Tortuosity for Double-Porosity Media

Theoretical estimates of tortuosity for the matrix and fracture components of the double-porosity medium may be obtained by noting that equation (tau) implies

^(1) = 12(1 + 1^(1))   for storage porosity that is spherical in shape, while

^(2) = 1,   for the fracture porosity, because $\phi^{(2)} = 1$ by assumption.

It is more difficult to estimate the overall tortuosity $\tau$, but a physically reasonable value can be obtained by considering the Hashin-Shtrikman bounds on electrical conductivity of a composite medium (Hashin and Shtrikman, 1962). These bounds show that for a two-component medium the effective conductivity will lie between the values $\sigma_{HS}^{\pm}$ given by the formula (Berryman, 1995)

1_HS^+2_m = 1-v^(2)_1+2_m + v^(2)_2+2_m,   where

_m = (_1,_2)  for  _HS^+   and

_m = (_1,_2)  for  _HS^-.   This notation means that $\sigma_{HS}^{+}$ is the upper bound, while $\sigma_{HS}^{-}$ is the lower bound.

Recalling that electrical tortuosity is related to formation factor F by $\tau= \phi F$, where $F = \sigma_f/\sigma$, we find that the tortuosity bounds for the double-porosity medium are:

1/^ +2/F_m = 1-v^(2)^(1)/^(1) + 2/F_m + v^(2)1+2/F_m.   We will assume that the overall tortuosity of the fractured double-porosity medium is in fact dominated by the fractures, in which case it is appropriate to assume that the actual electrical conductivity will be close to the upper bound $\sigma_{HS}^{+}$. In this case we choose F_m = 1 and, after rearranging the formula, we find

v^(2)^(1) + (3 - v^(2))^(1) (3-2v^(2))^(1) + 2v^(2)^(1).   Also, recall that the overall porosity is given by $\phi= (1-v^{(2)})\phi^{(1)} + v^{(2)}$.The formula (tauoverall) is expected to be valid for situations in which $v^{(2)} \ll 1$, and then (tauoverall) reduces approximately to $\tau\simeq \tau^{(1)}$.For applications to media in which such an assumption is not valid, the bounds in (HSboundsGs) should generally be used instead of (tauoverall).

Another physical constraint imposed by our model is that, if the drag/permeability coupling terms b₂₃ are neglected, then internal consistency of the theory may also require that $\rho_{23} \simeq 0$. Then, (rho23) can be used to solve for the effective $\tau$ that gives $\rho_{23} = 0$.Interestingly, the result in the limit $v^{(2)} \ll 1$ is again that $\tau\simeq \tau^{(1)}$. So these two approaches give very consistent results, and suggest that $\rho_{23} \simeq 0$may also be a physically reasonable approximation in many situations.