Browsing by Author "Azhari Ahmed Abdalla"

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Boundary Element Method For Porous Media Flow
(Neelain University, 2005) Azhari Ahmed Abdalla
ABSTRACT The focus point of this study is to develop BEM formulation to overcome the difﬁculties 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 resen/oirs. The contribution done in the theory sections is mainly a mathematical derivation of the standard boundary element method for Laplace'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 memod. 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 eiementlﬁnite 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 efﬁclent means to handle heterogeneity. Comparisons of the proposed Green element approach to standard ﬂnite 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 multl particle system. Also, the method is modiﬁed for estimating flow parameters for a speciﬁed porous media. A new method for the solution of the unsteady incompressible Navier-Stokes equations was presented . lll
Boundary Element Method For Porous Media Flow
(Al Neelain University, 2005-07) Azhari Ahmed Abdalla
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
some Aspects Of Flow Through Pomus Medla
(Neelain University, 2001) Azhari Ahmed Abdalla
ABSTRACT A mathematical modelling of ﬂow through porous media, from Darcy to Non- Darcy is presented. The nature of non—linear ﬂow in porous media is analyzed by means of the volumetric averaging approach, and physical explanation for the dispersion term is deduced. The effect of porosity on natural convective ﬂow and heat transfer ina saturated porous medium has been investigated using Galerkrin’s ﬁnite element method. A boundary value solution to axisymmetric creeping ﬂow ‘ past and through aporous prolate spheroid is presented. Solutions for the temperature ﬁeld caused by the ﬂow past aheated spheroid in saturated -porous media are presented inthe case where the spheroid is heldat constant temperature, and- Where the ﬂux at all points on the spheroid is held constant. Some applications of the equations of ﬂow through porous media are given in the groundwater ﬂow and oil recovery. Singular integral equation methods are used to obtain closed form solutions for ﬂow from single straight line fracture. ' An analytical solution is derived for two dimensional steady tunnel in afully saturated, homogeneous, isotropic, and semi- inﬁnite aquifer.