On the relative importance of global and squirt flow in cracked porous media

Abstract

A unified theory of global and squirt flow in cracked porous media was developed several years ago on the basis of a combination of the dynamic T-matrix approach to rock physics. The theory has been successfully used to model ultrasonic velocity and attenuation anisotropy measurements in real rocks under pressure. At the same time, it was recently pointed out that this theory, which contain an established theory of interconnected cracks as a special case contains an error related to fluid mass conservation. The error was recently corrected, and this paper represents an attempt to perform a systematic study of the implications of unified theory for the relative importance of global and squirt flow in cracked porous media characterized by different microstructures and fluid mobilities. Our numerical results suggest that squirt flow dominates over global flow and global flow appears to be more important at higher frequencies for more realistic models of microstructure. The attenuation peak of squirt flow move towards lower frequencies with the increasing fluid viscosity i.e. changing saturating fluid from water to oil, while the global flow attenuation peak move towards higher frequencies with increasing fluid viscosity. A previous observation of negative velocity dispersion in unified theory still remain, even if we use the correct effective wave number, when dealing with the phenomenon of wave-induced fluid flow in models of cracked porous media where global flow effects dominates. The attenuation peak of the global flow obtained using the correct wave number is always shifted to the left as compared to the approximate solution. At seismic frequencies global flow effects are not so important and needs very high permeability and low viscosity to have an effect.

Appendices

Appendix 1: Christoffel equation for viscoelastic media

From Hooke’s law the stress-strain relation can be written as using the matrix notation (Carcione 2007)

$$\begin{aligned} \sigma =\mathbf{C}\cdot \mathbf{e} \end{aligned}$$

(31)

where $\sigma$ is the stress tensor, e is the strain tensor and C is the stiffness tensor. The equation of motion in the absence of body forces can be written as (Carcione 2007)

$$\begin{aligned} \Gamma _\nabla \cdot \mathbf{u}=\rho \partial _{tt}^2 \mathbf{u}, \end{aligned}$$

(32)

where

$$\begin{aligned} \Gamma _\nabla =\nabla \cdot \mathbf{C}\cdot \nabla ^{T}, \end{aligned}$$

(33)

where u is the displacement vector and the symmetric gradient operator $\nabla$ using the matrix representation is given by Auld (1990) and Carcione (2007)

$$\begin{aligned} \nabla =\left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {\partial _1}&{} 0&{} 0&{} 0&{} {\partial _3}&{} {\partial _2} \\ 0&{} {\partial _2}&{} 0&{} {\partial _3}&{} 0&{} {\partial _1} \\ 0&{} 0&{} {\partial _3}&{} {\partial _2}&{} {\partial _1} &{} 0 \\ \end{array}}\right] . \end{aligned}$$

(34)

The strain displacement relation is given by Carcione (2007)

$$\begin{aligned} \mathbf{e}=\nabla ^{T}\cdot \mathbf{u}. \end{aligned}$$

(35)

A general plane wave solution for the displacement vector of the body waves is given by Carcione (2007)

$$\begin{aligned} \mathbf{u}=\mathbf{u}_0 \exp \left[ {i(\omega t-\mathbf{k}\cdot \mathbf{x})}\right] , \end{aligned}$$

(36)

where $\mathbf{u}_0$ represents a constant complex vector, $\omega$ is the angular frequency and k is the wave-number vector. The particle velocity is given by time derivative of Eq. (36) given by

$$\begin{aligned} \mathbf{v}=\partial _t \mathbf{u}=i\omega \mathbf{u}. \end{aligned}$$

(37)

In the absence of body forces $(\mathbf{f}=0)$, we consider plane waves propagating along the direction given by Carcione (2007)

$$\begin{aligned} \hat{\mathbf{k}}=l_1 \hat{\mathbf{e}}_1 +l_2 \hat{\mathbf{e}}_2 +l_3 \hat{\mathbf{e}}_3, \end{aligned}$$

(38)

where $l_{1}, l_{2}$ and $l_{3}$ are the direction cosines. The wave-number vector k can be written as (Carcione 2007)

$$\begin{aligned} \mathbf{k}=(k_1,k_2,k_3)=k(l_1, l_2, l_3)=k\hat{\mathbf{k}}, \end{aligned}$$

(39)

where $k$ is the magnitude of the wave-number vector. The spatial differential operator in Eq. (34) can be replaced by

$$\begin{aligned} \nabla \rightarrow -ik \left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {l_1}&{} 0&{} 0&{} 0&{} {l_3} &{} {l_2} \\ 0&{} {l_2} &{} 0&{} {l_3} &{} 0&{} {l_1} \\ 0&{} 0&{} {l_3}&{} {l_2}&{} {l_1} &{} 0 \\ \end{array}}\right] \equiv -ik\mathbf{L}. \end{aligned}$$

(40)

Now substituting the time derivative as $\partial _{t}\rightarrow i\omega$ and using Eq. (40) in Eq. (32), we get (Carcione 2007)

$$\begin{aligned} k^{2}\Gamma \cdot \mathbf{u}=\rho \omega ^{2}\mathbf{u}, \end{aligned}$$

(41)

where

$$\begin{aligned} \Gamma =\mathbf{L}\cdot \mathbf{C}\cdot \mathbf{L}^{T}, \end{aligned}$$

(42)

is the Christoffel matrix. The dispersion relation is given by (Carcione 2007)

$$\begin{aligned} \det (\Gamma -\rho V^{2}\mathbf{I}_3)=0. \end{aligned}$$

(43)

where

$$\begin{aligned} V=\frac{\omega }{k}, \end{aligned}$$

(44)

is the complex velocity.

Equation (43) is known as the Christoffel equation. Using Eq. (44), the components of the slowness and attenuation vectors can be expressed in terms of the complex velocity as (given by Carcione 1995)

$$\begin{aligned} \mathbf{s}=\hbox {Re}\left[ {\frac{1}{V}}\right] \hat{\mathbf{k}}, \end{aligned}$$

(45)

and

$$\begin{aligned} \alpha =-\omega \;\hbox {Im}\left[ {\frac{1}{V}}\right] \hat{\mathbf{k}}. \end{aligned}$$

(46)

Appendix 2: The effective permeability tensor

The effective permeability tensor $\mathbf{K}^{*}$of the fractured porous reservoir model, assuming that the distribution of fractures is same for all fracture families is given by Jakobsen (2007)

$$\begin{aligned} \mathbf{K}^{*}=\mathbf{K}^{(0)}+\mathbf{K}_1 \cdot (\mathbf{I}_2 +\mathbf{g}_d \cdot \mathbf{K}_1)^{-1}, \end{aligned}$$

(47)

where

$$\begin{aligned} \mathbf{K}_1 =\sum _r {v^{(r)}{\varvec{\tau }}^{(r)}}. \end{aligned}$$

(48)

Here, $\mathbf{K}^{(0)}$ is background or matrix permeability, $\mathbf{I}_2$ is the (Kronecker-delta) identity for second rank tensors, $v^{(r)}$ is the volume concentration for fractures of type $r$ and $\mathbf{g}_d$ is a tensor given by the strain Green’s function integrated over an ellipsoid determining the symmetry of the correlation function for the spatial distribution of fractures (see Jakobsen 2007; Shahraini et al. 2010). The $\tau ^{(r)}$ for a single fracture of type $r$ is given by Jakobsen (2007)

$$\begin{aligned} {\varvec{\tau }}^{(r)}=(\mathbf{K}^{(r)}-\mathbf{K}^{(0)})\cdot \left[ {\mathbf{I}_2 -\mathbf{g}^{(r)}\cdot (\mathbf{K}^{(r)}-\mathbf{K}^{(0)})}\right] ^{-1}. \end{aligned}$$

(49)

Here, $\mathbf{g}^{(r)}$ is a second-rank tensor given by the pressure gradient Green’s function integrated over a characteristic spheroid having the same shape as inclusions of type $r$ (see Jakobsen 2007; Shahraini et al. 2010), and $\mathbf{K}^{(r)}$ is a second-rank tensor of permeability coefficients for fractures of type $r$, which can be estimated using the cubic law given by Golf-Racht (1982)

$$\begin{aligned} \mathbf{K}^{(r)}=\frac{(a^{(r)})^{2}}{12}\mathbf{I}_2. \end{aligned}$$

(50)

Here $a$ represents the aperture of the fractures of type $r$.