Evaluating the Poroelastic Effect on Anisotropic, Organic-Rich, Mudstone Systems

Abstract

Understanding the poroelastic effect on anisotropic organic-rich mudstones is of high interest and value for evaluating coupled effects of rock deformation and pore pressure, during drilling, completion and production operations in the oilfield. These applications include modeling and prevention of time-dependent wellbore failure, improved predictions of fracture initiation during hydraulic fracturing operations (Suarez-Rivera et al. Presented at the Canadian Unconventional Resources Conference held in Calgary, Alberta, Canada, 15–17 November 2011. CSUG/SPE 146998 2011), improved understanding of the evolution of pore pressure during basin development, including subsidence and uplift, and the equilibrated effective in situ stress (Charlez, Rock mechanics, vol 2 1997; Katahara and Corrigan, Pressure regimes in sedimentary basins and their prediction: AAPG Memoir, vol 76, pp 73–78 2002; Fjær et al. Petroleum related rock mechanics. 2nd edn 2008). In isotropic rocks, the coupled poro-elastic deformations of the solid framework and the pore fluids are controlled by the Biot and Skempton coefficients. These are the two fundamental properties that relate the rock framework and fluid compressibility and define the magnitude of the poroelastic effect. In transversely isotropic rocks, one desires to understand the variability of these coefficients along the directions parallel and longitudinal to the principal directions of material symmetry (usually the direction of bedding). These types of measurements are complex and uncommon in low-porosity rocks, and particularly problematic and scarce in tight shales. In this paper, we discuss a methodology for evaluating the Biot’s coefficient, its variability along the directions parallel and perpendicular to bedding as a function of stress, and the homogenized Skempton coefficient, also as a function of stress. We also predict the pore pressure change that results during undrained compression. Most importantly, we provide values of transverse and longitudinal Biot’s coefficients and the homogenized Skempton coefficient for two important North American, gas-producing, organic-rich mudstones. These results could be used for petroleum-related applications.

Appendix

Equation (10) gives the difference between the poroelastic coefficients for directions normal to and parallel to the unique axis of a material with TI symmetry. This expression can be derived from the general Biot–Hooke law introduced by Biot and Willis (1957):

$$ \left( {\begin{array}{*{20}c} {\Updelta \sigma_{x} } \\ {\Updelta \sigma_{y} } \\ {\Updelta \sigma_{z} } \\ {\Updelta \tau_{yz} } \\ {\Updelta \tau_{zx} } \\ {\Updelta \tau_{xy} } \\ {\phi \Updelta p_{\text{f}} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} P & A & F & 0 & 0 & 0 & T \\ A & P & F & 0 & 0 & 0 & T \\ F & F & W & 0 & 0 & 0 & V \\ 0 & 0 & 0 & L & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & L & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & N & 0 \\ T & T & V & 0 & 0 & 0 & R \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\Updelta \varepsilon_{x} } \\ {\Updelta \varepsilon_{x} } \\ {\Updelta \varepsilon_{x} } \\ {\Updelta \gamma_{yz} } \\ {\Updelta \gamma_{zx} } \\ {\Updelta \gamma_{xy} } \\ {\Updelta \varepsilon } \\ \end{array} } \right) $$

(12)

Here we use the divergence $\varepsilon = \nabla \cdot \vec{u}_{\text{f}}$ of the fluid displacement field $\vec{u}_{\text{f}}$ as the strain parameter for the fluid phase, in accordance with Biot and Willis’ original formulation. Unlike Biot and Willis, we use the same sign convention for both stress, strain and pore pressure.

The last of the 7 equations in (12),

$$ \phi \Updelta p_{\text{f}} = 2T\Updelta \varepsilon_{r} + V\Updelta \varepsilon_{z} + R\Updelta \varepsilon $$

(13)

gives us

$$ \Updelta \varepsilon = \frac{1}{R}\left( {\phi \Updelta p_{\text{f}} - 2T\Updelta \varepsilon_{r} - V\Updelta \varepsilon_{z} } \right) $$

(14)

Combining this result with the first three equations in (12), and realizing that ε _x = ε _y = ε_z, we find

$$ \Updelta \sigma_{r} = \left( {P + A} \right)\Updelta \varepsilon_{r} + F\Updelta \varepsilon_{z} + \frac{T}{R}\left( {\phi \Updelta p_{\text{f}} - 2T\Updelta \varepsilon_{r} - V\Updelta \varepsilon_{z} } \right) $$

(15)

$$ \Updelta \sigma_{z} = 2F\Updelta \varepsilon_{r} + W\Updelta \varepsilon_{z} + \frac{V}{R}\left( {\phi \Updelta p_{\text{f}} - 2T\Updelta \varepsilon_{r} - V\Updelta \varepsilon_{z} } \right) $$

(16)

From Eqs. (15) and (16) and Eqs. (6) and (7) we see that

$$ \alpha_{r} = \phi \frac{T}{R} $$

(17)

$$ \alpha_{z} = \phi \frac{V}{R} $$

(18)

Given hydrostatic conditions, ∆σ _r = ∆σ _z, we find from Eqs. (15) and (16)

$$ \left[ {P + A - 2\frac{{T^{2} }}{R} - 2F + 2\frac{VT}{R}} \right]\Updelta \varepsilon_{r} = \left[ {W - \frac{{V^{2} }}{R} - F + \frac{VT}{R}} \right]\Updelta \varepsilon_{z} - \phi \Updelta p_{\text{f}} \frac{T - V}{R} $$

(19)

By considering this expression for the two situations, drained ($\Updelta p_{\text{f}} = 0$) and undrained, we find

$$ \left( {\frac{{\Updelta \varepsilon_{r} }}{{\Updelta \varepsilon_{z} }}} \right)_{\text{ud}} = \left( {\frac{{\Updelta \varepsilon_{r} }}{{\Updelta \varepsilon_{z} }}} \right)_{\text{dr}} - \left( {T - V} \right)\frac{\phi }{R}\left( {\frac{{\Updelta p_{\text{f}} }}{{\Updelta \varepsilon_{z} }}} \right)_{\text{ud}} \left[ {P + A - 2F - 2\frac{T}{R}\left( {T - V} \right)} \right]^{ - 1} $$

(20)

We now ignore higher than first order terms in anisotropy, which implies assuming that (T − V)² → 0 and replacing (P + A – 2F)/2 with the isotropic shear modulus G. Introducing the anisotropic poroelastic constants from Eqs. (17) and (18), we find

$$ \left( {\frac{{\Updelta \varepsilon_{r} }}{{\Updelta \varepsilon_{z} }}} \right)_{\text{ud}} \approx \left( {\frac{{\Updelta \varepsilon_{r} }}{{\Updelta \varepsilon_{z} }}} \right)_{\text{dr}} - \left( {\alpha_{r} - \alpha_{z} } \right)\left( {\frac{{\Updelta p_{\text{f}} }}{{\Updelta \varepsilon_{z} }}} \right)_{\text{ud}} \frac{1}{2G} $$

(21)

Reorganizing Eq. (21) gives us Eq. (10).

From the first three and the last of the 7 equations of (12) we may also show (after some algebra, and ignoring higher than first order terms in anisotropy) that an increment ∆ε _v in the volumetric strain is proportional to

$$ \Updelta \sigma - \frac{\phi }{3R}\left( {2T + V} \right)\Updelta p_{\text{f}} $$

(22)

The volumetric strain increment is proportional to the effective stress increment. Thus, by combining Eqs. (5), (17), (18) and (22) we may reproduce Eq. (9).