Finite Difference Scheme Based on the Lebedev Grid for Seismic Wave Propagation in Fractured Media

Abstract

Fractures in underground media are mostly vertical and orthogonal. Based on the assumption of long wavelengths, fractures can be considered infinitely thin planes embedded in a homogeneous medium. The fracture interface satisfies the conditions of displacement discontinuity and stress continuity, i.e. a linear slip boundary. The finite difference method is typically used to simulate the propagation of seismic waves at the fracture interface. In this study, a new finite difference scheme is proposed based on the velocity-stress equation, which can be used to simulate the propagation of seismic waves in vertical and orthogonal fracture media. The new finite difference scheme more closely resembles the conditions of actual vertical and orthogonal fractures. The Lebedev grid was adopted, and no interpolation was required for the calculation, which improved the accuracy. The numerical simulation of the new finite difference scheme reveals its long-term stability and accuracy. This scheme can be used for the analysis of reservoir fracture delineation. In addition, an explicit slip interface condition and the new finite difference scheme were used to comparatively model fractured media. Virtually identical results were obtained, which verifies the effectiveness of this model. Finally, based on the boundary conditions of linear slip, an improved Zoeppritz equation was established, and the effect of fracture compliance on the reflection and transmission coefficients was studied. This scheme can be used for the analysis of reservoir fracture traps.

Appendices

Appendix A

To verify the effectiveness of the new scheme, we compared the results obtained with the new scheme with those of the explicit application of boundary conditions. Explicit boundary conditions are difficult to achieve in the numerical simulation of orthogonal fractured media. However, it is relatively easy to implement explicit boundary conditions for a single vertical or horizontal fracture. In this appendix, we present the derivation of the explicit boundary conditions.

In two dimensions, the elastic (P-SV) wave equation in an isotropic homogeneous medium can be written as (Aki & Richards, 1980):

$$\frac{{\partial^{2} {\mathbf{u}}}}{{\partial t^{2} }} = {\mathbf{A}}\frac{{\partial^{2} {\mathbf{u}}}}{{\partial x^{2} }} + {\mathbf{B}}\frac{{\partial^{2} {\mathbf{u}}}}{\partial x\partial z} + {\mathbf{C}}\frac{{\partial^{2} {\mathbf{u}}}}{{\partial z^{2} }},$$

(11)

where ${\mathbf{u}} = \left( {\begin{array}{*{20}c} {u_{x} } \\ {u_{z} } \\ \end{array} } \right)$; ${\mathbf{A}} = \left( {\begin{array}{*{20}c} {\alpha^{2} } & 0 \\ 0 & {\beta^{2} } \\ \end{array} } \right)$; ${\mathbf{B}} = \left( {\begin{array}{*{20}c} 0 & {\alpha^{2} - \beta^{2} } \\ {\alpha^{2} - \beta^{2} } & 0 \\ \end{array} } \right)$; ${\mathbf{C}} = \left( {\begin{array}{*{20}c} {\beta^{2} } & 0 \\ 0 & {\alpha^{2} } \\ \end{array} } \right)$; $u_{x}$ and $u_{z}$ represent the normal and tangential displacements, respectively; $\alpha = \sqrt {\frac{\lambda + 2\mu }{\rho }}$ and $\beta = \sqrt {\frac{\mu }{\rho }}$ are the P- and S-wave velocities, respectively; $\lambda$ and $\mu$ are the Lamé coefficients; and $\rho$ is the density of the medium.

According to $\frac{{\partial^{2} {\mathbf{u}}}}{{\partial t^{2} }} = \frac{{\partial {\mathbf{v}}}}{\partial t}$ and ${\mathbf{v}} = \left( {\begin{array}{*{20}c} {v_{x} } \\ {v_{z} } \\ \end{array} } \right)$, where $v_{x}$ and $v_{z}$ represent the normal and tangential velocities, respectively, Eq. (11) can be expanded as follows:

$$\frac{{\partial v_{x} }}{\partial t} = \frac{\lambda + 2\mu }{\rho }\frac{{\partial^{2} u_{x} }}{{\partial x^{2} }} + \frac{\lambda + \mu }{\rho }\frac{{\partial^{2} u_{z} }}{\partial x\partial z} + \frac{\mu }{\rho }\frac{{\partial^{2} u_{x} }}{{\partial z^{2} }},$$

(12)

$$\frac{{\partial v_{z} }}{\partial t} = \frac{\mu }{\rho }\frac{{\partial^{2} u_{z} }}{{\partial x^{2} }} + \frac{\lambda + \mu }{\rho }\frac{{\partial^{2} u_{x} }}{\partial x\partial z} + \frac{\lambda + 2\mu }{\rho }\frac{{\partial^{2} u_{z} }}{{\partial z^{2} }},$$

(13)

We considered the following linear slip boundary conditions of vertical fracture:

$$\Delta u_{x} = S_{N} \sigma_{xx} , \, \Delta u_{z} = S_{T} \sigma_{xz} ,$$

(14)

Equation (14) can be rewritten as:

$$\Delta u_{x} {/}\Delta x = \partial u_{x} /\partial x = \frac{{S_{N} }}{\Delta x}\sigma_{xx} ,\,\Delta u_{z} {/}\Delta x = \partial u_{z} /\partial x = \frac{{S_{T} }}{\Delta x}\sigma_{xz} ,$$

(15)

Based on the application of the Navier and Cauchy equations to an isotropic background medium, we obtained:

$$\left\{ \begin{gathered} \sigma_{zz} = \lambda \frac{{\partial u_{x} }}{\partial x} + (\lambda + 2\mu )\frac{{\partial u_{z} }}{\partial z} \hfill \\ \sigma_{xx} = (\lambda + 2\mu )\frac{{\partial u_{x} }}{\partial x} + \lambda \frac{{\partial u_{z} }}{\partial z} \hfill \\ \sigma_{xz} = \mu \left( {\frac{{\partial u_{x} }}{\partial z} + \frac{{\partial u_{z} }}{\partial x}} \right) \hfill \\ \end{gathered} \right.,$$

(16)

According to Baku’s theory, the fractured medium is composed of isotropic background and fractured media.

$$\partial u_{x} {/}\partial x = \left( {\partial u_{x} {/}\partial x} \right)_{{{\text{iso}}}} + \left( {\partial u_{x} {/}\partial x} \right)_{{{\text{fracture}}}} ,$$

$$\partial u_{z} {/}\partial x = \left( {\partial u_{z} {/}\partial x} \right)_{{{\text{iso}}}} + \left( {\partial u_{z} {/}\partial x} \right)_{{{\text{fracture}}}} ,$$

(17)

The subscript $iso$ represents the isotropic background medium. Based on substituting Eqs. (15) and (16) into (17), we obtained:

$$\left\{ \begin{gathered} \frac{{\partial u_{x} }}{\partial x} = \left( {\frac{\lambda + 2\mu }{{4\mu (\lambda + \mu )}} + \frac{{S_{N} }}{\Delta x}} \right)\sigma_{xx} + \frac{ - \lambda }{{4\mu (\lambda + \mu )}}\sigma_{zz} \hfill \\ \frac{{\partial u_{z} }}{\partial z} = \frac{ - \lambda }{{4\mu (\lambda + \mu )}}\sigma_{xx} + \frac{\lambda + 2\mu }{{4\mu (\lambda + \mu )}}\sigma_{zz} \hfill \\ \frac{{\partial u_{x} }}{\partial z} + \frac{{\partial u_{z} }}{\partial x} = \left( {\frac{1}{\mu } + \frac{{S_{T} }}{\Delta x}} \right)\sigma_{xz} \hfill \\ \end{gathered} \right.,$$

(18)

By substituting Eq. (18) into Eqs. (12) and (13), we obtained the first derivative of the velocity in vertical fractured media with respect to time:

$$\frac{{\partial v_{x} }}{\partial t} = \frac{1}{\rho }\left( {1 + \frac{{\left( {\lambda + 2\mu } \right)S_{N} }}{\Delta x}} \right)\frac{{\partial \sigma_{xx} }}{\partial x} + \frac{1}{\rho }\left( {1 + \frac{{\mu S_{T} }}{\Delta x}} \right)\frac{{\partial \sigma_{xz} }}{\partial z},$$

(19)

$$\frac{{\partial v_{z} }}{\partial t} = \frac{1}{\rho }\left( {1 + \frac{{\lambda S_{N} }}{\Delta x}} \right)\frac{{\partial \sigma_{xx} }}{\partial z} + \frac{1}{\rho }\left( {1 + \frac{{\mu S_{T} }}{\Delta x}} \right)\frac{{\partial \sigma_{xz} }}{\partial x},$$

(20)

Taking the derivative of Eq. (18) with respect to time $t$, the expression of the first derivative of stress in the vertical fracture medium with respect to time was obtained:

$$\left\{ \begin{gathered} \frac{{\partial \sigma_{zz} }}{\partial t} = \lambda \frac{{\partial v_{x} }}{\partial x} + (\lambda + 2\mu )\frac{{\partial v_{z} }}{\partial z} \hfill \\ \frac{{\partial \sigma_{xx} }}{\partial t} = (\lambda + 2\mu )\frac{{\partial v_{x} }}{\partial x} + \lambda \frac{{\partial v_{z} }}{\partial z} \hfill \\ \frac{{\partial \sigma_{xz} }}{\partial t} = \mu \left( {\frac{{\partial v_{x} }}{\partial z} + \frac{{\partial v_{z} }}{\partial x}} \right) \hfill \\ \end{gathered} \right.,$$

(21)

Equations (19), (20), and (21) were combined to obtain the first-order velocity-stress equations for vertical fractured media:

$$\left\{ \begin{gathered} \frac{{\partial v_{x} }}{\partial t} = \frac{1}{\rho }\left( {1 + \frac{{\left( {\lambda + 2\mu } \right)S_{N} }}{\Delta x}} \right)\frac{{\partial \sigma_{xx} }}{\partial x} + \frac{1}{\rho }\left( {1 + \frac{{\mu S_{T} }}{\Delta x}} \right)\frac{{\partial \sigma_{xz} }}{\partial z} \hfill \\ \frac{{\partial v_{z} }}{\partial t} = \frac{1}{\rho }\left( {1 + \frac{{\lambda S_{N} }}{\Delta x}} \right)\frac{{\partial \sigma_{zz} }}{\partial z} + \frac{1}{\rho }\left( {1 + \frac{{\mu S_{T} }}{\Delta x}} \right)\frac{{\partial \sigma_{xz} }}{\partial x} \hfill \\ \frac{{\partial \sigma_{zz} }}{\partial t} = \lambda \frac{{\partial v_{x} }}{\partial x} + (\lambda + 2\mu )\frac{{\partial v_{z} }}{\partial z} \hfill \\ \frac{{\partial \sigma_{xx} }}{\partial t} = (\lambda + 2\mu )\frac{{\partial v_{x} }}{\partial x} + \lambda \frac{{\partial v_{z} }}{\partial z} \hfill \\ \frac{{\partial \sigma_{xz} }}{\partial t} = \mu \left( {\frac{{\partial v_{x} }}{\partial z} + \frac{{\partial v_{z} }}{\partial x}} \right) \hfill \\ \end{gathered} \right.,$$

(22)

Equation (22) is applicable to vertical fractured media. When $S_{N} = S_{T} = 0$, Eq. (22) is the isotropic first-order velocity–stress equation. Equation (14) was changed to $\Delta u_{x} = S_{T} \sigma_{xz}$ and $\Delta u_{z} = S_{N} \sigma_{zz}$; after the derivation, it can be applied to horizontal fractured media.

$$\left\{ \begin{gathered} \frac{{\partial v_{x} }}{\partial t} = \frac{1}{\rho }\left( {1 + \frac{{\left( {\lambda + 2\mu } \right)S_{N} }}{\Delta x}} \right)\frac{\Delta r}{{2\Delta x}}\left( {\frac{{\partial \sigma_{xx} }}{{\partial \overline{z}}} + \frac{{\partial \sigma_{xx} }}{{\partial \overline{x}}}} \right) + \frac{1}{\rho }\left( {1 + \frac{{\mu S_{T} }}{\Delta x}} \right)\frac{\Delta r}{{2\Delta z}}\left( {\frac{{\partial \sigma_{xz} }}{{\partial \overline{z}}} - \frac{{\partial \sigma_{xz} }}{{\partial \overline{x}}}} \right) \hfill \\ \frac{{\partial v_{z} }}{\partial t} = \frac{1}{\rho }\left( {1 + \frac{{\lambda S_{N} }}{\Delta x}} \right)\frac{\Delta r}{{2\Delta z}}\left( {\frac{{\partial \sigma_{zz} }}{{\partial \overline{z}}} - \frac{{\partial \sigma_{zz} }}{{\partial \overline{x}}}} \right) + \frac{1}{\rho }\left( {1 + \frac{{\mu S_{T} }}{\Delta x}} \right)\frac{\Delta r}{{2\Delta x}}\left( {\frac{{\partial \sigma_{xz} }}{{\partial \overline{z}}} + \frac{{\partial \sigma_{xz} }}{{\partial \overline{x}}}} \right) \hfill \\ \frac{{\partial \sigma_{zz} }}{\partial t} = \lambda \frac{\Delta r}{{2\Delta x}}\left( {\frac{{\partial v_{x} }}{{\partial \overline{z}}} + \frac{{\partial v_{x} }}{{\partial \overline{x}}}} \right) + (\lambda + 2\mu )\frac{\Delta r}{{2\Delta z}}\left( {\frac{{\partial v_{z} }}{{\partial \overline{z}}} - \frac{{\partial v_{z} }}{{\partial \overline{x}}}} \right) \hfill \\ \frac{{\partial \sigma_{xx} }}{\partial t} = (\lambda + 2\mu )\frac{\Delta r}{{2\Delta x}}\left( {\frac{{\partial v_{x} }}{{\partial \overline{z}}} + \frac{{\partial v_{x} }}{{\partial \overline{x}}}} \right) + \lambda \frac{\Delta r}{{2\Delta z}}\left( {\frac{{\partial v_{z} }}{{\partial \overline{z}}} - \frac{{\partial v_{z} }}{{\partial \overline{x}}}} \right) \hfill \\ \frac{{\partial \sigma_{x} }}{\partial t} = \mu \left( {\frac{\Delta r}{{2\Delta z}}\left( {\frac{{\partial v_{x} }}{{\partial \overline{z}}} - \frac{{\partial v_{x} }}{{\partial \overline{x}}}} \right) + \frac{\Delta r}{{2\Delta x}}\left( {\frac{{\partial v_{z} }}{{\partial \overline{z}}} + \frac{{\partial v_{z} }}{{\partial \overline{x}}}} \right)} \right) \hfill \\ \end{gathered} \right..$$

(23)

Appendix B

Six types of waves are generated in fractured media when P-waves from the background media are incident on the vertical fracture interface at an angle $\alpha$ including incident P-waves, reflected P-waves, reflected S-waves, transmitted P-waves, and transmitted S-waves (Fig. 

Fig. 10

Reflection and transmission on fracture interface

10).

Their plane wave expression potentials are as follows:

$$\begin{gathered} \varphi_{{p_{1} }} = A_{p1} \exp i\left[ {\omega t - l_{1} \left( {x\cos \alpha + z\sin \alpha } \right)} \right] \, \hfill \\ \varphi_{{p_{11} }} = A_{{p_{11} }} \exp i\left[ {\omega t - l_{11} \left( { - x\cos \alpha_{11} + z\sin \alpha_{11} } \right)} \right] \, \hfill \\ \psi_{{s_{1} }} = A_{{s_{1} }} \exp i\left[ {\omega t - l_{s1} \left( { - x\cos \beta_{1} + z\sin \beta_{1} } \right)} \right] \, \hfill \\ \varphi_{{p_{21} }} = A_{{p_{21} }} \exp i\left[ {\omega t - l_{21} \left( {x\cos \alpha_{21} + z\sin \alpha_{21} } \right)} \right] \, \hfill \\ \psi_{{s_{2} }} = A_{{s_{2} }} \exp i\left[ {\omega t - l_{s2} \left( {x\cos \beta_{2} + z\sin \beta_{2} } \right)} \right]{, } \hfill \\ \end{gathered}$$

(24)

where $\varphi$ represents the P-wave potentials and $\psi$ represents the S-wave potentials. $A$ represents the amplitude of the displacement caused by the incident wave, $\omega$ represents the circular frequency, $l$ represents the circular wave number, $x$ and $z$ are the spatial coordinates, and $\alpha$ and $\beta$ are the incident angles.

The boundary conditions at the interface are displacement discontinuity and stress continuity. Therefore, the boundary conditions of the vertically fractured media can be written as:

$$\left\{ \begin{gathered} \left[ {u_{x} } \right]^{ + } = \left[ {u_{x} } \right]^{ - } + S_{N} \sigma_{xx} \hfill \\ \left[ {u_{z} } \right]^{ + } = \left[ {u_{z} } \right]^{ - } + S_{T} \sigma_{xz} \hfill \\ \left[ {\sigma_{xx} } \right]^{ + } = \left[ {\sigma_{xx} } \right]^{ - } \hfill \\ \left[ {\sigma_{xz} } \right]^{ + } = \left[ {\sigma_{xz} } \right]^{ - } \hfill \\ \end{gathered} \right.,$$

(25)

where $u_{x}$ and $u_{z}$ represent the normal and tangential displacement, respectively.

In the two-dimensional plane, the displacement of the particle vibration during the propagation of the seismic wave can be expressed as:

$$\left\{ {\begin{array}{*{20}c} {u_{x} = \frac{\partial \varphi }{{\partial x}} - \frac{\partial \psi }{{\partial z}}} \\ {u_{z} = \frac{\partial \varphi }{{\partial z}} + \frac{\partial \psi }{{\partial x}}} \\ \end{array} } \right.,$$

(26)

In the two-dimensional case, we obtained the following equations according to the correlation between stress and strain described by Hooke’s law:

$$\sigma_{xx} = \lambda \frac{1}{{v_{p}^{2} }}\frac{{\partial^{2} \varphi }}{{\partial t^{2} }} + 2\mu \left( {\frac{{\partial^{2} \varphi }}{{\partial x^{2} }} - \frac{{\partial^{2} \psi }}{\partial x\partial z}} \right),$$

(27)

$$\sigma_{zx} = \mu e_{zx} = \mu \left( {\frac{{\partial u_{z} }}{\partial x} + \frac{{\partial u_{x} }}{\partial z}} \right) = \mu \left( {2\frac{{\partial^{2} \varphi }}{\partial x\partial z} + \frac{{\partial^{2} \psi }}{{\partial x^{2} }} - \frac{{\partial^{2} \psi }}{{\partial z^{2} }}} \right),$$

(28)

Subsequently, we analysed the four boundary conditions separately. Based on the tangential displacement discontinuity $\left[ {u_{x} } \right]^{ + } = \left[ {u_{x} } \right]^{ - } + S_{N} \sigma_{xx}$ and Eqs. (24)–(28), we obtained:

$$\begin{gathered} \frac{{A_{{p_{11} }} }}{{A_{p1} }}\frac{{v_{p1} }}{{v_{{p_{11} }} }}\cos \alpha_{11} + \frac{{A_{{s_{1} }} }}{{A_{p1} }}\frac{{v_{p1} }}{{v_{{s_{1} }} }}\sin \beta_{1} + \frac{{A_{{p_{21} }} }}{{A_{p1} }}\frac{{v_{p1} }}{{v_{{p_{21} }} }}\left( \begin{gathered} \cos \alpha_{21} - iS_{N} \lambda l_{21} \hfill \\ - iS_{N} 2\mu l_{21} \cos^{2} \alpha_{21} \hfill \\ \end{gathered} \right) \hfill \\ - \frac{{A_{{s_{2} }} }}{{A_{p1} }}\frac{{v_{p1} }}{{v_{{s_{2} }} }}\left( {\sin \beta_{2} + iS_{N} \mu l_{s2} \sin 2\beta_{2} \, } \right) = \cos \alpha , \hfill \\ \end{gathered}$$

(29)

Based on the discontinuity of the normal displacement $\left[ {u_{z} } \right]^{ + } = \left[ {u_{z} } \right]^{ - } + S_{T} \sigma_{xz}$, we obtained:

$$\begin{gathered} \frac{{A_{{p_{11} }} }}{{A_{p1} }}\frac{{v_{p1} }}{{v_{{p_{11} }} }}\sin \alpha_{11} - \frac{{A_{{s_{1} }} }}{{A_{p1} }}\frac{{v_{p1} }}{{v_{{s_{1} }} }}\cos \beta_{1} - \frac{{A_{{p_{21} }} }}{{A_{p1} }}\frac{{v_{p1} }}{{v_{{p_{21} }} }}\left( {\sin \alpha_{21} - iS_{T} \mu l_{21} \sin 2\alpha_{21} } \right) \hfill \\ - \frac{{A_{{s_{2} }} }}{{A_{p1} }}\frac{{v_{p1} }}{{v_{{s_{2} }} }}\left( {\cos \beta_{2} - iS_{T} \mu l_{s2} \cos 2\beta_{2} } \right) = - \sin \alpha , \hfill \\ \end{gathered}$$

(30)

Based on the normal stress continuity $\left[ {\sigma_{xx} } \right]^{ + } = \left[ {\sigma_{xx} } \right]^{ - }$, we obtained:

$$\begin{gathered} \frac{{A_{{p_{11} }} }}{{A_{p1} }}\frac{{v_{p1}^{2} }}{{v_{p11}^{2} }}\left( {2\mu \cos^{2} \alpha_{11} + v_{p11}^{2} } \right){ + }\frac{{A_{{s_{1} }} }}{{A_{p1} }}\frac{{v_{p1}^{2} }}{{v_{s1}^{2} }}\mu \sin 2\beta_{1} - \frac{{A_{{p_{21} }} }}{{A_{p1} }}\frac{{v_{p1}^{2} }}{{v_{p21}^{2} }}\left( {2\mu \cos^{2} \alpha_{21} + \lambda } \right) \hfill \\ + \frac{{A_{{s_{2} }} }}{{A_{p1} }}\frac{{v_{p1}^{2} }}{{v_{s2}^{2} }}\mu \sin 2\beta_{2} = - \left( {\lambda + \, 2\mu \cos^{2} \alpha } \right), \hfill \\ \end{gathered}$$

(31)

According to the tangential stress continuity $\left[ {\sigma_{xz} } \right]^{ + } = \left[ {\sigma_{xz} } \right]^{ - }$, we obtained:

$$\begin{gathered} \frac{{A_{{p_{11} }} }}{{A_{p1} }}\frac{{v_{p1}^{2} }}{{v_{p11}^{2} }}\sin 2\alpha_{11} - \frac{{A_{{s_{1} }} }}{{A_{p1} }}\frac{{v_{p1}^{2} }}{{v_{s1}^{2} }}\cos 2\beta_{1} + \frac{{\rho_{2} }}{{\rho_{1} }}\frac{{v_{s2}^{2} }}{{v_{s1}^{2} }}\frac{{A_{{p_{21} }} }}{{A_{p1} }}\frac{{v_{p1}^{2} }}{{v_{p21}^{2} }}\sin 2\alpha_{21} \hfill \\ + \frac{{\rho_{2} }}{{\rho_{1} }}\frac{{v_{s2}^{2} }}{{v_{s1}^{2} }}\frac{{A_{{s_{2} }} }}{{A_{p1} }}\frac{{v_{p1}^{2} }}{{v_{s2}^{2} }}\cos 2\beta_{2} = \sin 2\alpha , \hfill \\ \end{gathered}$$

(32)

The reflection coefficient of the P-wave, reflection coefficient of the S-wave, transmission coefficient of the P-wave, and transmission coefficient of the S-wave are defined as:

$$R_{{p_{11} }} = \frac{{A_{{p_{11} }} }}{{A_{{p_{1} }} }}\frac{{v_{{p_{1} }} }}{{v_{{p_{11} }} }},\,R_{{s_{1} }} = \frac{{A_{{s_{1} }} }}{{A_{{p_{1} }} }}\frac{{v_{{p_{1} }} }}{{v_{{s_{1} }} }}{,}\, \, T_{{p_{21} }} = \frac{{A_{{p_{21} }} }}{{A_{{p_{1} }} }}\frac{{v_{{p_{1} }} }}{{v_{{p_{21} }} }}{,}\, \, T_{{s_{2} }} = \frac{{A_{{s_{2} }} }}{{A_{{p_{1} }} }}\frac{{v_{{p_{1} }} }}{{v_{{s_{2} }} }},$$

(33)

In addition, $v_{{p_{1} }} = v_{{p_{11} }} = v_{{p_{21} }}$, $v_{{s_{1} }} = v_{{s_{2} }}$, and $\rho_{1} = \rho_{2}$. By synthesising Eqs. (30)–(33), we obtained Eq. (10).