101068.915 | 6235191.645 | 1378463.16 | 1769298.728 | 23557564.54 |
1681.5934 | 4335396.363 | 7226023.963 | 1694705.511 | 1759287.154 | 23918367.9 |
2112.0768 | 4462372.666 | 8257991.514 | 2010656.317 | 1770713.335 | 24217012.8 |
2652.2558 | 4463511.759 | 9301323.123 | 2326363.101 | 1736601.944 | 24472345.81 |
3330.0995 | 4300906.005 | 10331215.57 | 2641858.765 | 1709368.012 | 24697796.42 |
4180.8019 | 4056795.12 | 11304325.65 | 2957168.15 | 1701504.59 | 24897723.65 |
4684.3808 | 3917486.951 | 11762181.87 | 3272311.76 | 1711704.512 | 25077261.23 |
5880.9567 | 3569692.658 | 12616571.14 | 3587306.105 | 1744436.565 | 25240114.57 |
7383.6218 | 3191507.548 | 13383609.83 | 6295436.794 | 1821464.183 | 26234973.81 |
9271.202 | 2873762.477 | 14063387.95 | 9432920.282 | 2036512.441 | 26970809.17 |
11642.9004 | 2591188.809 | 14668995.81 | 12564795.27 | 2355209.644 | 27552034.83 |
14623.503 | 2366500.199 | 15216145.54 | 15691965 | 2740929.323 | 28084374.29 |
18369.9003 | 2197999.347 | 15726460.2 | 18814758.13 | 3212985.03 | 28594968.02 |
23079.3518 | 2076483.39 | 16205683.18 | 21933329.44 | 3695740.205 | 29097073.99 |
28999.9239 | 1985415.325 | 16662774.49 | 25047755.95 | 4195533.855 | 29595671.37 |
36443.5374 | 1918507.605 | 17104058.59 | 28158083.33 | 4703995.72 | 30093194.56 |
Pressure, pseudopressure, time, and pseudotime data for example 1.
Estimate permeability, skin factor, and drainage area by both conventional analysis and
Figure 2 presents a semilog pressure of pseudopressure versus pseudotime. The slope and intercept of the radial flow regime straight line in such plot are given below:
Semilog plot for example 1.
Use of Eqs. (27) and (28) allows finding reservoir permeability and pseudoskin factor, respectively:
To find the well drainage area, the Cartesian plot given in Figure 3 was built. Its slope,
Cartesian plot for example 1.
Figure 4 presents the pseudopressure and pressure derivative versus pseudotime log-log plot in which wellbore storage, radial flow regime, and late pseudosteady-state regimes are clearly observed. The following characteristic points were read from Figure 4:
∆ | |
Pseudopressure drop and pseudopressure derivative versus time log-log plot for example 1.
Permeability and pseudoskin factor are respectively estimated from Eqs. (42) and (44),
and well drainage area is found with Eq. (47):
Finally, the inertial factor and the non-Darcy flow coefficient are estimated with Eqs. (14) and (15):
The true skin factor is found with Eq. (9):
It can be seen that the simulated parameters closely match the results obtained from the examples.
Transient rate analysis is performed by recording the continuous changing flow rate under a constant bottom-hole pressure condition. This procedure is normally achieved in very low gas formations and shale gas systems.
The Laplace domain, the rate of solution for a well producing against a constant bottom-hole well-flowing pressure was given by [34]:
The solution for a bounded reservoir was presented by [5]:
For considerable longer times, Ref. [27] showed that the
where the dimensionless reciprocal rate and reciprocal rate derivative are given by:
Including pseudoskin effects in Eq. (49),
After replacing the dimensionless quantities and changing the logarithm base, it yields:
As for the case of pressure transient analysis, from a semilog plot of pseudopressure versus time (or pseudotime), its slope allows calculating the reservoir permeability and the intercept is used to find the pseudoskin factor, respectively:
Considering approximation for large time to the analytical Laplace inversion of Eq. (49), the following expression is obtained:
For
where,
Eq. (58) suggests that a plot of log(
and intercept at (
The reservoir area can be determined by solving the Eq. (62) for
Escobar et al. [9] extended the
Using a procedure similar to the pressure transient case, Escobar et al. [9] found an expression to estimate the pseudoskin factor:
For the estimation of reservoir area, Escobar et al. [9] also presented an equation that uses the starting time of the pseudosteady-state period,
As treated in pressure transient analysis, Eq. (41), the reciprocal rate derivative takes a value of 0.5 during radial flow. The intercept of this with the reciprocal rate derivative of Eq. (57) will provide:
in which numerical solution gives:
After replacing the dimensionless quantities, we obtain:
Refs. [13] and [14] presented rate transient analysis for long homogeneous and naturally fractured oil reservoirs using
Escobar et al. [9] presented an example for a homogeneous bounded reservoir. Figure 5 and Table 2 present the reciprocal rate and reciprocal rate derivative versus rigorous time for this exercise. Other relevant data for this example are given below:
Reciprocal rate and reciprocal rate derivative for example 2—homogeneous bounded reservoir. After Ref. [
Δ | Δ |
1/ | 1/ | ||||
---|---|---|---|---|---|
3.11E-04 | 43189.24959 | 179751.1071 | 4.72E-02 | 17360.95346 | 135561.954 |
3.89E-04 | 40834.91569 | 162199.4397 | 6.09E-02 | 16811.48278 | 135417.1412 |
4.67E-04 | 39060.52002 | 163534.7645 | 7.58E-02 | 16365.45213 | 135450.8332 |
5.45E-04 | 37655.94848 | 160065.4877 | 0.094767643 | 15935.44127 | 134654.7122 |
7.01E-04 | 35537.19664 | 157149.8777 | 0.122175588 | 15464.07533 | 130033.9989 |
8.72E-04 | 33849.9472 | 154708.1815 | 0.152075165 | 15058.4209 | 120446.6934 |
1.07E-03 | 32361.46026 | 152593.8776 | 0.176991479 | 14766.60107 | 110524.7186 |
1.35E-03 | 30871.32659 | 150543.0499 | 0.224830802 | 14266.98732 | 91414.11215 |
1.66E-03 | 29601.36073 | 148839.1511 | 0.284629956 | 13699.34978 | 71972.81156 |
1.97E-03 | 28617.55908 | 147555.5061 | 0.354395635 | 13079.44168 | 55862.58323 |
2.36E-03 | 27646.46771 | 146325.5009 | 0.450074281 | 12279.66271 | 41400.85561 |
2.80E-03 | 26785.03603 | 145257.0868 | 0.569672588 | 11348.65545 | 30197.97654 |
3.42E-03 | 25823.21027 | 144105.0598 | 0.709203946 | 10349.64392 | 22079.63438 |
4.04E-03 | 25068.84401 | 143224.7896 | 0.900561238 | 9119.41797 | 15300.5905 |
4.73E-03 | 24398.17164 | 142461.4954 | 1.139757852 | 7783.903591 | 10316.16432 |
5.54E-03 | 23753.19888 | 141745.6607 | 1.315168702 | 6930.721305 | 7965.441894 |
6.63E-03 | 23056.85049 | 140996.5214 | 1.522472435 | 6043.090594 | 6009.327954 |
7.87E-03 | 22423.58559 | 140330.625 | 1.801535152 | 5027.292996 | 4239.056471 |
9.12E-03 | 21908.28203 | 139803.0353 | 2.120463971 | 4077.321632 | 2935.619806 |
1.15E-02 | 21141.51732 | 139045.8455 | 2.630750081 | 2923.310968 | 1712.191843 |
1.49E-02 | 20332.18915 | 138280.5426 | 3.28455416 | 1917.14852 | 909.5234175 |
1.86E-02 | 19683.45322 | 137688.6861 | 4.241340618 | 1040.993708 | 385.7970016 |
2.34E-02 | 19065.59817 | 137131.4759 | 5.261912839 | 543.216581 | 162.6885651 |
3.02E-02 | 18406.42719 | 136508.2532 | 6.569520997 | 230.4829168 | 41.88651512 |
3.77E-02 | 17872.96521 | 135981.9447 | 9.12095155 | 27.98082076 | 0.548915189 |
Reciprocal rate, reciprocal rate derivative versus time data for example 2.
Find reservoir permeability, skin factor, and drainage radius for this example using the
The following characteristic points were read from Figure 5:
[ | |
(1/ |
Eqs. (65), (66), and (70) are used to obtain permeability, skin factor, and drainage.
Notice that the results closely match the permeability and external reservoir radius as presented by Ref. [9].
Finally, it is worth to mention that nowadays, conventional shale-gas reservoirs have become very attractive in the oil industry. Then, their characterization via well test analysis is very important. Shale-gas reservoir is normally tested under constant well-flowing pressure conditions—transient rate analysis—then, the recent studies performed in Refs. [17] and [22] should be read. If such wells are tested under constant rate conditions—pressure transient analysis—then the reader should refer to the works by Bernal et al. [3] and Escobar et al. [18].
A | Well drainage area, ft2 and Ac |
B | Volumetric factor, rb/MSCF |
C | Wellbore storage coefficient, bbl/psi |
ct | Total compressibility, 1/psi |
D | Turbulent flow factor, Mscf/D |
h | Formation thickness, ft |
hp | Perforated interval, ft |
I0, I1 | Bessel function |
k | Permeability, md |
K0,K1 | Bessel function |
m | Semilog slope |
m* | Cartesian slope |
m(P) | Pseudopropressure function, psi2/cp |
Mdecline | Slope of plot of log(q) versus time |
n | Flow exponent |
P | Pressure, psi |
PD | Dimensionless pressure |
Pwf | Well-flowing pressure, psi |
q | Gas flow rate, MSCF |
1/q | Reciprocal of the flow rate, D/Mscf |
r | Radius, ft |
re | External reservoir radius, ft |
rw | Radio del pozo, ft |
rweff | Effective wellbore radius, rwe−s, ft |
s’ | Apparent or pseudoskin factor |
sa | Total skin factor |
t | Time, hr |
tD*PD’ | Dimensionless pressure derivative |
tDpss | Exponential decline period |
t*(1/q)’ | Reciprocal rate derivative, D/Mscf |
tD*(1/qD)’ | Dimensionless reciprocal rate derivative |
tp | Horner or producing time |
tpss | Exponential decline period, hr |
tspss | Time to initiate pseudosteady state, hr |
u | Argument for a Bessel function |
Z | Gas supercompressibility factor |
α | Turbulence factor or inertial factor |
Δ | Change, drop |
φ | Porosity, fraction |
γ | Euler’s constant—1.781 or e0.5772 |
γ | gGas gravity |
λ | Mobility, md/cp |
μ | Viscosity, cp |
1 hr | One hour |
cr | Condition at critical point |
DA | Dimensionless referred to drainage area |
Da | Dimensionless referred to pseudotime |
D | Dimensionless |
De | Dimensionless referred to external |
e | External |
eff | Effective |
g | Gas |
i | Initial or intercept |
pss | Pseudosteady state |
r | Radial flow |
ref | Reference |
rpi | Intercept radial-pseudosteady |
t | Total |
ta(P) | Pseudotime, psi-hr/cp |
w | Well |
Describing the behavior of strongly correlated systems is a challenge for every area of physics. The many-body ground state that emerges in these systems can have exotic and unexpected properties. As a remarkable of these correlated systems was the discovery of superconductivity by Onnes in 1911 [1], which opens active areas of study in the field of condensed matter physics.
This state is a quantum phenomenon that is displayed by certain materials under particular magnetic and temperature regimes. The Meissner effect discovered in 1933 shows that superconductivity is more than just the disappearance of resistance and it is a true thermodynamic state of matter because it leads to the superconductors as perfect diamagnetism materials [2]. This effect distinguishes a superconductor from a perfect conductor.
To describe these phenomena, a lot of theoretical works were proposed. The first phenomenological theory which is known as London theory was proposed by the London brothers [3]. This theory described well the magnetic properties and magnetic penetration depth of Type I superconductors in weak magnetic fields. By extending the London theory, Ginzburg and Landau proposed a phenomenological theory, which enables the study of Type II superconductors in a strong magnetic field [4]. By assuming the second-order phase using physical intuition and using variational principle of quantum mechanics, the Ginzburg-Landau (GL) theory allowed the calculation of macroscopic quantities of the material in the superconducting state. The most beautiful and successful theory in condensed matter physics, i.e. the microscopic theory of superconductivity which is known as BCS theory was formulated by Bardeen et al. [5].
The starting point of this theory was an effective Hamiltonian of fermionic quasiparticle excitations that interact via a weak phonon-mediated potential close to the Fermi surface. This attractive interaction leads to bound electron pair states known as Cooper pairs. While the single electrons are fermions, not being able to occupy the same state due to Pauli exclusion principle, Cooper pairs are no longer obliged to obey the Fermi-Dirac statistics, and like bosons, they can enter the same state. Superconductors that can be described by the BCS theory are known as conventional superconductors.
The discovery of high-temperature superconductors (HTS) in 1986 [6] presented another challenge for theoretical physicist. Their superconductivity mechanism remains one of the most challenging problems in physics. It is believed that the BCS theory alone cannot explain properties of these classes of superconductors and new theoretical concepts will be required. This unexpected discovery opened a new area in the history of superconductivity, and experimental researchers started trying to find new compounds in this class of superconductors. In the following years, many different HTS compounds with high transition temperatures were found [7, 8, 9]. Unfortunately, most HTS compounds are ceramic in nature and thus their industrial application (e.g., cables and conductors) is difficult. The symmetry of the Cooper pairs in exotic superconductors like cuprates, heavy-fermion [10, 11], organic superconductors [12], and Sr2RuO4 [13] is different from conventional superconductors. Since the pairing symmetry of these superconductors is no longer s-wave, they are known as unconventional superconductors and the study of them has become a central issue in condensed matter physics. These superconductors are often characterized by the anisotropic character in the superconducting gap function with nodes along a certain direction in the momentum space [14, 15]. Since the pairing interaction has an important role in the superconducting gap structure, its determination is very important to explain the basic pairing mechanism. The experimental techniques for the determination of the nodal structure such as phase-sensitive experiments [16] and angle-resolved photoemission spectroscopy (ARPES) [17] are suitable for HTC cuprates but are difficult to apply to other unconventional superconductors with low transition temperature Tc. Therefore, the superconducting gap structure of most unconventional superconductors is still unclear.
The classification of the superconducting order parameter is determined by its behavior under symmetry transformations. From the viewpoint of symmetry, the full symmetry group of the crystal includes the gauge
Below the transition temperature, the
Superconductors that yield due to breaking additional symmetries besides the
By using the parity state in space of Cooper pairs, we can classify superconductors as those with even (spin singlet) and odd (spin-triplet) parities. The discovery of superconductors without inversion symmetry (noncentrosymmetric superconductors) such as
Motivated by new symmetry classes that appear in disordered superconductors, they have also gained great attention for theoretical studies [26, 27, 28]. In these systems, the single particle momentum is no longer a good quantum number and plain-wave eigenfunctions with momentum k should be replaced by position-dependent functions and pairing is between time-reversed states.
To find these functions, the generalized Hartree-Fock equations including the pairing potential of the superconducting state can be used, which leads to Bogoliubov-de Gennes (BdG) equations. The BdG equations are mostly used to describe the behavior of elementary excitation or quasiparticles that are created by Cooper pair breaking in the superconductors. Simultaneously, the properties and classification of the dirty superconductor can be determined by the BdG equations, where pairing symmetry is reflected.
When we apply magnetic field on the superconductor, depending on the strength of the magnetic field, the spin-polarized (strong magnetic field) or BCS states (weak magnetic field) will be favored. In the intermediate region of magnetic field, the Cooper pairs with opposite spins and net momentum are formed and Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is realized [29, 30]. In this state, both translational and rotational symmetries are broken. This state is attractive for particle and nuclear physicists, because it is realized in high-density quark matter, nuclear matter, and cold fermionic atom systems [31]. In addition to FFLO states, there are other types of fascinating superconductors with multiple broken symmetry, such as ferromagnetic superconductors. According to the BCS theory, Cooper pairs in s-wave superconductors could be destroyed by an exchange field, which is known as paramagnetic effect, and it can be observed within a specific temperature interval with the maximum paramagnetic Meissner temperature.
Although, ferromagnetism and superconductivity have opposed order much attention has been paid due to nontrivial phenomena which predicted theoretically or found experimentally [32, 33, 34]. Ferromagnetic superconductors are expected to have triplet pairings since triplet pairings and ferromagnetism can coexist. So far, several ferromagnetic superconductors in bulk materials, for example, UGe 2 [35], ZrZn 2 [36], and URhGe [37], have been identified.
As mentioned earlier, symmetry is a cornerstone in the theory of superconductivity, and the properties of the Cooper pairs, that is, the basic constituents of superconductors, are determined by symmetry. Mathematically, correlation of two electrons to make Cooper pair depends on the position, spin, and time coordinate of these electrons. In BCS theory, the time coordinate is usually ignored but the symmetry property of a paired state allows for the interesting possibility that the two electrons are not correlated at equal times and that they are instead correlated as the time separation grows. In fact, this is realized if the correlation function is odd in time. This novel type of superconducting correlation that is odd in relative time or frequency is known as odd-frequency pairing and it has attracted a lot of attention. The pairing mechanism of odd-frequency superconductors had been theoretically introduced by Berezinskii in the context of superfluid 3He [38]. This pairing mechanism was also predicted in ferromagnet/conventional superconductor junctions due to the breakdown of symmetry in spin space [33, 34]. The experimental results show that odd-frequency pairing exists near the interface in normal metal/superconductor junctions due to the violation of translational symmetry [39]. Consequently, the symmetry breaking more than U(1) is important for existence odd-frequency pairing. The emerging field of complexity in strongly correlated electronic systems may be connected to multiple symmetry breaking systems [40].
Quite analogously with noncentrosymmetric superconductors without inversion symmetry, where even- and odd-parity components are mixed, a mixing of even- and odd-frequency components should occur in principle, when the time reversal symmetry is broken, which affects many superconducting properties [41]. For example, the mixing of the conventional
In last few years, technological advancement allowed to build superconductors of mesoscopic size like single electron transistors, quantum wires, quantum dots, quantum Hall systems, normal metal-superconductor-ferromagnet hybrid structures, magnetic multilayer systems, carbon nanotubes, graphene, small metallic nanoparticles, and nanomechanical systems, which are considered for both experimental and theoretical investigations [43, 44, 45]. In comparing with natural systems, artificial systems are of more interest for investigations because their transport properties can be measured in a more controllable way.
The superconducting properties of mesoscopic superconductors, in which one or more dimensions are comparable with the coherence length, differ remarkably from those of bulk superconductors and their properties such as the critical temperature and the critical fields are affected by the size of the sample. On the other hand, thermal and quantum fluctuations, which are strongly enhanced in these systems, cause dissipation and prevent the formation of the long-range-ordered state.
The study of emergent phases, phase competition, phase transitions, interactions between particles, and dimensional confinement in superconductors, which are of basic interest in the fundamental physics of magnetism, superconductivity, magnetoelectric and spin-dependent transport, is the central goal of the researchers. Their activities may produce new intuition on several related and persistent open questions in condensed matter physics and the results of these studies may also have practical implications for potential applications in spintronics and superconducting micro- or nano-electronics devices. The surface states of topological insulators have attracted strong interest in the past few years. The unique properties of the surface states provide a favorable field to explore emergent phenomenon, such as Majorana fermion-like excitations and quantum anomalous Hall effect. They are also considered attractive candidates for spintronic devices with pure spin current that can be controlled by electric field.
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I am also a member of the team in charge for the supervision of Ph.D. students in the fields of development of silicon based planar waveguide sensor devices, study of inelastic electron tunnelling in planar tunnelling nanostructures for sensing applications and development of organotellurium(IV) compounds for semiconductor applications. I am a specialist in data analysis techniques and nanosurface structure. I have served as the editor for many books, been a member of the editorial board in science journals, have published many papers and hold many patents.",institutionString:null,institution:{name:"Sheffield Hallam University",country:{name:"United Kingdom"}}},{id:"54525",title:"Prof.",name:"Abdul Latif",middleName:null,surname:"Ahmad",slug:"abdul-latif-ahmad",fullName:"Abdul Latif Ahmad",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"20567",title:"Prof.",name:"Ado",middleName:null,surname:"Jorio",slug:"ado-jorio",fullName:"Ado Jorio",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Universidade Federal de Minas Gerais",country:{name:"Brazil"}}},{id:"47940",title:"Dr.",name:"Alberto",middleName:null,surname:"Mantovani",slug:"alberto-mantovani",fullName:"Alberto Mantovani",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"12392",title:"Mr.",name:"Alex",middleName:null,surname:"Lazinica",slug:"alex-lazinica",fullName:"Alex Lazinica",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/12392/images/7282_n.png",biography:"Alex Lazinica is the founder and CEO of IntechOpen. After obtaining a Master's degree in Mechanical Engineering, he continued his PhD studies in Robotics at the Vienna University of Technology. Here he worked as a robotic researcher with the university's Intelligent Manufacturing Systems Group as well as a guest researcher at various European universities, including the Swiss Federal Institute of Technology Lausanne (EPFL). During this time he published more than 20 scientific papers, gave presentations, served as a reviewer for major robotic journals and conferences and most importantly he co-founded and built the International Journal of Advanced Robotic Systems- world's first Open Access journal in the field of robotics. Starting this journal was a pivotal point in his career, since it was a pathway to founding IntechOpen - Open Access publisher focused on addressing academic researchers needs. Alex is a personification of IntechOpen key values being trusted, open and entrepreneurial. Today his focus is on defining the growth and development strategy for the company.",institutionString:null,institution:{name:"TU Wien",country:{name:"Austria"}}},{id:"19816",title:"Prof.",name:"Alexander",middleName:null,surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/19816/images/1607_n.jpg",biography:"Alexander I. Kokorin: born: 1947, Moscow; DSc., PhD; Principal Research Fellow (Research Professor) of Department of Kinetics and Catalysis, N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow.\r\nArea of research interests: physical chemistry of complex-organized molecular and nanosized systems, including polymer-metal complexes; the surface of doped oxide semiconductors. He is an expert in structural, absorptive, catalytic and photocatalytic properties, in structural organization and dynamic features of ionic liquids, in magnetic interactions between paramagnetic centers. The author or co-author of 3 books, over 200 articles and reviews in scientific journals and books. He is an actual member of the International EPR/ESR Society, European Society on Quantum Solar Energy Conversion, Moscow House of Scientists, of the Board of Moscow Physical Society.",institutionString:null,institution:{name:"Semenov Institute of Chemical Physics",country:{name:"Russia"}}},{id:"62389",title:"PhD.",name:"Ali Demir",middleName:null,surname:"Sezer",slug:"ali-demir-sezer",fullName:"Ali Demir Sezer",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/62389/images/3413_n.jpg",biography:"Dr. Ali Demir Sezer has a Ph.D. from Pharmaceutical Biotechnology at the Faculty of Pharmacy, University of Marmara (Turkey). He is the member of many Pharmaceutical Associations and acts as a reviewer of scientific journals and European projects under different research areas such as: drug delivery systems, nanotechnology and pharmaceutical biotechnology. Dr. Sezer is the author of many scientific publications in peer-reviewed journals and poster communications. Focus of his research activity is drug delivery, physico-chemical characterization and biological evaluation of biopolymers micro and nanoparticles as modified drug delivery system, and colloidal drug carriers (liposomes, nanoparticles etc.).",institutionString:null,institution:{name:"Marmara University",country:{name:"Turkey"}}},{id:"61051",title:"Prof.",name:"Andrea",middleName:null,surname:"Natale",slug:"andrea-natale",fullName:"Andrea Natale",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"100762",title:"Prof.",name:"Andrea",middleName:null,surname:"Natale",slug:"andrea-natale",fullName:"Andrea Natale",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"St David's Medical Center",country:{name:"United States of America"}}},{id:"107416",title:"Dr.",name:"Andrea",middleName:null,surname:"Natale",slug:"andrea-natale",fullName:"Andrea Natale",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Texas Cardiac Arrhythmia",country:{name:"United States of America"}}},{id:"64434",title:"Dr.",name:"Angkoon",middleName:null,surname:"Phinyomark",slug:"angkoon-phinyomark",fullName:"Angkoon Phinyomark",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/64434/images/2619_n.jpg",biography:"My name is Angkoon Phinyomark. I received a B.Eng. degree in Computer Engineering with First Class Honors in 2008 from Prince of Songkla University, Songkhla, Thailand, where I received a Ph.D. degree in Electrical Engineering. My research interests are primarily in the area of biomedical signal processing and classification notably EMG (electromyography signal), EOG (electrooculography signal), and EEG (electroencephalography signal), image analysis notably breast cancer analysis and optical coherence tomography, and rehabilitation engineering. I became a student member of IEEE in 2008. During October 2011-March 2012, I had worked at School of Computer Science and Electronic Engineering, University of Essex, Colchester, Essex, United Kingdom. In addition, during a B.Eng. 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