FLUID-SATURATED POROELASTIC ROCKS

In contrast to traditional elastic analysis, the presence in rock of a saturating pore fluid introduces the possibility of an additional control field and an additional type of strain variable. The pressure p_f in the fluid is a new field parameter that can be controlled. Allowing sufficient time for global pressure equilibration permits us to consider p_f to be a constant throughout the percolating (connected) pore fluid, while restricting the analysis to quasistatic processes. (But ultimately we are not interested in such quasi-static processes in this paper, as we are trying to reconcile laboratory wave data with the theory.) The change $\zeta$ in the amount of fluid mass contained in the pores [see Biot (1962) or Berryman and Thigpen (1985)] is a new type of strain variable, measuring how much of the original fluid in the pores is squeezed out during the compression of the pore volume while including the effects of compression or expansion of the pore fluid itself due to changes in p_f. It is most convenient to write the resulting equations in terms of compliances rather than stiffnesses, so the basic equation to be considered takes the following form for isotropic media:

$$\begin{eqnarray} \left( \begin{array} {c} e_{11} \\ e_{22} \\ e_{33} \\ -\zet... ...\\ \sigma_{22} \\ \sigma_{33} \\ - p_f \\ \end{array}\right). \end{eqnarray}$$ (1)

The constants appearing in the matrix on the right hand side will be defined in the following two paragraphs. It is important to write the equations this way rather than using the inverse relation in terms of the stiffnesses, because the compliances s_ij appearing in (1) are simply related to the drained elastic constants $\lambda_{dr}$ and G_dr in the same way they are related in normal elasticity, whereas the individual stiffnesses obtained by inverting the equation in (1) must contain coupling terms through the parameters $\beta$ and $\gamma$ that depend on the pore and fluid compliances. Thus, we find that

$$\begin{eqnarray} s_{11} = \frac{1}{E_{dr}} = \frac{\lambda_{dr}+G_{dr}}{G_{dr}(3\lambda_{dr}+2G_{dr})} \end{eqnarray}$$ (2)

and

$$\begin{eqnarray} s_{12} = - \frac{\nu_{dr}}{E_{dr}}, \end{eqnarray}$$ (3)

where the drained Young's modulus E_dr is defined by the second equality of (2) and the drained Poisson's ratio is determined by

$$\begin{eqnarray} \nu_{dr} = \frac{\lambda_{dr}}{2(\lambda_{dr}+G_{dr})}. \end{eqnarray}$$ (4)

When the external stress is hydrostatic so $\sigma= \sigma_{11} = \sigma_{22} = \sigma_{33}$, equation (1) telescopes down to

$$\begin{eqnarray} \left( \begin{array} {c} e \\ -\zeta\\ \end{array}\right) = \... ... \left( \begin{array} {c} \sigma\\ -p_f \\ \end{array}\right), \end{eqnarray}$$ (5)

where e = e₁₁ + e₂₂ + e₃₃, $K_{dr} = \lambda_{dr} + {2\over3}G_{dr}$ is the drained bulk modulus, $\alpha= 1 - K_{dr}/K_m$ is the Biot-Willis parameter (Biot and Willis, 1957) with K_m being the bulk modulus of the solid minerals present, and Skempton's pore-pressure buildup parameter B (Skempton, 1954) is given by

$$\begin{eqnarray} B = \frac{1}{1 + K_p(1/K_f - 1/K_m)}. \end{eqnarray}$$ (6)

New parameters appearing in (6) are the bulk modulus of the pore fluid K_f and the pore modulus $K_p^{-1} = \alpha/\phi K_{dr}$ where $\phi$ is the porosity. The expressions for $\alpha$ and B can be generalized slightly by supposing that the solid frame is composed of more than one constituent, in which case the K_m appearing in the definition of $\alpha$is replaced by K_s and the K_m appearing explicitly in (6) is replaced by $K_{\phi}$ (see Brown and Korringa, 1975; Rice and Cleary, 1976; Berryman and Wang, 1995). This is an important additional complication (Berge and Berryman, 1995), but -- for the sake of desired simplicity -- we will not pursue the matter further here.

Comparing (1) and (5), we find that

$$\begin{eqnarray} \beta= {{\alpha}\over{3K_{dr}}} \end{eqnarray}$$ (7)

and

$$\begin{eqnarray} \gamma= {{\alpha}\over{BK_{dr}}}. \end{eqnarray}$$ (8)

As we develop the ideas to be presented here, we will need to treat Eqs. (1)-(6) as if they are true locally, but perhaps not globally. In particular, if we assume overall drained conditions, then p_f = a constant everywhere. But, if we assume locally undrained conditions, then $p_f \simeq$ a constant in local patches, but these local constant values may differ from patch to patch. This way of thinking about the system is intended to mimic the behavior expected when a high frequency wave propagates through a system having highly variable (or just uniformly very low) fluid permeability everywhere.