Assessment of unsteady Brinkman’s model for flow in karst aquifers

Abstract

The Brinkman’s equation simplifies the numerical modeling of karst aquifers by allowing the use of a single transport equation to model the flow of fluids in both the free-flow and porous regions, in effect reducing the error arising from improper modeling of the interface between the two regions. Most equations available to model flow within karst aquifers deal with steady flow conditions. This may not be accurate in aquifers where unsteady conditions exist. We considered the effects of unsteady flow conditions in karst aquifers by assessing the addition of an unsteady flow term to the Brinkman’s equation. We solved the coupled mass conservation-transport equations that models unsteady fluid transport in karst aquifers and studied the effects of unsteady flow conditions on tracer transport in two different sample aquifers and compared to the results obtained from the steady flow Brinkman’s equation. The solution method adopted is sequential and it involves solving the unsteady Brinkman’s model first, followed by the advection-diffusion-adsorption equation using the cell-centered finite volume approach. The first example presented here is a simple aquifer model consisting of a single conduit surrounded by porous regions. The second example is a complicated structure consisting of complex geometrical caves embedded in a highly heterogeneous porous media. The results show that, inside the caves, the unsteady Brinkman’s model yielded lower tracer concentrations at early times when compared to the steady flow model. At longer times, both models produced almost similar results. In particular, the results obtained from the simplified example case (Example 1) indicate that the velocity profiles for unsteady flow within open conduits do not instantly yield a parabolic shape expected from the Brinkman’s equation, but gradually develops into one starting from a linear profile. Results obtained also show that the addition of unsteady flow term to the Brinkman’s model does not affect the flow of tracer within porous media in any significantly observable manner.

Appendix

A. Discretization of equations of flow

While discretizing the equations of flow, the perturbation is done such that the pressures are at the center of the grid blocks while the velocities are at the grid interfaces. The convective acceleration terms in the unsteady Brinkman’s equation introduces nonlinearity to the equation. Therefore, the discretized equations are written as residual functions and the Newton-Raphson method for solving nonlinear simultaneous equations is used to solve the problem.

The discretization for the conservation of mass (Eq. 9) in two-dimensions is given by

$$ {R}_C^{\gamma +1}=\frac{\phi \rho {c}_t}{\varDelta t}{p}_{h,i}^{\gamma +1}-\frac{\rho }{\varDelta y}{u}_{y_{h,i-\frac{1}{2}}}^{\gamma +1}+\frac{\rho }{\varDelta y}{u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1}-\frac{\rho }{\varDelta x}{u}_{x_{h-\frac{1}{2},i}}^{\gamma +1}+\frac{\rho }{\varDelta x}{u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}-\frac{\rho {q}_{sc}}{V_b}-\frac{\phi \rho {c}_t}{\varDelta t}{p}_{h,i}^n, $$

(16)

where h and i are the indices of the grid blocks in the x and y-directions, the superscript γ is the iteration index, while the superscript n is the index of time. ${R}_C^{\gamma +1}$, is the residual function for the conservation of mass.

The Brinkman’s equations representing the flow in x and y directions, respectively, are

$$ \frac{\rho }{\phi}\frac{\partial {u}_x}{\partial t}+\frac{\rho }{\phi^2}{u}_x\frac{\partial {u}_x}{\partial x}+\frac{\rho }{\phi^2}{u}_y\frac{\partial {u}_x}{\partial y}+\frac{\partial p}{\partial x}+\frac{\mu }{K_x}{u}_y-{\mu}_{eff}\left(\frac{\partial^2{u}_x}{\partial {x}^2}+\frac{\partial^2{u}_x}{\partial {y}^2}\right)=0, $$

(17)

and

$$ \frac{\rho }{\phi}\frac{\partial {u}_y}{\partial t}+\frac{\rho }{\phi^2}{u}_x\frac{\partial {u}_y}{\partial x}+\frac{\rho }{\phi^2}{u}_y\frac{\partial {u}_y}{\partial y}+\frac{\partial p}{\partial y}+\frac{\mu }{K_y}{u}_y-{\mu}_{eff}\left(\frac{\partial^2{u}_y}{\partial {x}^2}+\frac{\partial^2{u}_y}{\partial {y}^2}\right)=0 $$

(18)

The discretization of Eq. 17 will give the following equation

$$ {\displaystyle \begin{array}{c}{R}_{u_x}^{\gamma +1}=\frac{\rho }{\phi}\left[\frac{u_{x_{h+\frac{1}{2},i}}^{\gamma +1}-{u}_{x_{h+\frac{1}{2},i}}^n}{\varDelta t}\right]+\frac{\rho }{\phi^2}{u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}\left[\frac{u_{x_{\alpha}}^{\gamma +1}-{u}_{x_{\beta}}^{\gamma +1}}{\varDelta x}\right]+\frac{\rho }{\phi^2}{u}_{y_{h+\frac{1}{2},i}}^{\gamma +1}\left[\frac{u_{x_{\theta}}^{\gamma +1}-{u}_{x_{\tau}}^{\gamma +1}}{\varDelta y}\right]+\frac{p_{h+1,i}^{\gamma +1}-{p}_{h,i}^{\gamma +1}}{\varDelta x}+\frac{\mu }{K_x}{u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}\\ {}-\frac{\mu_{eff}}{\varDelta {x}^2}\left[{u}_{x_{h+\frac{3}{2},i}}^{\gamma +1}-2{u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}+{u}_{x_{h-\frac{1}{2},i}}^{\gamma +1}\right]-\frac{\mu_{eff}}{\varDelta {y}^2}\left[{u}_{x_{h+\frac{1}{2},i+1}}^{\gamma +1}-2{u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}+{u}_{x_{h+\frac{1}{2},i-1}}^{\gamma +1}\right]=0\end{array}}, $$

(19)

where ${R}_{u_x}^{\gamma +1}$ is the residual function for the Brinkman’s equation in the x-direction. The values of ${u}_{x_{\alpha}}^{\gamma +1}$, ${u}_{x_{\beta}}^{\gamma +1}$, ${u}_{x_{\theta}}^{\gamma +1}$, and ${u}_{x_{\tau}}^{\gamma +1}$ are evaluated using the upwinding technique

$$ {\displaystyle \begin{array}{l}{u}_{x_{\alpha}}^{\gamma +1}=\Big\{\begin{array}{c}{u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}\ \mathrm{if}\ {u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}>0\\ {}{u}_{x_{h+\frac{3}{2},i}}^{\gamma +1}\ \mathrm{if}\ {u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}<0\end{array},\\ {}{u}_{x_{\beta}}^{\gamma +1}=\Big\{\begin{array}{c}{u}_{x_{h-\frac{1}{2},i}}^{\gamma +1}\ \mathrm{if}\ {u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}>0\\ {}{u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}\ \mathrm{if}\ {u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}<0\end{array},\end{array}} $$

(20)

and

$$ {\displaystyle \begin{array}{l}{u}_{x_{\theta}}^{\gamma +1}=\Big\{\begin{array}{c}{u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}\ \mathrm{if}\ {u}_{y_{h+\frac{1}{2},i}}^{\gamma +1}>0\\ {}{u}_{x_{h+\frac{1}{2},i+1}}^{\gamma +1}\ \mathrm{if}\ {u}_{y_{h+\frac{1}{2},i}}^{\gamma +1}<0\end{array},\\ {}{u}_{x_{\tau}}^{\gamma +1}=\Big\{\begin{array}{c}{u}_{x_{h+\frac{1}{2},i-1}}^{\gamma +1}\ \mathrm{if}\ {u}_{y_{h+\frac{1}{2},i}}^{\gamma +1}>0\\ {}{u}_{x_{h+\frac{1}{2},i}}^{\gamma +1}\ \mathrm{if}\ {u}_{y_{h+\frac{1}{2},i}}^{\gamma +1}<0\end{array}.\end{array}} $$

(21)

The discretization of Eq. 18 will give the following equation

$$ {\displaystyle \begin{array}{c}{R}_{u_y}^{\gamma +1}=\frac{\rho }{\phi}\left[\frac{u_{y_{h,i+\frac{1}{2}}}^{\gamma +1}-{u}_{y_{h,i+\frac{1}{2}}}^n}{\varDelta t}\right]+\frac{\rho }{\phi^2}{u}_{x_{h,i+\frac{1}{2}}}^{\gamma +1}\left[\frac{u_{y_{\alpha}}^{\gamma +1}-{u}_{y_{\beta}}^{\gamma +1}}{\varDelta x}\right]+\frac{\rho }{\phi^2}{u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1}\left[\frac{u_{y_{\theta}}^{\gamma +1}-{u}_{y_{\tau}}^{\gamma +1}}{\varDelta y}\right]+\frac{p_{h,i+1}^{\gamma +1}-{p}_{h,i}^{\gamma +1}}{\varDelta x}+\frac{\mu }{K_y}{u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1},\\ {}-\frac{\mu_{eff}}{\varDelta {x}^2}\left[{u}_{y_{h+1,i+\frac{1}{2}}}^{\gamma +1}-2{u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1}+{u}_{y_{h-1,i+\frac{1}{2}}}^{\gamma +1}\right]-\frac{\mu_{eff}}{\varDelta {y}^2}\left[{u}_{y_{h,i+\frac{3}{2}}}^{\gamma +1}-2{u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1}+{u}_{y_{h,i-\frac{1}{2}}}^{\gamma +1}\right]=0\end{array}} $$

(22)

where ${R}_{u_y}^{\gamma +1}$ is the residual function for the Brinkman’s equation in the y-direction. The values of ${u}_{y_{\alpha}}^{\gamma +1}$, ${u}_{y_{\beta}}^{\gamma +1}$, ${u}_{y_{\theta}}^{\gamma +1}$, and ${u}_{y_{\tau}}^{\gamma +1}$ is evaluated using the upwinding technique

$$ {\displaystyle \begin{array}{l}{u}_{y_{\alpha}}^{\gamma +1}=\Big\{\begin{array}{c}{u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1}\ \mathrm{if}\ {u}_{x_{h,i+\frac{1}{2}}}^{\gamma +1}>0\\ {}{u}_{y_{h+1,i+\frac{1}{2}}}^{\gamma +1}\ \mathrm{if}\ {u}_{x_{h,i+\frac{1}{2}}}^{\gamma +1}<0\end{array},\\ {}{u}_{y_{\beta}}^{\gamma +1}=\Big\{\begin{array}{c}{u}_{y_{h-1,i+\frac{1}{2}}}^{\gamma +1}\ \mathrm{if}\ {u}_{x_{h,i+\frac{1}{2}}}^{\gamma +1}>0\\ {}{u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1}\ \mathrm{if}\ {u}_{x_{h,i+\frac{1}{2}}}^{\gamma +1}<0\end{array},\end{array}} $$

(23)

and

$$ {\displaystyle \begin{array}{l}{u}_{y_{\theta}}^{\gamma +1}=\Big\{\begin{array}{c}{u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1}\ \mathrm{if}\ {u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1}>0\\ {}{u}_{y_{h,i+\frac{3}{2}}}^{\gamma +1}\ \mathrm{if}\ {u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1}<0\end{array},\\ {}{u}_{y_{\tau}}^{\gamma +1}=\Big\{\begin{array}{c}{u}_{y_{h,i-\frac{1}{2}}}^{\gamma +1}\ \mathrm{if}\ {u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1}>0\\ {}{u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1}\ \mathrm{if}\ {u}_{y_{h,i+\frac{1}{2}}}^{\gamma +1}<0\end{array}.\end{array}} $$

(24)

B. Discretization of Equations for Tracer Transport

When discretizing the equations of tracer transport, the perturbation is done such that the tracer concentrations are obtained at the center of the grid blocks while the velocities are located at the grid interfaces.

The discretization of Eq. 15 gives

$$ {\displaystyle \begin{array}{l}{\left(\frac{\phi {D}_y}{\varDelta {y}^2}\right)}_{h,i+\frac{1}{2}}^{n+1}{c}_{h,i-1}^{n+1}+{\left(\frac{\phi {D}_x}{\varDelta {x}^2}\right)}_{h+\frac{1}{2},i}^{n+1}{c}_{h-1,i}^{n+1}\\ {}-\left[{\left(\frac{\phi {D}_x}{\varDelta {x}^2}\right)}_{h+\frac{1}{2},i}^{n+1}+{\left(\frac{\phi {D}_x}{\varDelta {x}^2}\right)}_{h-\frac{1}{2},i}^{n+1}+{\left(\frac{\phi {D}_y}{\varDelta {y}^2}\right)}_{h,i+\frac{1}{2}}^{n+1}+{\left(\frac{\phi {D}_y}{\varDelta {y}^2}\right)}_{h,i-\frac{1}{2}}^{n+1}+\frac{\phi_{h,i}{R}_{h,i}}{\varDelta t}\right]{c}_{h,i}^{n+1}\\ {}+{\left(\frac{\phi {D}_x}{\varDelta {x}^2}\right)}_{h-\frac{1}{2},i}^{n+1}{c}_{h+1,i}^{n+1}+{\left(\frac{\phi {D}_y}{\varDelta {y}^2}\right)}_{h,i-\frac{1}{2}}^{n+1}{c}_{h,i+1}^{n+1}\\ {}-\frac{u_{x_{h+\frac{1}{2},i}}^{n+1}}{\varDelta x}c{\prime}_{h+\frac{1}{2},i}^{n+1}+\frac{u_{x_{h-\frac{1}{2},i}}^{n+1}}{\varDelta x}c{\prime}_{h-\frac{1}{2},i}^{n+1}-\frac{u_{y_{h,i+\frac{1}{2}}}^{n+1}}{\varDelta y}c{\prime}_{h,i+\frac{1}{2}}^{n+1}+\frac{u_{y_{h,i-\frac{1}{2}}}^{n+1}}{\varDelta y}c{\prime}_{h,i-\frac{1}{2}}^{n+1}=-\frac{\phi_{h,i}{R}_{h,i}}{\varDelta t}{c}_{h,i}^n-{\overset{\cdotp }{c}}_s\end{array}}. $$

(25)

The term c^′ is set using the upwinding technique. One example of upwinding technique is shown in Eq. 26

$$ c{\prime}_{h+\frac{1}{2},i}^{n+1}=\Big\{{\displaystyle \begin{array}{c}c{\prime}_{h,i}^{n+1}\ \mathrm{if}\ {u}_{x_{h+\frac{1}{2},i}}^{n+1}>0\;\\ {}c{\prime}_{h+1,i}^{n+1}\ \mathrm{if}\ {u}_{x_{h+\frac{1}{2},i}}^{n+1}<0\end{array}}. $$

(26)

The discretized advection-diffusion-adsorption equations in (25) are solved only after the coupled flow equations (Eqs. 16, 19, and 22) have been solved for pressures and the velocities at the new time. This decoupling of the tracer transport equations from the flow equations ensures that two separate smaller-sized problems are solved instead of a larger nonlinear system of equations. The discretized equations in (25) are linear and thus do not require a Newton-Raphson iterative solver.