Page 47 - Percolation Models for Transport in Porous Media With

38 CHAPTER2. ONE-PHASE FLOW IN ROCKS

7.0
Figure 11: Plots of the conductivity of the medium against its strained state

where a* is the minimal radius of a conducting capillary.
Since in the three-dimensional case 11 = 0.9±0.1, it is possible to find K for an
exponential function analytically, using formula (2.1), if 11 is set to equall. In this
case we get rc = 4a*,K = 2.81r7a!/l • Setting 7 = 2.85,1 = 0.5 ·10- m, a*=
2
3
0.5 ·10- 5 m, we find K = 18ttm 2 • Using the relationships (2.39), (2.40), and (2.43)
we can find the coefficient of permeability as a function of pressure P

Usually actual experiments measure the quantity !J.KfK, where !J.K = K(p)-
K(O). The theoretical relationship f:!.K/K calculated for g 0 = 10 3 ,C = 15. 10 8
Pais presented in fig. 11 (curve 1). The plots of !J.KfK and SE/'E against
pressure, measured experimentally for a sandstone core in [48] (curves 3 and 4).
are also presented in this figure. The theoretical dependence !J.E/E (curve 2) has
been calculated using formulas (2.4) - (2.6) for )..' fu' = 7 · 10- 6 m, II = 0, other
parameters the same. Curve 5 corresponds to the difference !J.Kd Kt calculated
using formulas (2.42), (1.9), and (1.11) for the case of anisotropic loading of a
grained medium under constant pressure p = 200 · 10 5 Pa.
It can be noticed from fig.11 that a satisfactory agreement between theoretical
and experimental data takes place for the given set of parameters of the model
of a nonlinearly elastic porous medium. Note that the value of Young's modulus
E ~ 10 3 MPa differs from its actual characteristic values for typical reservoir
rocks of the sandstone type by approximately an order of magnitude. Therefore
the presented calculations correspond to strongly contractible rocks with large
porosity.
The remarkable phenomenon of high sensitivity of the medium with E ~ 10 4
MPa to relatively low pressures with small deformations can be explained, obvi-
ously, by significant impact that a film of a "bounded" fluid has on the flow in
thin capillaries. This phenomenon will be studied in detail in the next chapter.
The change of the component of the permeability tensor K 1 , found theoreti-