Body waves in poroelastic media saturated by two immiscible fluids

A study of body waves in elastic porous media saturated by two immiscible Newtonianfluids is presented. We analytically show the existence of three compressionalwaves and one rotationalwave in an infinite porous medium. The first and second compressional waves are analogousto the fast and slow compressionalwaves in Biot's theory. The third compressionalwave is associated with the pressure difference between the fluid phases and dependent on the slope of capillary pressure-saturationrelation. Effect of a second fluid phase on the fast and slow waves is numerically investigatedfor Massilion sandstone saturated by air and water phases. A peak in the attenuationof the first and second compressionalwaves is observed at high water saturations. Both the first and second compressional waves exhibit a drop in the phase velocity in the presence of air. The results are compared with the experimental data available in the literature. Although the phase velocity of the first compressional nd rotationalwaves are well predicted by the theory, there is a discrepancy between the experimental and theoretical values of attenuationcoefficients. The causes of discrepancy are explained based on experimental observations of other researchers. Introduction Biot's theory ofwave propagation in linear, elastic, saturated porous media has been the basis of many velocity and attenuation analyses [Blot, 1956a,b]. Biot demonstrated the presence of two compressional and one rotational wave in a porous medium saturated by a single fluid phase. The first compressional wave (also referred to as fast wave) is analogous to the compressional wave in elastic media. The second compressional wave is slower and highly attenuated. Because of its highly dissipative behavior, this wave is very difficult to observe. The second compressional wave was first observed by Plozza [1980] (also see Bcroqnan [1980]). However, we must note that neither of these waves propagate as a wave in the fluid or in the matrix alone, both travel jointly in the matrix and pore fluid. All waves are dispersed and attenuated. Attenuation arises because of the relative movement of solid and fluid phases. At low frequencies, attenuation is related to the permeability of solid matrix. Since laminar flow assumption does not hold for high frequencies, the theory needs modifications [Biot, 1956b]. We refer to Corapcioglu and Tmzcay [1996] for an extensive review of Biot's theory. In addition to the three types of body waves, other types of waves exist close to the interfaces. Analogous to elastic media, Dcr1⁄2sicv•cx [1962] and Jones [1961] showed the existence of surface waves in saturated porous media. They examined surface waves by considering the coupling of the transverse and the fast compressional waves. Later Tajuddin [1984] presented a study of Rayleigh waves considering all three types of body waves. Love waves which appear due to •Now at Izmir Institute of Technology, Cankaya, Izmir, Turkey. Copyright 1996 by the American Geophysical Union. Paper number 96JB02297. 0148-0227/96/9 6JB-02297 $09.00 stratification of the Earth were investigated by Deresisvdcz [1961, 1964, 1965] and Chattopadhyay and Ds[1983]. In contrast to porous media saturated by a single fluid, wave propagation in porous media saturated by multiphase fluids received limited attention from the researchers. The general trend is to extend Biot's formulation developed for saturated medium by replacing model parameters with the ones modified for the fluid-fluid or fluid-gas mixtures [Doraszd•o, 1974; Mo•hizula; 1982; Murphy, 1984; PrMs st a/., 1992]. This approach results in two compressional waves and has been shown to be successful in predicting the first compressional and rotational wave velocities for practical purposes. Brutsasrt [1964] who extended Biot's theory appears to be the first one to predict three compressional waves. The third compressional wave arises due to presence of a second fluid phase in the pores. Brutsasrt andLuthin [1964] provided experimental data which agrees with the results of Brutsasrt's [1964] theory. The third compressional wave was also predicted by Garg and Nayœsh [1986] and Santos st al. [1990]. We should note that in these studies, although same type of governing equations were employed, the expressions for material constants differed. Miksis [1988] developed a model for wave attenuation in partially saturated rocks based on the local three-phase physics in the pore space. Tuncay [1995] derived the governing equations and constitutive relations of elastic porous media saturated by two Newtonian fluids by employing the volume-averaging technique. In this study, starting from the governing equations, we investigate the types of waves and their characteristics in infinite isotropic porous media saturated by two fluids. We show that three compressional waves and one rotational wave exist. All waves are dispersed and attenuated. First and second compressional waves are similar to those in saturated media. We show that the third compressional wave occurs because of the existence of capillary pressure. Because of the very low velocity and high attenuation ofthe third compressional wave, 25,149 25,150 TUNCAY AND CORAPCIOGLU: WAVE PROPAGATION IN POROUS MEDIA an experimental observation does not appear to be possible. Finally, we compare the results of model with experimental data obtained in terms of phase velocities and attenuation coefficients of the first compressional and rotational waves. Final Set of Governing Equations We start by introducing the governing equations for elastic porous media saturated by two immiscible Newtonian fluids, for example, water and air. In the derivation, we assume that there is no mass exchange between the phases and the phases are at rest. Furthermore, the solid phase is assumed to be isotropic, experiencing small deformations and providing all shear resistance of the porous medium. Momentum transfer terms are expressed in terms of intrinsic and relative permeabilities assuming the validity of Darcy's law. At very low saturations (i.e., residual saturations), the theory may not be valid since not all phases of the medium are interconnected. Furthermore, the theory employed in this study is limited to low-frequency wave propagation. Let L, ] and •-be the characteristic lengths of the macroscopic scale, averaging volume, and pore scale, respectively. The required condition for the volume averaging is •' << ] << L. In this study we assume that this requirement is satisfied, and furthermore, if ), is the wavelength of the wave, we assume ] < < X. Therefore we limit the present study to low-frequency waves. In the case of two pore fluids, the low-frequency limit should be lower than that of a single fluid case since the size of averaging volume would be larger to include the presence of two pore fluids. In the case of a single pore fluid, the size of the averaging volume is determined by the pore structure only. However, when two fluids are present in the pores, the orientation of fluid phases is an important factor on the size of averaging volume. In addition to assuming an isotropic elastic porous matrix saturated by two immiscible Newtonian fluids, validity of Darcy's Law, and low-frequency wave propagation, we furthermore assumed the validity of the relationship between capillary pressure and a fluid saturation, and negligibility of cross permeabilities known as the Yuster effect. The governing equations are obtained in terms of displacements of solid and fluid phases as [ Tu•cay, 1995] (@s;•-=V((z,,+ 7 )V'•s+ Zl:V'T+ al•V'uz) (1)