Elastoplastic Modelling of Porous Limestones with Porosity Dependency

Abstract

In this study, three typical limestones, including Tavel limestone, Indiana limestone, and Lixhe chalk, were selected from a large number of porous limestones. These limestones with different porosities have been largely studied in previous experimental investigations because of the complexity of mechanical behavior. According to previous experimental studies, porous limestones present two basic plastic mechanisms: plastic shear as a response at low confining pressures and plastic pore collapse at high confining pressures. In related to the plastic mechanisms, two types of plastic volumetric deformation are revealed: plastic compaction induced by pore collapse, and plastic dilatancy by plastic shearing. In this paper, a micromechanics-based plastic model is extended to describe the elastoplastic behavior of porous limestones. The plastic criterion of porous rock is explicitly dependent on the porosity in addition to being directly based on the relevant mechanical properties of solid matrix at the microscopic scale. An additional plastic hardening law for the solid matrix is proposed, in which two plastic deformation mechanisms are considered in hardening law of the solid matrix, including hardening effect caused by the local equivalent plastic deformation and weakening effect caused by the increase in porosity. Three typical porous limestones with different porosity are selected to validate the proposed model on both hydrostatic and triaxial compression tests. By comparing numerical predictions and experimental data, it is shown that the presented model can correctly describe the mechanical behavior of porous rocks.

Highlights

• Limestones with different porosities have been largely studied in previous experimental investigations because of the complexity of mechanical behavior.

• In related to the plastic mechanisms, two types of plastic volumetric deformation are revealed: plastic compaction induced by pore collapse, and plastic dilatancy by plastic shearing.

• A micromechanics-based plastic model is extended to describe the elastoplastic behavior of porous limestones.

• The plastic criterion of porous rock is explicitly dependent on the porosity in addition to being directly based on the relevant mechanical properties of solid matrix at the microscopic scale.

Appendices

Appendix A: Macroscopic Yield Criterion of Porous Material with Mises–Schleicher Solid Matrix [Adapted from Han et al. (2020)]

Consider now a porous material constituted by an isotropic solid matrix and spherical voids. The porosity is denoted by $\phi$. The Mises–Schleicher criterion (Schleicher 1925) for the local strength of solid matrix is written as

$$\begin{aligned} \varphi (\sigma ) = \sigma _{eq}^2 + 3\alpha {\sigma _0}{\sigma _m} - \sigma _0^2 \le 0, \end{aligned}$$

(A.1)

${\sigma _m} = tr(\bar{\bar{\sigma }})/3$ is the mean stress and ${\sigma _{eq}} = \sqrt{{{3} \!{/} \!{2}}{{\bar{\bar{\sigma }} }_d}:{{\bar{ \bar{\sigma }}}_d}}$ with ${\bar{\bar{\sigma }} }_d = \bar{\bar{\sigma }} - {\sigma _m}\bar{ \bar{\delta }}$ is the deviatoric stress of the local stress tensor in the solid matrix. Two coefficients ${\sigma _0}\ge 0$ and ${\alpha }\ge 0$ are related to the uniaxial tension strength $\sigma _t$ and the absolute value of uniaxial compression strength $\sigma _c$ by

$$\begin{aligned} \sigma _0 = \sqrt{\sigma _t\sigma _c}\ {;} \ \alpha = \frac{{\sigma _c - \sigma _t}}{{\sqrt{\sigma _t\sigma _c}} }. \end{aligned}$$

(A.2)

The parameter $\alpha$ is generally called the friction coefficient, and it controls the ratio between the uniaxial compression strength and the uniaxial tension strength. The parameter ${\sigma _0}$ is physically related to the material cohesion. Due to its nonlinear form, the Mises–Schleicher criterion is more suitable than the linear Drucker–Prager criterion to account for the strength dissymmetry between compression and tension.

Denote $\bar{\bar{\Sigma }}$ as the macroscopic stress tensor applied to the representative volume element (RVE) of the porous material. Adopting homogenization technique, Shen et al. (2015) derived a closed form of the macroscopic yield criterion of the porous material as a function of porosity and plastic parameters of solid matrix obeyed to the Mises–Schleicher criterion at the microscopic scale. Following the results given in Shen et al. (2015), the macroscopic criterion is written as follows:

$$\begin{aligned}&\Phi = \frac{{{{\Sigma _{eq}^2} {/} {\sigma _0^2}}}}{{B - 3\alpha C{{{\Sigma _m}} {/} {{\sigma _0}}}}} + 2\Gamma \cosh \left( {A\ln \left( {1 - 3\alpha \frac{{{\Sigma _m}}}{{{\sigma _0}}}} \right) } \right) \nonumber \\&\quad - 1 - {\Gamma ^2} = 0 \end{aligned}$$

(A.3)

$$\begin{aligned}&\text {with}\left\{ {\begin{array}{*{20}{c}} {B = \frac{{{{\left( {1 - \phi } \right) }^2}}}{{{{(1 - \Gamma )}^2}}}}\\ {C = \frac{{\left( {1 - \phi } \right) }^s}{{{{(1 - \Gamma )}^2}}}}\\ {\Gamma = {{\left( {{\alpha ^2}{{\left( {W\left( {\phi p} \right) + 1} \right) }^2} - {\alpha ^2}} \right) }^A}}\\ {A = \mathrm{sign}\left( {{\Sigma _m}} \right) \left( {\frac{4}{{9\alpha }} + \frac{{\alpha {{\left( {W\left( {\phi p} \right) + 1} \right) }^2} - \alpha }}{{18}}} \right) ,} \end{array}} \right. \end{aligned}$$

(A.4)

${\Sigma _m} = tr(\bar{\bar{\Sigma }} )/3$ is the macroscopic mean stress and ${\Sigma _{eq}} = \sqrt{{{3} \!{/ } \!{2}}{{\bar{\bar{\Sigma }} }_d}:{{\bar{ \bar{\Sigma }}}_d}}$ with ${\bar{\bar{\Sigma }} _d} = \bar{\bar{\Sigma }} - {\Sigma _m}\bar{\bar{\delta }}$ is the macroscopic deviatoric stress. s is a parameter controlling the shape of strength surface. W denotes the “Lambert W” function which satisfies ${W(x){e^{W(x)}} = x}$. W(x) has two branches: the upper branch ${W_0}(x) \ge - 1$ and the lower branch ${W_{ - 1}}(x) \le - 1$ (for $- {e^{ - 1}} \le x < 0$ ). Furthermore, one gets $W\left( {\phi p} \right) = {W_{ - 1}}\left( {\phi {p_ - }} \right)$ for compression zone and $W\left( {\phi p} \right) = {W_0}\left( {\phi {p_ + }} \right)$ for tension zone. The coefficients $p_ -$ and $p_ +$ are functions of $\alpha$ as follows:

$$\begin{aligned} \left\{ \begin{array}{c} \begin{array}{*{20}{c}} {{p_ + } = {z_ + }\exp ({z_ + }),}&{}{{z_ + } = \frac{{ - \alpha + \sqrt{{\alpha ^2} + 1} }}{\alpha }} \end{array}\\ \begin{array}{*{20}{c}} { {p_ - } = {z_ - }\exp ({z_ - }),}&{}{{z_ - } = \frac{{ - \alpha - \sqrt{{\alpha ^2} + 1} }}{\alpha }}. \end{array} \end{array} \right. \end{aligned}$$

(A.5)

Following a classical approach in the context of porous materials, the hardening variable $\bar{\sigma }$ is introduced in (A.3) to replace $\sigma _0$. The macroscopic plastic yield function is adopted in the following form (Han et al. 2020):

$$\begin{aligned}&\Phi = \frac{{\Sigma _{eq}^2}}{{\Theta {{\bar{\sigma }}^2}}} + 2\Gamma \cosh \left( {A\ln \left( {1 - 3\alpha \frac{{{\Sigma _m}}}{{\bar{\sigma }}}} \right) } \right) \nonumber \\&\quad - 1 - {\Gamma ^2} = 0 \end{aligned}$$

(A.6)

$$\begin{aligned}&\text {with}\left\{ {\begin{array}{*{20}{c}} {\Theta = {{\left( {\frac{{1 - \phi }}{{1 - \Gamma }}} \right) }^2} - 3\alpha \frac{{{{\left( {1 - \phi } \right) }^s}}}{{{{(1 - \Gamma )}^2}}}\frac{{{\Sigma _m}}}{{\bar{\sigma }}}}\\ {\Gamma = {{\left( {{\alpha ^2}{{\left( {W\left( {\phi p} \right) + 1} \right) }^2} - {\alpha ^2}} \right) }^A}}\\ {A = \mathrm{sign}\left( {{\Sigma _m}} \right) \left( {\frac{4}{{9\alpha }} + \frac{{\alpha {{\left( {W\left( {\phi p} \right) + 1} \right) }^2} - \alpha }}{{18}}} \right) .} \end{array}} \right. \end{aligned}$$

(A.7)

The yield criterion given in (A.6) was adapted to describe the plastic deformation of porous chalk under hydrostatic and triaxial compressive stress, and extended to describe the plastic deformation of porous sandstone under high confining pressure (Han et al. 2020).

Appendix B: Values of Parameters for Oil-Saturated Lixhe Chalk (Han et al. 2020)

Table 5 Values of parameters for oil-saturated Lixhe chalk