Closed-Form Expressions of Plane-Wave Reflection and Transmission Coefficients at a Planar Interface of Porous Media with a Normal Incident Fast P-Wave

Abstract

A porous medium is composed of a rock skeleton and pore fluids, and seismic wave propagation in it will produce complex and diverse variations influenced by pores and fluids filling. It is very important to carry out the study of closed-form expressions of the plane-wave reflection and transmission coefficients at a planar interface between porous media for analyzing the properties of pores and its fluids and eventually revealing underground oil-bearing reservoirs. In this paper, based on the relationships among seismic wave functions, displacements and stresses in porous media, an exact equation of plane-wave reflection and transmission coefficients with a normal incident fast P-wave is first derived. Considering the characteristics of the parameters in a coefficient matrix of an exact equation, the closed-form expressions with clear geophysical meaning are further derived, which include three parts, the rock skeleton term, the fluid–solid coupling term and the pore fluid term. Through the establishment of two porous media models, the influence of each term in the approximate expression on the reflection characteristics of a fast P-wave is analyzed. Different approximate expressions of the reflection coefficient of a fast P-wave can be selected for oil and gas prediction of different reservoirs, which lays a foundation for the identification of gas, oil and brine.

Appendix

The two-dimensional system of linear algebraic equations (2D SLAE) relating to the reflection coefficients of $r_{1}$ and $r_{2}$ are obtained in Eq. 32. Solving this equation, the reflection coefficients can be derived as

$$ \begin{aligned} N & = \left( {e_{21} d_{12} - e_{12} d_{21} } \right) + \left( {e_{11} d_{12} - e_{12} d_{11} } \right) \\ & \quad + \frac{{\gamma_{12} }}{{\gamma_{22} }}\left[ {\left( {e_{21} d_{22} - e_{22} d_{21} } \right) + \left( {e_{11} d_{22} - e_{22} d_{11} } \right)} \right], \\ \end{aligned} $$

(44)

$$ r_{1} = \frac{1}{N}\left\{ {\begin{array}{*{20}l} {\left( {e_{21} d_{12} - e_{12} d_{21} } \right) - \left( {e_{11} d_{12} - e_{12} d_{11} } \right) + } \hfill \\ {\quad \frac{{\gamma_{12} }}{{\gamma_{22} }}\left[ {\left( {e_{21} d_{22} - e_{22} d_{21} } \right) - \left( {e_{11} d_{22} - e_{22} d_{11} } \right)} \right]} \hfill \\ \end{array} } \right\}, $$

(45)

$$ r_{2} = \frac{2}{N}\left\{ {e_{11} d_{21} - e_{21} d_{11} } \right\}. $$

(46)

Taking Eqs. 13, 14 and 33 into 32, combining the definitions of P-wave impedances in Eqs. 34a and 34b, and considering Eq. 27, the following approximate equations can be obtained:

$$ e_{21} d_{12} - e_{12} d_{21} \approx \gamma_{12} Ip_{{s_{21} }} Ip_{{f_{12} }} + \left( {\alpha_{1} Ip_{{s_{21} }} Ip_{{f_{12} }} - \alpha_{2} Ip_{{s_{12} }} Ip_{{f_{21} }} } \right) + \alpha_{2} \left( {\alpha_{2} - \alpha_{1} } \right)\left( {\alpha_{1} { + }\gamma_{12} } \right)Ip_{{f_{12} }} Ip_{{f_{21} }} , $$

(47)

$$ e_{11} d_{12} - e_{12} d_{11} \approx \gamma_{12} Ip_{{s_{11} }} Ip_{{f_{12} }} , $$

(48)

$$ e_{21} d_{22} - e_{22} d_{21} \approx \gamma_{22} Ip_{{s_{21} }} Ip_{{f_{22} }} , $$

(49)

$$ e_{11} d_{22} - e_{22} d_{11} \approx \gamma_{22} Ip_{{s_{11} }} Ip_{{f_{22} }} \quad + \left( {\alpha_{2} Ip_{{s_{11} }} Ip_{{f_{22} }} - \alpha_{1} Ip_{{s_{22} }} Ip_{{f_{11} }} } \right) + \alpha_{1} \left( {\alpha_{1} - \alpha_{2} } \right)\left( {\alpha_{2} { + }\gamma_{22} } \right)Ip_{{f_{11} }} Ip_{{f_{22} }} , $$

(50)

$$ e_{11} d_{21} - e_{21} d_{11} \approx \quad - \left\{ {\left[ {\alpha_{1} Ip_{{f_{11} }} Ip_{{s_{21} }} - \alpha_{2} Ip_{{f_{21} }} Ip_{{s_{11} }} } \right] + \alpha_{1} \alpha_{2} \left( {\alpha_{2} - \alpha_{1} } \right)Ip_{{f_{11} }} Ip_{{f_{21} }} } \right\}. $$

(51)

Bringing Eqs. 47–51 into Eqs. 44–46, the closed-form expressions with explicitly geophysical meaning can finally be derived as

$$ r_{1} \approx \frac{{\left( {Ip_{{s_{21} }} - Ip_{{s_{11} }} } \right) + \left( {X_{2} - X_{1} } \right){ + }\left( {Y_{2} - Y_{1} } \right)}}{{\left( {Ip_{{s_{21} }} + Ip_{{s_{11} }} } \right) + \left( {X_{2} { + }X_{1} } \right) + \left( {Y_{2} + Y_{1} } \right)}}, $$

(52)

$$ r_{2} \approx - \frac{1}{{\gamma_{12} }} \cdot \frac{{\left( {Z_{2} - Z_{1} } \right) + Z_{3} }}{{\left( {Ip_{{s_{21} }} + Ip_{{s_{11} }} } \right) + \left( {X_{2} { + }X_{1} } \right) + \left( {Y_{2} + Y_{1} } \right)}}. $$

(53)

The parameters of $X_{k} ,Y_{k} ,k = 1,2$ and $Z_{k} ,k = 1,2,3$ are defined as

$$ X_{1} = \frac{{\alpha_{2} \cdot Ip_{{f_{22} }} }}{{\gamma_{22} \cdot \left( {Ip_{{f_{12} }} + Ip_{{f_{22} }} } \right)}} \cdot Ip_{{s_{11} }} - \frac{{\alpha_{1} \cdot Ip_{{f_{11} }} }}{{\gamma_{22} \cdot \left( {Ip_{{f_{12} }} + Ip_{{f_{22} }} } \right)}} \cdot Ip_{{s_{22} }} , $$

(54)

$$ X_{2} = \frac{{\alpha_{1} \cdot Ip_{{f_{12} }} }}{{\gamma_{12} \cdot \left( {Ip_{{f_{12} }} + Ip_{{f_{22} }} } \right)}} \cdot Ip_{{s_{21} }} - \frac{{\alpha_{2} \cdot Ip_{{f_{21} }} }}{{\gamma_{12} \cdot \left( {Ip_{{f_{12} }} + Ip_{{f_{22} }} } \right)}} \cdot Ip_{{s_{12} }} . $$

(55)

$$ Y_{1} = \alpha_{1} \cdot \left( {\alpha_{1} - \alpha_{2} } \right) \cdot \left( {1 + \frac{{\alpha_{2} }}{{\gamma_{22} }}} \right) \cdot \frac{{Ip_{{f_{22} }} }}{{\left( {Ip_{{f_{12} }} + Ip_{{f_{22} }} } \right)}} \cdot Ip_{{f_{11} }} , $$

(56)

$$ Y_{2} = \alpha_{2} \cdot \left( {\alpha_{2} - \alpha_{1} } \right) \cdot \left( {1 + \frac{{\alpha_{1} }}{{\gamma_{12} }}} \right) \cdot \frac{{Ip_{{f_{12} }} }}{{\left( {Ip_{{f_{12} }} + Ip_{{f_{22} }} } \right)}} \cdot Ip_{{f_{21} }} . $$

(57)

$$ Z_{1} = \frac{{\alpha_{2} \cdot Ip_{{f_{21} }} }}{{Ip_{{f_{12} }} + Ip_{{f_{22} }} }} \cdot Ip_{{s_{11} }} , $$

(58)

$$ Z_{2} = \frac{{\alpha_{1} \cdot Ip_{{f_{11} }} }}{{Ip_{{f_{12} }} + Ip_{{f_{22} }} }} \cdot Ip_{{s_{21} }} , $$

(59)

$$ Z_{3} = \alpha_{1} \cdot \alpha_{2} \cdot \left( {\alpha_{2} - \alpha_{1} } \right) \cdot \frac{{Ip_{{f_{11} }} \cdot Ip_{{f_{21} }} }}{{Ip_{{f_{12} }} + Ip_{{f_{22} }} }}. $$

(60)