Inertial coefficients

It is easy to understand that the inertial coefficients appearing in the kinetic energy T must depend on the densities of solid and fluid constituents $\rho_s$ and $\rho_f$, and also on the volume fractions v⁽¹⁾, v⁽²⁾ and porosities $\phi^{(1)}$, $\phi^{(2)}$ of the matrix material and fractures, respectively. The total porosity is given by $\phi= v^{(1)}\phi^{(1)} + v^{(2)}\phi^{(2)}$ and the volume fraction occupied by the solid material is therefore $1-\phi$.

For a single porosity material, there are only three inertial coefficients and the kinetic energy can be written as

2T = B<>u & B<>U ¯_11 & ¯_12 ¯_12 & ¯_22 B<>u B<>U ,   where $\dot{\bf U}$ is the velocity of the only fluid present. Then, it is easy to see that, if $\dot{\bf u} = \dot{\bf U}$,the total inertia $\bar{\rho}_{11} + 2\bar{\rho}_{12} + \bar{\rho}_{22}$ must equal the total inertia present in the system $(1-\phi)\rho_s + \phi\rho_f$. Furthermore, Biot (1956) has shown that $\bar{\rho}_{11} + \bar{\rho}_{12} = (1-\phi)\rho_s,$ and that $\bar{\rho}_{22} + \bar{\rho}_{12} = \phi\rho_f$. These three equations are not linearly independent and therefore do not determine the three coefficients. So we make the additional assumption that $\bar{\rho}_{22} = \tau\phi\rho_f$, where $\tau$ (Note: This $\tau$ without subscripts should not be confused with the stress tensor introduced earlier in the paper.) was termed the structure factor by Biot (1956), but has more recently been termed the electrical tortuosity (Brown, 1980; Johnson et al., 1982), since $\tau= \phi F$, where F is the electrical formation factor. Berryman (1980) has shown that

= 1 + r(1 - 1),   follows from interpreting the coefficient $\bar{\rho}_{11}$ as resulting from the solid density plus the induced mass due to the oscillation of the solid in the surrounding fluid. Then, $\bar{\rho}_{11} = (1-\phi)(\rho_s + r\rho_f)$, where r is a factor dependent on microgeometry that is expected to lie in the range $0 \le r \le 1$, with $r = {{1}\over{2}}$ for spherical grains. For example, if $\phi= 0.2$ and r = 0.5, equation (tau) implies $\tau= 3.0$, which is a typical value for tortuosity of sandstones.

For double porosity, the kinetic energy may be written as

2T = B<>u & B<>U^(1) & B<>U^(2) _11 & _12 & _13 _12 & _22 & _23 _13 & _23 & _33 B<>u B<>U^(1) B<>U^(2) .   We now consider some limiting cases: First, suppose that all the solid and fluid material moves in unison. Then, in complete analogy to the single porosity case, we have the result that $\rho_{11} + \rho_{22} + \rho_{33} + 2\rho_{12} + 2\rho_{13} + 2\rho_{23}$must equal the total inertia of the system $(1-\phi)\rho_s + \phi\rho_f$.Next, if we suppose that the two fluids can be made to move in unison, but independently of the solid, then we can take $\dot{\bf U} = \dot{\bf U}^{(1)} = \dot{\bf U}^{(2)}$, and telescope the expression for the kinetic energy to

2T = B<>u & B<>U _11 & (_12+_13) (_12+_13) & (_22+2_23+_33) B<>u B<>U .   We can now relate the matrix elements in (allfluidcase) directly to the barred matrix elements appearing in (singleinertia), which then gives us three equations for our six unknowns. Again these three equations are not linearly independent, so we still need four more equations.

Next we consider the possibility that the fracture fluid can oscillate independently of the solid and the matrix fluid, and furthermore that the matrix fluid velocity is locked to that of the solid so that $\dot{\bf u} = \dot{\bf U}^{(1)}$. For this case, the kinetic energy telescopes in a different way to

2T = B<>u & B<>U^(2) (_11+2_12+_22) & (_13+_23) (_13+_23) & _33 B<>u B<>U^(2) .   This equation is also of the form (singleinertia), but we must be careful to account properly for the parts of the system included in the matrix elements. Now we treat the solid and matrix fluid as a single unit, so

_11 + 2_12 + _22 = (1-)_s + (1-v^(2))^(1)_f + (^(2)-1)v^(2)^(2)_f,  

_13 + _23 = - (^(2)-1)v^(2)^(2)_f,   and

_33 = ^(2)v^(2)^(2)_f,   where $\tau^{(2)}$ is the tortuosity of fracture porosity alone and v⁽²⁾ is the volume fraction of the fractures in the system.

Finally, we consider the possibility that the matrix fluid can oscillate independently of the solid and the fracture fluid, and furthermore that the fracture fluid velocity is locked to that of the solid so that $\dot{\bf u} = \dot{\bf U}^{(2)}$. The kinetic energy telescopes in a very similar way to the previous case with the result

2T = B<>u & B<>U^(1) (_11+2_13+_33) & (_12+_23) (_12+_23) & _22 B<>u B<>U^(1) .   We imagine that this thought experiment amounts to analyzing the matrix material alone without fractures being present. The equations resulting from this identification are completely analogous to those in (firstfrac)-(thirdfrac), so we will not show them explicitly here.

We now have nine equations in the six unknowns and six of these are linearly independent, so the system can be solved. The result of this analysis is that the off-diagonal terms are given by

2_12/_f = (^(2)-1)v^(2)^(2) - (^(1)-1)(1-v^(2))^(1) - (-1),  

2_13/_f = (^(1)-1)(1-v^(2))^(1) - (^(2)-1)v^(2)^(2) - (-1),   and

2_23/_f = (-1)- (^(1)-1)(1-v^(2))^(1) - (^(2)-1)v^(2)^(2).   The diagonal terms are given by

_11 = (1-)_s + (-1)_f,  

_22 = ^(1)(1-v^(2))^(1)_f,   and $\rho_{33}$ is given by (thirdfrac).

Estimates of the three tortuosities $\tau$, $\tau^{(1)}$, and $\tau^{(2)}$may be obtained using (tau), or direct measurements may be made using electrical methods as advocated by Brown (1980) and Johnson et al. (1982). Appendix A explains one method of estimating $\tau$ for the whole medium when the constituent tortuosities and volume fractions are known.