# 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

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