Petroleum reservoir engineering applications of the dual reciprocity boundary element method and the Green element method

Transcription

1 Petroleum reservoir engineering applications of the dual reciprocity boundary element method and the Green element method R. Archer^, R.N. Horne^, O. Onyejekwe^ W Department of Petroleum Engineering, Stanford University, W Department of Civil Engineering, University of Durban-Westville, Durban 4000, South Africa horne@pangea.stanford.edu Abstract Petroleum reservoir engineering requires solutions to the transient diffusion and convection-diffusion equations. Traditionally these calculations are carried out using finite difference methods. Prior workers have used perturbation-based boundary element methods for single phase flow in heterogeneous media. The current work applies the Dual Reciprocity Boundary Element Method (DRBEM) and the Green Element Method (GEM) to flow in heterogeneous media and to the convection-diffusion equation. GEM is a domain-based scheme incorporating the singular integral theory of BEM in an element-by-element fashion. It is very suitable for large problems such as those that can be generated in reservoir engineering because it produces sparse matrices. The performance of DRBEM and GEM are compared for convection-diffusion problems. Numerical experiments showed GEM to be more accurate and more stable than DRBEM at high Peclet number. Both methods allow reservoir heterogeneity to be treated more efficiently than with perturbation-based methods. 1 Introduction Boundary element methods have been used in petroleum engineering for a variety of purposes. Masukawa and Home [6], and Numbere and Tiab [7] used steady state BEM for streamline tracking.

2 526 Boundary Elements Kikani and Home [3] solved transient problems using a Laplace space BEM to model well tests in arbitrarily shaped reservoirs. Koh and Tiab [4] used BEM to describe the flow around tortuous horizontal wells. These studies were all limited to homogeneous, or piecewise homogeneous, reservoirs. To handle heterogeneity Lafe and Cheng [5], and Sato and Home [13], [14] used perturbation-based methods. El Harrouni et al. [2] proposed the use of a transformed form of Darcy's law combined with DRBEM to handle heterogeneity. Petroleum reservoir engineers study the movement offluidthrough porous media. Both single-phase and multiphase flow occurs. The mathematical models that are run by reservoir engineers can be divided into two classes. The first set of models are those run to match historical data in an attempt to determine the properties of the reservoir materials. The second set of models are run in a predictive mode to determine the future hydrocarbon production from a reservoir. Well test analysis is an example of a history matching process. When a well test is conducted, a well is subjected to a change in its flow rate. The resulting pressure response is then measured. This response is compared to analytical or numerical models to estimate reservoir parameters such as permeability. Analytical models are only applicable to reservoirs that are homogeneous or have only very limited heterogeneity. A well test in a single-phase reservoir is governed When numerical models are used they must be highly accurate because reservoir features and parameters are identified by matching both the pressure and the more sensitive pressure derivative to the measured data. Tracer tests are also used to determine reservoir properties. In a tracer test tagged fluid is injected into one well. The recovery of the tracer is then measured at a well to assess the properties of the interwell region. The equation governing tracer flow is: (1) 8C V (D V C) - y VC = (2) Predictive simulations usually consider multiphase flow. Each phase, such as oil and water, is governed by its own mass balance equation. These equations are coupled by a nonlinear function known as relative permeability. This quantifies how the flow of one phase is hindered

3 Boundary Elements 527 by the saturation of the other phase. The mass balance equations are: (3) These equations neglect the effects of gravity and capillary pressure. A typical relative permeability relationship is: (4) (6) 2 Treatment of Heterogeneity None of the equations presented in the introduction are suitable for solution using BEM in the form in which they are written. To proceed with the BEM the equations need to be rewritten with a V^ operator on the left hand side. Two approaches are applicable to Eqn. (1). The first introduces a change of variable, inspired by El Harrouni et of. [2]: where f=t- ^ Under this change of variable Eqn. (1) becomes: W where V2% = 6'% (9) 6' = Y^Z (10) v9 The second approach to treating heterogeneity rewrites Eqn. (1) in terms of the original variable but introduces a second term to the right hand side of the equation inspired by Onyejekwe[9]: (11)

4 528 Boundary Elements 3 Singularity Programming Sato [12] introduced the use of singularity programming in conjunction with boundary element methods to compute pressure transients. Unlike the current study Sato's work solution was performed in Laplace space. Singularity programming decomposes the solution into singular and nonsingular components: PD^PD+PD (12) Dake [1] gives the singular solution for the pressure response of a well flowing at a given rate in an infinite homogeneous reservoir with permeability, k$\ Out of regard for numerical stability, dimensionless variables are used in the implementation of singularity programming. The choice of dimensionless variables follows Sato [12]: PD = ^^ (14) Pini 9D = - 7ropini Using this choice of variables the singular solution is: (18) To apply singularity programming in a heterogeneous reservoir, careful consideration must be made of the appropriate singular solution. The boundary element solution scheme will be used to solve for the nonsingular solution. The dimensionless pressure satisfies: (20)

5 Boundary Elements 529 The singular part of the pressure solution given in Eqn. (13) satisfies: (21) Subtracting Eqn. (21) from Eqn. (20) gives: Noting that pp p^f + p*p Eqn. (22) can be simplifed to: lc T - The right hand side of Eqn. (23) couples the singular and nonsingular solutions. This does not present a problem however since the singular solution is known and can be easily incorporated into the boundary element solution. 4 Green Element Method The Green Element Method (GEM) is an element-by-element approach to implementing the boundary element method. This method retains the singular integral theory of BEM however the implementation is more similar to thefiniteelement method. GEM is described in Taigbenu et al. [15]. GEM has been applied many problems including flow in the unsaturated zone [10], the convection-diffusion equation [16] and mass transfer with chemical reactions [8]. The integral equation solved by the Green Element Method is formed in the same manner as for a classical boundary element method. The appropriate integral equation for the well testing problem Eqn. (11) is: a,, o where (24) X = A (25) G = ln(r - n) (26)

6 530 Boundary Elements K = (27) rc At this point GEM departs from the classical implementation of the boundary element method. Both the domain and boundary integrals in Eqn. (24) are implemented as a summation over element integrals. In this form Eqn. (24) becomes: IL- Further progress can be made by introducing Lagrange basis functions which describe the pressure at any point within an element in terms of the nodal pressures: Substituting Eqn. (29) into Eqn. (29) gives: (29) M e=\ *$jpi + LijV - V&lnKjpi + Ufji - = 0 (30) where *% = Jr. (31) G(r,n)Sljdr (32) (34) These element integrals can be evaluated analytically for rectangular and triangular elements. The unknown internal fluxes are eliminated by expressing them as finite difference approximations in terms of the nodal pressures. The matrix equation GEM generates is sparse which leads to a considerable reduction in computation time.

7 Boundary Elements Dual Reciprocity Boundary Element Method The essence of the dual reciprocity approach is the decomposition of the right hand side of the equation into a weighted sum of approximating functions. In the case of the well testing problem this is expressed as: V*p = 5>j-V*W (35) 3 The transient diffusion and convection-diffusion equations can be handled in this manner. For the details of the dual reciprocity approach see Partridgefll]. 6 Applications 6.1 Tracer Flow Tracer tests in petroleum reservoirs are typically conducted at high Peclet numbers. Numerical experiments were performed to assess the suitability of GEM and DRBEM for modeling tracer tests. The boundary and initial conditions considered were: C(z,0)=0 (36) ) = l (37) Figure 1 shows the results obtained from GEM, DRBEM and a finite difference method. The analytical solution to the problem is also plotted. The Peclet number is These results show that both finite differences and DRBEM were subject to spurious numerical oscillations while GEM was not. GEM also maintained good accuracy at this high Peclet number. 6.2 Well Testing in Heterogeneous Reservoirs Figure 2 shows a permeability distribution in a two-dimensional reservoir. The permeability values vary from 5md to 300md. The pressure response from a well test conducted in this reservoir, computed using GEM and singularity programming, is shown in Figure 3. The upper curves show the variation of pressure with time. The lower curves show the derivative of the pressure with respect to the logarithm

8 532 Boundary Elements of time. This derivative is an important part of well test interpretation since it has distinct shapes that are characteristic of certain reservoir features. The dots represent the GEM simulation and the solid line represents the analytical solution to the same well testing problem if the reservoir were homogeneous with a permeability of 50md. The relatively small departure of the heterogeneous solution from the homogeneous one is unsuprising due to the diffusive nature of the pressure transport mechanism. When GEM and singularity programming were used to simulate the well test in a homogeneous reservoir an excellent match was obtained to the analytical solution, even on a coarse ten element by ten element grid. 6.3 Multiphase Flow GEM has been applied to flow of oil and water in one dimension. The approach was an IMPES (implicit pressure, explicit saturation) scheme. The IMPES equation is formed by adding the mass balance equations for the oil and water phases together to give: V (A,Vp) = 0 (38) where + ^A (39) The IMPES solution proceeds by solving Eqn. (38) using GEM for the pressure field at a given timestep, with the saturations held at their values from the prior timestep. Velocities can then be calculated and fluids can be moved through the system to update the saturation values. The heterogeneous nature of Eqn. (38) can be handled by rearranging it using the methods in Section 2. If Eqn. (38) is nonlinear, for example if the viscosity were pressure-dependent, then the equation could be solved using the Newton-Raphon method. GEM is particularly amenable to this. Figure 4 show compares the GEM simulation, afinitedifference simulation and an analytical solution. The problem being simulated is known as the Buckley-Leverett problem. Water is injected into one end of a one dimensional oil reservoir and fluids are produced from the other end of the reservoir. In this case the injected water forms

9 Boundary Elements 533 a piston-like shock. GEM predicted this more accurately that the finite difference model used. 7 Conclusions Three problems of interest to petroleum reservoir engineers have been solved using DRBEM and GEM. The equation governing singularity programming for well testing in heterogeneous reservoirs has been presented and solved using GEM. This enables highly accurate solutions to be computed on coarse grids. The performance of DRBEM and GEM was compared for tracer transport problems. Numerical experiments showed GEM has better stability than DRBEM. GEM was applied to a one dimensional multiphase flow problem. The saturation shock that occurs in the analytical solution to this problem was reproduced by the GEM solution more accurately than in a finite difference solution of the same problem. These results show that boundary element schemes that can handle the terms resulting from the heterogeneity of petroleum reservoirs, are valuable tools for reservoirs engineers. GEM is particularly promising because of the sparse nature of the matrix equation it forms and its applicability to nonlinear problems. 8 Nomenclature C Ct D G k M N p S t u V concentration compressibility diffusion coefficient Green's function permeability number of elements number of nodes pressure saturation time transformed pressure variable velocity Superscript e ini ns Subscript D o r w element initial nonsingular singular dimensionless local node numbers oil relative water

10 534 Boundary Elements Greek r 7 /* <t> 0 boundary JL <l>n viscosity porosity domain Acknowledgements This work was supported by the Stanford University Research Consortium for Innovation in Well Testing (SUPRI-D). References [1] Dake, L. P.: "Fundamentals of Reservoir Engineering", Elsevier Science, Amsterdam, 1978 [2] El Harrouni, K., Ouazar, D., Wrobel, L. C. and Cheng, A. H- D.: "Global Interpolation Function Based DRBEM Applied to Darcy's Flow in Hetereogeneous Media", Engineering Analysis with Boundary Elements, 1996, [3] Kikani, J. and Home, R. N.: "Pressure-Transient Analysis of Arbitrarily Shaped Reservoirs With the Boundary Element Method", SPE Formation Evaluation, 1992, [4] Koh, L. S. and Tiab, D.: "A Boundary Element Algorithm for Modelling 3D Horizontal Wells Problems Using 2D Grids", SPE Petroleum Computer Conference, New Orleans, LA, July 1993, [5] Lafe, O. E., and Cheng, A.H-D.: "A Perturbation Boundary Element Code for Steady State Groundwater Flow in Hetereogeneous Aquifers", Water Resources Research, 1987, 23(6), [6] Masukawa, J. and Home, R. N.: "Application of the Boundary Integral Method to Immiscible Displacement Problems", SPE Reservoir Engineering, August 1988,

11 Boundary Elements 535 [7] Numbere, D. T. and Tiab, D.: "An Improved Streamline- Generating Technique that Uses the Boundary (Integral) Element Method", SPE Reservoir Engineering, August 1988, [8] Onyejekwe, O. O.: "A Green Element Treatment of Isothermal Flow with Second Order Reaction", Int. Comm. Heat Mass Transfer, 24, 1997, [9] Onyejekwe, O. O.: "A Boundary Element-Finite Element Equations Solution to Flow in Heterogeneous Porous Media", Transport in Porous Media, 31, 1998, [10] Onyejekwe, O. O.: "Boundary Integral Procedures for Unsaturated Flow Problems", Transport in Porous Media, 31, 1998, [11] Partridge, P. W., Brebbia, C. A. and Wrobel, L. C.: "The Dual Reciprocity Boundary Element Method", Elsevier Applied Science, 1992 [12] Sato, K.: Accelerated Perturbation Boundary Element Model for Flow Problems in Heterogeneous Reservoirs, Ph.D. Dissertation, Stanford U., 1992 [13] Sato, K. and Home, R. N.: "Perturbation Boundary Element Method for Heterogeneous Reservoirs: Part 1 - Steady-State Flow Problems", SPE Formation Evaluation, 1993, [14] Sato, K. and Home, R. N.: "Perturbation Boundary Element Method for Heterogeneous Reservoirs: Part 2 - Transient Flow Problems", SPE Formation Evaluation, 1993, [15] Taigbenu, A. E.: "The Green Element Method", International Journal for Numerical Methods in Engineering, 1995, 38, [16] Taigbenu, A. E. and Onyejekwe, O. O.: "Transient ID Transport Equation Simulated by a Mixed Green Element Formulation", International Journal for Numerical Methods in Fluids, 25, 1997,

12 Watercut, % & 8 8 cal schemes for a very convection dominated case 3 well test in heterogeneous Dimensionless 04 distance Figure 1: Comparison of numeri- )3 10* 10' = Time, hours ^igure 3: Pressure respom Concentration b o p p p p ^. ^ ~ X 1 if'"'" III Pressure change, pressure derivative, psi ' s*