Reference (3)

Solid-Phase - solid1d-mush

The solid1d-mush model also uses the shooting method approach and is in many regards identical to the original Love.jl model by Hamish Hay (see original repository). In addition to the solid1d model, it includes the effects of porosity making it a poro-viscoelastic model.

The governing equations are an extension of the solid1d model. The additional complexity is handled through two extra variables: the 7th and 8th $y$-functions, representing pore pressure and Darcy flux (relative tangential displacement), respectively.

Porous Transitions and Elementary Solutions

The solver initially treats the system as a $6 \times 6$ problem. When it encounters a porous layer, it updates the elementary solutions by embedding the $3 \times 1$ coefficient set into a $4 \times 1$ matrix and adding a fourth elementary solution vector, $\pmb{y}^{(4)}_n(r_i)$.

At the initial solid-to-porous transition ($r_i$), the new elementary vector is defined as: $\pmb{y}^{(4)}_{n,m}(r_i) = (0, 0, 0, 0, 0, 0, 1, 0)^T$

Continuity and Source Terms

By turning back to the governing differential equation: $\frac{d\pmb{y}_{n,m}(r)}{dr} = \pmb{A}_n (r) \pmb{y}_{n,m}(r) - \pmb{f}_{n,m}(r)$

We impose continuity across an infinitesimally thin layer ($\Delta r = 0$) where $\pmb\Pi_n(r_{i}) = \pmb{1}$. We allow $\pmb{y}_{n,m}(r_i^+)$ to be an $8 \times 1$ vector and $\pmb{y}_{n,m}(r_i^-)$ to be $6 \times 1$. To preserve continuity while introducing pore pressure, we include a source term $\pmb{f}_{n,m}(r_i^+)$:

\[\begin{pmatrix} \pmb{1}_{(8\times8)} \end{pmatrix} \pmb{y}_{n,m}(r_i^+) - \pmb{f}_{n,m}(r_i^+) = \begin{pmatrix} \pmb{1}_{(8\times6)} \\ \pmb{0}_{(2\times6)} \end{pmatrix} \pmb{y}_{n,m}(r_i^-)\]

Since the elementary solutions are linearly independent, the modal solution at the interface is: $\pmb{y}_n(r_i^+) = a_1 \mathbf{y}^{(1)}(r_i^+) + a_2 \mathbf{y}^{(2)}(r_i^+) + a_3 \mathbf{y}^{(3)}(r_i^+) + a_4 \mathbf{y}^{(4)}(r_i^+)$ $\pmb{y}_{n,m}(r_i^+) = \{ \text{solution of } 6\times 6 \text{ system} \} + (0, 0, 0, 0, 0, 0, a_4, 0)^T$

To preserve continuity, the source term must exactly cancel the introduced pressure: $\pmb{f}_{n,m}(r_i^+) = (0, 0, 0, 0, 0, 0, a_4, 0)^T$

Understanding these source/sink steps is key for including effects like porosity or seismic activity in the relaxation-based solver.

Removing Porous Functions

If the medium becomes solid again before the surface, we reverse the process. We use a sink term $\pmb{f}_{n,m}(r_i^-)$ at the upper interface to remove excess pore pressure:

\[\begin{pmatrix} \pmb{1}_{(8\times6)} \\ \pmb{0}_{(2\times6)} \end{pmatrix}^T \pmb{y}_{n,m}(r_i^+) = \begin{pmatrix} \pmb{1}_{(8\times8)} \end{pmatrix} \pmb{y}_{n,m}(r_i^-) - \pmb{f}_{n,m}(r_i^-)\]

We must constrain the Darcy flux to be zero at the interface (upper boundary condition). While a sink term can remove excess pore pressure, it cannot "force" a boundary condition like zero Darcy flux if the propagation has already diverged. Instead, we find: $\pmb{f}_{n,m}(r_i^-) = (0, 0, 0, 0, 0, 0, y_7(r_i^-), 0)^T$

A simpler numerical approach is to manually set $y_7 = 0$ for all $r > r_i^-$, as it no longer interacts with the $6 \times 6$ system.

Final Solution

The remaining steps are identical to the solid1d model:

1. Propagate the elementary solution vectors ($8 \times 1$ or $6 \times 1$) to the surface.
2. Determine the coefficients $\pmb{C}$ (now including $a_4$ for the porous component).
3. Apply $\pmb{C}$ to obtain the final modal solution.