Computational Simulation of Chemical Dissolution-Front Instability Problems Associated with Radially Divergent Flow in Fluid-Saturated Porous Media

Abstract

This paper proposes a computational simulation procedure for simulating chemical dissolution-front instability problems, in which radially divergent flow is involved in fluid-saturated porous media. In the proposed computational simulation procedure, a combination of the finite element and finite difference methods is used to simulate a chemical dissolution-front instability problem involving radially divergent flow, while a new algorithm is used to apply a small perturbation to the problem. Particular attention is paid on simulating low-order modes of an unstable circular chemical dissolution-front propagating in a fluid-saturated porous medium, in which dissolvable materials only occupy a small part, so that the final porosity is remarkably smaller than unity when dissolvable materials are completely dissolved in the chemical dissolution system. To verify the proposed computational simulation procedure, analytical solutions for a benchmark chemical dissolution-front instability problem involving radially divergent flow are derived in a purely mathematical manner. The related computational simulation results have demonstrated that: (1) the proposed computational simulation procedure is correct and useful for simulating chemical dissolution-front instability problems, which are associated with both stable and unstable chemical dissolution systems involving radially divergent flow in fluid-saturated porous media; (2) the simulated shapes of the second-order, third-order and fifth-order modes associated with an unstable chemical dissolution-fronts are respectively an ellipse, a star of three angles and a star of five angles in the unstable chemical dissolution system involving radially divergent flow in the fluid-saturated porous medium; (3) although the heterogeneity of a porous medium can affect the propagation speed of a chemical dissolution-front, it does not affect the low-order mode shape in the unstable chemical dissolution system involving radially divergent flow in the fluid-saturated heterogeneous porous medium.

Appendix: Derivation of Analytical Perturbation Solutions of the Considered Problem

According to linear stability theory (Chadam et al. 1986; Zhao et al. 2008b, 2013b), total solutions of the considered problem can be written as a summation of base solutions and perturbation solutions in the following forms:

$$S(\overline{r},\,\theta ,\,\overline{t}) = \overline{R}_{S} + \delta F(\overline{t})\cos (\overline{m}\theta )$$

(51)

$$\overline{p}_{uT} (\overline{r},\,\theta ,\,\overline{t}) = \overline{p}_{uB} (\overline{r},\,\overline{t}) + \delta \,\hat{p}_{u} (\overline{r})F(\overline{t})\cos (\overline{m}\theta )$$

(52)

$$\overline{p}_{dT} (\overline{r},\,\theta ,\,\overline{t}) = \overline{p}_{dB} (\overline{r},\,\overline{t}) + \delta \,\hat{p}_{d} (\overline{r})F(\overline{t})\cos (\overline{m}\theta )$$

(53)

$$\overline{C}_{uT} (\overline{r},\,\theta ,\,\overline{t}) = \overline{C}_{uB} (\overline{r},\overline{t}) + \delta \,\hat{C}_{u} (\overline{r})F(\overline{t})\cos (\overline{m}\theta )$$

(54)

where $\overline{C}_{uT}$ is the total solution for the dimensionless acid concentration in the upstream regions; $\overline{p}_{dT}$ and $\overline{p}_{uT}$ are the total solutions for the dimensionless pore-fluid pressures in the downstream and upstream regions; $\hat{C}_{u}$ is the perturbation solution for the dimensionless acid concentration in the upstream region; $\hat{p}_{d}$ and $\hat{p}_{u}$ are the perturbation solutions for the dimensionless pore-fluid pressures in the downstream and upstream regions; $F(\overline{t})$ is a dimensionless time-dependent function (e.g. $F(\overline{t}) = e^{{\overline{\omega }\overline{t}}}$) involved in the applied perturbation;$\,\overline{t}$ is the dimensionless time; $\overline{m}$ is the dimensionless wavenumber of the applied perturbation; $\overline{\omega }$ is the dimensionless growth rate of the applied perturbation; $\delta$ is the amplitude of the applied perturbation and $\delta < < 1$ in the linear stability analysis.

Equations (51) to (54) indicate that if the considered chemical dissolution system is in a stable state, then the applied small perturbation decays with time, so that the total solutions of the considered problem are equal to the base analytical solutions of the considered problem. In this case, the considered chemical dissolution system can be served as a benchmark problem, which can be used to verify the computational simulation procedure for solving chemical dissolution-front instability problems involving radially divergent flow in fluid-saturated porous media. This is the first purpose of deriving analytical solutions in this study.

Substitution of Eqs. (51) to (54) into Eqs. (32) and (33) yields the first-order dimensionless perturbation equations of the chemical dissolution-front instability problem associated with radially divergent flow as follows:

$$\overline{r}^{2} \frac{{\partial^{2} \hat{p}_{u} }}{{\partial \overline{r}^{2} }} + \overline{r}\frac{{\partial \hat{p}_{u} }}{{\partial \overline{r}}} - \overline{m}^{2} \hat{p}_{u} = 0$$

(55)

$$\frac{{\partial^{2} \hat{C}_{u} }}{{\partial \overline{r}^{2} }} + (1 - \overline{R}_{0} Zh)\frac{1}{{\overline{r}}}\frac{{\partial \hat{C}_{u} }}{{\partial \overline{r}}} - \overline{m}^{2} \frac{1}{{\overline{r}^{2} }}\hat{C}_{u} + \frac{{\partial \overline{C}_{uB} }}{{\partial \overline{r}}}\frac{{\partial \hat{p}_{u} }}{{\partial \overline{r}}} = 0\quad \left( {{\text{In }}\,{\text{the}}\,{\text{ upstream }}\,{\text{region}}} \right)$$

(56)

$$\overline{r}^{2} \frac{{\partial^{2} \hat{p}_{d} }}{{\partial \overline{r}^{2} }} + \overline{r}\frac{{\partial \hat{p}_{d} }}{{\partial \overline{r}}} - \overline{m}^{2} \hat{p}_{d} = 0\quad \left( {{\text{In}}\,{\text{ the}}\,{\text{ downstream }}\,{\text{region}}} \right)$$

(57)

For the first-order perturbation equations (i.e. Eqs. 55 to 57), the corresponding boundary conditions can be written as follows:

$$\hat{C}_{u} = 0,\quad \frac{{\partial \hat{p}_{u} }}{{\partial \overline{r}}} = 0,\quad \left( {\overline{r} = \overline{R}_{0} } \right)$$

(58)

$$\frac{{\partial \hat{p}_{d} }}{{\partial \overline{r}}} = 0\quad \left( {\overline{r} \to \infty } \right)$$

(59)

The interface conditions associated with the first-order perturbation equations can be expressed as:

$$\hat{C}_{u} (\overline{R}_{S} ) + \frac{{\partial \overline{C}_{uB} (\overline{R}_{S} )}}{{\partial \overline{r}}} = 0,\quad \hat{p}_{u} (\overline{R}_{S} ) + \frac{{\partial \overline{p}_{uB} (\overline{R}_{S} )}}{{\partial \overline{r}}} = \hat{p}_{d} (\overline{R}_{S} ) + \frac{{\partial \overline{p}_{dB} (\overline{R}_{S} )}}{{\partial \overline{r}}}\quad \left( {{\text{at}}\,\,\,\overline{r} = \overline{R}_{S} + \Delta \overline{r}} \right)$$

(60)

$$\frac{{\partial \hat{C}_{u} (\overline{R}_{S} )}}{{\partial \overline{r}}} + \frac{{\partial \overline{C}_{uB}^{2} (\overline{R}_{S} )}}{{\partial \overline{r}^{2} }} = - \overline{\omega }\varepsilon (\phi_{f} - \phi_{0} ),$$

$$\frac{{\partial \hat{p}_{u} (\overline{R}_{S} )}}{{\partial \overline{r}}} + \frac{{\partial^{2} \overline{p}_{uB} (\overline{R}_{S} )}}{{\partial \overline{r}^{2} }} = \psi^{*} (\phi_{0} )\left[ {\frac{{\partial \hat{p}_{d} (\overline{R}_{S} )}}{{\partial \overline{r}}} + \frac{{\partial^{2} \overline{p}_{dB} (\overline{R}_{S} )}}{{\partial \overline{r}^{2} }}} \right]\quad \left( {{\text{at}}\quad \overline{r} = \overline{R}_{S} + \Delta \overline{r}} \right)$$

(61)

where $\Delta \overline{r} = \delta F(\overline{t})\cos (\overline{m}\theta )$ and $\overline{\omega }$ is the dimensionless growth rate of the applied perturbation.

After Eqs. (55) and (57) are solved, the analytical perturbation solution for the dimensionless pore-fluid pressure can be written as:

$$\hat{p}_{u} (\overline{r}) = A_{1} \overline{r}^{{\overline{m}}} + A_{2} \overline{r}^{{ - \overline{m}}} \quad \left( {\overline{R}_{0} \le \overline{r} \le \overline{R}_{S} } \right)$$

(62)

$$\hat{p}_{d} (\overline{r}) = B_{1} \overline{r}^{{ - \overline{m}}} \quad \left( {\overline{R}_{S} \le \overline{r}} \right)$$

(63)

where

$$A_{1} = - \frac{{[1 - \psi^{*} (\phi_{0} )]\overline{R}_{0} Zh\overline{R}_{S}^{{\overline{m} - 1}} }}{{\psi^{*} (\phi_{0} )(\overline{R}_{S}^{{2\overline{m}}} + \overline{R}_{0}^{{2\overline{m}}} ) + (\overline{R}_{S}^{{2\overline{m}}} - \overline{R}_{0}^{{2\overline{m}}} )}}$$

(64)

$$A_{2} = A_{1} \overline{R}_{0}^{{2\overline{m}}}$$

(65)

$$B_{1} = \frac{{(\overline{R}_{S}^{{2\overline{m}}} - \overline{R}_{0}^{{2\overline{m}}} )[1 - \psi^{*} (\phi_{0} )]\overline{R}_{0} Zh\overline{R}_{S}^{{\overline{m} - 1}} }}{{\psi^{*} (\phi_{0} )[\psi^{*} (\phi_{0} )(\overline{R}_{S}^{{2\overline{m}}} + \overline{R}_{0}^{{2\overline{m}}} ) + (\overline{R}_{S}^{{2\overline{m}}} - \overline{R}_{0}^{{2\overline{m}}} )]}}$$

(66)

After Equation (56) is solved, the analytical perturbation solution for the dimensionless acid concentration of the considered problem can be expressed as follows:

$$\hat{C}_{u} = D_{1} \overline{r}^{{\alpha_{1} }} + D_{2} \overline{r}^{{\alpha_{2} }} + \frac{{E_{1} }}{{\overline{m}\overline{R}_{0} Zh}}\left[ {\overline{r}^{{(\overline{m} + \overline{R}_{0} Zh)}} + \overline{R}_{0}^{{2\overline{m}}} \overline{r}^{{( - \overline{m} + \overline{R}_{0} Zh)}} } \right]$$

(67)

where

$$\alpha_{1} = \frac{{\overline{R}_{0} Zh + \sqrt {(\overline{R}_{0} Zh)^{2} + 4\overline{m}^{2} } }}{2},\quad \alpha_{2} = \frac{{\overline{R}_{0} Zh - \sqrt {(\overline{R}_{0} Zh)^{2} + 4\overline{m}^{2} } }}{2}$$

(68)

$$E_{1} = \frac{{\overline{R}_{0} ZhA_{1} \overline{m}}}{{\overline{R}_{S}^{{\overline{R}_{0} Zh}} - \overline{R}_{0}^{{\overline{R}_{0} Zh}} }}$$

(69)

$$\begin{aligned} D_{2} & = \frac{{E_{1} }}{{\overline{m}\overline{R}_{0} Zh(\overline{R}_{0}^{{\alpha_{2} }} \overline{R}_{S}^{{\alpha_{1} }} - \overline{R}_{0}^{{\alpha_{1} }} \overline{R}_{S}^{{\alpha_{2} }} )}}\left[ {\overline{R}_{S}^{{( - \overline{m} + \overline{R}_{0} Zh)}} (\overline{R}_{S}^{{2\overline{m}}} + \overline{R}_{0}^{{2\overline{m}}} )\overline{R}_{0}^{{\alpha_{1} }} - 2\overline{R}_{0}^{{(\overline{m} + \overline{R}_{0} Zh)}} \overline{R}_{S}^{{\alpha_{1} }} } \right] \\ & \quad - \frac{1}{{(\overline{R}_{0}^{{\alpha_{2} }} \overline{R}_{S}^{{\alpha_{1} }} - \overline{R}_{0}^{{\alpha_{1} }} \overline{R}_{S}^{{\alpha_{2} }} )}}\left[ {\frac{{\overline{R}_{0} Zh\overline{R}_{S}^{{(\overline{R}_{0} Zh - 1)}} }}{{\overline{R}_{S}^{{(\overline{R}_{0} Zh)}} - \overline{R}_{0}^{{(\overline{R}_{0} Zh)}} }}} \right] \\ \end{aligned}$$

(70)

$$D_{1} = - D_{2} \overline{R}_{0}^{{\alpha_{2} - \alpha_{1} }} - \frac{{2E_{1} }}{{\overline{m}\overline{R}_{0} Zh}}\overline{R}_{0}^{{(\overline{m} + \overline{R}_{0} Zh) - \alpha_{1} }}$$

(71)

Consequently, the following dimensionless growth rate of the applied perturbation to the considered chemical dissolution-front instability problem associated with radially divergent flow in the fluid-saturated porous medium can be written as:

$$\begin{aligned} \overline{\omega } & = - \frac{1}{{\varepsilon (\phi_{f} - \phi_{0} )}}\left\{ {D_{1} \alpha_{1} \overline{R}_{S}^{{\alpha_{1} - 1}} + D_{2} \alpha_{2} \overline{R}_{S}^{{\alpha_{2} - 1}} } \right. \\ & \quad + \frac{{E_{1} \overline{R}_{S}^{{ - \overline{m} + \overline{R}_{0} Zh - 1}} }}{{\overline{m}\overline{R}_{0} Zh}}\left[ {(\overline{m} + \overline{R}_{0} Zh)\overline{R}_{S}^{{(2\overline{m})}} + ( - \overline{m} + \overline{R}_{0} Zh)\overline{R}_{0}^{{(2\overline{m})}} } \right] \\ & \quad \left. { - \frac{{\overline{R}_{0} Zh(\overline{R}_{0} Zh - 1)\overline{R}_{S}^{{(\overline{R}_{0} Zh - 2)}} }}{{\overline{R}_{S}^{{(\overline{R}_{0} Zh)}} - \overline{R}_{0}^{{(\overline{R}_{0} Zh)}} }}} \right\} \\ \end{aligned}$$

(72)

If $\overline{\omega } = 0$, then the characteristic equation for the critical Peclet number of the considered problem can be written as:

$$\begin{aligned} & D_{1} \alpha_{1} \overline{R}_{S}^{{\alpha_{1} - 1}} + D_{2} \alpha_{2} \overline{R}_{S}^{{\alpha_{2} - 1}} \\ & \quad + \frac{{E_{1} \overline{R}_{S}^{{ - \overline{m} + pe - 1}} }}{{\overline{m}Pe}}\left[ {(\overline{m} + Pe)\overline{R}_{S}^{{(2\overline{m})}} + ( - \overline{m} + Pe)\overline{R}_{0}^{{(2\overline{m})}} } \right] \\ & \quad - \frac{{Pe(Pe - 1)\overline{R}_{S}^{(Pe - 2)} }}{{\overline{R}_{S}^{(Pe)} - \overline{R}_{0}^{(Pe)} }} = 0 \\ \end{aligned}$$

(73)

where $Pe$ is the Peclet number of the chemical dissolution system and can be expressed as:

$$Pe = \overline{R}_{0} Zh = \frac{{V_{in} R_{0} }}{{\phi_{f} D}}$$

(74)

Solution of Eq. (73) yields the critical Zhao number, which can be used to assess whether a chemical system is in a stable state or in an unstable state.