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).

The chapter starts by explaining how petroleum reservoirs are formed and gives a brief introduction to various concepts from geology to non-geologists. Next, we discuss the continuum hypothesis and how flow through subsurface porous media is modeled on different spatial scales. An essential part is to develop a description of petrophysical properties like porosity and permeability. We explain how this is achieved in MRST, and outline a few examples of models that give realistic representations of reservoir rocks. This includes the popular SPE10 benchmark and a model of a shallow-marine formation.

The preface briefly describes the purpose of the book and how it differs from many other books.

The chapter explains the need for modeling subsurface flow to solve important societal challenges. We introduce the basic processes involved in primary, secondary, and tertiary petroleum recovery, and explain the ingredients used in reservoir simulation. Finally, we outline the scope of the book and introduce the companion software MRST, which is used widely throughout.

The two-point flux-approximation (TPFA) scheme is robust in the sense that it generally gives a linear system that has a solution regardless of the variations in K and the geometrical and topological complexity of the grid. The resulting solutions will also be monotone, but the scheme is only consistent for certain combinations of grids and permeability tensors K. This implies that a TPFA solution will not necessarily approach the true solution when we increase the grid resolution. It also means that the scheme may produce different solutions depending upon how the grid is oriented relative to the main flow directions. In this chapter, we first explain the lack of consistency for TPFA, before we introduce a few consistent schemes implemented in MRST, including the mimetic finite-difference method and one example of a multipoint flux approximation method (MPFA-O). These can all be written on a general mixed hybrid form, which is motivated by mixed finite-element methods. We explain how you can specify different methods that reduce to known methods on simple grids by adjusting the inner product in the mixed hybrid formulation.

This chapter presents flow diagnostics methods you can use to delineate volumetric communications and improve your understanding of how flow patterns in the reservoir are affected by geological heterogeneity and respond to engineering controls. Using these methods, you can answer questions such as: to what region does a given injector provide pressure support? Which injection and production wells are in communication? Which parts of the reservoir affect this communication? How much does each injector support the recovery from a given producer? Do any of the wells have backflow? What is the sweep and displacement efficiency within a given drainage, sweep, or well-pair region? Which regions are likely to remain unswept? Flow diagnostics also provide several measures of the dynamic heterogeneity of a reservoir model, i.e., the variation in flow paths and their associated travel or residence times. We present several examples that demonstrate how you can use flow diagnostics to analyze interwell communication, improve well placement and sweep efficiency, and pre- and postprocess multiphase flow simulations.

Generating a coarser volumetric description of the reservoir rock is a common task in reservoir engineering. This chapter discusses how to partition a fine grid model into a smaller set of coarse blocks. After the partition, the coarse blocks will each consist of a finite collection of cells from the underlying fine model. Through a series of examples, we demonstrate a variety of different partition methods. Whereas the simplest methods only utilize the geometry or topology of the grid, the more advanced methods can compute partitions that adapt to petrophysical properties, fluid contacts, flow fields, near-well regions, or underlying geological properties like depositional environments, flow units, rock types, etc.

The chapter introduces you to mathematical modeling of flow in porous media. We start by explaining Darcy's law, which together with conservation of mass comprises the basic models for single-phase flow. We then discuss various special cases, including incompressible flow, constant compressibility, weakly compressible flow, and ideal gases. We then continue to discuss additional equations required to close the model, including equations of state, boundary and initial conditions. Flow in and out of wells take place on a smaller spatial scale and is typically modeled using special analytical submodels. We outline basic inflow–performance relationships for the special cases of steady and pseudo-steady radial flow, and develop the widely used Peaceman well model. We also introduce streamlines, time-of-flight, and tracer partitions that all can be used to understand flow patterns better. Finally, we introduce basic finite-volume discretizations, including the two-point flux approximation method, and show how such schemes can be implemented very compactly in MATLAB if we introduce abstract, discrete differentiation operators that are agnostic to grid geometry and topology.

A reservoir simulator consists of a large set of models and parameters to describe the geology, fluid behavior, wells and surface facilities, and rules and conditions that describe and control the production process. On the numerical side, you have a combination of discretization methods, nonlinear solvers, linearizations, linear solvers, preconditioners, stability and convergence checks, and algorithms for automated time-step selection. The AD-OO framework in MRST is developed to encapsulate all these details, such that on one side it offers industry-grade simulator capabilities, and on the other side a flexible framework for rapid development of new proof-of-concept implementations. This chapter outlines the design philosophy behind AD-OO and explains many of its ingredients in detail. By reading this chapter, you will get an in-depth introduction to all the details you need to make a full-fledge reservoir simulator. The chapter ends with a discussion of three cases: a simple pressure depletion, multisegment representation of instrumented wells, and the full SPE 9 benchmark.

The chapter explains how you can generate grid models to represent subsurface reservoirs. We outline a number of elementary grid types: structured/rectilinear grids, fictitious domains, Delaunay triangulations, and Voronoi grids. We then explain stratigraphic grids that are commonly used to model real subsurface formations, including in particular corner-point and perpendicular bisector (PEBI) grids. We explain how such grids are represented in MRST using a data structure for general unstructured grids, and we discuss how to compute geometric properties like volumes, face areas, face normals, etc. We end the chapter by presenting an overview of alternative gridding techniques, including composite grids, multiblock grids, and control-point and boundary conformal grids.