UP-SCALING MODEL FOR GEOMECHANICS OF RESERVOIRS

Elasticity of layered materials Next, to determine the overall drained (or undrained) bulk and shear moduli of the reservoir, assume a typical building block of the random system is a small (relative to the size of the reservoir) ``grain'' of laminate material whose elastic response for a transversely isotropic (hexagonal) system can be described locally by:  

$$\left(\begin{array} {c} \sigma_{11} \\ \sigma_{22} \\ \sig... ...} \\ e_{33} \\ e_{23} \\ e_{31} \\ e_{12}\end{array}\right),$$ (18)

where $\sigma_{ij}$ are the usual stress components for i,j=1-3 in Cartesian coordinates, with 3 (or z) being the axis of symmetry (the lamination direction for such a layered material). Displacement u_i is then related to strain component e_ij by $e_{ij} = (\partial u_i/\partial x_j + \partial u_j/\partial x_i)/2$.This definition introduces some convenient factors of two into the 44,55,66 components of the matrix of stiffness coefficients shown in (18). For definiteness I also assume that the matrix of stiffness coefficients in (18) arises from the lamination of N isotropic constituents having bulk and shear moduli K_n, $\mu_n$, in the N > 1 layers present in each building block. It is important that the thicknesses d_n always be in the same proportion in each of these laminated blocks, so that $f_n = d_n/\sum_{n'} d_{n'}$. But the order in which layers were added to the blocks is not important, as Backus's formulas (Backus, 1962) for the constants show. For the overall quasistatic (long wavelength) behavior of the system I am studying, Backus's results [also see Postma (1955), Berryman (1998; 2004b), Milton (2002)] state that  

$$\begin{array} {ll} c_{33} = \left<\frac{1}{K+4\mu/3}\right\g... ...2}{K+4\mu/3}\right\gt, & c_{12} = c_{11} - 2c_{66}.\end{array}$$ (19)

This bracket notation can be correctly viewed as a line integral along the symmetry axis x₃. The bulk modulus K_n and shear modulus $\mu_n$ displayed in these averages can be either the drained or the undrained moduli for the individual layers. For the undrained case, the results are inherently assumed either to apply at very high frequencies, such as ultrasonic frequencies in laboratory experiments, or to situations wherein each layer is physically isolated so that fluid increments cannot move from one porous layer to the next. The bulk modulus for each laminated grain is that given by the compressional Reuss average K_R of the corresponding compliance matrix s_ij [the inverse of the usual stiffness matrix c_ij, whose nonzero components are shown in (18)]. The result is $e = e_{11}+e_{22}+e_{33} = \sigma/K_{\rm eff}$, where $1/K_{\rm eff} = 1/K_R = 2s_{11} + 2s_{12} + 4s_{13} + s_{33}$.Even though $K_{\rm eff} = K_R$ is the same for every grain, since the grains themselves are not isotropic, the overall bulk modulus K^* of the random polycrystal does not necessarily have the same value as K_R for the individual grains (Hill, 1952). Hashin-Shtrikman bounds on K^* for random polycrystals whose grains have hexagonal symmetry (Peselnick and Meister, 1965; Watt and Peselnick, 1980) show in fact that the K_R value lies outside the bounds in many situations (Berryman, 2004).

Bounds for random polycrystals Voigt and Reuss bounds: hexagonal symmetry: For hexagonal symmetry, the nonzero stiffness constants are: c₁₁, c₁₂, c₁₃ = c₂₃, c₃₃, c₄₄ = c₅₅, and c₆₆ = (c₁₁-c₁₂)/2. The Voigt (1928) average for bulk modulus of hexagonal systems is well-known to be  

$$K_V = \left[2(c_{11}+c_{12}) +4c_{13}+c_{33}\right]/9.$$ (20)

Similarly, for the overall shear modulus G^*, I have  

$$G_V = \frac{1}{5}\left(G_{\rm eff}^v + 2c_{44} + 2c_{66}\right),$$ (21)

where the new term appearing here is essentially defined by (21) and given explicitly by  

$$G_{\rm eff}^v = (c_{11} + c_{33} - 2c_{13} - c_{66})/3.$$ (22)

The quantity $G_{\rm eff}^v$ is the energy per unit volume in a grain when a ``pure uniaxial shear'' strain of unit magnitude [i.e., $(e_{11},e_{22},e_{33}) = (1,1,-2)/\sqrt{6}$], whose main compressive strain is applied to the grain along its axis of symmetry (Berryman, 2004a; 2004b). Note that the concept of ``pure uniaxial shear'' strain (or stress) is based on the observation that if a uniaxial principal strain (or stress) of magnitude 3 is applied along the symmetry axis, it can be decomposed according to (0,0,3)^T = (1,1,1)^T - (1,1,-2)^T into a pure compression and a pure shear contribution, which is then called for the sake of brevity the ``pure uniaxial shear.'' The Reuss (1929) average K_R for bulk modulus can also be written in terms of stiffness coefficients as  

$$\frac{1}{K_R - c_{13}} = \frac{1}{c_{11} - c_{66} - c_{13}} + \frac{1}{c_{33} - c_{13}}.$$ (23)

The Reuss average for shear is  

$$G_R = \left[\frac{1}{5}\left(\frac{1}{G_{\rm eff}^r} + \frac{2}{c_{44}} + \frac{2}{c_{66}}\right)\right]^{-1},$$ (24)

that defines $G_{\rm eff}^r$ - i.e., the energy per unit volume in a grain when a pure uniaxial shear stress of unit magnitude [i.e., $(\sigma_{11},\sigma_{22},\sigma_{33}) = (1,1,-2)/\sqrt{6}$], whose main compressive pressure is applied to a grain along its axis of symmetry. For each grain having hexagonal symmetry, two product formulas found by Berryman (2004a) hold: $3K_RG_{\rm eff}^v = 3K_VG_{\rm eff}^r = \omega_+\omega_-/2 = c_{33}(c_{11}-c_{66})-c_{13}^2$.The symbols $\omega_\pm$ stand for the quasi-compressional and quasi-uniaxial-shear eigenvalues for the crystalline grains. Thus, it follows that  

$$G_{\rm eff}^r = K_RG_{\rm eff}^v/K_V$$ (25)

is a general formula, true for hexagonal symmetry. Hashin-Shtrikman bounds: It has been shown elsewhere (Berryman, 2004a; 2004b) that the Peselnick-Meister-Watt (Peselnick and Meister, 1965; Watt and Peselnick, 1980) bounds for bulk modulus of a random polycrystal composed of hexagonal (or transversely isotropic) grains are given by  

$$K_{PM}^\pm = \frac{K_V(G_{\rm eff}^r + \zeta_\pm)} {(G_{\rm ... ...K_RG_{\rm eff}^v + K_V\zeta_\pm} {G_{\rm eff}^v + \zeta_\pm},$$ (26)

where $G_{\rm eff}^v$ ($G_{\rm eff}^v$) is the uniaxial shear energy per unit volume for a unit applied shear strain (stress). The second equality follows directly from the product formula (25). Parameters $\zeta_\pm$ are defined by  

$$\zeta_{\pm} = \frac{G_\pm}{6}\left(\frac{9K_\pm+8G_\pm}{K_\pm+2G_\pm}\right).$$ (27)

In (27), values of $G_\pm$ (shear moduli of isotropic comparison materials) are given by inequalities  

$$0 \le G_- \le \min(c_{44},G_{\rm eff}^r,c_{66}),$$ (28)

and  

$$\max(c_{44},G_{\rm eff}^v,c_{66}) \le G_+ \le \infty.$$ (29)

The values of $K_\pm$ (bulk moduli of isotropic comparison materials) are then given by algorithmic equalities  

$$K_\pm = \frac{K_V(G_{\rm eff}^r-G_\pm)}{(G_{\rm eff}^v - G_\pm)},$$ (30)

derived by Peselnick and Meister (1965) and Watt and Peselnick (1980). Also see Berryman (2004a). Bounds $G_{\rm hex}^\pm$ (+ is upper bound, - is the lower bound) on the shear moduli for random polycrystals of hexagonal crystals are then given by  

$$\frac{1}{G_{\rm hex}^\pm + \zeta_\pm} = \frac{1}{5}\Big[\fr... ...\frac{2}{c_{44}+\zeta_\pm} + \frac{2}{c_{66}+\zeta_\pm}\Big],$$ (31)

where $\gamma_\pm$ and $\delta_\pm$ are given by  

$$\gamma_\pm = \frac{1}{K_\pm + 4G_\pm/3}, \quad\hbox{and}\quad \delta_\pm = \frac{5G_\pm/2}{K_\pm + 2G_\pm}.$$ (32)

K_V is the Voigt average of the bulk modulus as defined previously. 1.4

TABLE 1. Input Parameters for Weber Sandstone Model of Double-Porosity System.

$$\phi^{(1)} \phi^{(2)}$$ K_s K_s⁽¹⁾ K_d⁽¹⁾ G_d⁽¹⁾ K_s⁽²⁾ K_d⁽²⁾ G_d⁽²⁾

Note: Porosity $\phi$ is dimensionless.

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