The effect of the nanogranular nature of shale on their poroelastic behavior

Abstract

Natural composite materials are highly heterogeneous porous materials, with porosities that manifest themselves at scales much below the macroscale of engineering applications. A typical example is shale, the transverse isotropic sealing formation of most hydrocarbon bearing reservoirs. By means of a closed loop approach of microporomechanics modeling, calibration and validation of elastic properties at multiple length scales of shale, we show that the nanogranular nature of this highly heterogeneous material translates into a unique poroelastic signature. The self-consistent scaling of the porous clay stiffness with the clay packing density minimizes the anisotropy of the Biot pore pressure coefficients; whereas the intrinsic anisotropy of the elementary particle translates into a pronounced anisotropy of the Skempton coefficients. This new microporoelasticity model depends only on two shale-specific material parameters which neatly summarize clay mineralogy and bulk density, and which makes the model most appealing for quantitative geomechanics, geophysics and exploitation engineering applications.

Appendix

1.1 Hill concentration tensor

Within the framework of microporomechanics, the Hill concentration tensor (or P-tensor) characterizes the interaction between the different phases, and it is expressed as [40]:

$${\mathbb{P}}_{ijkl}^{0}= \left.-\left(\partial^{2}\left(\int_{\Upomega}G_{ik}^{0}(\underline{x}-\underline{x}^{\prime}){\rm d}\Upomega \right)\right/\partial x_{j}\partial x_{j}\right)_{\left(ij\right) \left(kl\right)}$$

(55)

where the Green’s function $G_{ik}^{0}(\underline{x}-\underline{x}^{\prime})$ expresses the displacement at point $\underline{x}$ in a linear elastic solid medium of stiffness ${{\mathbb{C}}}^{0},$ resulting from a unit force applied at point $\underline{x}^{\prime}$ in the medium. The P-tensor is determined by the stiffness properties of the matrix phase and the shape of the inclusion phase. Integral expressions of the P-tensor for ellipsoidal inclusions in a transversely isotropic media were developed by Laws [56], which in a slightly different form reads [71]:

$$P_{ijkl}=\frac{1}{16\pi}\left({\mathcal{M}}_{kijl}+{\mathcal{M}}_{kjil}+ {\mathcal{M}}_{lijk}+{\mathcal{M}}_{ljik}\right) \label{Pijkl}$$

(56)

where,

$${\mathcal{M}}_{kijl}=\int_{S\left(\omega \right)}\frac{a_{1}a_{2}a_{3}}{\left(a_{1}^{2}\omega_{1}^{2}+a_{2}^{2}\omega_{2}^{2}+a_{3}^{2}\omega_{3}^{2}\right)^{3/2}}\Upgamma_{kj}^{-1}\left({\varvec{\omega}}\right) \omega_{i}\omega_{l}dS\left({\varvec{\omega}}\right)$$

(57)

Parameters a ₁, a ₂, a ₃ relate to the shape of the ellipsoid, ${\rm d}S\left({\varvec{\omega}}\right)$ is the surface element of a unit ellipsoid of components ω₁, ω₂, ω₃, and $\Upgamma_{kj}\left({\varvec{\omega}}\right) =C_{ijkl}\omega_{j}\omega_{l}$ represents the Christoffel matrix denoting the stiffness of the anisotropic matrix. For the case of a transverse isotropic media and spherical inclusions, (57) can be specialized by taking advantage of spherical symmetries. The unit vector ${\varvec{\omega}}$ is expressed in spherical coordinates θ ∈ [0,π]; ϕ ∈ [0, 2 π] by the following transformations:

$$\omega_{1} =\sin \theta \cos \phi$$

(58)

$$\omega_{2} =\sin \theta \sin \phi$$

(59)

$$\omega_{3} =\cos \theta$$

(60)

The non-zero terms of (56) are reduced to line integrals in $\xi =\cos \theta, {\rm d}\xi =-\sin \theta {\rm d}\theta,$ whose evaluations are achieved numerically using standard software packages such as Matlab. The final expressions of the elements of the P-tensor, which is heavily used in the multi-scale model of shale herein developed, are the following (see [40], except for a misprint corrected in the expressions below):

$$\begin{aligned}P_{11}&= \frac{1}{16}\int\limits_{-1}^{1}\frac{1}{D_{1}}\left(\xi^{2}-1\right)\left(-8\xi^{4}C_{33}C_{44}\right.\\\, & -3\xi^{4}C_{12}C_{33}-2\xi^{4}C_{13}^{2}+3\xi^{4}C_{12}C_{44}\\ \, & -5\xi^{4}C_{11}C_{44}+5\xi^{4}C_{11}C_{33}-4\xi^{4}C_{13}C_{44}\\\,&+6\xi^{4}C_{44}^{2}-6\xi^{2}C_{44}^{2} +4\xi^{2}C_{13}C_{44}\\\, &+3\xi^{2}C_{12}C_{33}-5\xi^{2}C_{11}C_{33}-6\xi^{2}C_{12}C_{44}\\\, & +2\xi^{2}C_{13}^{2} +10\xi^{2}C_{11}C_{44}+3C_{12}C_{44}\\\, & -5C_{11}C_{44}){\rm d}\xi\end{aligned}$$

(61)

$$\begin{aligned}P_{12} &= -\frac{1}{16}\int\limits_{-1}^{1}\frac{1}{D_{1}}\left(\xi -1\right)^{2}\left(\xi +1\right)^{2}\left(C_{12}C_{44}\right.\\ &\quad -\xi^{2}C_{12}C_{44}+\xi^{2}C_{12}C_{33}+C_{11}C_{44}\\& \quad -\xi^{2}C_{11}C_{44}+\xi^{2}C_{11}C_{33}-2\xi^{2}C_{13}^{2}\\&\quad\left.-\;4\xi^{2}C_{13}C_{44}-2\xi^{2} C_{44}^{2}\right){\rm d}\xi\\\end{aligned}$$

(62)

$$P_{13}=-\frac{1}{4}\left(C_{13}+C_{44}\right) \int\limits_{-1}^{1}\frac{\xi^{2}\left(\xi^{2}-1\right)}{D_{2}}{\rm d}\xi$$

(63)

$$P_{33}=\frac{1}{2}\int\limits_{-1}^{1}\frac{\xi^{2}\left(-C_{11}+C_{11}\xi^{2}-C_{44}\xi^{2}\right)}{D_{2}}{\rm d}\xi$$

(64)

$$\begin{aligned}P_{44}=&-\frac{1}{16}\int\limits_{-1}^{1}\frac{1}{D_{1}}\left(3\xi^{2}C_{11}^{2}-2\xi^{6}C_{13}^{2}-C_{11}^{2}\right.\\\,&-4\xi^{6}C_{11}C_{44}-8\xi^{6}C_{13}C_{44}-4\xi^{6}C_{33}C_{44}\\\, & +3\xi^{6}C_{11}C_{33}-3\xi^{4}C_{11}^{2}+C_{11}C_{22}\\\,&+\xi^{6}C_{11}^{2}-\xi^{6}C_{11}C_{12} -\xi^{6}C_{12}C_{33}\\\, &+4\xi^{4}C_{12}C_{13}-2\xi^{2}C_{12}C_{13}+2\xi^{6}C_{11}C_{13}\\\, & +2\xi^{4}C_{13}^{2}+\xi^{4}C_{12}C_{33}+8\xi^{4}C_{11}C_{44}\\\, &-3\xi^{4}C_{11}C_{33}-4\xi^{2}C_{11}C_{44} +8\xi^{4}C_{13}C_{44}\\\, &-4\xi^{4}C_{11}C_{13}+3\xi^{4}C_{11}C_{12}-3\xi^{2}C_{11}C_{12}\\\, &\left.-2\xi^{6}C_{12}C_{13}+2\xi^{2}C_{11}C_{13}\right){\rm d}\xi\\\end{aligned}$$

(65)

where

$$D_{1}=\left(\xi^{2}C_{11}-C_{11}-2\xi^{2}C_{44}-\xi^{2}C_{12}+C_{12}\right) \left(D_{2}\right)$$

(66)

$$\begin{aligned}D_{2}=&-\xi^{4}C_{33}C_{44}+2\xi^{2}C_{13}C_{44}-\xi^{2}C_{11}C_{33}\\\, &-2\xi^{4}C_{13}C_{44}+\xi^{4}C_{11}C_{33}+2\xi^{2}C_{11}C_{44}\\\, &+\xi^{2}C_{13}^{2} -\xi^{4}C_{11}C_{44}-\xi^{4}C_{13}^{2}-C_{11}C_{44}\\\end{aligned}$$

(67)

1.2 Elastic constants from UPV measurements

The inversion of UPV measurements to elasticity constants of transversely isotropic media is achieved through the following relations [6]:

$$C_{11}^{\rm UPV} =\rho V_{\rm P1}^{2}$$

(68)

$$C_{33}^{\rm UPV} =\rho V_{\rm P3}^{2}$$

(69)

$$C_{66}^{\rm UPV} =\frac{1}{2}\left(C_{11}^{\rm UPV}-C_{12}^{\rm UPV}\right) =\rho V_{S1}^{2}$$

(70)

$$C_{44}^{\rm UPV} = \rho V_{\rm S3}^{2}$$

(71)

$$C_{13}^{\rm UPV} = -C_{44}+\alpha \sqrt{\left(C_{11}+C_{44}-2\rho V_{45}^{2}\right) \left(C_{33}+C_{44}-2\rho V_{45}^{2}\right)}$$

(72)

where:

V _P1 :

pure longitudinal mode in bedding direction

V _P3 :

pure longitudinal mode normal to bedding

V _S1 :

pure shear mode polarized normal to axis of symmetry

V _S3 :

pure shear modes polarized normal to axis of symmetry

V ₄₅ :

quasi-longitudinal or quasi-shear wave measured at 45° from axis of symmetry

α :

(+ 1) for quasi-longitudinal (qP) wave, (−1) for quasi-shear (qS) wave.

1.3 Experimental information on shale validation data set #2 (VDS-II)

Table 9 provides a summary of the relevant experimental information and testing conditions for the shale specimens documented by Jones and Wang [51], Hornby [47], Jakobsen and Johansen [50], Domnesteanu et al. [29], and Dewhurst and Siggins [27]. The macroscopic elasticity and material composition data of these references is used for the validation of the microporomechanical model at the macroscopic scale (level II).

Table 9 Summary of shale specimens and experimental procedures for the VDS-II data set, which was collected from open literature

1.4 Mineral densities

Table 10 provides the density of minerals used for translating the mineralogy data in the CDS, VDS-I, and VDS-II data sets into volume fractions.

Table 10 Density information of some minerals present in shale