BIOT-GASSMANN EQUATIONS

Consider a porous medium whose connected pore space is saturated with a single-phase viscous fluid. The fraction of total volume occupied by fluid is the porosity , which is assumed to be uniform on some appropriate length scale. Bulk modulus and density of the fluid are K_f and $\Gr_f$,respectively. Bulk and shear moduli of the drained porous frame are K_d and .For simplicity, assume the frame is composed of a single granular constituent whose bulk and shear moduli and density are K_m, $\Gm_m$, and $\Gr_m$. Frame moduli may be measured on drained samples, or may be estimated using one of a variety of methods from the theory of composites (Berryman, 1980b,c; Berryman and Milton, 1988).

For long wavelength acoustic pulses ($\Gl \gt\gt h$ where h is a typical pore size) propagating through such a porous medium, we define average values of local displacements in the solid and also in the saturating fluid. The average displacement vector in the solid frame is ,while that in the pore fluid is $\vecu_f$.A more useful way of quantifying fluid displacement is to introduce the average displacement of fluid relative to frame which is $\vecw = \Gv(\vecu_f - \vecu)$.For small strains, frame dilatation is

e = .

Similarly, average fluid dilatation is

e_f = _f,

which includes fluid flow terms as well as dilatation. The increment of fluid content is defined by

= -= (e-e_f).

With these definitions, Biot (1956a,b;1962) introduces a quadratic strain-energy functional of the independent variables e and for an isotropic, linear porous medium

2E = He^2 - 2Ce+ M^2 - 4I_2,   where, if $e_{xy} = {1\over2}(u_{x,y}+u_{y,x}), \ldots$, then I₂ = e_xxe_yy+e_yye_zz+e_zze_xx-e_xy²-e_yz²-e_zx² is the second strain invariant (Berryman and Thigpen, 1985). Elementary bounds on coefficients in the equations of poroelasticity are presented by Thigpen and Berryman (1985). Thermodynamic and mechanical stability require non-negativity of E, which implies that $H\ge0$, $M\ge0$, $HM-C^2 \ge 0$,and $\Gm\ge0$. Then, components of the average stress tensor for the saturated porous medium are

_ij = [(H-2)e-C]_ij + 2e_ij,   and fluid pressure p_f is

p_f = M- C e.  

Two coupled equations of motion for small disturbances in fluid-saturated media may be derived easily from the energy functional E with these definitions of stresses and pressures. With time dependence of the form $\exp(-i\Gw t)$,Biot's equations of poroelasticity are, using the later notation of Biot (1962),

^2+ (H-)e - C+ ^2(+_f) = 0,  

Ce - M+ ^2(_f+ q) = 0,   where

= _f + (1-)_m

and

q() = _f[/+ iF()/].

Tortuosity $\Gt \ge 1$ is a pure number related to frame inertia which has been measured (Brown, 1980; Johnson et al., 1982) and can also be estimated theoretically (Berryman, 1980a). Kinematic viscosity of the saturating fluid is ; permeability of the porous frame is ; $F(\Gx)$ is a dynamic viscosity factor proposed by Biot (1956b), where $\Gx = (\Gw h^2/\Gc)^{1\over2}$. The dynamic parameter h is a characteristic length generally associated with steady-flow hydraulic radius of the pores, or with a typical pore size.

The coupled equations (Biot1) and (Biot2) give rise to three distinct modes of wave propagation: two compressional waves (with speeds v₊ and v_o for the faster and slower of the two wave speeds, respectively) and a single shear wave (with speed v_s having two polarizations).

Coefficients appearing in Biot's equations of poroelasticity must be known before quantitative predictions can be made. Results of Brown and Korringa (1975) may be used to show that these coefficients are given for general isotropic porous media by

H = K_u + 43,   where (using the definitions that follow) the undrained modulus K_u is determined by

K_s K_uK_s-K_u = K_s K_dK_s-K_d + K_K_f (K_-K_f),   while the coefficients C and M are given by

C = (K_u - K_d)/  and  M = C/,   with

= 1 - K_d/K_s.   The three bulk moduli characteristic of the drained porous frame are defined by Brown and Korringa (1975) through the expressions:

1K_d = - 1V(Vp_d)_p_f,  

1K_s = - 1V(Vp_f)_p_d,   and

1K_ = - 1V_(V_p_f)_p_d,   where V is total sample volume, $V_\Gv = \Gv V$ is pore volume, $p_c = -{1\over3} Tr(\Gt) = -{1\over3}(\Gt_{xx}+\Gt_{yy} +\Gt_{zz})$ is external (confining) pressure, p_f is pore pressure, and p_d = p_c - p_f is differential pressure. Brown and Korringa (1975) state that, although these three bulk moduli have simple physical interpretations, this ``does not necessarily help in knowing their values.'' Observing the change in pore volume (an internal variable) required by (Kpore) is clearly more difficult than observing the change in total sample volume (an external variable) required by the other two moduli. Nevertheless, all three moduli may, in principle, be measured in quasi-static measurements, thus determining their values directly. However, it is commonly the case that acoustic measurements are made on compressional and shear wave speeds in fluid saturated porous media. It would therefore be beneficial to have a means of deducing these constants from wave data.

Constant K_d is the (jacketed) bulk modulus of the drained porous frame, and is commonly measured since it is precisely what one would think of as the bulk modulus of the dry (or drained) rock in nonporous elasticity. However, values of the two remaining constants K_s (the unjacketed modulus) and $K_\Gv$ (the unjacketed pore modulus) are not generally known or measured unless the porous frame is homogeneous on the microscopic scale. For this special circumstance [which is also the only one considered explicitly by Gassmann (1951)] with a single type of elastic solid composing the frame, these two moduli both equal the bulk modulus K_m of the single granular constituent

K_s = K_= K_m.   Thus, Gassmann's equation is equivalent to

1M = K_f + -K_m,   = 1 - K_d/K_m,   while Brown and Korringa's more general result (undrained) may be rearranged to show that

1M = K_f + K_s - K_,   = 1 - K_d/K_s.   Note the important fact (to be used later) that the theory clearly shows 1/M is always a linear function of the fluid compressibility $\Gb_f = 1/K_f$.Gassmann's result (Gassmanns) has also been derived within the context of Biot's theory of poroelasticity by Biot and Willis (1957) and by Geertsma (1957). Geertsma and Smit (1962) discuss practical aspects of applications of the theory to rocks. Rice (1975) and Rice and Cleary (1976) also obtain a general result essentially equivalent to that of Brown and Korringa (BandKs).

The more general constants of Brown and Korringa, K_s and $K_\Gv$,must somehow be related to material properties of multiple solid constituents of the porous frame. Equation (BandKs) provides a possible method of determining the constants as we will show: by saturating the same porous medium with different pore fluids, it should therefore be possible to deduce the values of the frame moduli from the variations in M with fluid compressibility.

Before attacking the inversion problem, we first show an example illustrating the success of the theory in predicting the results of certain wave propagation experiments.

 

plona
Figure 1 Ultrasonic velocities (slow compressional - o, shear - s, fast compressional - + ) in water-saturated porous glass. Theoretical curves from Biot's theory as described in the text. Data from Plona (1980), Johnson and Plona (1982), and Plona and Johnson (1984).