*4.2. Balance Equations*

We considered the body as a solid skeleton with fully saturated interstitial fluid. It was assumed that there was no mass transfer, and thermal gradients, inertia, and gravity were neglected. Based on Biot's theory, the momentum balance reads

$$
\nabla \cdot \boldsymbol{\sigma} = \mathbf{0} \quad \text{in} \quad \Omega\_{\prime} \tag{5}
$$

with *σ* the total stress, which is decomposed into the effective stress *σe* and the pore fluid pressure *p*,

$$
\boldsymbol{\sigma} = \boldsymbol{\sigma}\_{\boldsymbol{\varepsilon}} - p\mathbf{I}, \tag{6}
$$

with **I** being the unit tensor.

Equations (5) and (6) can be written with respect to the reference configuration, using the transformation of **P** = *Jσ* · **<sup>F</sup>**−*T*, read

$$\begin{aligned} \nabla\_0 \cdot \mathbf{P} &= \mathbf{0} \quad \text{in} \quad \Omega\_0 \qquad &\text{(momentum balance)},\\ \mathbf{P} &= \mathbf{P}\_\mathbf{c} - Jp\mathbf{F}^{-T} \quad &\text{(total first Piola-Kirchhoff stress)}, \end{aligned}$$

where **P***e* is the effective first Piola-Kirchhoff stress.

Conservation of mass for an incompressible fluid yields the mass balance in the reference configuration,

$$f + \nabla\_0 \cdot \mathbf{Q} = 0,\tag{7}$$

with ˙ *J* = *J*div**u˙** and **Q** = −**K** · <sup>∇</sup>0*μ<sup>f</sup>* the seepage flux obeying the Darcy's relation in the presence of the concentration gradient. In the equation of the seepage flux, **K** is the permeability tensor back-transformed to the reference configuration, and *μf* is the chemical potential, defined as

$$
\mu^f = p - \pi,\tag{8}
$$

with *p* being the hydrostatic pressure, and *π* the osmotic potential.

The fracture process behaviour is governed by a traction separation law. Here, we assumed stress continuity from the gel to the discontinuity, the local momentum balance being described as

$$\mathbf{P} \cdot \mathbf{n}\_{\Gamma\_d} = f||\mathbf{F}^{-T} \cdot \mathbf{n}\_{\Gamma\_d}||\mathbf{t}\_d - f(\boldsymbol{\mu}\_d^f + \boldsymbol{\tau}\_d)\mathbf{F}^{-T} \cdot \mathbf{n}\_{\Gamma\_{d'}}\tag{9}$$

in which **<sup>n</sup>**Γ*d* is the normal of the discontinuity Γ*d*, **t***d* is the traction, and *μfd* and *πd* are the chemical potential and osmotic pressure, respectively, within the discontinuity.

The local mass balance was obtained by integrating the continuous mass balance across the discontinuity.

$$\mathbf{n}\_{\Gamma\_d} \cdot (\mathbf{Q}\_{\Gamma\_d}^+ - \mathbf{Q}\_{\Gamma\_d}^-) = f \boldsymbol{\mu}\_n \frac{\partial}{\partial \mathbf{s}} (k\_d \frac{\partial \boldsymbol{\mu}\_d^f}{\partial \mathbf{s}}) - f \boldsymbol{\mu}\_n (\frac{\partial \boldsymbol{v}}{\partial \mathbf{s}}) - f \boldsymbol{\mu}\_n. \tag{10}$$

where *kd* is the conductivity in the discontinuity

$$k\_d = \frac{(u\_n)^3}{12\mu},\tag{11}$$

with *μ* being the viscosity of the fluid, *un* = **u**˜ · **<sup>n</sup>**Γ*d*the opening of the discontinuity.
