The Multiscale Finite-Volume Method on Stratigraphic Grids

Abstract

Finding a pressure solution for large and highly detailed reservoir models with fine-scale heterogeneities modeled on a meter scale is computationally demanding. One way of making such simulations less compute intensive is to employ multiscale methods that solve coarsened flow problems using a set of reusable basis functions to capture flow effects induced by local geological variations. One such method, the multiscale finite-volume (MsFV) method, is well studied for 2D Cartesian grids but has not been implemented for stratigraphic and unstructured grids with faults in 3D. We present an open-source implementation of the MsFV method in 3D along with a coarse partitioning algorithm that can handle stratigraphic grids with faults and wells. The resulting solver is an alternative to traditional upscaling methods, but can also be used for accelerating fine-scale simulations. To achieve better precision, the implementation can use the MsFV method as a preconditioner for Arnoldi iterations using GMRES, or as a preconditioner in combination with a standard inexpensive smoother. We conduct a series of numerical experiments in which approximate solutions computed by the new MsFV solver are compared with fine-scale solutions computed by a standard two-point scheme for grids with realistic permeabilities and geometries. On one hand, the results show that the MsFV method can produce accurate approximations for geological models with pinchouts, faults, and non-neighboring connections, but on the other hand, they also show that the method can fail quite spectacularly for highly anisotropic systems in a way that cannot efficiently be mitigated by iterative approaches The MsFV method is, in our opinion, not yet sufficiently robust to be applied as a black-box solver for models with industrystandard complexity. However, extending the method to realistic grids is an important step on the way towards fast and accurate multiscale solution of large-scale reservoir models. In particular, our open-source implementation provides an efficient framework suitable for further experimentation with partitioning algorithms and MsFV variants.

The multiscale restriction smoothed basis method for fractured porous media (F-MsRSB)

“…While it is possible to extend conservative multiscale methods based on a dual-grid formulation to stratigraphic and other types of unstructured grids [21,30,51,63], it has proved to be di cult, if possible, to develop satisfactory dual-primal partitions for a grid with complex geometries. Moreover, localization errors induced by strong permeability contrasts across block boundaries introduce instabilities in the corresponding multipoint coarse-scale stencil.…”

“…They lead to degenerate cells with faces of zero area resulting in a complex grid geometry. With the addition of heterogeneity, it becomes quite a challenging test case for multiscale methods [14,21]. In order to improve the e ciency of F-MsRSB preprocessing steps (such as computing fracture-matrix transmissibility) for this challenging grid geometry, first the CI factors (Eq.…”

“…Once the coarse-scale system is solved, its solution is interpolated into the original fine-scale resolution using the sub-resolution of the basis functions. Among the proposed multiscale methods, multiscale finite-volume (MSFV) methods not only provide mass-conservative solutions at fine-scale, which is a crucial property for convergent solution of transport equations, but also enable relatively simple inclusion of the type of multiphase flow equations seen in contemporary reservoir models [6,[17][18][19][20][21][22].…”

Optimizing oil and gas field management through a fractal reservoir study model

“…Based on the synthetic seismogram profile (Møyner and Lie 2014) obtained, the definition of tops and bottoms of the reservoir layers is achieved and the model development begins. Horizons and faults are continuously identified and mapped to be used at a later period during the grid construction.…”

Constrained pressure residual multiscale (CPR-MS) method for fully implicit simulation of multiphase flow in porous media

“…The MSFV has been applied to a wide range of applications from compressible and three-phase flows to faults and fractures [5][6][7][8][9][10][11][12][13][14][15]. Its recent advancements involve scalable and conservative iterative strategy which has been benchmarked against the commercial Algebraic MultiGrid (AMG) solvers [16][17][18][19].…”