Reflection of Plane Waves at an Interface of Water/Patchy Saturated Porous Media with Underlying Solid Substrate

Abstract

The problems with elastic wave reflection at the interface between the water/patchy saturated porous media and the underlying solid matrix are considered. When porous media is saturated by biphasic fluids, the patchy saturation theory can explain the wave dispersion and attenuation better than the classical Biot theory at mesoscopic scales. Based on patchy saturation theory, this paper focuses on the reflection of elastic waves by patchy saturated porous media inclusion in layered media. Using numerical calculation, the reflection coefficients with different excitation frequencies, water saturations, and porous media depths are discussed. The results show that the differences between two kinds of patchy saturated porous media cases in the oil-water model are lower than those between the same two cases in the gas-water model. Low water saturation and high water saturation have significant effects on the reflection coefficient.

APPENDIX A

The elements of the matrix [M] in Eq. (34) are given by:

$$\begin{gathered} {{M}_{{11}}} = {{\sigma }_{i}}{\kern 1pt} {\kern 1pt} ,\,\,\,\,{{M}_{{12}}} = (1 + {{\alpha }_{{p1}}})\sigma _{{p1}}^{{}}, \\ {{M}_{{13}}} = (1 + {{\alpha }_{{p2}}})\sigma _{{p2}}^{{}},\,\,\,\,{{M}_{{14}}} = (1 + {{\alpha }_{s}})l_{s}^{{}}, \\ {{M}_{{15}}} = - (1 + {{\alpha }_{{p1}}})\sigma _{{p1}}^{{}}, \\ \end{gathered} $$

$$\begin{gathered} {{M}_{{16}}} = - (1 + {{\alpha }_{{p2}}})\sigma _{{p2}}^{{}}, \\ {{M}_{{17}}} = (1 + {{\alpha }_{s}})l_{s}^{{}},\,\,\,\,{{M}_{{18}}} = {{M}_{{19}}} = 0; \\ \end{gathered} $$

$$\begin{gathered} {{M}_{{21}}} = - {{\rho }_{w}}{{\omega }^{2}}, \\ {{M}_{{22}}} = {{M}_{{25}}} = (C(\omega ) + M(\omega ){{\alpha }_{{p1}}})k_{{p1}}^{2}, \\ {{M}_{{23}}} = {{M}_{{26}}} = (C(\omega ) + M(\omega ){{\alpha }_{{p2}}})k_{{p2}}^{2}, \\ \end{gathered} $$

$${{M}_{{24}}} = {{M}_{{27}}} = {{M}_{{28}}} = {{M}_{{29}}} = 0;$$

$$\begin{gathered} {{M}_{{31}}} = {{\rho }_{w}}{{\omega }^{2}},\,\,\,\,{{M}_{{32}}} = {{M}_{{35}}} = 2G(\omega )l_{{p1}}^{2} \\ - \,\,(H(\omega ) + C(\omega ){{\alpha }_{{p1}}})k_{{p1}}^{2}, \\ \end{gathered} $$

$${{M}_{{33}}} = {{M}_{{36}}} = 2G(\omega )l_{{p2}}^{2} - (H(\omega ) + C(\omega ){{\alpha }_{{p2}}})k_{{p2}}^{2},$$

$$\begin{gathered} {{M}_{{34}}} = - 2G(\omega )l_{s}^{{}}\sigma _{s}^{{}}, \\ {{M}_{{37}}} = 2G(\omega )l_{s}^{{}}\sigma _{s}^{{}},\,\,\,\,{{M}_{{38}}} = {{M}_{{39}}} = 0; \\ \end{gathered} $$

$$\begin{gathered} {{M}_{{41}}} = 0,\,\,\,\,{{M}_{{42}}} = - 2l_{{p1}}^{{}}\sigma _{{p1}}^{{}},\,\,\,\,{{M}_{{43}}} = - 2l_{{p2}}^{{}}\sigma _{{p2}}^{{}}, \\ {{M}_{{44}}} = {{M}_{{47}}} = k_{s}^{2} - 2l_{s}^{2},\,\,\,\,{{M}_{{45}}} = 2l_{{p1}}^{{}}\sigma _{{p1}}^{{}}, \\ \end{gathered} $$

$${{M}_{{46}}} = 2l_{{p2}}^{{}}\sigma _{{p2}}^{{}},{{M}_{{48}}} = {{M}_{{49}}} = 0;$$

$$\begin{gathered} {{M}_{{51}}} = 0,\,\,\,\,{{M}_{{52}}} = \sigma _{{p1}}^{{}}{{e}^{{i\sigma _{{p1}}^{{}}h}}},\,\,\,\,{{M}_{{53}}} = \sigma _{{ps}}^{{}}{{e}^{{i\sigma _{{p2}}^{{}}h}}}, \\ {{M}_{{54}}} = l_{s}^{{}}{{e}^{{i\sigma _{s}^{{}}h}}},\,\,\,\,{{M}_{{55}}} = - \sigma _{{p1}}^{{}}{{e}^{{ - i\sigma _{{p1}}^{{}}h}}}, \\ \end{gathered} $$

$$\begin{gathered} {{M}_{{56}}} = - \sigma _{{p2}}^{{}}{{e}^{{ - i\sigma _{{p2}}^{{}}h}}},\,\,\,\,{{M}_{{57}}} = l_{s}^{{}}{{e}^{{ - i\sigma _{s}^{{}}h}}}, \\ {{M}_{{58}}} = - \sigma _{{tp}}^{{}}{{e}^{{i\sigma _{{tp}}^{{}}h}}},\,\,\,\,{{M}_{{59}}} = - l_{{ts}}^{{}}{{e}^{{i\sigma _{{ts}}^{{}}h}}}; \\ \end{gathered} $$

$$\begin{gathered} {{M}_{{61}}} = 0,\,\,\,\,{{M}_{{62}}} = {{\alpha }_{{p1}}}\sigma _{{p1}}^{{}}{{e}^{{i\sigma _{{p1}}^{{}}h}}}, \\ {{M}_{{63}}} = {{\alpha }_{{p2}}}\sigma _{{p2}}^{{}}{{e}^{{i\sigma _{{p2}}^{{}}h}}},\,\,\,\,{{M}_{{64}}} = {{\alpha }_{s}}l_{s}^{{}}{{e}^{{i\sigma _{s}^{{}}h}}}, \\ {{M}_{{65}}} = - {{\alpha }_{{p1}}}\sigma _{{p1}}^{{}}{{e}^{{ - i\sigma _{{p1}}^{{}}h}}}, \\ \end{gathered} $$

$$\begin{gathered} {{M}_{{66}}} = - {{\alpha }_{{p2}}}l_{{p2}}^{{}}{{e}^{{ - i\sigma _{{p2}}^{{}}h}}},\,\,\,\,{{M}_{{67}}} = {{\alpha }_{s}}l_{s}^{{}}{{e}^{{ - i\sigma _{s}^{{}}h}}}, \\ {{M}_{{68}}} = 0,{{M}_{{69}}} = 0; \\ \end{gathered} $$

$$\begin{gathered} {{M}_{{71}}} = 0,\,\,\,\,{{M}_{{72}}} = l_{{p1}}^{{}}{{e}^{{i\sigma _{{p1}}^{{}}h}}},\,\,\,\,{{M}_{{73}}} = l_{{p2}}^{{}}{{e}^{{i\sigma _{{p2}}^{{}}h}}}, \\ {{M}_{{74}}} = - \sigma _{s}^{{}}{{e}^{{i\sigma _{s}^{{}}h}}},\,\,\,\,{{M}_{{75}}} = {{l}_{{p1}}}{{e}^{{ - i\sigma _{{p1}}^{{}}h}}}, \\ \end{gathered} $$

$$\begin{gathered} {{M}_{{76}}} = l_{{p2}}^{{}}{{e}^{{ - i\sigma _{{p2}}^{{}}h}}},\,\,\,\,{{M}_{{77}}} = \sigma _{s}^{{}}{{e}^{{ - i\sigma _{s}^{{}}h}}}, \\ {{M}_{{78}}} = - l_{{tp}}^{{}}{{e}^{{i\sigma _{{tp}}^{{}}h}}},\,\,\,\,{{M}_{{79}}} = \sigma _{{ts}}^{{}}{{e}^{{i\sigma _{{ts}}^{{}}h}}}; \\ \end{gathered} $$

$$\begin{gathered} {{M}_{{81}}} = 0,\,\,\,\,{{M}_{{82}}} = (2G(\omega )l_{{p1}}^{2} \\ - \,\,(H(\omega ) + C(\omega ){{\alpha }_{{p1}}})k_{{p1}}^{2}){{e}^{{i\sigma _{{p1}}^{{}}h}}}, \\ \end{gathered} $$

$${{M}_{{83}}} = (2G(\omega )l_{{p2}}^{2} - (H(\omega ) + C(\omega ){{\alpha }_{{p2}}})l_{{p2}}^{2}){{e}^{{i\sigma _{{p2}}^{{}}h}}},$$

$$\begin{gathered} {{M}_{{84}}} = - 2G(\omega )l_{s}^{{}}\sigma _{s}^{{}}{{e}^{{i\sigma _{s}^{{}}h}}}, \\ {{M}_{{85}}} = (2G(\omega )l_{{p1}}^{2} - (H(\omega ) + C(\omega ){{\alpha }_{{p1}}})l_{{p1}}^{2}){{e}^{{ - i\sigma _{{p1}}^{{}}h}}}, \\ \end{gathered} $$

$${{M}_{{86}}} = (2G(\omega )l_{{p2}}^{2} - (H(\omega ) + C(\omega ){{\alpha }_{{p2}}})k_{{p2}}^{2}){{e}^{{ - i\sigma _{{p2}}^{{}}h}}},$$

$$\begin{gathered} {{M}_{{87}}} = 2G(\omega )l_{s}^{{}}\sigma _{s}^{{}}{{e}^{{ - i\sigma _{s}^{{}}h}}}, \\ {{M}_{{88}}} = \left( {{{\lambda }_{e}}k_{{tp}}^{2} + 2{{\mu }_{e}}\sigma _{{tp}}^{2}} \right){{e}^{{i\sigma _{{tp}}^{{}}h}}}, \\ {{M}_{{89}}} = 2{{\mu }_{e}}l_{{ts}}^{{}}\sigma _{{ts}}^{{}}{{e}^{{i\sigma _{{ts}}^{{}}h}}}; \\ \end{gathered} $$

$$\begin{gathered} {{M}_{{91}}} = 0,\,\,\,\,{{M}_{{92}}} = - 2G(\omega )l_{{p1}}^{{}}\sigma _{{p1}}^{{}}{{e}^{{i\sigma _{{p1}}^{{}}h}}}, \\ {{M}_{{93}}} = - 2G(\omega )l_{{p2}}^{{}}\sigma _{{p2}}^{{}}{{e}^{{i\sigma _{{p2}}^{{}}h}}}, \\ \end{gathered} $$

$$\begin{gathered} {{M}_{{94}}} = G(\omega )(k_{s}^{2} - 2l_{s}^{2}){{e}^{{i\sigma _{s}^{{}}h}}}, \\ {{M}_{{95}}} = 2G(\omega )l_{{p1}}^{{}}\sigma _{{p1}}^{{}}{{e}^{{ - i\sigma _{{p1}}^{{}}h}}}, \\ {{M}_{{96}}} = 2G(\omega )l_{{p2}}^{{}}\sigma _{{p2}}^{{}}{{e}^{{ - i\sigma _{{p2}}^{{}}h}}}, \\ \end{gathered} $$

$$\begin{gathered} {{M}_{{97}}} = G(\omega )(k_{s}^{2} - 2l_{s}^{2}){{e}^{{ - i\sigma _{s}^{{}}h}}}, \\ {{M}_{{98}}} = 2{{\mu }_{e}}l_{{tp}}^{{}}\sigma _{{tp}}^{{}}{{e}^{{i\sigma _{{tp}}^{{}}h}}}, \\ {{M}_{{99}}} = - {{\mu }_{e}}(k_{{ts}}^{2} - 2l_{{ts}}^{2}){{e}^{{i\sigma _{{ts}}^{{}}h}}}; \\ \end{gathered} $$

where ${{k}_{{p1}}}$, ${{k}_{{p2}}}$, ${{k}_{s}}$ and ${{k}_{{tp}}}$, ${{k}_{{ts}}}$ are the wave numbers of the p1-, p2-, S-waves in the porous media and P-, S-waves in the solid, respectively. They are given by

$$\begin{gathered} k_{{p1}}^{2} = l_{{p1}}^{2} + \sigma _{{p1}}^{2},\,\,\,\,k_{{p2}}^{2} = l_{{p2}}^{2} + \sigma _{{p2}}^{2}, \\ k_{s}^{2} = l_{s}^{2} + \sigma _{s}^{2},\,\,\,\,k_{{tp}}^{2} = l_{{tp}}^{2} + \sigma _{{tp}}^{2}, \\ k_{{ts}}^{2} = l_{{ts}}^{2} + \sigma _{{ts}}^{2}. \\ \end{gathered} $$

The matrix [B] in Eq. (48) are given by