A three-scale micro-mechanical model for elastic–plastic damage modeling of shale rocks

Abstract

This paper is devoted to study the elastic–plastic damage behavior of heterogeneous shale rocks. The representative microstructure of this kind of rocks is first studied in order to define the representative elementary volume for the implementation of homogenization procedure. Three relevant material scales are considered. Inter-particle pores are distributed at the nanoscopic scale. Fine grains of calcite and kerogen are immersed at the microscopic scale. Large grains of minerals are embedded at the mesoscopic scale. Effective elastic properties of shale rocks are first determined by using a three-step linear homogenization procedure. The plastic damage behavior is estimated by developing a three-step nonlinear homogenization method. The effective plastic behavior of porous clay matrix with nanoscopic pores is described by an analytical model. The effects of small and large grains of various mineral inclusions are investigated by using a two-step incremental model. The damage due to progressive debonding of mineral inclusions is taken into account. After the implementation of the proposed model, comparisons between numerical results and experimental data are presented.

Appendices

Appendices: Algorithm for local integration of the clay phase

The algorithm for local integration of the elastoplastic damage clay phase is presented in this paragraph for a step (n+1) or at time $t_{n+1}= t_n+dt_{n+1}$ as follows:

• 1. Input data: ${\tilde{\varvec{\varepsilon}}}_{0,n},~{\tilde{\varvec{\varepsilon}}}_{0,n}^p,~\varepsilon _{n}^p\;{\text {and}}\;~\Delta {\tilde{\varvec{\varepsilon}}}_0$;

• 2. Calculating the deformation at step $n+1$ : ${\tilde{\varvec{\varepsilon}}}_{0,n+1}= {\tilde{\varvec{\varepsilon}}}_{0,n} +\Delta {\tilde{\varvec{\varepsilon}}}_0$;

• 3. Initialize (elastic prediction): For $i = 0$

  $$ \left\{ {\begin{array}{*{20}l} {\tilde{\varepsilon }_{{0,n + 1}}^{{p,0}} = {\text{ }}\tilde{\varepsilon }_{{0,n}}^{p} } \hfill \\ {\tilde{\sigma }_{{n + 1}}^{0} = \mathbb{C}_{0} :(\tilde{\varepsilon }_{{0,n + 1}} - \tilde{\varepsilon }_{{0,n + 1}}^{{p,0}} )} \hfill \\ {\Phi (\tilde{\sigma }_{{n + 1}}^{i} ,\varepsilon _{{n + 1}}^{{p,i}} ,f) = \Phi _{{n + 1}}^{i} {\text{ }}} \hfill \\ \end{array} } \right. $$

• 4. If $\Phi _{n+1}^{i} \le$ 0, then go to step 7; else, go to step 5 for plastic correction:

• 5. $\delta (\Delta \lambda )= \frac{\Phi _{n+1}^{i}}{{\frac{\partial \Phi }{\partial {\tilde{\varvec{\sigma }}}}:{\mathbb {C}}_0:\frac{\partial G}{\partial {\tilde{\varvec{\sigma }}}}-\frac{\partial \Phi }{\partial f}\left[ \frac{\partial G}{\partial {\tilde{\sigma }}_m}(1-f)-3\alpha _2 \frac{{\tilde{\varvec{\sigma }}}:\frac{\partial G}{\partial {\tilde{\varvec{\sigma }}}}}{\bar{\sigma }+3(\alpha _2-\alpha )\frac{{\tilde{\sigma }}_m}{1-f}} \right] -\frac{\partial \Phi }{\partial \bar{\sigma }}\frac{\partial \bar{\sigma }}{\partial \varepsilon ^p}\frac{{\tilde{\varvec{\sigma }}}:\frac{\partial G}{\partial {\tilde{\varvec{\sigma }}}}}{(1-f)\left[ \bar{\sigma }+3(\alpha _2-\alpha )\frac{{\tilde{\sigma }}_m}{1-f}\right] } }}$

• 6. Calculate new values for each iteration:

  $$\begin{aligned} {\left\{ \begin{array}{ll} {\tilde{\varvec{\sigma }}}_{n+1}^{i+1} = {\tilde{\varvec{\sigma }}}_{n+1}^{i} - \delta (\Delta \lambda ) {\mathbb {C}}_0 :\frac{\partial G_{n+1}^{i}}{\partial {\tilde{\varvec{\sigma }}}}({\tilde{\varvec{\sigma }}}_{n+1}^{i},\varepsilon _{n+1}^{p,i},f) \\ \varepsilon _{n+1}^{p,i+1}= \varepsilon _{n+1}^{p,i} + \delta (\Delta \lambda ) \frac{{\tilde{\varvec{\sigma }}}:\frac{\partial G}{\partial {\tilde{\varvec{\sigma }}}}}{(1-f)\left[ \bar{\sigma }+3(\alpha _2-\alpha )\frac{{\tilde{\sigma }}_m}{1-f}\right] } \\ \Delta \lambda ^{i+1} = \lambda ^{i} + \delta (\Delta \lambda ) \\ {\tilde{\varepsilon }}_{n+1}^{p,i+1}= {\tilde{\varepsilon }}_{n+1}^{p,i} + \delta (\Delta \lambda ) \frac{\partial G}{\partial {\tilde{\varvec{\sigma }}}} \\ \end{array}\right. } \end{aligned}$$

  Set $i= i + 1$ and return to step 4

• 7. End of Algorithm

Algorithm for the implementation of the proposed three-scale model