Modeling Flow in Geometrically Complex Reservoirs Using Hexahedral Multiblock Grids

Abstract

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.

Implementation and Application of a Hybrid Multipoint Flux Approximation for Reservoir Simulation on Corner-Point Grids

“…4 These methods have been extended to two-dimensional unstructured grids [5][6][7] and to three-dimensional hexahedral grids [8][9][10][11] that account for complex geological features such as faults, pinch-outs and deviated wells. 12,13 These works demonstrate that there are many practical situations in which neglecting the effect of permeability anisotropy and grid nonorthogonality results in large errors in flow predictions.…”

“…Special treatment of multipoint flux approximations are required in the presence of complex geologic and geometric features such as faults and pinch-outs. 12,13 Flow through faults depends not only on the permeability of the layers and the fault geometry, but also on the properties of the fault itself. Faults are typically surrounded by a damaged zone that may drastically reduce the flow across the fault.…”

Unstructured Cut-Cell Grids for Modeling Complex Reservoirs

“…Hexahedral multiblock grids have shown good promise for accurate reservoir simulations (Arbogast et al 2000;Jenny et al 2002;Lee et al 2002aLee et al , 2003. Unfortunately, the generation of the multiblock structure for complex reservoir geometries requires significant user expertise and has proved challenging to automate.…”

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“…In this paper, we present a new hybrid algorithm for complex 3D geometries based on locally structured, globally unstructured multiblock grids [30,31]. Compared with fully unstructured grids, such grids have numerous algorithmic advantages and can still honor very complex 3D geometries.…”