Convective Stability of a Net Mass Flow Through a Horizontal Porous Layer with Immobilization and Clogging

Abstract

Solutal convection in a horizontal layer filled with porous media with horizontal seepage of a mixture is investigated considering the solute immobilization and clogging. A flow through porous media is modelled within the standard Darcy–Boussinesq model, and the immobilization process is described by the mobile/immobile media (MIM) approach. To describe the clogging process, the present model takes into account and the dependence of media permeability on porosity within the Carman–Kozeny equation. The presence of immobile (or adsorbed) particles of the solute decreases the porosity of media, and porous media become less permeable. The variation of porosity is modelled by a linear function of solute concentration in the immobile phase. We consider the case of high solute concentrations, in which the immobilization is described by the nonlinear MIM (mobile/immobile media) model. As a result, it was shown that the immobilization leads to the stabilization of the homogeneous filtration regime and to slowing down of the perturbation dynamics. The stability maps were plotted in a wide range of system parameters. The results showed that for some specific value of clean media porosity the system becomes most unstable and dynamics of perturbations (frequency of oscillations) is most intensive. This value corresponds to the minimal effect of porosity change to variation of permeability due to the immobilization.

Appendices

Appendix A: Verification of the Numerical Calculations for the Case with Constant Permeability

The equation system for neutral perturbations (13) for the case of $\kappa =\kappa _0=\kappa ^0=\mathrm{const}$ reads

$$\begin{aligned} -i\omega \left( C+Q\right)&= (\partial _y^2-k^2-ik \mathrm{Pe})C-\frac{ik}{\kappa }\varPsi ,\nonumber \\ (\partial _y^2-k^2) \varPsi&= -ik \mathrm{Rp}\kappa C,\nonumber \\ -i\omega Q&=aC\left( q_0-q^0\right) -\left( b+ac^0\right) Q,\nonumber \\ \varPsi \mid _{y=0,1}&=0, \quad C,Q \mid _{y=0,1}=0. \end{aligned}$$

(14)

For the limiting case of the saturated porous media ($b\ll 1$ and $Q=q_{C0}=\mathrm{const}$), Eq. (14) after substitution (9) becomes

$$\begin{aligned} -i\omega C&= (\partial _y^2-k^2-ik \mathrm{Pe})C-\frac{ik}{\kappa }\varPsi ,\nonumber \\ (\partial _y^2-k^2) \varPsi&= -ik \mathrm{Rp}\kappa C,\nonumber \\ Q&=\mathrm{const}, \end{aligned}$$

(15)

and using the perturbations in form of $C,\varPsi \sim \sin (n\pi y)$ the well-known solution Prats (1966) can be obtained

$$\begin{aligned} \omega&= k \mathrm{Pe} \nonumber \\ \mathrm{Rp}&= \frac{(n^2\pi ^2+k^2)^2}{k^2}. \end{aligned}$$

(16)

While for the limiting case of the convection with small initial concentration ($C_0\ll \phi _0,q_0$), Eq. (14) together with (10) correspond to

$$\begin{aligned} -i\omega \left( C+Q\right)&= (\partial _y^2-k^2-ik \mathrm{Pe})C-\frac{ik}{\kappa }\varPsi ,\nonumber \\ (\partial _y^2-k^2) \varPsi&= -ik \mathrm{Rp}\kappa C,\nonumber \\ -i\omega Q&=aq_{C0}C-bQ. \end{aligned}$$

(17)

The Eq. (14) can be reduced to one complex equation for $Q,C,\varPsi \sim \sin (n\pi y)$ in the form of

$$\begin{aligned} -i\omega \left( 1+\frac{aq_{C0}}{b-i\omega }\right) -k^2\frac{\mathrm{Rp}}{n^2\pi ^2+k^2}+ikPe=-\pi ^2n^2-k^2, \end{aligned}$$

(18)

which is coincided with Eq. (10) in Maryshev (2015) up to replacement $aq_{C0}\rightarrow a$.

Appendix B: The Case Without Adsorption

We shall rewrite solution Eq. (8) for the case $a=0$ (the immobilization effect is excluded)

$$\begin{aligned} q^0&= 0,\nonumber \\ c^0&= y,\nonumber \\ p^0&= -x-\frac{\mathrm{Rp}}{2\mathrm{Pe}}y^2, \nonumber \\ u^0&= \kappa (\phi ^0)\mathrm{Pe}=\mathrm{const},\nonumber \\ \phi ^0&= \phi _0. \end{aligned}$$

(19)

Then, equations for normal neutral perturbations (13) are

$$\begin{aligned} -i\omega \left( C+Q\right)&= (\partial _y^2-k^2-ik \mathrm{Pe})C-\frac{ik}{\kappa _0}\varPsi \nonumber \\ (\partial _y^2-k^2) \varPsi&= -ik \mathrm{Rp}\kappa _0 C,\nonumber \\ q-i\omega Q&=-ayQ. \end{aligned}$$

(20)

The last equation has only trivial solution: $Q=0$. Thus, substitution of $C, \varPsi \sim \sin {\pi y}$ gives

$$\begin{aligned} (i\omega -\pi ^2-k^2-ik \mathrm{Pe})C-\frac{ik}{\kappa _0}\varPsi&=0\nonumber \\ ik \mathrm{Rp}\kappa _0 C-(\pi ^2+k^2) \varPsi&=0. \end{aligned}$$

(21)

Finally,

$$\begin{aligned} \omega&= k \mathrm{Pe} \nonumber \\ \mathrm{Rp}&= \frac{(\pi ^2+k^2)^2}{k^2}. \end{aligned}$$

(22)

which is a well-known solution Prats (1966).

Appendix C: The Case Without Desorption

For the case $b=0$ (desorption is absent), solution Eq. (8) reads

$$\begin{aligned} q^0&= q_{C0}=\mathrm{const},\nonumber \\ c^0&= y,\nonumber \\ p^0&= -x-\frac{\mathrm{Rp}}{2\mathrm{Pe}}y^2, \nonumber \\ u^0&= \kappa (\phi ^0)\mathrm{Pe},\nonumber \\ \phi ^0&= \phi _0-C_0q_{C0}. \end{aligned}$$

(23)

Then, equations for normal neutral perturbations (13) become

$$\begin{aligned} -i\omega \left( C+Q\right)&= \left( \partial _y^2-k^2-ik \frac{\kappa ^0}{\kappa _0} \mathrm{Pe}\right) C-\frac{ik}{\kappa _0}\varPsi \nonumber \\ (\partial _y^2-k^2) \varPsi&= -ik \mathrm{Rp}\kappa _0 C,\nonumber \\ -i\omega Q&=-bQ. \end{aligned}$$

(24)

The last equation has only trivial solution: $Q=0$. Thus, substitution of $C, \varPsi \sim \sin {\pi y}$ gives

$$\begin{aligned} \omega&= \frac{\kappa ^0}{\kappa _0} k \mathrm{Pe} ,\nonumber \\ \mathrm{Rp}&= \frac{\kappa ^0}{\kappa _0}\frac{(\pi ^2+k^2)^2}{k^2}. \end{aligned}$$

(25)