Most practical reservoir simulation studies are performed using the so-called black oil model, in which the phase behavior is represented using solubilities and formation volume factors. We extend the multiscale finite-volume (MSFV) method to deal with nonlinear immiscible three-phase compressible flow in the presence of gravity and capillary forces (i.e., black oil model). Consistent with the MSFV framework, flow and transport are treated separately and differently using a sequential implicit algorithm. A multiscale operator splitting strategy is used to solve the overall mass balance (i.e., the pressure equation). The black-oil pressure equation, which is nonlinear and parabolic, is decomposed into three parts. The first is a homogeneous elliptic equation, for which the original MSFV method is used to compute the dual basis functions and the coarse-scale transmissibilities. The second equation accounts for gravity and capillary effects; the third equation accounts for mass accumulation and sources/ sinks (wells). With the basis functions of the elliptic part, the coarse-scale operator can be assembled. The gravity/capillary pressure part is made up of an elliptic part and a correction term, which is computed using solutions of gravity-driven local problems. A particular solution represents accumulation and wells. The reconstructed fine-scale pressure is used to compute the finescale phase fluxes, which are then used to solve the nonlinear saturation equations. For this purpose, a Schwarz iterative scheme is used on the primal coarse grid. The framework is demonstrated using challenging black-oil examples of nonlinear compressible multiphase flow in strongly heterogeneous formations.
Summary A semianalytical method is presented for the approximate modeling of the productivity of nonconventional wells in heterogeneous reservoirs. The approach is based on Green's functions and represents an extension of a previous model applicable for homogeneous systems. The new method, referred to as the s-k* approach, models permeability heterogeneity in terms of an effective skin s that varies along the well trajectory and a constant background permeability k*. The skin is computed through local, weighted integrations of the permeability in the near-well region and is then incorporated into the semianalytical solution method. The overall method, which can also model effects due to wellbore hydraulics, is quite efficient in comparison to detailed finite-difference calculations. Results for the performance of nonconventional wells in three-dimensional heterogeneous reservoirs are computed using the s-k* approach and compared to finite-difference calculations resolved on the geostatistical fine grid. The new method is shown to provide an accurate estimate of wellbore pressure and production rate, as a function of position along the wellbore, for various well configurations and heterogeneous permeability fields. The possible use of the overall approach in a simulation while drilling (SWD) tool, in which the well path and trajectory are "optimized" using real-time data, is also discussed. Introduction Nonconventional wells (e.g., horizontal, deviated, or multilateral) have become quite common throughout the oil industry. In designing or optimizing the length and placement of such wells, it is important to estimate accurately the well productivity. One approach for determining this well productivity is to simulate the reservoir performance using a finite-difference simulator. This is the most rigorous approach available, though it is also the most demanding in terms of time and data requirements. An alternate approach for modeling the productivity of nonconventional wells operating under primary production is to employ a semianalytical solution technique. Early work along these lines included single horizontal wells (of infinite conductivity) aligned parallel to one side of a box-shaped reservoir. Solution methods were successive integral transforms1,2 and the use of instantaneous Green's functions,3–6 resulting in infinite series expressions. More complex geometries were considered later7–9 as numerical integration became more feasible. A number of works (see Ouyang,10 and citations therein) include coupling of wellbore hydraulics (i.e., finite-conductivity wells) with reservoir flow. The method we apply in this study9 has one of the most general treatments of wellbore hydraulics. All of the semianalytical techniques mentioned above have the advantage of limited data requirements and high degrees of computational efficiency. These techniques are, however, limited to homogeneous systems or at most strictly layered systems.11,12 This represents a substantial limitation because the productivity of nonconventional wells can be significantly impacted by fine-scale heterogeneities in the near-well region. Fine-scale heterogeneity can be incorporated into detailed simulation models, though the resulting models are complex to build and require substantial computation time to run. The purpose of this paper is to extend an existing semianalytical approach to approximately account for heterogeneity in the near-well region. This will enable us to apply the semianalytical approach to more realistic heterogeneous systems. We accomplish this by introducing an effective skin s into the semianalytical model and then estimating this effective skin as a function of position along the wellbore. The skin is computed via local, weighted integrations of the permeability field in the near-well region. This skin differs significantly from skin in the usual sense, as it is here due to intrinsic heterogeneity in the permeability field rather than from formation damage or stimulation. Away from the wellbore, the reservoir is modeled in terms of the large-scale effective permeability k* The overall method is highly efficient and approximates both near-well effects (through s) and global effects (through k*) with reasonable accuracy. The approach presented here combines and extends formulations developed in two separate earlier studies. These studies addressed the development of a semianalytical well model9 and the approximation of the effects of heterogeneity in the region near a vertical well.13 The semianalytical well model is applicable for very general well configurations and also accounts for pressure drop in the wellbore due to friction, gravitational, and acceleration effects. These can be important in long horizontal wells. The approximate heterogeneity model applied here was developed for the modeling of vertical wells in heterogeneous, two-dimensional areal systems. Both single-well and two-well systems were considered. The basic approach was shown to provide accurate estimates for well productivity, relative to fine-grid simulation results, for many geostatistical realizations over a range of geostatistical parameters. As will be shown below, our new method successfully builds upon both the semianalytical well model and the approximate heterogeneity model. Another technique for approximately modeling the effects of heterogeneity on horizontal wells was previously developed.14 This method, based on a network modeling type of approach, differs considerably from the procedure presented here in that our methodology has as its basis a semianalytical solution technique. The earlier method does, however, display accurate results for a range of problems similar to those considered here. This paper proceeds as follows. We first describe the overall method in some detail. Then, we present numerical results for horizontal and multilateral wells in heterogeneous three-dimensional systems. These results are in many cases compared with detailed finite-difference calculations to assess their level of accuracy. Our new description is shown to provide an accurate estimate of production rate, as a function of position along the wellbore, for a variety of well configurations and for different heterogeneous permeability fields. Finally, we discuss how the overall approach could represent a component of a simulation while drilling (SWD) capability, in which the well path and trajectory are "optimized" using real-time data coupled with our new well model.
Summary A multiscale finite-volume (MSFV) framework for reservoir simulation is described. This adaptive MSFV formulation is locally conservative and yields accurate results of both flow and transport in large-scale highly heterogeneous reservoir models. IMPES and sequential implicit formulations are described. The algorithms are sensitive to the specific characteristics of flow (i.e., pressure and total velocity) and transport (i.e., saturation). To compute the fine-scale flow field, two sets of basis functions - dual and primal - are constructed. The dual basis functions, which are associated with the dual coarse grid, are used to calculate the coarse scale transmissibilities. The fine-scale pressure field is computed from the coarse grid pressure via superposition of the dual basis functions. Having a locally conservative fine scale velocity field is essential for accurate solution of the saturation equations (i.e., transport). The primal basis functions, which are associated with the primal coarse grid, are constructed for that purpose. The dual basis functions serve as boundary conditions to the primal basis functions. To resolve the fine-scale flow field in and around wells, a special well basis function is devised. As with the other basis functions, we ensure that the support for the well basis is local. Our MSFV framework is designed for adaptive computation of both flow and transport in the course of a simulation run. Adaptive computation of the flow field is based on the time change of the total mobility field, which triggers the selective updating of basis functions. The key to achieving scalable (efficient for large problems) adaptive computation of flow and transport is the use of high fidelity basis functions with local support. We demonstrate the robustness and computational efficiency of the MSFV simulator using a variety of large heterogeneous reservoir models, including the SPE 10 comparative solution problem.
Summary A hexahedral multiblock grid (MBG) formulation for the modeling of two-phase reservoir flows is developed and applied. Multiblock approaches are well suited to the modeling of geometrically complex features while avoiding many of the complications of fully unstructured techniques. Important implementation issues are discussed, including the accurate treatment of grid nonorthogonality and full tensor permeability (through use of a 27-point finite difference stencil), the treatment of exceptional cases arising when five blocks intersect, and a new well model that allows for the accurate resolution of near-well effects. Solution efficiency issues, including the use of tensor splitting and the linear solution technique, are also discussed. The actual grid generation step is accomplished using a commercial package. Results for a variety of cases involving flow through geologically complex systems and reservoirs with horizontal and deviated wells are presented. Comparison with analytical results demonstrates the high level of accuracy of the 27-point formulation and the new well model. The method is also applied to a realistic example involving flow through a heterogeneous faulted system. This example illustrates the types of geometrically complex systems that can be modeled using the hexahedral multiblock approach. Introduction The next generation of reservoir simulators must be capable of accurately representing a variety of complicated effects. Of particular importance is the accurate modeling of the effects of geometrically complex features on reservoir flow. Such features, whether geological (e.g., faults, pinchouts, and inclined beddings) or well-related (multilateral wells of general orientation), can have a significant impact on reservoir performance. Because these features are difficult to model using traditional finite difference methods, sophisticated gridding and discretization techniques are required if their effects are to be captured accurately in flow simulators. Several approaches for modeling these geometrically complex features could be pursued. The most general approaches involve the use of fully unstructured grids (triangular grids in two dimensions and tetrahedral or prismatic grids in three dimensions) or structured grids coupled with locally unstructured grids. For a discussion of various grids of this type, see Aziz.1 These techniques, though very general, can result in complex linear systems that may be quite time-consuming to solve. In addition, several technical issues must be solved before highly complex geometrical features can be gridded and modeled in practice (e.g., highly robust unstructured gridding procedures will be required). An alternate, though somewhat less general, approach for modeling geometrically complex features involves the use of hexahedral MBGs. These grids are locally structured (meaning they possess a logical i, j, k ordering locally) but are globally unstructured. As such, they are adept at capturing many types of features, such as faults (the local grid is oriented with the fault) or deviated wells (the grid near the well is approximately radial). Hexahedral MBGs result in locally structured linear systems, which are more computationally efficient to solve than fully unstructured systems. In addition, these approaches provide a very natural framework for parallel computation. In this paper, we develop and apply a hexahedral multiblock formulation for the solution of two-phase reservoir flow problems. A commercial gridding package, GridPro,2 is applied for the actual grid generation step. We introduce a 27-point stencil to handle grid nonorthogonality and full tensor permeabilities. We present approaches for the solution of the resulting linear system and for the treatment of the cross terms in the finite difference stencil. A well model, in which the well is represented in terms of a three dimensional (3D) "hole" through the grid, is also developed. We present results for several problems including flow along a pinchout, flow around a deviated well in a heterogeneous reservoir, and flow in a complex faulted system. Hexahedral multiblock methods have been applied previously in many areas of science and engineering,3,4 including reservoir simulation. For a general review of multiblock gridding approaches, see the book by Thompson et al.4 and references therein. Within the context of reservoir simulation, hexahedral multiblock methods have been developed and applied previously within a finite difference context by Edwards5 and within a mixed finite element context by Arbogast et al.6 and Wheeler et al.7 Edwards5 used a multiblock approach to model 2D sand/shale systems. Arbogast et al.6 introduced Lagrange multiplers to couple blocks, while Wheeler et al.7 applied mortar space techniques to enforce flux continuity between the various blocks. Our work here differs from previous efforts in that we develop a 3D multiblock simulator, based on a 27-point flux continuous finite volume formulation and including a well model, and couple it with a highly sophisticated commercial grid generation tool. This enables us to apply and test the multiblock approach for several complex, realistic reservoir simulation problems. This paper proceeds as follows. We first present the governing equations and the general MBG approach. We then describe our 27-point flux continuous finite difference stencil, well model, and linear solution techniques. The scalability of the linear solver is demonstrated. Results for flow along a pinchout are then presented. Next, we illustrate the accuracy of the well model through simulations of homogeneous systems, for which comparison against an analytical solution is possible. Then, we demonstrate the performance of the well model for a deviated well in a heterogeneous reservoir. The last example involves flow through a heterogeneous faulted reservoir with single injection and production wells. Finally, we draw conclusions and describe the future directions for the development of the MBG approach. Governing Equations and Solution Approach In this section we present the equations for immiscible displacement and describe the multiblock approach used for their solution. Issues involving the discretization scheme, linear solution, and treatment of exceptional cases are discussed.
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