Adaptive Collocation Method for the Transport Problem Induced by Irregular Well Patterns

  • Vietchau Nguyen
  • Georges Abi-Ghanem
Conference paper

Abstract

In this communication, we present a numerical method of solution to the transport problem associated with a multiple source/sink flow field. Consider the miscible displacement equation:
$$\frac{{\partial c}} {{\partial t}} + \frac{\partial } {{\partial x}}\left( {v_x c} \right) + \frac{\partial } {{\partial y}}\left( {v_y c} \right) = \nabla \cdot \left( {D\left( {x,y} \right)\nabla c} \right),\nabla = \frac{{(\partial }} {{\partial x}},\frac{{\partial )}} {{\partial y}},$$
(1)
where c is the concentration, D is the variable dispersion coefficient for applications under study, and the flow field is defined by
$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v}=\left[{\begin{array}{*{20}{c}}{{v_x}}\\{{v_y}}\end{array}} \right] = \frac{1}{{2\pi \phi h}}\sum\limits_{m = 1}^n {\frac{{{q_m}}}{{{{(x - {x_m})}^2} + {{(y - y)}^2}}}} \left[ {\begin{array}{*{20}{c}}{x - {y_m}} \\{y - {y_m}}\end{array}}\right],$$
(2)
where is the porosity, h is the thickness of the flow field, and qm is the pumping rate at well (xm,ym), m = 1, ..., N Equation (2) is obtained by first solving the Laplace equation for the hydraulic head, then applying Darcy, s law and the principle of superposition to the N-well system. Note that equation (1) becomes advection-dominated in the neighborhood of the (xm, ym)-well, and consequently leads to numerical instability of the solution.

Keywords

Porosity Uranium 

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References

  1. 1.
    M.B. Allen “How Upstream Collocation Works”, Intl. J. Dim. Meth. Eng., to appear.Google Scholar
  2. 2.
    M.A. Celia and G.F. Pinder (1982) “Transport Simulation Using Three Dimensional Alternating Collocation”, in Finite Elements in Water Resources IV, R.P. Holtz et al ed., Hanover, Germany, 14.9–14.9.Google Scholar
  3. 3.
    J. Douglas and T. Dupont (1971) “Alternating-Direction Galerkin Methods on Rectangles”, Symp. on the Numerical Solution of PDEs-II, B. Hubbard ed., Academic, New York, 133–213.Google Scholar
  4. 4.
    V.V. Nguyen (1983) “Collocation Simulator for a System of Coupled Transport Operators”, in IMACS Trans. on Scientific Computation VoL III, W.F. Ames ed., North-Holland, Amsterdam, The Netherlands.Google Scholar
  5. 5.
    V.V. Nguyen et al. (1983) “Numerical Simulation of Uranium In- Situ Mining”, Chemical Engng. Sci., 38 (11): 1885–1862.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Vietchau Nguyen
    • 1
  • Georges Abi-Ghanem
    • 1
  1. 1.EWA, Inc.MinneapolisUSA

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