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.

Why doesn’t one single, solitary structural discontinuity form and cut across a laboratory test specimen or an outcrop, rather than forming a network? Why isn’t the San Andreas Fault just a single, continuous strand? Why are echelon arrays formed by the different structure types, such as joints, faults, or deformation bands?

Planar breaks in rock are one of the most spectacular, fascinating, and important features in structural geology. Joints control the course of river systems, the extrusion of lava flows and fire fountains, and modulate groundwater flow. Joints and faults are associated with bending of rock strata to form spectacular folds as seen in orogenic belts from British Columbia to Iran, as well as seismogenic deformation of continental and oceanic lithospheres. Anticracks akin to stylolites accommodate significant volumetric strain in the fluid-saturated crust. Deformation bands are pervasive in soft sediments and in porous rocks such as sandstones and carbonates, providing nuclei for fault formation on the continents. Faults also form the boundaries of the large tectonic plates that produce earthquakes—and related phenomena such as mudslides in densely populated regions such as San Francisco, California—in response to tectonic forces and heat transport deep within the Earth. Faults, joints, and deformation bands have been recognized on other planets, satellites, and/or asteroids within our Solar System, attesting to their continuing intrigue and importance to planetary structural geology and tectonics.

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.