Analysis of Dynamic Fracture Compliance Based on Poroelastic Theory. Part I: Model Formulation and Analytical Expressions

Abstract

The presence of bedding-parallel fractures at any scale in a rock will considerably add to its compliance and elastic anisotropy. Those properties will be more significantly affected when there is a relatively high degree of connectivity between the fractures and the corresponding interconnected pores. This contribution uses linear poroelasticity to reveal the characteristics of the full frequency-dependent compliance of an infinitely extended fracture model assuming the periodicity of the fractured structures. The fracture compliance tensor is complex-valued due to the wave-induced fluid flow between fractures and pores. The interaction between the adjacent fractures is considered under fluid mass conservation throughout the whole pore space. The quantitative effects of fracture (volume) density (the ratio between fracture thickness and spacing) and host rock porosity are analyzed by the diffusion equation for a relatively low-frequency band. The model in this paper is equivalent to the classical dry linear slip model when the bulk modulus of fluid in the fractures tends to zero. For the liquid-filled case, the model becomes the anisotropic Gassmann’s model and sealed saturated linear slip model at the low-frequency and high-frequency limits, respectively. Using the dynamic compliance definition, we can effectively distinguish the saturating fluids in the fractures with the same order magnitude of bulk modulus (e.g., water and oil) using the compliance ratio method. Additionally, the modified dynamic model can be simplified as acceptable empirical formulas if the strain on the fractures induced by the incoming waves is small enough.

Appendices

Appendix 1

1.1 Frequency Dependence of Im[Z _n]

The frequency characteristics of Im[Z _n] (attenuation) are associated with the pressure equilibrium process. In this paper, the wave attenuation is controlled two diffusion systems corresponding two characteristic lengths that are fracture spacing and fracture thickness, respectively.

When ω → 0, Eq. (3) can be written as

$$G = \frac{ - 1}{{\left( {2 + \sqrt {\frac{i\omega }{{D_{\text{m}} }}} h_{\text{m}} } \right)\frac{{\frac{{\kappa_{\text{c}} }}{{D_{\text{c}} }}h_{\text{c}} }}{{\frac{{\kappa_{\text{m}} }}{{D_{\text{m}} }}h_{\text{m}} }} + \left( {2 + \sqrt {\frac{i\omega }{{D_{\text{c}} }}} h_{\text{c}} } \right)}},$$

(26)

Using the values in Table 1 for the physical parameters introduced this paper leads to the inequality

$$\xi = \frac{{\frac{{\kappa_{\text{c}} }}{{D_{\text{c}} }}h_{\text{c}} }}{{\frac{{\kappa_{\text{m}} }}{{D_{\text{m}} }}h_{\text{m}} }} \ll 1,$$

(27)

We can then derive the imaginary part of the compliance

$$\begin{aligned} \text{Im} [Z_{\text{n}} ] &= \text{Im} \left[ { - \frac{{h_{\text{c}} }}{{\psi M_{\text{c}} \alpha_{\text{c}} }}\frac{{ - 2\left( {\frac{1}{2}\sqrt {\frac{i\omega }{{D_{\text{c}} }}} h_{\text{c}} + 1} \right)}}{{\left( {2 + \sqrt {\frac{i\omega }{{D_{\text{m}} }}} h_{\text{m}} } \right)\xi + \left( {2 + \sqrt {\frac{i\omega }{{D_{\text{c}} }}} h_{\text{c}} } \right)}}} \right] \hfill \\ &= \omega \frac{{h_{\text{c}} }}{{\psi M_{\text{c}} \alpha_{\text{c}} }}\frac{1}{{4D_{\text{m}} \xi h_{\text{m}} }} \hfill \\ \end{aligned}$$

(28)

This means the low-frequency asymptote of the compliance, proportional to frequency. Additionally, the intermediate segment of Im[Z _n] regime becomes broader for smaller fracture thickness. When h _c → 0 for a fixed (but finite) frequency,

$$\frac{2}{{h_{\text{c}} \sqrt {\frac{i\omega }{{D_{\text{c}} }}} }}\left( {{\text{e}}^{{\sqrt {\frac{i\omega }{{D_{\text{c}} }}} h_{\text{c}} }} - 1} \right) \gg \frac{{4\kappa_{\text{c}} M_{\text{c}} }}{{i\omega \eta h_{\text{c}} }}\left( {{\text{e}}^{{\sqrt {\frac{i\omega }{{D_{\text{c}} }}} \frac{{h_{\text{c}} }}{2}}} - 1} \right)^{2} .$$

(29)

Then G = −1/2, $\psi = \frac{{K_{\text{m}}^{\text{sat}} C_{\text{c}} }}{{\alpha_{\text{m}} M_{\text{m}} C_{\text{c}} - \alpha_{\text{c}} M_{\text{c}} C_{\text{m}} }}$, $\lim_{{h_{\text{c}} \to 0}} M_{\text{c}} = \lim_{{h_{\text{c}} \to 0}} C_{\text{c}}$ and a _c = 1. So Im[Z _n] is given by

$$\text{Im} [Z_{\text{n}} ] \approx \text{Im} \left[ { - \frac{{h_{\text{c}} }}{{M_{\text{c}} \alpha_{\text{c}} }}\frac{1}{\psi }\frac{ - 1}{{h_{\text{c}} \sqrt {\frac{i\omega }{{D_{\text{c}} }}} }}\left( {h_{\text{c}} \sqrt {\frac{i\omega }{{D_{\text{c}} }}} + \frac{{h_{\text{c}}^{2} \frac{i\omega }{{D_{\text{c}} }}}}{2}} \right)} \right].$$

(30)

Thereby, Eq. (30) reduces to a formula in a frequency-dependent form

$$\text{Im} [Z_{\text{n}} ] = \sqrt \omega \frac{{\alpha_{\text{m}} M_{\text{m}} - C_{\text{m}} }}{{K_{\text{m}}^{\text{sat}} C_{\text{c}} }}h_{\text{c}}^{2} \frac{1}{{2\sqrt {2D_{\text{c}} } }}.$$

(31)

That is, the imaginary part of the compliance is proportional to ω ^1/2 in the intermediate frequency band. Finally, when ω → ∞, the following inequality can be obtained

$$\frac{2}{{h_{\text{c}} \sqrt {\frac{i\omega }{{D_{\text{c}} }}} }}\left( {e^{{\sqrt {\frac{i\omega }{{D_{\text{c}} }}} h_{\text{c}} }} - 1} \right) \gg \frac{{4\kappa_{\text{c}} M_{\text{c}} }}{{i\omega \eta h_{\text{c}} }}\left( {e^{{\sqrt {\frac{i\omega }{{D_{\text{c}} }}} \frac{{h_{\text{c}} }}{2}}} - 1} \right)^{2}.$$

(32)

Thus, we find immediately that

$$\text{Im} [Z_{\text{n}} ] = \text{Im} \left[ {G\frac{2}{{h_{\text{c}} \sqrt {\frac{i\omega }{{D_{\text{c}} }}} }}{\text{e}}^{{h_{\text{c}} \sqrt {\frac{i\omega }{{D_{\text{c}} }}} }} } \right],$$

(33)

meanwhile, when ω → ∞

$$G \approx - \frac{1}{{\left( {\frac{{\kappa_{\text{c}} \sqrt {1/D_{\text{c}} } }}{{\kappa_{\text{m}} \sqrt {1/D_{\text{m}} } }} + 1} \right){\text{e}}^{{\sqrt {i\omega /D_{\text{c}} } r_{\text{c}} H}} }},$$

(34)

therefore,

$$\begin{aligned} \text{Im} [Z_{\text{n}} ] &= \text{Im} \left[ { - \frac{{h_{\text{c}} }}{{M_{\text{c}} \alpha_{\text{c}} }}\frac{1}{\psi }\text{Im} \left[ {\frac{1}{{\left( {\frac{{\kappa_{\text{c}} \sqrt {1/D_{\text{c}} } }}{{\kappa_{\text{m}} \sqrt {1/D_{\text{m}} } }} + 1} \right)e^{{\sqrt {i\omega /D_{\text{c}} } r_{\text{c}} H}} }}\frac{2}{{h_{\text{c}} \sqrt {\frac{i\omega }{{D_{\text{c}} }}} }}{\text{e}}^{{\sqrt {i\omega /D_{\text{c}} } h_{\text{c}} }} } \right]} \right] \hfill \\ &= \frac{2}{\sqrt \omega }\frac{{h_{\text{c}} }}{{M_{\text{c}} \alpha_{\text{c}} }}\frac{1}{\psi }\frac{1}{{h_{\text{c}} \sqrt {\frac{1}{{D_{\text{c}} }}} \left( {\frac{{\kappa_{\text{c}} \sqrt {1/D_{\text{c}} } }}{{\kappa_{\text{m}} \sqrt {1/D_{\text{m}} } }} + 1} \right)}} \hfill \\ \end{aligned}$$

(35)

The imaginary part of the compliance is proportional to ω ^−1/2.

Appendix 2

2.1 Fluid Flux Ratio Normal to Fracture

As for the derivation of Eqs. (1–3), fluid flow between the fractures and the background pores can be seen as an inter-coupling process controlled simultaneously by the hydrologic properties of fractures and of the background. Combining Eqs. (1–3) and Darcy’s law $V_{\text{flow}} = - \nabla P\frac{\kappa }{\eta }$, we can obtain the fluid flux equation

$$S = - \frac{{\kappa_{\text{c}} }}{\eta }A_{2} \left( {{\text{e}}^{{\sqrt {\frac{i\omega }{{D_{\text{c}} }}} h_{\text{c}} /2}} - 1} \right)^{2} + \frac{{\kappa_{\text{m}} }}{\eta }B_{1} \left( {{\text{e}}^{{\sqrt {\frac{i\omega }{{D_{\text{m}} }}} h_{\text{m}} /2}} - 1} \right)^{2} ,$$

(36)

Then the flux ratio between fracture and matrix $\varsigma = \frac{{S_{\text{c}} }}{{S_{\text{m}} }}$ is

$$\varsigma = - \frac{{\kappa_{\text{c}} }}{{\kappa_{\text{m}} }}\frac{{A_{2} }}{{B_{1} }}\frac{{\left( {{\text{e}}^{{R_{\text{c}} h_{\text{c}} /2}} - 1} \right)^{2} }}{{\left( {{\text{e}}^{{R_{\text{m}} h_{\text{m}} /2}} - 1} \right)^{2} }},$$

(37)

In the low-frequency limit, the flux ratio is h _c/h _m ≪ 1 according to Eq  (36). The pore pressure gradient is a constant in the whole pore structure. The flux ratio is $\sqrt {D_{\text{c}} /D_{\text{m}} } \gg 1$ in the high-frequency limit, indicating that the WIFF only occurs near the fractures planes. Therefore, we can conclude that the total flux mainly comes from host rock in the low-frequency limit while from fracture in the high-frequency limit.