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Detailed dynamic poroelastic modeling of the experimental results is beyond the scope of this chapter. However, we can gain some insights into the observed behavior of fractured samples using a simple, isotropic, quasi‐static model (i.e., isotropic Gassmann model).

Figure 5.6 Shear modulus and related attenuations determined from SHRB tests during scCO₂ injection experiments on Carbon Tan sandstone cores: (a) Carbon Tan #1 elastic moduli; (b) Carbon Tan #2 elastic moduli; (c) Carbon Tan #1 attenuations; (d) Carbon Tan #2 attenuations.

In the following, we consider only small stress and displacement perturbations caused by seismic waves. For the porous, intact matrix of the sandstone samples, we assume the following constitutive equations for an isotropic homogeneous poroelastic medium (e.g., Pride et al., 2002):

(5.3)

(5.4)

Figure 5.7 X‐ray CT images of scCO₂ invasion into intact and fractured sandstone cores. Brighter colors indicate higher CO₂ saturation. Average apertures determined from CT images are indicated for open, sheared fractures. (Note that at the top of the Frac IIb images, unintended shifts of the experimental setup between an initial calibration scan and subsequent scans during the scCO₂ injection experiment resulted in false images of scCO₂ along the bottom edge of the core, which disappear in the subsequent images.)

In 5.3 and 5.4, repeated indices indicate summation, and, _i ≡ ∂/∂x _i . u _i is the local average solid frame displacement vector. w _i ≡ φ (U _i ‐ u _i ) is the relative fluid volume displacement vector defined via u _i , local average fluid displacement vector in the pore space U _i , and porosity φ . δ _ij indicates an identity tensor. τ _ij is the total stress tensor, and p _f is the fluid pressure (positive sign for compression). G is the solid frame shear modulus, and K _U is the undrained bulk modulus. C and M are the Biot's coupling and fluid storage moduli, respectively. Note that these parameters are related via C = αM, K _U = K _D +α ² M, K _D =(1‐αB) K _U , where α is the Biot‐Willis coefficient, B is the Skempton coefficient, and K _D is the drained bulk modulus. For the fracture part, assuming a plane, permeable, and compliant fracture, the boundary conditions for the fracture‐normal displacement, stress, and pressure can be stated as (Nakagawa & Schoenberg, 2007)

(5.5)

(5.6)

The superscripts “⁺” and “^–” indicate the opposing surfaces of the fracture, and subscript “n” indicates the direction perpendicular (normal) to the fracture plane. The effect of fluid flow parallel to the fracture is neglected. The thickness of the fracture h is assumed to be very small compared wih the diameter of the sample. Also note that the effective stress coefficient of the open, permeable fracture α ^F can be assumed to be 1. η _D and η _M are the specific drained normal fracture compliance and the specific fracture storage compliance. For an open fracture, η _M can be computed via η _M = h/M ^F ~ h/K _f , where M ^F is the storage modulus of the material within the fracture, and K _f is the bulk modulus of the fluid contained in the fracture (the fracture porosity φ ^F is 1).

From here on, we will use Cartesian coordinates with the 3 axis aligned with the core axis. First, we compute low‐frequency Young's modulus for a jacketed cylindrical core with a radius a and a height H, containing a single fracture along its axis. Conservation of fluid mass in the core requires that the fluid volume exchanged between the fracture and the matrix be in balance:

(5.7)

The total stress in the radial directions ( τ _n , τ ₁₁, and τ ₂₂) and the wave‐induced relative fluid displacement (Darcy flux) in the axial direction (w ₃) is zero. By introducing these values and equations (5.5)–(5.7) into equations (5.3) and (5.4), we can eliminate all the variables in the ratio between the total stress τ ₃₃ and the axial strain u _{3, 3} = ε ₃₃. This ratio, the core‐parallel Young's modulus E _para , is given by

(5.8)

Note that this indicates that large fracture compliances result in K _* → K _U − α ² M = K _D , that is, E _para → E _D (drained Young's modulus). In contrast, when the fracture compliances are very small, K _* → K _U and E _para → E _U (undrained Young's modulus). What this equation reveals is that for large drained normal fracture compliance η _D , substitution of fluids within the fracture, which increases η _M , may not result in significant changes in the Young's modulus.

Next, we examine the case when a core contains a fracture perpendicular to the axis. The conservation of fluid mass requires

(5.9)

Also, the total stress and Darcy flux in the radial directions ( τ ₁₁, τ ₂₂,w ₁,w ₂) are zero, and τ ₃₃ = τ _n . Similar to the core‐parallel case, using equations (5.3) through (5.6) and equation (5.9), the total stress τ ₃₃ and the total axial strain in the sample can be related, after somewhat lengthy manipulations, via

(5.10)

Note that is a Skempton‐coefficient‐like parameter, providing the ratio between −p _f and τ ₃₃. Also, the effect of fluid substitution in the fractured core affects the effective Young's modulus only through this parameter. Small fracture compliances result in , that is, E _norm → E _U (undrained Young's modulus). Large drained fracture compliance results in . Therefore,

(5.11)

Equation (5.11) indicates, in contrast with the fracture parallel to the core axis, η _M can still have an impact on the Young's modulus of the entire core sample.

Figure 5.8a and b shows predictions for the Young's modulus of the fractured samples during scCO₂ injection, based upon equations (5.8) and (5.10), respectively, for a range of η _D values. Based upon the CT images of scCO₂ distribution from the experiment, the substitution of the fluids was assumed only for the fracture by changing η _M as a function of the scCO₂ saturation of the entire core. In contrast, for Figure 5.9, the substitution was assumed to happen throughout the sample. The poroelastic parameters of the rock were evaluated from the experimentally determined undrained Young's modulus and shear modulus of the rock at the beginning of the injection experiment, and a mineral bulk modulus of quartz was assumed (K _s = 38 GPa). Note that the experimentally determined dry elastic moduli were not used, because saturating the sandstone with water resulted in chemical softening of the rock matrix (both Young's and shear moduli decreased). Parameters used in the analysis are summarized in Table 5.2.

Figure 5.8 Fluid substitution modeling of the impact of fracture compliance using isotropic Gassmann models. Compare with the experimental results in Figure 5.5a and b. For (a) the core‐parallel fracture case, fluid substitution is assumed to happen only within the fracture. For (b) the core‐perpendicular case, it is assumed for the entire core. For better agreement between the models and the experiment, the shear modulus was slightly increased for the both cases, and the results are shown in dotted lines.

In Figure 5.8, qualitatively consistent with the experiment, large η _D resulted in Young's modulus values that did not depend upon the presence of scCO₂ in the fracture. The model, however, does not show good quantitative agreement with the experiment, mostly underestimating the effect of the fracture and scCO₂ injection. This may be attributed to the additional effect of compliant interfaces between the sample and the metal bars as mentioned earlier, but also to the use of an isotropic and homogeneous Gassmann model for representing possibly anisotropic rock. Particularly, small errors in the isotropic shear modulus can have a large impact on the fluid substitution. In Figure 5.8a, in broken lines, examples are shown for the case with a shear modulus, which is ~2% larger than the experimental value, resulting in better agreement of the Young's modulus values for the intact sample.