Flow of Non-Newtonian Fluids within a Double Porosity Reservoir under Pseudosteady State Interporosity Transfer Conditions

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1 SPE MS Flow of Non-Newtonian Fluids within a Double Porosity Reservoir under Pseudosteady State Interporosity Transfer Conditions J. R. Garcia-Pastrana, A. R. Valdes-Perez, and T. A. Blasingame, Texas A&M University Copyright 2017, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Latin America and Caribbean Petroleum Engineering Conference held in Buenos Aires, Argentina, May This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright. Abstract Non-Newtonian fluids have been characterized over the decades, and such characterizations may be used to model a new approach in engineering disciplines. Non-Newtonian fluids are classified as nontime dependent and time-dependent fluids. This paper focuses on the non-time dependent classification, specifically pseudoplastic fluids. The ranges in these fluids allow the proposed model to be validated. The new reservoir model accounts for non-newtonian behavior within a double-porosity reservoir. This model demonstrates an interporosity transfer function for pseudosteady state, based on a new parameter: dimensionless matrix contribution (D). This parameter differentiates our method from previous efforts based on the pseudosteady state interporosity flow for Newtonian fluids introduced in the 1960's. We derive the partial differential equation for a non-newtonian flow within a double porosity reservoir under pseudosteady state interporosity transfer conditions. The solution presented is for an infinite acting reservoir (assuming the corresponding initial, inner and outer conditions). The objective of this paper is to deliver and provide tools that may help to characterize double porosity reservoir under the condition that a non-newtonian fluid is present, and the interporosity transfer conditions between a matrix system and a fracture system are in pseudosteady-state. Introduction The primary purpose of this work is to develop tools to characterize a naturally fractured reservoir (NFR) that produces heavy oil. Non-Newtonian (i.e., shear-thinning) fluids are sometimes injected or produced from petroleum reservoirs. The proposed solution model may assist with characterizing fluids classified as non- Newtonian. However each reservoir has unique properties and should be managed carefully. The literature describes efforts using the diffusivity equation to characterize reservoir properties, such as reservoir size, flow capacity and heterogeneities. Naturally fractured reservoirs are highly heterogeneous systems. Traditionally, they have been idealized as double porosity reservoirs. The governing equation of the fluid for this type of reservoir is modeled considering flow regime and the geometrty of the reservoir. Many double porosity model applications have been published in the literature, but they can be classified as Pseudosteady-state Interporosity Transfer (Barenblatt and Zheltov 1960; Warren and Root 1963), or Transient Interporosity Transfer (Najurieta 1980;

2 2 SPE MS De Swaan, 1978). A third class of double porosity models take the Transient Interporosity Transfer as a basin, and introduce an interporosity skin to reach an apparent Pseudosteady-state Interporosity Transfer condition (Moench 1984; Cinco-Ley et al. 1985). This paper focuses on the Pseudosteady-state Interporosity Transfer, and the Warren and Root (1963) model will be used as reference. The fluid is considered as pseudoplastic within the non-newtonian classification. Therefore, its behavior is described by a power-law function (Chirstopher et al 1965). Within the power law function there is a parameter that describes how non-newtonian a fluid can be. This parameter is the non-newtonian flow behavior index (for pseudoplastic fluids 0 < n < 1). Ikokuet al. (1979) presented a model characterizing homogeneous reservoirs with the presence of a non-newtonian (pseudoplastic) fluid. Two main approaches were performed using the power-law equation in a double porosity reservoir, Escobar et al (2011) and Olarewaju (1992). Escobar modeled the non-newtonian effect at early and late times, but he implicitly assumed a "Newtoniantransference" during the interaction period between matrix and fracture using the terms similar to the Warren and Root interporosity interaction coefficient (λ) and storativity ratio (ω). Olarewaju (1992) modeled the Transient InterporosityTransfer also accounting for the interaction between matrix and fracture as "Newtonian transference". Proposed Model The physical principles used for the model are the law of conservation of mass, a transport equation, and an equation of state. A scheme of the proposed model is shown in Fig.1. Figure 1 Schematic representation of the proposed model.

3 SPE MS 3 Except for the modification in Darcy's law, the typical assumptions for a double porosity reservoir (Warren and Root, 1963)are used. The motion expression used is the one proposed by Christopher et al. (1965) and the fluid of interest is considered tobe pseudoplastic within the classification of the non- Newtonian fluids. The resulting partial differential equation is defined as follows, and the source term, which adds fluids to the fracture system is defined by: During the derivation of eq. 1 and eq.2, a linearization is proposed to obtain the pressure difference between the fracture and the matrix elevated to 1. The validation of eq.1 and eq.2 arises when the fluid is Newtonian (n=1) both equations become the Warren and Root model. The detailed derivation is given in Appendix A. General Solution The solution obtained in the dimensioneless form and in the Laplace domain for an infinite acting reservoir with constant flowrate is defined by, (1) (2) (3) Appendix B shows the development of the solution in more detail. Discussion The Gaver-Stehfest method (Stehfest 1970) is used to invert Eq. 3 numerically, and. the results are presented in the following graphs. Fig. 2 shows the dimensionless pressure derivative with different values of the non- Newtonian flow behavior index (n) ranging from 0.1 to 1. The slope (v) observed at early times is equal to the slope at late times, and its value, which is obtained from a log-log plot, may be used to solve n in Eq. B.5.

4 4 SPE MS Figure 2 Proposed model for several values of the n-parameter. To calculate such parameters, the effective viscosity (μ etf ) has to be known. It is clear that the slope is caused by the effect of the flow behavior index as Ikoku et al (1979) demonstrated. The slope also impacts the interporosity transfer between the matrix and the fracture (valley). This effect is caused by the dimensionless matrix contribution (D), which accounts for the non-newtonian behavior. Notice that when n=1, the model converges to the Warren and Root's (1963) well-known solution for the infinite acting case. The following analysis is based on the variations for the dimensionless matrix contribution (D) ranging from to Figs. 3-5 show the response of this parameter for different values of the non- Newtonian flow behavior index The dimensionless matrix contribution is defined by a ratio between flowrates (matrix to total system) and an area ratio (wellbore to matrix blocks). The numbers assigned to D are intended to show the response in a log-log plot. Figure 3 Dimensionless matrix contribution (n = 0.1).

5 SPE MS 5 Figure 4 Dimensionless matrix contribution (n = 0.5). Figure 5 Dimensionless matrix contribution (n = 0.75). The physical meaning behind this parameter is that when the value is very low, the total flowrate is higher than the matrix flowrate. When the value is very high, the wellbore area is bigger than the matrix blocks. The values of storativity ratio (ω) and the interface interporosity coefficient (λ) are kept constant, and there are no wellbore storage effects. The derivative plots are divided in early, middle and late times. The early time region is when the fracture system is expanding. The middle time region occurs when the pesudosteady state interporosity transfer conditions between the fracture system and the matrix system are in progress (valley). The late time region is when the total system is expanding.

6 6 SPE MS The interface interporosity coefficient, the storativity ratio and the dimensionless matrix characterize the shape and time response of the dip in the log-log plot. The first two are the same parameters defined by Warren and Root (1963), and the third is related to the non-newtonian behavior. Thes three parameters affect either the response or delay for the occurrence of the dip. The model shows a significant change in the slope of the valley following the trend of the early times and the late times. A comparison between Escobar et al (2011) model and the proposed model is shown in Fig.6. The dotted black line represents Escobar's findings ("Newtonian transference") whereas the continuous red line represents the solution of the proposed model. Both models merge to the Warren and Root model infinite acting case when n=1. The difference arises when the effect of the non-newtonian fluid is present in the pseudosteady state interporosity transfer conditions due to the dimensionless matrix. Figure 6 Comparison between Escobar model and the proposed model. The valley in Escobar's derivative response remains as an interporosity within a Newtonian behavior while the model has a significant slope in the valley when the flow behavior index changes its value. This trend is a continuous effect given by the non-newtonian flow behavior index and the dimensionless matrix contribution. These two parameters are closely related. The model demonstrates that a non-newtonian behavior is present in the response at early, middle and late times. Conclusions and Recommendations Conclusions 1. The interporosity transfer depends on the non-newtonian flow behavior index (n). This value depends on the storativity ratio (ω), the interface interporosity parameter (λ) and the dimensionless matrix contribution factor (D). The latter is a function of the non-newtonian flow behavior index (n). 2. The proposed model has two validations first, the model collapses to the Warren-Root solution when the flow behavior index n=1; and second, if ω = 1 and/or λ = 0, then the model collapses into the Ikoku et al (1979) non-newtonian model.

7 SPE MS 7 3. For the matrix contribution factor D, if there is no matrix flow then the parameter is 0, which means that only one system is present (i.e., the fracture). Recommendations 1. A numerical model/solution can be developed to validate the analytical solutions presented in this paper. 2. The non-newtonian interporosity transfer should be considered in other fracture and/or matrix blocks geometries and regimes in the interporosity transfer. Nomenclature D = Dimensionless matrix contribution, dimensionless H = Variable of consistency [Pa s n ] I 0 = Modified Bessel Functions of the first kind, zero order, dimensionless I 1 = Modified Bessel Functions of the first kind, first order, dimensionless I v = Modified Bessel Functions of the first kind, v order, dimensionless K 0 = Modified Bessel Functions of the first kind, zero order, dimensionless = Modified Bessel Functions of the first kind, first order, dimensionless K 1 K v = Modified Bessel Functions of the first kind, v order, dimensionless c o = Fluid compressibility, (M/Lt 2 ) -1 [Pa -1 ] or [psia -1 ] c r = Formation compressibility, M/Lt 2 ) -1 [Pa -1 ] or [psia -1 ] c t = Total compressibility, M/Lt 2 ) -1 [Pa -1 ] or [psia -1 ] h = Net pay thickness, L [m] or [ft]. k = Permeability, L 2 [md] or [m 2 ] k f = Fracture permeability, L 2 [md] or [m 2 ] k m = Matrix permeability, L 2 [md] or [m 2 ] n = Flow behavior index, dimensionless p = Pressure, M/Lt 2 [Pa] or [psi] p f p i p m p fd p md = Pressure in the fracture, M/Lt 2 [Pa] or [psia] = Initial pressure, M/Lt 2 [Pa] or [psia] = Pressure in the matrix, M/Lt 2 [Pa] or [psia] = Dimensionless pressure in the fracture, dimensionless = Dimensionless pressure in the matrix, dimensionless q = Flowrate, L 3 /t [m3/sec] or [ft 3 /s] q m = Matrix flowrate, L 3 /t [m3/sec] or [ft 3 /s] r = Radial distance, L [m] or [ft] r w = Wellbore radius, L [m] or [ft] r ed = Dimensionless external radius drainage, dimensionless r D = Dimensionless radius, dimensionless s = Skin factor, dimensionless t = Time, t [sec] t D = Dimensionless time, dimensionless u, = Laplace transform variable v r = Radial velocity, L/t [m/s] or [ft/s] ΔL = Length, L [m] or [ft] Δp = Pressure differential, M/Lt 2 [Pa] or [psi] = Shear rate, t -1 [s -1 ]

8 8 SPE MS μ eff = Effective viscosity, M/Lt [cp] or [lb m /ft s] λ = Interporosity flow parameter, dimensionless ϕ = Porosity, fraction = Fracture Porosity, fraction ϕ f ϕ m = Matrix Porosity, fraction ρ = Density, M/L 3 [kg/m 3 ] or [lb m /ft 3 ] ρ o = Initial Density, M/L 3 [kg/m 3 ] or [lb m /ft 3 ] ω = Storativity ratio, dimensionless τ = Shear stress M/Lt 2 [N/m 2 ] or [lb f /ft 2 ] References Cinco-Ley, H., Samaniego V., F., and Kucuk, F. (1985). The Pressure Transient Behavior for Naturally Fractured Reservoirs With Multiple Block Size. Society of Petroleum Engineers. doi: /14168-MS. Chirstopher, R.H and Middleman S., (1965), Power-Law Flow through a Packed Tube. Department of Chemical Engineering, University of Rochester, N.Y. De Swaan O., A. (1976). Analytic Solutions for Determining Naturally Fractured Reservoir Properties by Well Testing. Society of Petroleum Engineers. doi: /5346-PA. Escobar, F. et al Pressure and Pressure Derivative Analysis for Non-Newtonian Pseudoplastic Fluids in DoublePorosity Formations. Ciencia-Teconologia. Colombia. Garcia Jorge R Flow of non-newtonian fluids within a double porosity reservoir under pseudosteady-state interporosity transfer conditions, Master in Science Thesis, Texas A&M University. College Station, TX. Ikoku, C. U., & Ramey, H. J. (1979). Transient Flow of Non-Newtonian Power-Law Fluids in Porous Media. Society of Petroleum Engineers. doi: /7139-PA. Moench, A.F Double-Porosity Models for a Fissured Groundwater Reservoir with Fracture Skin. Water Resources Research, Vol. 20, No. 7, Najurieta, H. L. (1980). A Theory for Pressure Transient Analysis in Naturally Fractured Reservoirs. Society of Petroleum Engineers. doi: /6017-PA. Stehfest, H Algorithm Numerical Inversion of Laplace Transforms. Communication, ACM 13 (1): Warren, J. E., & Root, P. J. (1963). The Behavior of Naturally Fractured Reservoirs. Society of Petroleum Engineers. doi: /426-PA Olarewaju, J. S. (1992). A Reservoir Model of Non-Newtonian Fluid Flow. Society of Petroleum Engineers. (SPE MS).

9 SPE MS 9 Appendix A Partial Differential Equation The continuity equation for a "double porosity" reservoir system is given by: Christopher et al. (1965) proposed the following velocity power law equation (A.1) (A.2) Substituting eq. A.2 in eq. A.2, and assuming the permeability and the effective viscosity are constant after expanding the left hand side is given by, (A.3) Assuming that c t = C f + c r, and applying the chain rule on both sides, after expanding and arraying (A.4) Ikoku et al (1979) proposed the following linearization (A.5) Substituting eq A.5 in A.4 (A.6) The volumetric flow coming from the matrix network is defined (A.7) Recalling eq. A.2 but in terms of the pressure differential going from the matrix to the fracture (A.8) After substituting eq A.8 in eq A.7 and arraying terms (A.9) where

10 10 SPE MS (A.10) Notice that eq.a.9 has a pressure difference elevated to a power, which will present a very significant challenge in our quest to find a solution. In order to eliminate this situation, the following linearization is presented After arraying terms and substituting eq. A.11 in eq. A.9, the source term is given by, (A.11) (A.12) Formulation in Terms of Dimensionless Variables To be able to transform eq.a.6 and eq. A.12 into dimensionless form the following dimensionless variables are used, Dimensionless Pressure: Fracture network Dimensionless Pressure: Matrix system Dimensionless Time: (A.13) (A.14) (A.15) Where the total expansion term for the reservoir is given as: Dimensionless Radius: (A.16) (A.17) Solving p f, p m, t and respectively r in order to transform Eqs. A.6 and A.12: (A.18) (A.19) (A.20) (A.21)

11 SPE MS 11 After substituting the dimensionless variables, the dimensionless form of Eq. A.6 and Eq. A.12 is given by, (A.22) and (A.23) Where the storativity ratio (ω), the interface flow coefficient (λ), and the dimensionless matrix contribution (D) are defined as, (A.24) (A.25) respectively. (A.26)

12 12 SPE MS Appendix B Solution Constant Rate Infinite-Acting Reservoir To provide a solution suitable for well test analysis the following initial and boundary conditions are given in dimensionlessvariables. The initial condition, uniform pressure distribution, is given by: The inner boundary condition, constant flow rate, is given by: The outer boundary condition, inifinite acting reservoir, is: Taking the Laplace transform of Eqs. A.22 and A.23, and Eqs. B.1, B.2 and B.3, the general solution in Laplace domain is given by: Where (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) and the interporosity flow function, g is defined by: The constants C1 and C2 depend upon the inner and outer boundary conditions, which for this case, infinite acting reservoir, are defined as: (B.7) (B.8) (B.9) (B.10) Substituting Eq. B.9 and Eq. B.10 in B.4, the solution in Laplace domain for an infinite acting reservoir with constant flowrate at the well is defined by,

13 SPE MS 13 (B.11)