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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) left-parenthesis w Subscript n Superscript plus Baseline minus w Subscript n Superscript minus Baseline right-parenthesis 2 a upper H plus w Subscript k comma k Baseline pi a squared upper H equals 0 period

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) left-parenthesis w Subscript n Superscript plus Baseline minus w Subscript n Superscript minus Baseline right-parenthesis pi a squared plus w Subscript k comma k Baseline pi a squared upper H equals 0 period

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 epsilon 33 Superscript t o t a l Baseline equals u Subscript 3 comma 3 Baseline plus left-parenthesis u Subscript n Superscript plus Baseline minus u Subscript n Superscript plus Baseline right-parenthesis slash upper H can be related, after somewhat lengthy manipulations, via

(5.10)

Note that upper B overTilde 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 upper B overTilde right-arrow upper B slash 3 equals alpha upper M slash 3 upper K Subscript upper U , that is, E _norm → E _U(undrained Young's modulus). Large drained fracture compliance results in . Therefore,

(5.11) StartFraction 1 Over upper E Subscript n o r m Baseline EndFraction right-arrow StartFraction 1 Over upper E Subscript upper D Baseline EndFraction minus StartFraction alpha Over 3 upper K Subscript upper D Baseline EndFraction plus StartFraction eta Subscript upper M Baseline Over upper H EndFraction period

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

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