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A transport equation in porous media with an oblique, evolutionary boundary condition
Abstract
In this thesis we consider a linear partial differential equation that models contaminant transport with sources in a fractured porous medium. The geometry we study consists of a single fracture bounded by semi-infinite porous blocks. A general mathematical model describing contaminant flow in fractures is developed, and a careful scaling analysis is used to compare this model with some of the known models studied in the literature. A simplified model which serves as the focus for this paper is$$\eqalign{{u}\sb{t}=u\sb{yy}&-\lambda u+f(x,y,t),\quad x,y>0,\ \ t>0\cr {u}\sb{t}=-{u}\sb{x}+\gamma u\sb{y}&-\lambda u\ {\rm on}\ y=0;\quad u(0,0,t)=u\sb0(t),\ \ t>0\cr}$$with zero initial data. In the fracture, this simplified model assumes convection, decay, linear adsorption and loss to the porous matrix. In the porous medium, the model includes diffusion, decay, linear adsorption and sources. The governing equations are represented by a degenerate, linear, parabolic equation in a quarter-space with an oblique, evolutionary differential equation as a boundary condition. We solve the problem using Laplace transforms to obtain the Green's function and determine how contaminant sources at the fracture inlet and in the porous media are propagated in time. The one-dimensional diffusion assumed in the problem is common among models in the literature and results in the omission of a no-flux boundary condition along $x=0$ (which in some physical problems may be incorrect). In this paper we also show that this simplified problem is the outer problem for the correctly posed singular perturbation problem$$\eqalign{u\sb{t}&=\sqrt{\epsilon u}\sb{xx}+u\sb{yy}-\lambda u,\quad x,y>0,\ \ t>0\cr u\sb{t}&=\epsilon u\sb{xx}-u\sb{x}+u\sb {y}-\lambda u\ {\rm on}\ y=0\cr u\sb{x}&=0\ {\rm on}\ x=0\cr u(0,0,t)&=u\sb0(t),\quad t>0\cr}$$with zero initial data, which includes two-dimensional diffusion and a no-flux boundary condition along $x=0.$ Solution surfaces for the simplified problem are obtained in two ways: first, in the exact integral solution, the infinite limits of integration are mapped to a finite region to avoid truncation difficulties and a direct quadrature scheme is applied; second, the problem is solved numerically. An implicit, finite difference scheme is applied to the diffusion problem in the matrix while an explicit difference scheme is used to approximate the differential equation along the boundary. Finally, consistency and convergence of the scheme is proved subject to conditions on the discretization.
Subject Area
Mathematics|Hydrology
Recommended Citation
Homp, Michelle Reeb, "A transport equation in porous media with an oblique, evolutionary boundary condition" (1997). ETD collection for University of Nebraska-Lincoln. AAI9805509.
https://digitalcommons.unl.edu/dissertations/AAI9805509