Geophysical Monitoring for Geologic Carbon Storage. Группа авторов

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СКАЧАТЬ _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)СКАЧАТЬ