Solid–fluid interaction in porous materials with internal erosion

Abstract

Various applications in science and engineering involve porous materials where the fluid erodes the solid either chemically or mechanically and transports the finer particles through the larger pore spaces of the residual solid. In soil mechanics, the process is called suffusion. In rocks, erosion is mainly due to chemical dissolution. Irrespective of the manner in which the solid erodes, the much finer particles mix with the pure fluid to form a thick fluid whose mass density is greater than that of the pure fluid but less than that of the intact solid. As the solid loses mass, its porosity increases and its mechanical properties degrade, thus impacting the deformation and fluid flow responses of the system. This paper formulates the complex kinematics and conservation equations governing the solid–fluid interaction with internal erosion. We use the classic $\varvec{u}/p$ formulation in which $\varvec{u}$ is the displacement of the residual solid and p is the pressure in the thick fluid. We focus on the case of chemical erosion in rocks where the eroded particles are so small that the interface between them and the pure fluid may be neglected. We then present numerical examples demonstrating the flow and deformation processes in porous materials subjected to internal erosion.

Appendix A. Tangent operator

In this appendix, we summarize the expressions for the tangent operators used with Newton iteration to solve the nonlinear coupled system. Denoting the permeability matrix as $\varvec{\upkappa }$, the submatrix ${{\textbf {K}}}_{ij}$ in the tangent matrix for a Newton-Raphson iteration loop takes the following forms:

$$\begin{aligned} {{\textbf {K}}}_{11}&= \frac{\partial {\mathcal {\varvec{R}}}_u}{\partial {\varvec{d}}} = \int _{\mathcal {B}}{{\textbf {B}}}^{{\textsf {T}}}\frac{\partial \{\varvec{\upsigma }'\}}{\partial {\{\varvec{\upepsilon}}\}}{{\textbf {B}}}\,dV+\int _{\mathcal {B}}\frac{\partial {{\textbf {N}}}_u^{{\textsf {T}}}{\dot{m}}^s}{\partial {\varvec{d}}}\frac{{\varvec{q}}}{\widetilde{\phi }^f_n}\, dV\,, \nonumber \\&=\int _{\mathcal {B}}{{\textbf {B}}}^{{\textsf {T}}}{{\textbf {C}}}^e{{\textbf {B}}}\,dV\nonumber \\&\quad + \int _{\mathcal {B}} {{\textbf {N}}}_u^{{\textsf {T}}}\phi _n^{sr}A(B-C)\exp (-A\varepsilon _v)\frac{{\varvec{q}}}{\widetilde{\phi }_n^f}{\textbf{1}}^{{{\textsf {T}}}}{{\textbf {B}}}\,dV. \end{aligned}$$

(72a)

$$\begin{aligned} {{\textbf {K}}}_{12}&= \frac{\partial {\mathcal {\varvec{R}}}_u}{\partial {p}} = - \int _{\mathcal {B}}{{\textbf {B}}}^{{\textsf {T}}}{\varvec{b}}{{\textbf {N}}}_p \,dV +\int _{\mathcal {B}}\frac{\partial {{\textbf {N}}}_u^{{\textsf {T}}}{\dot{m}}^s({\varvec{v}}_f-{\varvec{v}})}{\partial p}\, dV\,,\nonumber \\&= -\int _{\mathcal {B}}{{\textbf {B}}}^{{\textsf {T}}}{\varvec{b}}{{\textbf {N}}}_p \,dV + \int _{\mathcal {B}}\frac{{{\textbf {N}}}_{u}^{{\textsf {T}}}{\dot{m}}^s{{\textbf {E}}}}{\widetilde{\phi }^f_n}\,dV\,, \end{aligned}$$

(72b)

$$\begin{aligned} {{\textbf {K}}}_{21}&= \frac{\partial {\mathcal {\varvec{R}}}_p}{\partial {\varvec{d}}} = \int _{\mathcal {B}}{{\textbf {N}}}_p^{{\textsf {T}}}{\varvec{b}}{{\textbf {B}}}\,dV\nonumber \\&\quad + \left( \frac{1}{\rho _{sr}} - \frac{1}{{\rho }_{se}}\right) \int _{\mathcal {B}}\frac{\partial {{\textbf {N}}}_p^{{\textsf {T}}}{\dot{m}}^s}{\partial {\varvec{d}}}\, dV\,,\nonumber \\&=\int _\mathcal {B}{{\textbf {N}}}_p^{{\textsf {T}}}{\varvec{b}}{{\textbf {B}}}\,dV + \left( \frac{1}{\rho _{sr}} - \frac{1}{{\rho }_{se}}\right) \nonumber \\&\quad \int _{\mathcal {B}}{{\textbf {N}}}_p^{{\textsf {T}}}\phi _n^{sr}A(B-C)\exp (-A\varepsilon _v){\bf{1}}{{\textbf {B}}}\, dV\,, \end{aligned}$$

(72c)

$$\begin{aligned} {{\textbf {K}}}_{22}&= \frac{\partial {\mathcal {\varvec{R}}}_p}{\partial {p}}=\int _{\mathcal {B}}\frac{{\Delta } t}{\mu _f}{{\textbf {E}}}\varvec{\upkappa }{{\textbf {E}}}\,dV + \int _{\mathcal {B}}\frac{1}{{\mathcal {M}}_n}{{\textbf {N}}}_p^{{{\textsf {T}}}}{{\textbf {N}}}_p\,dV \nonumber \\&\quad + \int _{\mathcal {B}}\left( \frac{\psi ^f_n}{K_f} + \frac{\psi ^{se}_n}{{K}_{se}}\right) {\Delta } t{{\textbf {N}}}_p^{{{\textsf {T}}}}{\varvec{q}}{{\textbf {E}}}\,dV\nonumber \\&\quad -\int _{\mathcal {B}}\left( \frac{\psi ^f_n}{K_f} + \frac{\psi ^{se}_n}{{K}_{se}}\right) \frac{{\Delta } t}{\mu _n}{{\textbf {N}}}_p^{{{\textsf {T}}}}\nabla p\cdot \varvec{\upkappa } {{\textbf { E}}}\,dV\,. \end{aligned}$$

(72d)