POROELASTICITY AND ESHELBY

The equations of quasistatic poroelasticity, as presented for example by Rice and Cleary (1976), may be written concisely in the form:

_pq = S_pqrs<_rs>,  

= (m-m_0)/_0 = K [13_qq + 1Bp_f].   Commonly understood terms appearing in these equations are the strains $\varepsilon_{ij}$,the solid stress $\sigma_{ij}$, the fluid pressure p_f, the elastic compliance tensor S_ijkl of the drained porous frame, and the increment of fluid content $\zeta$ (which is related to the initial m₀ and current m fluid mass contents, and to the initial density $\rho_0$ of the fluid). Applying well-known definitions from Biot and Willis (1957), the effective stress (for total volume strain) appearing in (strain) is

<_pq> = _pq + p_f _pq,   where the coefficient $\alpha= 1 - K/K_s'$ is the Biot-Willis parameter, K is the bulk modulus of the solid frame (jacketed modulus), and K_s' is the unjacketed solid modulus. The coefficient B is Skempton's pore-pressure buildup coefficient (Skempton, 1954; Green and Wang, 1986; Hart and Wang, 1995), given by

1B = 1 + _0 K (1K_f - 1K_s''),   where $\phi_0$ is the initial porosity, K_f is the bulk modulus of the pore fluid, and K_s'' is the unjacketed pore modulus. The equation for the change in porosity $\phi$ is

- _0 = K [13_qq + p_f] - _0K_s''p_f.   In other work the present author has often used the alternative notation K_s = K_s' and $K_\phi= K_s''$ for the same two unjacketed bulk moduli.

Starting from these basic equations of poroelasticity I want to formulate methods of computing the effective coefficients in composite poroelastic media when these media are themselves composed of simpler (generally microhomogeneous) poroelastic media. The corresponding problem in elasticity has been studied extensively for at least the last 40 years. It is desirable to try to make the transition from composite elastic media to composite poroelastic media as elegantly as it can possibly be done. One way in which this might be accomplished within effective medium theory is through the use of similar techniques applied to the full poroelastic equations such as was done in Berryman (1992). Another way to reach the same goal is to find new extensions to poroelasticity of some of the classic results like Eshelby (1957) that make the analysis virtually the same as that in the elastic case.

I restrict discussion here to poroelasticity, but the modifications necessary for application to thermoelasticity are not difficult. In my notation, a superscript i refers to the inclusion phase, while superscripts h and * refer to host and composite media, respectively. In this initial analysis, the composite is a very simple one, being an infinite medium of host material with a single ellipsoidal inclusion of the ith phase. The basic result of Eshelby (1957) is then of the form

^(i)_pq = T_pqrs^*_rs,   where $\varepsilon^{(i)}$ is the uniform induced strain in the inclusion, $\varepsilon^{*}$ is the uniform applied strain of the composite at infinity, and T is the fourth-rank tensor relating these two strains. The summation convention for repeated indices is assumed in expressions such as (Eshelby).

After considering two thought experiments - one when there is no fluid present in the pores and another when a saturating fluid is present and both the confining and pore pressures are chosen so that a uniform expansion of the host medium and inclusion occur (Berryman and Milton, 1991; Berryman and Berge, 1998), I find that the final form of the generalization of Eshelby's formula to poroelasticity is given by

^(i)_pq - e_pq(p_f) = T_pqrs[^*_rs-e_rs(p_f)].   The full analysis shows that, if the pore fluid pressure vanishes (e.g., p_f = 0 in the absence of a pore fluid), then the uniform strain e disappears from (genEshelby) and it reduces exactly to (Eshelby) as it should. In the other limiting case, if the pore pressure has been specified to be a nonzero constant, then the uniform strain e in (genEshelby) can be easily computed. So, if the strain at infinity happens to be chosen to be equal to this uniform strain, then from (genEshelby) the inclusion strain also takes the value at infinity as it should. Since the equation for $\varepsilon^{(i)}$ is necessarily linear, these two cases are enough to determine the behavior for arbitrary values of $\varepsilon^{*}$ and p_f. (Think of the two thought experiments as independent boundary conditions on the linear equations that then determine the coefficients.) In poroelasticity, the strain e_pq can be determined in advance from the applied fluid pressure p_f and the properties of the host and inclusion. In particular, I find that

e_pq = (^(h)-^(i)K^(h)-K^(i)) p_f3_pq.   The formulas presented in the following work form one set of useful applications of this generalization of Eshelby's formula.