Boundary Element Method For Porous Media Flow

Abstract

The focus point of this study is to develop BEM formulation to overcome the difflculties caused by nonlinearity and heterogeneity in the solution of partial differential equations governing the porous media flow .This dissertation consist of two major parts ,theory of BEM , and applications of BEM to two important fields of porous media ,water flow in aquifers and oil flow in reservoirs. The contribution done in the theory sections is mainly a mathematical derivation of the standard boundary element method for Lapiace's equation and the step by step formulation to the BEM , starting from its correspondence differential equation ,beside the developed form of BEM based on GEM and DRBEM that has presented. to handle both heterogeneous and nonlinearity. In the applications section a novel boundary integral solution was applied for determining : (1) Water table elevation in an unconfined homogeneous aquifer subjected to recharge and dewatering from a stream as well as fluctuations induced by constant and continuous recharge in a two stream unconfined-aquifer system. (2) Changes in water table exposed to a transient boundary condition and space- dependent recharge. This technique was compared with the closed form solution obtained in [111] and excellent results were obtained. (3) Characteristics of the flow through heterogeneous unsaturated porous aquifer . (4) Solution of reservoir engineering problems. This work adapted the most recent developments in boundary element methods to reservoir engineering problems. The transient pressure (diffusion) and convection- diffusion equations were solved in heterogeneous media using the Dual Reciprocity Boundary Element Method (DRBEM) and the Green Element Method (GEM). Numerical experiments showed that DRBEM is more accurate than a standard finite difference method. However like finite difference methods, DRBEM is subject to spurious oscillation at high Peclet numbers. DRBEM also requires the solution of a dense system of equations. GEM, which is a hybrid boundary elementlfinite element method, overcomes these disadvantages. The method was found to produce very accurate solutions to convection-diffusion problems and only shows small oscillations in the solution at very high Peclet numbers. A further important advantage is the sparse nature of the matrix system. GEM is also amenable to solving transient nonlinear problems, which makes it the basis for a new technique for multiphase flow simulation. This work explores the advantages of a hybrid boundary element method known as the Green element method for modeling pressure transient tests. Boundary element methods are a natural choice for the problem because they are based on Green's functions, which are an established part of well test analysis. The classical boundary element method is limited to single phase flow in homogeneous media. This works presents formulations which give computationally efflcient means to handle heterogeneity. Comparisons of the proposed Green element approach to standard finite difference simulation show that both methods are able to model the pressure change in the well over time. When pressure derivative is considered however the finite difference method produces very poor results which would give misleading interpretations. The Green element method in conjunction with singularity programming reproduces the derivative curve very accurately. Boundary element method was applied for solving Stokes flow equations on multi particle system. Also, the method is modified for estimating flow parameters for a specified porous media. A new method for the so/ut/‘an of the unsteady incompressible Navier-Sta/res equations was presented