The semi-discrete equations derived above are discretized with a discrete set of times, and the numerical solution is obtained at a current time. In what follows, the discrete approximation at time \(t^L\) is indicated by superscript L and the time increment is defined as \(\varDelta t\equiv t^{L+1}-t^{L}\).
The time discretized forms of the momentum balance and the mass balance equations for the fluid phase, Eqs. (
30) and (
31), can be written as
$$\begin{aligned}&(\varvec{M}+\varvec{M}_{\text {s}})\frac{\partial \varvec{v}_i^\text {f}}{\partial t}\bigg |^{L+\theta }+(\varvec{A}+\varvec{A}_{\text {s}})\varvec{v}_i^{\text {f}, \, L+\theta }+(\varvec{G}_{\text {s},\,i}-\varvec{G}_i)\varvec{p}^{\text {f}, \, L+\theta }\nonumber \\&\quad +(\varvec{Q}_{(i)}+\varvec{Q} _{\text {s}, {{(i)}}})(\varvec{v}_i^{\text {f}, \, L+\theta }-\varvec{v}_i^{s,L+\theta })+\varvec{S}_{\text {c},\, ij}^\text {s}\varvec{v}_{j}^{s,L+\theta }+\varvec{S}_{\text {c},\, ij}^\text {f}\varvec{v}_{j}^{\text {f}, \, L+\theta }\nonumber \\&\quad \quad =(\varvec{F}+\varvec{F}_{\text {s}})\varvec{b}_i^{\text {f}, \, L}, \nonumber \\ \end{aligned}$$
(56)
$$\begin{aligned}&\varvec{C}^\text {s}_{i}\varvec{v}_i^{s, L+\theta }+\varvec{C}^\text {f}_{i}\varvec{v}_i^{\text {f}, \, L+\theta }+\varvec{M}_{\text {p},\,i}\frac{\partial \varvec{v}_i^\text {f}}{\partial t}\bigg |^{L+\theta }+\varvec{A}_{\text {p},\,i}\varvec{v}_i^{\text {f}, \, L+\theta }+\varvec{G}_{\text {p}}\varvec{p}^{\text {f}, \, L+\theta }\nonumber \\&\quad +\varvec{Q}_{\text {p},\,i}(\varvec{v}_i^{\text {f}, \, L+\theta }-\varvec{v}_i^{s, L+\theta })=\varvec{F}_{\text {p},\,i}\varvec{b}_i^{\text {f}, \, L},\nonumber \\ \end{aligned}$$
(57)
which are simultaneously solved for the fluid phase velocities
\(v_i^{\text {f}, \, L+1}\) and fluid pressure
\(p^{\text {f}, \, L+1}\). Here, the range of
\(\theta \) is set to be
\(0\le \theta \le 1\). Also, the nodal variables (time derivative, velocities, pressure) are approximated as
$$\begin{aligned}&\frac{\partial \varvec{v}_i^\text {f}}{\partial t}\bigg |^{L+\theta }\approx \frac{\varvec{v}_i^{\text {f}, \, L+1}-\varvec{v}_i^{\text {f}, \, L}}{\varDelta t}, \end{aligned}$$
(58)
$$\begin{aligned}&\varvec{v}_i^{\text {f}, \, L+\theta }\approx \varvec{v}_i^{\text {f}, \, L+\frac{1}{2}}=\frac{\varvec{v}_i^{\text {f}, \, L+1}+\varvec{v}_i^{\text {f}, \, L}}{2}, \end{aligned}$$
(59)
$$\begin{aligned}&\varvec{v}_i^{\text {s}, \, L+\theta }\approx \varvec{v}_i^{\text {s}, \, L}, \end{aligned}$$
(60)
$$\begin{aligned}&\varvec{p}^{\text {f}, \, L+\theta }\approx \varvec{p}^{\text {f}, \, L+1}, \end{aligned}$$
(61)
where
\(\theta \) has been set at 1 in Eqs. (
58) and (
61), 1/2 in Eq. (
59) and 0 in Eq. (
60). To explicitly approximate the advection velocities
\(\bar{v}_i^\text {f}\), we adopt the Adams-Bashforth method [
12], as
$$\begin{aligned} \varvec{\bar{v}}_i^\text {f}=\frac{3}{2}\varvec{v}_i^{\text {f}, \, L}-\frac{1}{2}\varvec{v}_i^{\text {f}, \, L-1}. \end{aligned}$$
(62)
Also, the continuity term
\(\varvec{C}_{i}^\text {f}\varvec{v}_i^{\text {f}, \, L+\theta }\) is implicitly represented, while the interaction terms,
\(\varvec{Q}_i\varvec{v}_i^{\text {f}, \, L+\theta }, \, \varvec{Q}_{\text {s},\,i}\varvec{v}_i^{\text {f}, \, L+\theta }\) and
\(\varvec{Q}_{\text {p},\,i}\varvec{v}_i^{\text {f}, \, L+\theta }\), are evaluated explicitly as
$$\begin{aligned}&\varvec{C}_{i}^\text {f}\varvec{v}_i^{\text {f}, L+\theta }=\varvec{C}_{i}^\text {f}\varvec{v}_i^{\text {f}, L+1}, \end{aligned}$$
(63)
$$\begin{aligned}&\varvec{Q}_{i}\varvec{v}_i^{\text {f}, L+\theta }=\varvec{Q}_i\varvec{v}_i^{\text {f}, \, L}, \end{aligned}$$
(64)
$$\begin{aligned}&\varvec{Q}_{\text {s},\,i}\varvec{v}_i^{\text {f}, L+\theta }=\varvec{Q}_{\text {s},\,i}\varvec{v}_i^{\text {f}, \, L}, \end{aligned}$$
(65)
$$\begin{aligned}&\varvec{Q}_{\text {p},\,i}\varvec{v}_i^{\text {f}, L+\theta }=\varvec{Q}_{\text {p},\,i}\varvec{v}_i^{\text {f}, \, L}. \end{aligned}$$
(66)
where
\(\theta \) has been set at 1 in Eq. (
63) and 0 in Eqs. (
64), (
65) and (
66). Using Eqs. (
56)-(
66) with some straightforward calculations, we obtain the final versions of Eqs. (
56) and (
57) as
$$\begin{aligned}&(\varvec{M}+\varvec{M}_{\text {s}})\frac{\varvec{v}_i^{\text {f}, \, L+1}}{\varDelta t}+\left( \varvec{A}+\varvec{A}_{\text {s}}+\varvec{S}_{\text {c},\, ij}^\text {f}\right) \frac{\varvec{v}_{j}^{\text {f}, \, L+1}}{2}+\left( \varvec{G}_{\text {s},\,i}-\varvec{G}_i\right) \varvec{p}^{\text {f}, \, L+1}\nonumber \\&\quad =(\varvec{M}+\varvec{M}_{\text {s}})\frac{\varvec{v}_i^{\text {f}, \, L}}{\varDelta t}-\left( \varvec{A}+\varvec{A}_{\text {s}}+\varvec{S}_{\text {c},\,ij}^\text {f}\right) \frac{\varvec{v}_{j}^{\text {f}, \, L}}{2}-\varvec{S}_{\text {c},\,ij}^\text {s}\varvec{v}_{j}^{\text {s}, \, L}\nonumber \\&\quad \quad +(\varvec{Q}_{(i)}+\varvec{Q}_{\text {s},\,({i})})(\varvec{v}_i^{\text {s}, \, L}-\varvec{v}_i^{\text {f}, \, L})+(\varvec{F}+\varvec{F}_{\text {s}})\varvec{b}_i^{\text {f}, \, L}, \nonumber \\ \end{aligned}$$
(67)
$$\begin{aligned}&\left( \varvec{C}^\text {f}_{i}+\frac{\varvec{M}_{\text {p},\,i}}{\varDelta t}+\frac{\varvec{A}_{\text {p},\,i}}{2}\right) \varvec{v}_i^{\text {f}, \, L+1}+\varvec{G}_{\text {p}}\varvec{p}^{\text {f}, \, L+1}=\left( \frac{\varvec{M}_{\text {p},\,i}}{\varDelta t}-\frac{\varvec{A}_{\text {p},\,i}}{2}\right) \varvec{v}_i^{\text {f}, \, L}\nonumber \\&\quad \quad +\varvec{Q}_{\text {p},\,i}(\varvec{v}_i^{\text {s}, \, L}-\varvec{v}_i^{\text {f}, \, L})-\varvec{C}^\text {s}_{i}\varvec{v}_i^{\text {s}, \, L}+\varvec{F}_{\text {p},\,i}\varvec{b}_i^{\text {f}, \, L},\nonumber \\ \end{aligned}$$
(68)
from which the fluid phase velocity and pressure are determined for the next time step.
The time-discretized form of the momentum balance equation for the solid phase (
52) can be written as
$$\begin{aligned} \varvec{M}^{\text {s}, \, L+\theta }\varvec{a}_{i}^{\text {s}, \, L+\theta }=\varvec{F}_{\text {int}, \, i}^{\text {s}, \, L+\theta }+\varvec{F}_{\text {ext}, \, i}^{\text {s}, \, L+\theta }+\varvec{Q}_{(i)}\left( \varvec{v}^{\text {f}, \, L+\theta }_{i}-\varvec{v}^{\text {s}, \, L+\theta }_{i}\right) . \end{aligned}$$
(69)
Using the nodal pressure,
\(\varvec{p}^{\text {f}, \, L+1}\), obtained from Eqs. (
67) and (
68), this is solved for
\(\varvec{a}_{i}^{\text {s}, \, L}\) explicitly with
\(\theta =0\). After the nodal acceleration of the solid skeleton,
\(\varvec{a}_{i}^{\text {s}, \, L}\), is obtained, the nodal velocity vector in the
\(x_i\)-direction is updated as
$$\begin{aligned} \varvec{v}_{i}^{\text {s}, \, L+1}=\varvec{v}_{i}^{\text {s}, \, L}+\varDelta t\varvec{a}_{i}^{\text {s}, \, L}, \end{aligned}$$
(70)
and then the velocity and position vectors of each solid material point can be updated by the following formulae:
$$\begin{aligned}&v_{i}^{p, \, L+1}=v_{i}^{p, \, L}+\varDelta t\sum _{\alpha =1}^{N^n} a_{i\alpha }^{\text {s}, \, L}N_{\alpha }(\varvec{x}^{p, \, L}), \end{aligned}$$
(71)
$$\begin{aligned}&x_{i}^{p, \, L+1}=x_{i}^{p, \, L}+\varDelta t\sum _{\alpha =1}^{N^n}v_{i\alpha }^{\text {s}, \, L+1}N_{\alpha }(\varvec{x}^{p, \, L}). \end{aligned}$$
(72)
Prior to updating the state valuables of each solid material point, we adopt the MUSL procedure [
32] to “refine” the nodal velocity vector by an additional mapping process. Then, the deformation gradient of a solid material point can be computed as
$$\begin{aligned} F_{ij}^{p, \, L+1}=\left( \delta _{ik}+\varDelta t \sum _{\alpha =1}^{N^n}\frac{\partial N_{\alpha }(\varvec{x}^{p, \, L})}{\partial x_k}v_{i \alpha }^{\text {s}, \, L+1}\right) F_{kj}^{p, \, L},\quad \end{aligned}$$
(73)
and its determinant
\(J^{p, \, L+1}=\det \varvec{F}^{p, \, L+1}\) is used to update its volume and the porosity as, respectively,
$$\begin{aligned} \varOmega ^{p, \, L+1}&=J^{p, \, L+1} \varOmega ^{p, \, 0}, \end{aligned}$$
(74)
$$\begin{aligned} n^{p, \, L+1}&=1-\frac{(1-n^{p, \, 0})}{J^{p, \, L+1}}. \end{aligned}$$
(75)
According to Eq. (
4), the permeability
\(k^{p, \, L+1}\) can also be calculated using
\(n^{p, \, L+1}\) as
$$\begin{aligned} k^{p, \, L+1}=\frac{D_{50}^2\left( n^{p, \, L+1}\right) ^3}{150\left( 1-n^{p, \, L+1}\right) ^2}. \end{aligned}$$
(76)
Here, the upper limit value of the solid porosity
\(n^{p, \, L+1}\) has to be set to suppress numerical instability. In this study, it will be set at
\(n^{p, \, L+1}=0.99\) in the numerical examples.