The multiscale restricted-smoothed basis (MsRSB) method is the current state-of-the-art within multiscale methods. MsRSB is very robust and versatile can be used either as an approximate coarse-scale solver having mass-conservative subscale resolution or as an iterative fine-scale solver that will provide mass-conservative solutions for any given tolerance. The performance of the method has been demonstrated on incompressible two-phase flow, compressible two- and three-phase black-oil models, as well as compositional models. It has also been demonstrated that the method can utilize combinations of multiple prolongation operators; e.g., corresponding to coarse grids with different resolutions, adapting to geological features, adapting to wells, or moving displacement fronts. This chapter explains the basic ideas of the MsRSB method, including methods to construct coarse partitions, prolongation, and restriction operators; reduction of the fine-scale flow equations to a coarse-scale system; and formulation as part of a two-level iterative solver. We outline the key functions in the module and show various examples of how the method can be used as an iterative solver for incompressible and compressible flow on 2D rectilinear grids, unstructured grids, and 3D stratigraphic grids.

The upr module in the MATLAB Reservoir Simulation Toolbox (MRST) can construct unstructured Voronoi grids that conform to polygonal boundaries and geometric constraints in arbitrarydimensions prescribed inside the reservoir volume. The resulting volumetric tessellations are usually realized as locally orthogonal, perpendicular bisector (PEBI) grids, in which cell faces can be aligned to accurately preserve objects of codimension one (curves in 2D and surfaces in 3D) and/or cell centroids can be set to follow curves in 2D or 3D. This enables you to accurately model faults, let grid cells follow horizontal and multilateral well paths, or create lower-dimensional or volumetric representations of fracture networks. The module offers methods for improving grid quality, like configurable policies for treating intersecting geometric object and handling conflicts among constraints, methods for locating and removing conflicting generating points, as well as force-based and energy-minimization approaches for optimizing the grid cells. You can use \mcode{upr} to create a consistent hierarchy of grids that represent the reservoir volume, the constraining geometric objects (surfaces and curves), as well as their intersections. The hierarchy is built such that the cell faces of a given (sub)grid conform to the cells of all bounding subgrids of one dimension lower.

We explain how you can use discontinuous Galerkin methods to formulate implicit higher-order discretizations of transport equations on stratigraphic and polytopal grids and outline how this is implemented in the dg module of the MATLAB Reservoir Simulation Toolbox (MRST).

Surfactant and polymer flooding, alone or in combination, are common and effective chemical EOR methods. This chapter reviews the main physical mechanisms and presents how the corresponding mathematical flow models are implemented as an add-on module to MRST to provide a powerful and flexible tool for investigating flooding processes in realistic reservoir scenarios. Using a so-called limited-compositional models, surfactant and polymer are both assumed to be transported in the water phase only, but also adsorbed within the rock. The hydrocarbon phases are described with the standard three-phase black-oil equations. The resulting flow models also take several physical effects into account, such as chemical adsorption, inaccessible pore space, permeability reduction, effective solution viscosities, capillary pressure alteration, relative permeability alteration, and so on. The new simulator is implemented using the object-oriented, automatic differentiation (AD-OO) framework from MRST, and can readily utilize features such as efficient iterative linear solvers with constrained pressure residual (CPR) preconditioners, efficient implicit and sequential solution strategies, advanced time-step controls, improved spatial discretizations, etc. We describe how the computation of fluid properties can be decomposed into state functions for better granularity and present several numerical examples that demonstrate the software and illustrate different physical effects. We also discuss the resolution of trailing chemical waves and validate our implementation against a commercial simulator.

One of the most challenging tasks in reservoir engineering is to homogenize data from a fine to a coarser model in a systematic and robust manner. This chapter reviews a variety of such upscaling methods. Simple averaging is sufficient for additive properties but only correct in special cases for nonadditive properties like permeability. The correct effective permeability depends on the applied flow field. In flow-based upscaling, one solves local flow problems with various types of boundary conditions to determine effective permeabilities or transmissibilities. We outline the most common methods, and discuss methods that reduce the influence of the prescribed boundary conditions by computing flow solutions on larger domains. Computations are achieved by imposing boundary conditions derived from a global flow solution. A number of cases compare the accuracy of different upscaling methods, and we discuss how flow diagnostics can be used for quality control. The last example summarizes major parts of the book by going all the way from geological horizons via flow simulation to upscaled models with flow diagnostics quality control.

This chapter introduces the basic equations used to describe multiphase flow. It also introduces key concepts such as saturation, wettability, relative permeability, and capillary pressure. Combining the multiphase extension of Darcy's law with mass conservation of fluid phases or chemical components gives a system of parabolic PDEs. The chapter derives the so-called fractional flow formulation and discusses several special cases of two-phase flow equations. The chapter ends with a discussion of various analytical and semi-analytical 1D solutions, including the classical Buckley–Leverett problem.

This chapter teaches you how to simulate incompressible, two-phase flow using a sequential formulation that splits the equation system into an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. We discuss fluid objects, the sequential solution procedure, and explicit and implicit transport solvers in some detail. The second part of the chapter is devoted to a number of simulation examples that highlight typical flow behavior. Examples include gravity segregation, homogeneous quarter five-spots, heterogeneous quarter five-spots with viscous fingering, and buoyant migration of CO2 in a sloping aquifer. Furthermore, we discuss water coning, gravity override, capillary fringes, and a simplified simulation of the Norne field model. We end the chapter by a discussion of various sources of numerical errors, including splitting and grid-orientation errors.

This chapter explains how the mathematical models from Chapter 4 are implemented and integrated to form a full simulator. To this end, we introduce data structures to represent fluid behavior, the reservoir state, boundary conditions, source terms, and wells. We then explain in detail how the two-point flux approximation (TPFA) scheme is implemented in MRST for general unstructured grids. We also outline the basic solver used to compute time-of-flight and tracer partitions. We end the chapter by presenting a few examples that demonstrate how to set up simulations in MRST and set appropriate boundary conditions, source terms, or well models. The examples include the famous quarter-five spot problem, a corner-point grid with four intersecting faults, and a model of a shallow-marine reservoir (SAIGUP).