Abstract
This chapter develops representations in state-space notation of the porous-media flow equations derived in Chap. 1. For single-phase flow, the states are grid-block pressures, and for two-phase flow they are grid-block pressures and saturations. The inputs are typically bottom-hole pressures or total well flow rates, the outputs are typically bottom-hole pressures in those wells were the flow rates were prescribed, and phase rates in those wells were the bottom-hole pressures were prescribed. The 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.
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Notes
- 1.
The dependent variables follow from the physics of the problem. In case of multi-phase flow through porous media they are typically pressures, component masses or phase saturations; see Chap. 1. Here we use only a single dependent variable, but in general multiple dependent variables will occur, in which case multiple differential equations are required to describe the problem.
- 2.
Typically, most of the values \( \bullet_{i} \) are equal to zero in a single equation. For example in the case of one-dimensional single-phase flow modeled with first-order finite differences, the only three non-zero values \( \bullet_{i} \) in the ith equation in a system of n equations with 1 < i < n are given by: \( \hat{e}_{i} \left( { \bullet_{i} ,{{d\left( { \bullet_{i} } \right)} \mathord{\left/ {\vphantom {{d\left( { \bullet_{i} } \right)} {dt}}} \right. \kern-0pt} {dt}}} \right) = \hat{f}_{i} \left( { \bullet_{i - 1} , \bullet_{i} , \bullet_{i + 1} ,\psi_{i} } \right). \)
- 3.
An alternative name for the input matrix is distribution matrix because it distributes the inputs u over the states x.
- 4.
Sometimes this form of generalized state equations is referred to as a descriptor system .
- 5.
System equations expressed as \( {\mathbf{g}}\left( \ldots \right) = \bf{0} \) are sometimes referred to as equations in residual form.
- 6.
The random model errors are also referred to as random input, or as a stochastic forcing term.
- 7.
Usually simply referred to as Jacobians.
- 8.
In the systems-and-control literature this is known as a control-affine nonlinear equation. An affine function is a linear function plus a translation. Control affine functions are an important topic of study in nonlinear control theory.
- 9.
In a numerical implementation, the inverse V −1 of the diagonal matrix V can be computed very efficiently by just taking the reciprocals of the diagonal elements.
- 10.
In grid blocks that are not penetrated by a well the prescribed flow rates are of course equal to zero.
- 11.
I.e. we do not consider kinetic energy at a macroscopic level. We do take into account energy dissipation, which is the change of mechanical energy into thermal energy, or heat, and which at an atomic level can be interpreted as kinetic energy again.
- 12.
Elevation-related potential energy plays an important role in well-bore flow. Most reservoirs have enough potential energy, at least initially, to lift the oil to surface naturally in the production wells . This lift effect is in most cases caused by the difference in density between oil and water, such that in an oil-filled well that drains a hydrostatically-pressured reservoir the oil will be lifted to surface because of elevation-related potential energy.
- 13.
For a detailed derivation of pressure-related potential energy for the case of pressure-dependent rock and fluid compressibilities, see Hubbert (1940).
- 14.
Energy delivered to a system through mechanical or hydraulic action is often referred to as work . Work (or energy) per unit time is then called power . In strict SI units time is expressed in s (seconds), energy and work in J (Joule) and power in W (Watt), such that 1 W is equal to 1 J/s.
- 15.
In comparison, Eq. (2.127) was also a function of the well-bore pressures \( {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{p} }}_{well} \) and \( {\bar{\mathbf{p}}}_{well} \).
- 16.
The general expression for the inverse of a 2 × 2 block matrix is given by \( \left[ {\begin{array}{*{20}c} {{\mathbf{V}}_{11} } & {{\mathbf{V}}_{12} } \\ {{\mathbf{V}}_{21} } & {{\mathbf{V}}_{22} } \\ \end{array} } \right]^{ - 1} = \left[ {\begin{array}{*{20}c} {{\tilde{\mathbf{V}}}_{1}^{ - 1} } & { - {\mathbf{V}}_{11}^{ - 1} {\mathbf{V}}_{12} {\tilde{\mathbf{V}}}_{2}^{ - 1} } \\ { - {\mathbf{V}}_{22}^{ - 1} {\mathbf{V}}_{21} {\tilde{\mathbf{V}}}_{1}^{ - 1} } & {{\tilde{\mathbf{V}}}_{2}^{ - 1} } \\ \end{array} } \right] \), where \( {\tilde{\mathbf{V}}}_{ 1} = {\mathbf{V}}_{ 1 1} { - }{\mathbf{V}}_{ 1 2} {\mathbf{V}}_{ 2 2}^{ - 1} {\mathbf{V}}_{ 2 1} \) and \( {\tilde{\mathbf{V}}}_{ 2} \, = \,{\mathbf{V}}_{ 2 2} { - }{\mathbf{V}}_{ 1 1}^{ - 1} {\mathbf{V}}_{ 1 2} \) are the Schur complements of V 11 and V 22 respectively; see e.g. Friedland (1986), pp. 479–481. Using the property that equally sized diagonal matrices commute, we can derive Eq. (2.151) from this more general expression.
- 17.
This distinction is not very clear cut. For example to compute the oil and water ‘input’ rates, we make use of the fractional flows around the wells which are a direct function of the saturations, i.e. of state variables. In this sense the rates also contain indirect output information on the saturations around the wells.
- 18.
Here, b is an arbitrary right-hand side.
- 19.
Even if the reservoir is above bubble point such that it contains only oil and water and no free gas, the flow in the well bore will be three-phase because associated gas will be released from the oil as the well-bore pressure decreases at increasing elevations above the reservoir. Alternatively, the lift table entries can be chosen as tubing-head pressure, oil rate, gas-oil ratio, and water-cut. Whatever the choice of the table entries, it is assumed that the fluid properties at standard conditions and the well-bore geometry do not change during the reservoir simulation.
- 20.
See the footnote on page 46.
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Jansen, J.D. (2013). System Models. In: A Systems Description of Flow Through Porous Media. SpringerBriefs in Earth Sciences. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00260-6_2
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