Summary of results for example 4
Conventional well test interpretation models do not work in reservoirs containing non-Newtonian fluids such as completion and stimulation treatment fluids: polymer solutions, foams, drilling muds (this should not be considered as a reservoir fluid, since before testing we should clean the well to remove all the drilling invasion fluids, however it obeys the power-law), etc., and some paraffinic oils and
As a special kind of non-Newtonian fluid, Bingham fluids (or plastics) exhibit a finite yield stress at zero shear rates. There is no gross movement of fluids until the yield stress,
The Bingham plastic concept has been found to approximate closely many real fluids existing in porous media, such as paraffinic oils, heavy oils, drilling muds and fracturing fluids, which are suspensions of finely divided solids in liquids. Laboratory investigations have indicated that the flow of heavy-oil in some fields has non-Newtonian behavior and approaches the Bingham type.
Pseudoplastic and dilatant fluids have no yield point. The slope of shear stress versus shear rate decreases progressively and tends to become constant for high values of shear stress for pseudoplastic fluids. The simplest model is power law,
Dilatants fluids are similar to pseudoplastic except that the apparent viscosity increases as the shear stress increases. The power-law model also describes the behavior of dilatant fluids but
Currently, unconventional reservoirs are the most impacting subject in the oil industry. Shale reservoirs, coalbed gas, tight gas, gas hydrates, gas storage, geothermal energy, coal – conversion to gas, coal-to-gas, in-situ gasification and heavy oil are considered unconventional reservoirs. In the field of well testing, several analytical and numerical models taking into account Bingham, pseudoplastic and dilatant non-Newtonian behavior have been introduced in the literature to study their transient nature in porous media for a better reservoir characterization. Most of them deal with fracture wells, homogeneous and double-porosity formations and well test interpretation is conducted via the straight-line conventional analysis or type-curve matching and recently some studies involving the pressure derivative have also been introduced.
When it is required to conduct a treatment with a non-Newtonian fluid in an oil-bearing formation, this comes in contact with conventional oil which possesses a Newtonian nature. This implies the definition of two media with entirely different mobilities. If a pressure test is run in such a system, the interpretation of data from such a test through the use of conventional straight-line method may be erroneous and may not provide a way for verification of the results obtained.
The purpose of this chapter is to provide the most updated state-of-the-art on transient analysis of Non-Newtonian fluids and to present both conventional and modern methodologies for well test interpretation in reservoirs saturated with such fluids. Especial interest is given to the use of the pressure and pressure derivative for both homogeneous and double-porosity formations.
2. Non-Newtonian fluids in transient pressure analysis
Non-Newtonian fluids are often used during various drilling, workover and enhanced oil recovery processes. Most of the fracturing fluids injected into reservoir-bearing formations behave non-Newtonianly and these fluids are often approximated by Newtonian fluid flow models. In the field of well testing, several analytical and numerical models taking into account Bingham and pseudoplastic non-Newtonian behavior have been introduced in the literature to study the transient nature of these fluids in porous media for a better reservoir characterization. Most of them deal with fracture wells and homogeneous formations and well test interpretation is conducted via the straight-line conventional analysis or type-curve matching. Only a few studies consider pressure derivative analysis. However, there exists a need of a more practical and accurate way of characterizing such systems.
Many studies in petroleum and chemical engineering and rheology have focused on non-Newtonian fluid behavior though porous formations, among them, we can name [6, 9, 10, 18, 20, 23]. Several numerical and analytical models have been proposed to study the transient behavior of non-Newtonian fluid in porous media. Since all of them were published before the eighties, when the pressure derivative concept was inexistent; interpretation technique was conducted using either conventional analysis or type-curve matching.
It is worth to recognize that Ikoku has been the researcher who has contributed the most to non-Newtonian power-law fluids modeling, as it is demonstrated in the works of [9,10,11,13]. All of these models have been used later for other researchers for further development of test interpretation techniques. For instance, reference  presented a study of a pressure fall-off behavior after the injection of a non-Newtonian power-law fluid.  presented a study using the elliptical flow on transient analysis interpretation in Polymer flooding EOR since polymer solutions also exhibit non-Newtonian rheological behavior such as in-situ shear-thinning and shear-thickening effects.
 used for the first time the pressure-derivative concept for well test analysis of non-Newtonian fluids, and later on,  presented the first extension of the
As far as non-Newtonian fluid flow through naturally fractured reservoirs is concerned only a study presented by  is reported in the literature. He presented the analytical solution for the transient behavior of double-porosity infinite formations which bear a non-Newtonian pseudoplastic fluid and his analytical solution also considers wellbore storage effects and skin factor; therefore,  used the analytical solution without wellbore storage and skin introduced by  was used to develop an interpretation technique using the pressure and pressure derivative, so expressions to estimate the Warren and Root parameters  (dimensionless
3. Pseudoplastic infinite-acting radial flow regime in homogeneous formations
Interpretation of pressure tests for non-Newtonian fluids is performed differently to conventional Newtonian fluids. During radial flow regime, Non-Newtonian fluids exhibit a pressure derivative curve which is not horizontal but rather inclined. As shown by , the smaller the value of
A partial differential equation for radial flow of non-Newtonian fluids that follow a power-law relationship through porous media was proposed . Coupling the non-Newtonian Darcy's law with the continuity equation, they derived a rigorous partial differential equation:
This equation is nonlinear. For analytical solutions, a linearized approximation was also derived by :
The dimensionless quantities were also introduced by  as
The dimensionless pressure derivative during radial flow regime is governed by:
 presented the following expression to estimate the permeability,
 presented more practical expressions for the determination of both permeability and skin factor:
4. Well pressure behavior in non-Newtonian/Newtonian interface
In many activities of the oil industry, engineers have to deal with completion and stimulation treatment fluids such as polymer solutions and some heavy crude oils which obey a non-Newtonian power-law behavior. When it is required to conduct a treatment with a non-Newtonian fluid in an oil-bearing formation, this comes in contact with conventional oil which possesses a Newtonian nature. This implies the definition of two media with entirely different mobilities. If a pressure test is run in such a system, the interpretation of data from such a test through the use of conventional straight-line method may be erroneous and may not provide a way for verification of the results obtained. Then,  proposed a solution for the system sketched in Figure 3 which was solved numerically by .
 presented for the first time the pressure derivative behavior for the mentioned system, Figure 4. Notice in that plot that the pressure derivative shows an increasing slope as the flow behavior index decreases. Also, the derivative has no slope during infinite-acting Newtonian behavior, as expected.
During the non-Newtonian region, region 1 in Figure 3, Equations 13 to 15 work well. For the Newtonian region, region 2, the permeability and skin factor are estimated with the equations presented by Tiab (1993) as:
Suffix 2 denotes the non-Newtonian region.
 also found an expression to estimate the non-Newtonian permeability using the time of intersection of the non-Newtonian and Newtonian radial lines,
The radius of the injected non-Newtonian fluid bank is calculated using the following correlation (not valid for
Later,  found that Equations 13, 14, 15 and 22 also worked for dilatant systems. This is the case when 2 <
n = 0.6
A value of 6.228x10-5 hr/(ft3-
Using a time of 0.0008 hr which corresponds to the intersect point formed between the non-Newtonian and Newtonian radial flow regime lines in Equation 19, a non-Newtonian effective fluid permeability of 96 md is re-estimated.  obtained a permeability of the non-Newtonian zone of 101 md and
5. Hydraulically fractured wells
 linearized the partial-differential equation for the problem of a well intercepted by a vertical fracture. Their dimensionless pressure solution is given below:
 presented two interpretation methodologies: type-curve matching and conventional straight-line for characterization of fall-off tests in vertically hydraulic wells with a pseudoplastic fluid. They indicated that at early times, a well-defined straight line with slope equal to 0.5 on log-log coordinates will be evident, then,
Where the characteristic viscosity,
And the derivative of Equation 24 is:
And the dimensionless fractured conductivity is;
 presented an expression which relate the half fracture length,
However, there is no proof that Equation 28 works for Non-Newtonian systems. Using Equation 23,  presented pressure and pressure derivative curves for vertically infinite-conductivity fractured wells. See Figure 7. They extended the
The expression governing the late-time pseudosteady-state flow regime is:
The point of intersection of the pressure derivatives during linear flow and pseudosteady-state (mathematical development is not shown here) allows to obtain the well drainage area by means of the following expression:
Using Equation 15, a value of 0.23 is found for
A fracture conductivity of 868.5 md‐ft was calculated using Equation 29. It is important to clarify that this equation is valid for the Newtonian case. This value was used in Equation 28 to find a dimensionless fracture conductivity of 1.73.
6. Finite-homogeneous reservoirs
The analytical solution in the Laplace space domain for the closed reservoirs under constant-rate case is given as:
For the case of constant-pressure external boundary, the boundary condition given by Equation 35 is changed to:
And the analytical solution for such case is:
Using the solution provided by ,  presented pressure and pressure derivative plots for such behaviors as shown in Figs. 9 and 10. In these plots it is seen for closed systems in both pseudoplastic and dilatant cases, that the late-time pressure derivative behavior always displays a unit-slope line as for Newtonian fluids. As for Newtonian behavior, the late-time pressure derivative decreases in both dilatant or pseudoplastic cases.
There is no pressure derivative expression for open boundary systems. Then, for pseudoplastic fluids the following correlation was also developed ,
Equating Equation 41 to 39 and solving for reservoir drainage area, such as:
For dilatant fluids the correlation found is:
In a similar fashion as for the pseudoplastic case,
7. Heterogeneous reservoirs
In the well interpretation area of the Petroleum Engineering discipline a homogeneous reservoir is conceived to possess a single porous matrix while a heterogeneous reservoir has a porous matrix and either vugs or fractures. A common term used for heterogeneous systems is naturally-fractured reservoirs. However, this term is not recommended to be used since the fractures may result for either a mechanic process or a chemical process (matrix dissolution). Therefore, a more convenient term used in this book is double porosity systems in which the well is fed by the fractures and the fractures are fed by the matrix. By the same token, in a double-permeability system the well is fed by both fractures and matrix and the fractures are also fed by the matrix. This last one, however, has little application in the oil industry.
The governing well pressure solution in the Laplacian domain for a double-porosity system with a non-Newtonian fluid excluding wellbore storage and skin effects was provided by  as:
The Laplacian parameter,
 implemented the
Figure 12 shows a log-log plot of the dimensionless pressure and pressure derivative for a double- porosity system with constant interporosity flow parameter, constant
The infinite-acting radial flow regime is identified by a straight line which slope increase as the value of the flow behavior index decreases. See Figure 14. The first segment of such line corresponds to the fracture-network dominated period, and, the second one -once the transition effects are no longer present-, responds for a equivalent homogeneous reservoir. An expression for the slope is given  as:
Also, the slope of the pressure derivative during radial flow regime is related to the flow behavior index by:
As observed in Figure 12, as the dimensionless storage coefficient decreases the transition period is more pronounced no matter the value of the interporosity flow parameter. Therefore, a correlation for 0
Another way to estimate
A final correlation to estimate ω valid for 0 1 with an error lower than 0.4 % is given as follows:
The interporosity flow parameter also plays an important role in the characterization of double porosity systems. From Figure 13, it is observed that the smaller the value of
Equation 51 is valid for 1x10-4 <
Which is valid for 1x10-4 <
During the infinite-acting radial flow regime the following points were arbitrarily read:
With these points a slope is estimated to be
As a final remark, I would like to comment that some crude oils or other type of fluids used in the oil industry may display a non-Newtonian Bingham-type behavior. It is common to deal with Non Newtonian fluids during fracturing and drilling operations and oil recovery processes, as well. When a reservoir contains a non-Newtonian fluid, such as those injected during EOR with polymers flooding or the production of heavy-oil, the interpretation of a pressure test for these systems cannot be conducted using the conventional models for Newtonian fluid flow since it will lead to erroneous results due to a completely different behavior.
The problem considered now, presented in reference , involves the production of a Bingham fluid from a fully penetrating vertical well in a horizontal reservoir of constant thickness; the formation is saturated only with the Bingham fluid. The basic assumptions are: (a) Isothermal, isotropic and homogeneous formation, (b) Single-phase horizontal flow without gravity effects, (c) Darcy’s law applies, and (d) Constant fluid properties and formation permeability.
The governing flow equation can be derived by combining the modified Darcy’s law with the continuity equation and is expressed in a radial coordinate system as:
The density of the Bingham fluid,
The initial condition is:
At the wellbore inner boundary,
 solved numerically Equation 55 and provided an interpretation technique for this type of fluids using the pressure and pressure derivative log-log plot. For a Bingham-type non-Newtonian fluid, this behavior changes by observing that there is a point where the dimensionless pressure derivative is high and this increases with an increase of
This chapter comprises the most updated state-of-the-art for well test interpretation in reservoirs having a non-Newtonian fluid. Extension of the
|Volumetric factor, RB/STB|
|System total compressibility, 1/psi|
|Wellbore storage, bbl/psi|
|Dimensionless fracture conductivity|
|Formation thickness, ft|
|Consistency (Power-law parameter), cp*sn-1|
|Group defined by Equation 3|
|Minimum pressure gradient, Psi/ft|
|Dimensionless pressure gradient|
|Flow consistency parameter|
|Fracture conductivity, md-ft|
|Flow behavior index (power-law parameter)|
|Flow/injection rate, STB/D|
|Pressure derivative, psi|
|Dimensionless pressure derivative|
|Distance from well to non-Newtonian/Newtonian front/interface|
|Half-fracture length, ft|
|Shear rate, s-1|
|Shear stress, N/m|
|Dimensionless interposity parameter|
|Effective viscosity for power-law fluids, cp*(s/ft)n-1|
|Bingham plastic coefficient, cp|
|Characteristic viscosity, cp/ft1-n|
|Shear stress, N/m|
|Dimensionless storativiy coefficient|
|Dimensionless based on area|
Based on area
|Dimensionless based on half-fracture length|
Based on area
|Intersect of linear and pseudosteady-state lines|
|Radial (any point on radial flow)|
|Intersect of radial and linear lines|
|Intersect of radial and pseudosteady-state lines|
|Intersect of radial and steady-state lines|