Quasistatic constitutive relations for isotropic materials

Taking the trace of equations (solidaveragestress) and (fluidaveragestress) gives the following results. The constitutive relations for dilatations and porosity are

- p_sK_m = B<>u_s - 1-   and

- p_fK_f = u_f + ,   where the partial averages $\bar{\bf u}_s$ and $\bar{\bf u}_f$ are related to the full volume averages by $\left<{\bf u}_s\right\gt = (1-\phi)\bar{\bf u}_s$ and $\left<{\bf u}_f\right\gt = \phi\bar{\bf u}_f$.One assumption implicit in (solids) and (fluidf) is that $\phi$ changes much more slowly in space than the displacement variables $\bar{\bf u}_s$ and $\bar{\bf u}_f$. This assumption allows us to remove the factors involving the porosity from the divergence terms, but we will show in a later publication that this assumption is not crucial to our analysis.

It is important at this point to understand the interpretations of all the symbols appearing in the last two equations. First, the variable $\delta p_f$ is just the change in the average fluid pressure throughout the fluid phase. The change in average solid pressure $\delta p_s$ is related to the macroscopic confining pressure change $\delta p_c$ by the averaging relation $\delta p_c = (1-\phi)\delta p_s + \phi\delta p_f$.Thus, $\delta p_s$ is just the average change in solid pressure experienced by the solid. Since $\delta p_c$ and $\delta p_f$ may be viewed as the pressures we can control, $\delta p_s$ is the (solid volume) weighted average of the confining pressure after subtracting that part of the confining pressure supported by the fluid pressure. The change in porosity is given by $\delta\phi$.The porosity change occurs naturally in these expressions because

= 1V_I_f B<>n_fu_fdS = -1V_I_s B<>n_su_sdS.   This result is demonstrated in the next section.

The divergence of the average solid displacement $\nabla\cdot\bar{\bf u}_s$ is properly interpreted as the dilatation of the porous solid frame (not the dilitation of the solid alone). This interpretation is not obvious, but it follows from the fact that the term arises from the external surface integral [c.f., Eq.(gradofave)]

<u_s> = 1V _E_s B<>n_s u_sdS,   which is exactly the surface integral needed to define the overall behavior of the porous solid frame. Thus, in terms of the definitions of Brown and Korringa (1975),

B<>u_s = VV = e = - p_dK^* - p_fK_m.   This interpretation is the same one reached by Pride et al. (1992) using a combination of the standard thought experiments (jacketed and unjacketed) of Biot and Willis (1957). To check that this is so, we can easily show that

1- = - [1-K^* - 1K_m] (p_s - p_f),   using either approach when a single constituent is present so that the Brown and Korringa unjacketed constants satisfy $K_m = K_\phi$.Note that (porositycheck) can also be written as

= - [1-K^*-1K_m]p_d,   emphasizing that porosity is constant if differential pressure is constant -- a general result for microhomogeneous porous frames, but not true otherwise. Thus, the left hand side of (solids) is just the solid dilatation $\delta V_s/V_s$, while the two terms on the right hand side are $\delta V/V + \delta(1-\phi)/(1-\phi)$.

Similarly, it is important to understand that the expression $\nabla\cdot \bar{\bf u}_f$ is not just a fluid dilatation, but also includes the effects of fluid motion in and out of the volume. In fact, this is already apparent from (fluidf) since the strict fluid dilatation satisfies

-V_fV_f = p_fK_f,   yet (fluidf) contains an additional term related to changes in porosity. The correct physical interpretation of $\nabla\cdot \bar{\bf u}_f$ is provided by its relation to the increment of fluid content

= (B<>u_s - B<>u_f),   where $\zeta$ is defined as

V_-V_fV = (VV - V_fV_f) +   and has the interpretation (Biot, 1973; Berryman and Thigpen, 1985) of the relative change in fluid mass per unit volume of initial fluid mass. Note that (fluiddil) and (zetadefined) are in agreement with (zetarelation) if the averaging equation (fluidf) is also satisfied.

The equations (solids), (fluidf), and (BandKe) are sufficient to arrive at the standard form of the equations relating e and $\zeta$ to the macroscopic pressures $\delta p_c$ and $\delta p_f$ for a single constituent porous medium given by

e - = 1/K^* & 1/K_m - 1/K^* 1/K_m - 1/K^* & 1/K^* + /K_f - (1+)/K_m -p_c - p_f .   These equations are completely consistent with the results of Pride et al. (1992) as can be demonstrated by substituting the definitions given above into their formulas (48) and (49), and then doing a straightforward (though somewhat tedious) $2\times 2$ matrix inversion. The relationship between these matrix elements and the coefficients H, C, and M in Biot's equations are given by Berryman (1992a). They are: $H = K^* + \alpha C + 4 G/3$,$C = \alpha M$, and $1/M = (\alpha- \phi)/K_m + \phi/K_f$, where the Biot-Willis parameter is defined by $\alpha= 1 - K^*/K_m$.