Problems of Multiphase Fluid Filtration

The following sections are included:

• Introduction

• Contents

In definition, filtration means the motion of a flow in a porous medium. The medium is considered to be porous if it has a great deal of pores whose sizes are small as compared to the characteristic sizes of the medium under consideration. The ratio of the pore volume to that of the total, m = Y_por/V_tot, can serve as a quantitative characteristic of porosity…

We will preliminarily transform equations (1.16). Instead of W_i, we will use its value from the generalized Darcy law (1.14). Then,

Equations of two-phase flow of an incompressible fluid in a porous medium without taking account of capillary forces are typically derived from the Darcy law in which P₁ = P₂ = P and from the law of conservation of mass of each phase. We will treat these equations as a limiting form of equations of two-phase flow in a porous medium (2.1) when P_k →0. By definition, q = −k₁ grad Φ − k₂ grad Φ₂. In terms of and R, this can be rewritten as follows: q = −M grad P − N grad R and, consequently, grad grad R. Substitute this expression into the first equation in (2.3), with a simultaneous replacing of R by s on the right-hand side of this equation…

In the previous chapters we have dealt with different formulations of two-phase incompressible fluid nitration problems which are based on the Buckley-Leverett (P₁ = P₂) and Musket-Leverett (P₁ ≠ P₂) models. Not aspiring to a comprehensive analysis of these models, we would like to mention some of their peculiarities showing up to a full extent only when these models are used in computational experiment…

Let the solution of the two-phase filtration problem be sought for in the domain Q_T = D × [0 < t ≤ T] and γ be the boundary of the domain D. Generally speaking, the domain D is arbitrary, because its concrete nature is always connected with a concrete problem. Let the solution of the problem be sought for using the net-point method. In this case, independent of the chosen stream model, the choice of the sought functions and a concrete algorithm for numerical implementation, we will unavoidably face the principal difficulties which are caused just by the arbitrariness of the domain D. Some of these difficulties are connected with grid construction. An attempt to introduce a grid with x_h ∈ γ, i.e. the grid whose boundary nodes lie on γ, immediately leads to a non-uniform grid. An attempt to keep a uniform step brings about a grid whose boundary nodes do not lie on …

The net-point method in one of the most widely used and very often the only possible method when solving numerically two-phase incompressible fluid filtration problems. In this case, a transition is made from the domains of determination of the solution to the initial problem D and Q to the discrete grid domains D_h and Q_h. Correspondingly, the functions of a continuous argument are replaced by the functions of a discrete argument with the determination domains D_h and Q_h. Differential equations of the initial problem are used in place of difference ones. The limiting conditions of the initial problem are changed in an analogous manner. As a result, for each concrete problem we get a closed system of linear (nonlinear) algebraic equations whose solution is regarded as an approximate solution of the initial differential problem…

Consider the parabolic problem:

We can now proceed to consider difference schemes for the general model problem of filtration of a two-phase incompressible fluid with making allowance for capillary forces. Let the solution of the mixed Cauchy problem be sought for in a cylinder Q = {D × (0 < t ≤ T)} with the boundary s:

We will continue our analysis of difference schemes for two-phase incompressible fluid filtration problems. Here these problems are considered in a full nonhnear statement. The principle of construction and study of difference schemes, which have been presented in the previous chapters, serve as a basis. We start with a one-dimensional problem of oil displacement by water under the action of capillary forces alone (the impregnation problem)…

Using one-dimensional problems, we illustrate the considerations leading us to the formulation of the method for solving multiphase incompressible fluid filtration problems. We consider the one-dimensional problem of filtration of a two-phase incompressible fluid in the , R statement:

In Chapter 3, when studying the Cauchy problem for the Buckley-Leverett equation

In Chapter 4, in the context of studying some properties of mathematical models of biphasic incompressible fluid filtration the technological chain related to the computational experiment should be mentioned. Namely: from an actual phenomenon to a mathematical model of this phenomenon, next - to a numerical algorithm and to a program coding this algorithm and, finally, to computations proper, an analysis of the results, the refinement, if necessary, of the mathematical model and so on. It is the implementation of this chain on a computer that determines the technology of the computational experiment…

The principles and methods presented in this book were implemented in the PAP “Oil” (on BESM-6) [64] and those for the filtration problems studied in [89, 90] are realized in the PAP “Filter”. Practically, the same principles and methods, but with a coarser modular structure, underlie the PAPs that were constructed at the National Scientific-Research Institute of Oil [80 - 83]…

The following sections are included:

• References