Elastic wave propagation and attenuation in a double-porosity dual-permeability medium

Abstract

To account for large-volume low-permeability storage porosity and low-volume highpermeability fracture/crack porosity in oil and gas reservoirs, phenomenological equations for the poroelastic behavior of a double porosity medium have been formulated and the coefficients in these linear equations identified. This generalization from a single porosity model increases the number of independent inertial coefficients from three to six, the number of independent drag coefficients from three to six, and the number of independent stress-strain coefficients from three to six for an isotropic applied stress and assumed isotropy of the medium. The analysis leading to physical interpretations of the inertial and drag coefficients is relatively straightforward, whereas that for the stress-strain coefficients is more tedious. In a quasistatic analysis, the physical interpretations are based upon considerations of extremes in both spatial and temporal scales. The limit of very short times is the one most pertinent for wave propagation, and in this case both matrix porosity and fractures are expected to behave in an undrained fashion, although our analysis makes no assumptions in this regard. For the very long times more relevant to reservoir drawdown, the double porosity medium behaves as an equivalent single porosity medium. At the macroscopic spatial level, the pertinent parameters (such as the total compressibility) may be determined by appropriate field tests. At the mesoscopic scale pertinent parameters of the rock matrix can be determined directly through laboratory measurements on core, and the compressiblity can be measured for a single fracture. We show explicitly how to generalize the quasistatic results to incorporate wave propagation effects and how effects that are usually attributed to squirt flow under partially saturated conditions can be explained alternatively in terms of the double-porosity model. The result is therefore a theory that generalizes, but is completely consistent with, Biot's theory of poroelasticity and is valid for analysis of elastic wave data from highly fractured reservoirs.

Quasi‐static analysis of elastic behavior for some systems having higher fracture densities

“…The approach this works well as has been described previously in a different physical context by Berryman and Wang [46]. Our cubic polynomial equation is…”

“…Similarly, once we have all three eigenvalues, the three corresponding eigenvectors are also very easy to compute. In general [46], if λ = x1, x2, or x3, then an eigenvector associated with one of these λ values is proportional to: T is an eigenvector of (59), when the value of λ is chosen to be one of the three eigenvalues.…”

Using a Poroelastic Theory to Reconstruct Subsurface Properties: Numerical Investigation

“…The attenuation mechanism that prevails at low frequency comes from the mesoscopic scale (Pride et al, 2004), and it is due to fluid flow that occurs at boundaries between any medium heterogeneities whose sizes are between the grain sizes and the seismic wavelengths. This is particularly true for layered media (Gurevich et al, 1997;Pride et al, 2002) or when the medium contains 1) inclusions of different materials such as composite medium or double porosity medium (Berryman & Wang, 2000;Pride et al, 2004;Santos et al, 2006), or 2) different fluids (Santos et al, 1990) or patches of different saturation (Johnson, 2001). …”

A low-frequency asymptotic model of seismic reflection from a high-permeability layer

“…A comprehensive overview of the Biot's theory is presented in [40]. Further extensions accounting for local heterogeneities including double-porosity or layered media have been developed in [3,10,11,29,31,41,42]. The reflection and transmission coefficients predicted by the Biot's theory for a wave crossing a planar interface have been calculated in [17,28].…”