Poroelastic modeling of seismic boundary conditions across a fracture (2006)

Poroelastic modeling of seismic boundary conditions across a fracture M.A. Schoenberg, Consultant, 5 Mountain Rd., West Redding CT06896 S. Nakagawa*, Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley CA94720 Summary A fracture within a porous background is modeled as a thin porous layer with increased compliance and finite permeability. For small layer thickness, a set of boundary conditions can be derived that relate particle velocity and stress across a fracture, induced by incident poroelastic waves. These boundary conditions are given via phenomenological parameters that can be used to examine and characterize the seismic response of a fracture. One of these parameters, here it is called membrane permeability, is shown through several examples to control the scattering amplitude of the slow P waves for very low-permeability fractures, which in turn controls the intrinsic attenuation of the waves. Introduction Sedimentary rock is often permeated by fractures, fault and joints occurring in sets with a common alignment. When background permeability is small, the degree of fracturing, the physical nature of the fractures, and their orientation, are vital data for understanding the fluid flow in rocks. In non-permeable media, specifying how the elastic moduli of a layer approach zero as the cumulative thickness of the layer approaches zero yields a set of phenomenological parameters which describe the additional compliance of the equivalent medium due to the presence of fractures (Schoenberg, 1980). The relation between acoustics and flow in permeable media has been described successfully by Biot theory, and this same approach will be used to derive the acoustic behavior of fractures in permeable rocks. In permeable media, a fracture may also be treated as a thin layer. Associated with it is a set P of parameters specifying the elastic and flow properties of the layer as a function of the layer thickness. As in the case of non-permeable layers, natural boundary conditions (open pores for the flow condition, no displacement discontinuities at the layer boundaries) are applied. Then allowing the thickness to approach zero leaves a subset P 0 of the parameter set which specifies, by way of the actual boundary conditions across the fracture, the average behavior of the fracture. It is still an open question whether such a surface may be sounded in a way that allows the parameters of P 0 which control the flow in the fracture to be estimated. Theory Our objective is to derive a set of boundary conditions for a thin poroelastic layer embedded within a homogeneous isotropic background poroelastic medium. The governing equations of poroelastic wave propagation can be stated as τ = G ( ∇ u + u ∇ ) + ⎡ ( K U − 2 G / 3 ) ∇ ⋅ u + C ∇ ⋅ w ⎤ 1 − p f = C ∇ ⋅ u + M ∇ ⋅ w −∇ p f = − ω f ( ρ u + ρ w ) ( ρ u + ρ w ) , ρ ≡ i η f f / ω k ( ω ) where u is the local average solid frame displacement, w≡ φ ( U - u ) is the relative fluid volume displacement defined via u , local fluid displacement in the pore space U and porosity φ . τ is the total stress and p f is the fluid pressure. G is the frame shear modulus and K U is the undrained bulk modulus. ρ is the bulk density, ρ f is the fluid modulus, and the parameter ρ is defined via fluid viscosity η f and the frequency-dependent permeability k ( ω ) (e.g., Johnson et al., 1987). Definitions of C and M and their relationships to other parameters can be found in standard poroelasticity literature (e.g., Pride, 2003). Solving these equations assuming a plane harmonic wave field results in four plane waves (fast and slow P waves and two S waves). Consider an interface across which certain stress and displacement (velocity) components are conserved. We assume this interface to be normal to the 3 rd direction and the wave propagation parallel to the 1, 3 plane. For a homogeneous medium, we can assume a plane harmonic wave field proportional to exp i ω ( ξ 1 x 1 - t ). The plane wave displacement and stress are introduced into Eqs.(1). With substitutions ∂/∂ x 1 → i ωξ 1 and ∂/∂ x 2 → 0 and by eliminating variables that can be discontinuous across the interface, the following coupled first-order differential equations are derived (Note: in this paper, an uncoupled shear wave with out-of-plane motions is not discussed) ⎡ u 1 ⎤ ∂ ⎢ − p f ⎥ ⎥ = − i ω ⎢ ∂ x 3 ⎢ τ 13 ⎥ ⎣ Q YX ⎢ u 3 ⎥ ⎢ w 3 ⎥ ⎡ u 1 ⎤ Q XY ⎤ ⎢ − p f ⎥ ⎢ u 3 ⎥ ⎢ w 3 ⎥