Multiscale Poromechanical Modeling of Shales Incorporating Microcracks

Abstract

A probabilistic multiscale model for poroelastic properties of organic-rich shale is extended to include the effect of microcracks. The approach is based on physics-based modeling of poroelastic properties and uncertainty propagation from nano- to macro-scale using homogenization schemes. The multiscale thought model used in this work improves on an existing thought model by adding another level to model embedded microcracks. From the scale of single clay particle to the scale of embedded microcracks, the model parameters are calibrated using experimental characterization at different length scales. To quantify the crack density and microcrack orientations, an inverse optimization approach is utilized which uses the UPV measurements at the macro-scale. Major model parameters representing compositional and mechanical properties at each length scale are modeled as a random variable. Probabilistic description of both scalar random variables and matrix-valued random variables is constructed using the maximum entropy principle (MaxEnt). Using such statistical description of model input parameters with Monte Carlo simulation, probabilistic descriptions of model outputs are obtained at each length scale. The results of stochastic upscaling are validated against available experimental values. Finally, a global sensitivity analysis is performed to identify the model input parameters which are most influential to the statistical fluctuation of upscaled poroelastic properties. The presented multiscale framework provides an enhanced understating regarding the effect of uncertainties associated with microcrack density and orientation, and other subscale features on homogenized poroelastic properties, thus further improves predictive capability for shale multiscale behavior.

Appendices

Appendix A: Calibration and Validation Data Sets

See Tables 10, 11, 12, 13

Table 10 Calibration data set 1 (CDS1) (Abedi et al. 2016a)

Table 11 Calibration data set 2 (CDS2) (Monfared and Ulm 2016)

Table 12 Validation data set 1 (VDS1) (Abedi et al. 2016a)

Table 13 Validation data set 2 (VDS2) (Monfared and Ulm 2016)

Appendix B

Description of volume fractions and stiffness tensors at Level I:

Volume fraction of phases at Level I reads:

$$\eta^{\text{c}} + \eta^{\text{k}} + \phi^{\text{I}} = 1$$

(33)

where $\eta^{\text{c}}$ is the volume fraction of clay, $\eta^{\text{k}}$ is the volume fraction of kerogen, and $\phi^{\text{I}}$ is the porosity at Level I which is distributed proportionally between clay and kerogen based on their volume fraction. These volume fractions can be obtained from their counterparts at Level II [see Eq. (37)].

To obtain the stiffness tensor of porous kerogen (or porous clay), one needs to employ Eq. (2) for $\varOmega_{\text{porous-kerogen}} = \varOmega_{\text{kerogen}} \text{U }\varOmega_{\text{pores}}$ (or $\varOmega_{\text{porous-clay}} = \varOmega_{\text{clay}} \text{U }\varOmega_{\text{pores}}$). For this two-phase composite, volume fraction of porosity is equal to $\phi^{\text{I}}$, and volume fraction of kerogen (or clay) is equal to $\left( {1 - \phi^{\text{I}} } \right)$.

The stiffness tensor of kerogen reads:

$$\left[\kern-0.15em\left[ {C^{{{\text{kerogen}}}} } \right]\kern-0.15em\right] = 3K\left[\kern-0.15em\left[ J \right]\kern-0.15em\right] + 2G \left[\kern-0.15em\left[ K \right]\kern-0.15em\right],$$

(34)

where $K$ and $G$ are the bulk and shear moduli of kerogen. $\left[\kern-0.15em\left[ J \right]\kern-0.15em\right]$ and $\left[\kern-0.15em\left[ K \right]\kern-0.15em\right]$ are the spherical and deviatoric tensor: ${\left[\kern-0.15em\left[ J \right]\kern-0.15em\right]}_{ijkl} = \frac{1}{3}\left( {\delta_{ij} \delta_{kl} } \right)$ and $\left[\kern-0.15em\left[ K \right]\kern-0.15em\right]{\text{ }} = {\text{ }}\left[\kern-0.15em\left[ I \right]\kern-0.15em\right] - \left[\kern-0.15em\left[ J \right]\kern-0.15em\right]{\text{ }}$. $\delta_{ij}$ represents Kronecker delta symbol ($\delta_{ij} = 1$ if $i = j$, $\delta_{ij} = 0$ otherwise)

Description of volume fractions and stiffness tensors at Level II:

Volume fraction of phases at Level II reads:

$$f^{\text{c}} + f^{\text{k}} + f^{\text{inc}} + \phi^{\text{II}} = 1,$$

(35)

$$f^{\text{inc}} = f^{\text{calcite}} + f^{\text{quartz}} ,$$

(36)

where $f^{\text{c}}$,$f^{\text{k}}$,$f^{\text{inc}}$, and $\phi^{\text{II}}$ are volume fractions of clay, kerogen, inclusions and pore space at Level II, respectively. Self-consistent texture implies that $\phi^{\text{I}} = \phi^{\text{II}}$. Moreover, it is assumed that solid-phase inclusions consist of either calcite, with volume fraction of $f^{\text{calcite}}$, or quartz, with volume fraction of $f^{\text{quartz}}$.

Volume fractions of clay and kerogen at Level I, i.e., $\eta^{\text{c}}$ and $\eta^{\text{k}}$, can be obtained from their counterparts at Level II, i.e., $f^{\text{c}}$ and $f^{\text{k}}$, as follows:

$$\eta^{m} = \frac{{f^{m} }}{{f^{\text{c}} + f^{\text{k}} + \phi_{{{\text{k}} + {\text{c}}}}^{\text{II}} }}; \, \{ m = {\text{ c }}\left( {\text{for clay}} \right),{\text{ k }}\left( {\text{for kerogen}} \right)\} ,$$

(37)

where $\phi_{{{\text{k}} + {\text{c}}}}^{\text{II}}$ denotes the portion of $\phi^{\text{II}}$ that belongs to clay and kerogen phases. The stiffness tensor of each solid phase in inclusion reads:

$$\left[\kern-0.15em\left[ {C_{{{\text{inc}}}}^{m} } \right]\kern-0.15em\right] = 3K_{{{\text{inc}}}}^{m} \left[\kern-0.15em\left[ J \right]\kern-0.15em\right] + 2G _{{{\text{inc}}}}^{m} \left[\kern-0.15em\left[ K \right]\kern-0.15em\right](m = {\text{quartz}},{\text{ calcite}}),$$

(38)

where $K_{\text{inc}}^{m}$ and $G_{\text{inc}}^{m}$ are the bulk and shear moduli of inclusion $m$, respectively ($K_{\text{inc}}^{\text{quartz}}$ = 37.9 GPa; $G_{\text{inc}}^{\text{quartz}}$ = 44.3 GPa and $K_{\text{inc}}^{\text{calcite}}$ = 58.2 GPa; $G_{\text{inc}}^{\text{calcite}}$ = 28.3 GPa (Mavko et al. 2009)).

Effect of weakened interface:

Interfacial transition zone (ITZ) is utilized to model an imperfect interface between matrix and inclusions. Introduction of ITZ alters the relationship of volume fractions at Eq. (36) to:

$$f^{inc} = f^{calcite} + f^{quartz} + f^{itz}$$

(39)

One should note that the ratio $\frac{{f^{\text{calcite}} }}{{f^{\text{quartz}} }}$ is constant before and after the introduction of ITZ. The effect of weakened interface can be integrated into the multiscale model through a three-step homogenization scheme known as generalized self-consistent method (Christensen and Lo 1979). In the first step of homogenization, the effective elasticity of solid matrix, i.e., a self-consistent mixture of quartz, and calcite is obtained. Next, the weakened interface is introduced to the homogenized solid matrix obtained from previous step by performing a generalized self-consistent homogenization. Finally, the porosity is introduced to the effective medium obtained from second step by implementing a self-consistent homogenization. Following relationships for relevant volume fractions involved in the three-step homogenization are considered:

$${\text{Step1}}:f^{m,1} = \frac{{f^{m} }}{{f^{\text{quartz}} + f^{\text{calcite}} }}\,(m = {\text{quartz}},{\text{ calcite}})$$

(40)

$${\text{Step2}}:f^{{{\text{inc}},2}} = \left( {\frac{{2 - \Delta_{\text{itz}} }}{2}} \right)^{3} , \,f^{{{\text{itz}},2}} = 1 - \left( {\frac{{2 - \Delta_{\text{itz}} }}{2}} \right)^{3} ,$$

(41)

$${\text{Step3}}:f^{{{\text{inc}},3}} = \frac{{f^{\text{inc}} }}{{f^{\text{inc}} + \phi^{\text{inc}} }};\,\phi^{{{\text{inc}},3}} = 1 - f^{{{\text{inc}},3}} (\phi^{\text{inc}} = \phi^{\text{II}} - \phi_{{{\text{k}} + {\text{c}}}}^{\text{II}} ),$$

(42)

where $\Delta_{\text{itz}}$ denotes the thickness of ITZ in micrometer. In the right-hand side of Eq. (41), the number 2 represents the average sum of radius of solid inclusion grain and the thickness of ITZ in micrometer (see Fig. 1), which is reported in Monfared and Ulm (2016).

Appendix C

Level I:

$${\text{Porous Clay}}\; \left[ \alpha \right]^{\text{porous-clay}} = \phi^{\text{I}} \left[ I \right]:\left[\kern-0.15em\left[ {A} \right]\kern-0.15em\right]^{{\phi^{\text{I}} }} ,$$

(43)

$$\frac{1}{{N^{\text{porous-clay}} }} = \left[ I \right]:S^{\text{clay}} :\left( {\left[ \alpha \right]^{\text{porous-clay}} - \phi^{\text{I}} \left[ I \right]} \right),$$

(44)

$${\text{Porous Kerogen}}\;\alpha^{\text{porous-kerogen}} = 1 - \frac{{K^{\text{porous-kerogen}} }}{{K^{\text{kerogen}} }},$$

(45)

$$\frac{1}{{N^{\text{porous-kerogen}} }} = \frac{{\alpha^{\text{porous-kerogen}} - \phi^{I} }}{{K^{\text{kerogen}} }}.$$

(46)

$${\text{Homogenized Level I}} \left[ \alpha \right]^{I} = (\varphi^{\text{clay}} + \eta^{\text{clay}} )\left[ \alpha \right]^{\text{porous-clay}} + (\varphi^{\text{kerogen}} + \eta^{\text{kerogen}} )[\alpha^{\text{porous-kerogen}} ],$$

(47)

$$\frac{1}{{N^{I} }} = (\varphi^{\text{clay}} + \eta^{\text{clay}} )\frac{1}{{N^{\text{porous-clay}} }} + (\varphi^{\text{kerogen}} + \eta^{\text{kerogen}} )\frac{1}{{N^{\text{porous-kerogen}} }},$$

(48)

Level II:

$${\text{Porous inclusion}}\; \alpha^{\text{pourous-inc}} = 1 - \frac{{K^{\text{porous-inc}} }}{{K^{\text{inc}} }},$$

(49)

$$\frac{1}{{N^{\text{porous-inc}} }} = \frac{{\alpha^{\text{porous-inc}} - \phi^{\text{II}} }}{{K^{\text{inc}} }},$$

(50)

$${\text{Homogenized Level II}}\;\left[ \alpha \right]^{\text{II}} = f^{\text{porous-inc}} [\alpha^{\text{porous-inc}} ] + \left( {1 - f^{\text{porous-inc}} } \right) \left[ \alpha \right]^{\text{I}} ,$$

(51)

$$\frac{1}{{N^{\text{II}} }} = f^{\text{porous-inc}} \frac{1}{{N^{\text{porous-inc}} }} + \left( {1 - f^{\text{porous-inc}} } \right)\frac{1}{{N^{\text{I}} }},$$

(52)

In the above equations, $\varphi^{\text{clay}} = \phi^{\text{I}} \frac{{\eta^{\text{c}} }}{{\eta^{\text{c}} + \eta^{\text{k}} }}$, $\varphi^{\text{kerogen}} = \phi^{\text{I}} \frac{{\eta^{\text{k}} }}{{\eta^{\text{c}} + \eta^{\text{k}} }}$, and $f^{\text{porous-inc}} = f^{\text{inc}} + \phi^{\text{inc}}$. $S^{\text{clay}}$ and $\left[ I \right]$ represent compliance tensor for clay and identity matrix, respectively. Also, superscript “inc” stands for inclusion.

Appendix D

A description of Hamilton function reads:

$${\mathcal{H}}\left( \lambda \right) = \lambda_{0} + \lambda_{1} \mu_{X} + \lambda_{2} \left( {1 + \delta_{X}^{2} } \right) \mu_{X}^{2} + \mathop \int \limits_{{S_{X} }} \exp \left( { - \lambda_{0} - \lambda_{1} x - \lambda_{2} x^{2} } \right) {\text{d}}x.$$

(53)

Appendix E

Following flowchart represents a schematic of the calibration and validation process