This text forms part of material taught during a course in advanced
reservoir simulation at Delft University of Technology over the
past 10 years. The contents have also been presented at various
short courses for industrial and academic researchers interested in
background knowledge needed to perform research in the area of
closed-loop reservoir management, also known as smart fields,
related to e.g. model-based production optimization, data
assimilation (or history matching), model reduction, or upscaling
techniques. Each of these topics has connections to
system-theoretical concepts.
The introductory part of the course, i.e. the systems description
of flow through porous media, forms the topic of this brief
monograph. The main objective is to present the classic reservoir
simulation equations in a notation that facilitates the use of
concepts from the systems-and-control literature. Although the
theory is limited to the relatively simple situation of horizontal
two-phase (oil-water) flow, it covers several typical aspects of
porous-media flow.
The first chapter gives a brief review of the basic equations to
represent single-phase and two-phase flow. It discusses the
governing partial-differential equations, their physical
interpretation, spatial discretization with finite differences, and
the treatment of wells. It contains well-known theory and is
primarily meant to form a basis for the next chapter where the
equations will be reformulated in terms of systems-and-control
notation.The second chapter develops representations in state-space
notation of the porous-media flow equations. The systematic use of
matrix partitioning to describe the different types of inputs leads
to a description in terms of nonlinear ordinary-differential and
algebraic equations with (state-dependent) system, input, output
and direct-throughput matrices. Other topics include generalized
state-space representations, linearization, elimination of
prescribed pressures, the tracing of stream lines, lift tables,
computational aspects, and the derivation of an energy balance for
porous-media flow.
The third chapter first treats the analytical solution of linear
systems of ordinary differential equations for single-phase flow.
Next it moves on to the numerical solution of the two-phase flow
equations, covering various aspects like implicit, explicit or
mixed (IMPES) time discretizations and associated stability issues,
Newton-Raphson iteration, streamline simulation, automatic
time-stepping, and other computational aspects. The chapter
concludes with simple numerical examples to illustrate these and
other aspects such as mobility effects, well-constraint switching,
time-stepping statistics, and system-energy accounting.
The contents of this brief should be of value to students and
researchers interested in the application of systems-and-control
concepts to oil and gas reservoir simulation and other applications
of subsurface flow simulation such as CO2 storage, geothermal
energy, or groundwater remediation.
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