On the propagation of a disturbance in a smoothly varying heterogeneous porous medium saturated with three fluid phases

Abstract

From the equations governing the deformation of a porous medium containing three fluid phases, I derive expressions for the phase velocity of the various modes of displacement. These expressions are valid for a medium with smoothly varying heterogeneity. There is a single mode of transverse displacement, similar in nature to an elastic shear wave. The four-phase velocities of the longitudinal modes of displacement are derived from the solutions of a quartic equation. The coefficients of the polynomial equation are written as linear sums of the determinants of 4 × 4 matrices. The matrices contain various combinations of the parameters from the governing equations. The three-phase expressions are compared to two-phase estimates for the case in which one of the fluid saturations vanishes. Also, in a numerical illustration, velocity variations of around 10% are associated with the cyclic injection of carbon dioxide and water into an oil-saturated reservoir.

The Seismic Response to Injected Carbon Dioxide: Comparing Observations to Estimates Based Upon Fluid Flow Modeling

“…For the case considered here, the passage of an elastic wave, we linearize the problem by assuming that the saturation changes induced by the wave are small. Following the approach taken in Vasco (2013) and Vasco and Datta-Gupta (2016), suitably generalized for the case of N f fluids, one can produce N f + 1 governing equations, where the first equation is indexed by n = 1, 2, … , N f and actually represents N f equations:…”

“…As shown in Vasco (2013) and Vasco and Datta-Gupta (2016, p. 291), for a reservoir with smoothly varying properties, we may derive an asymptotic solution in terms of the ratio of the seismic wavelength to the length scale of the spatial variations in material and fluid properties, 𝜀, in the form of a propagating wave…”

“…The poroelastic parameters are analogous to those found in the study of a single fluid phase in a poroelastic medium (Pride, ; Wang, ): K u (undrainded bulk modulus), C ns and C sj (Biot coupling moduli), and M nj (fluid storage coefficients). Their relationship to the properties of the medium, specifically the relative permeability functions, the capillary pressure curves, the fluid, and rock properties, are given in Vasco () for three fluid phases and are easily generalized to the case of N f fluids. The other constants are related to the frequency, the solid and fluid fractions, and the solid and fluid properties, ρ s is the solid density, ρ n is the density of the n th fluid, μ n is the viscosity of the n th fluid, and k n ( ω ) is the dynamic permeability discussed in the appendix (see equation ).…”

“…The roots of the polynomial determine the squared slownesses and the phase velocities of the fast and slow compressional waves in the poroelastic medium follow from the expression The poroelastic properties of the reservoir, the properties of the fluids, the frequency, and the fluid saturations are necessary in order to calculate all of the parameters in equation . As noted in Vasco (), there are 12 fundamental classes of parameters that are needed in order to define the coefficients.…”

“…for a solid matrix with shear modulus G. The poroelastic parameters are analogous to those found in the study of a single fluid phase in a poroelastic medium (Pride, 2005;Wang, 2000): K u (undrainded bulk modulus), C ns and C sj (Biot coupling moduli), and M nj (fluid storage coefficients). Their relationship to the properties of the medium, specifically the relative permeability functions, the capillary pressure curves, the fluid, and rock properties, are given in Vasco (2013) for three fluid phases and are easily generalized to the case of N f fluids. The other constants are related to the frequency, the solid and fluid fractions, and the solid and fluid properties,…”