Numerical validation of various mixing rules used for up-scaled geo-physical properties
Creators: name: Bruining, Johannes
name: Slob, Evert
name: Plug, Willem-Jan
name: Plug, Willem-Jan
Publisher: name: XVI International Conference on Computational Methods in Water Resources
Year: 2006

description: Our objective is to interpret laboratory measurements of effective dielectric 
permittivities as a function of water saturation during drainage and imbibition 
processes in a porous medium. Mixing rules are widely used to express effective 
properties of coefficients, e.g. diffusion coefficient, Darcy permeability, electric 
conductivity and the dielectric permittivity, appearing in heterogeneous diffusion 
like processes. Here we confine our interest to the electrostatic behaviour of a 
mixture because most of the mixing rules were derived to compute the effective 
dielectric permittivity. The validation of the so called mixing rules is obtained by 
comparing them with the effective properties computed from the numerical solution of 
the steady state diffusion equation, including boundary conditions. One of the 
issues concerns the very large number of grid blocks required to obtain convergence. 
The basic case considers a checkerboard permittivity distribution. From simulations 
we can conclude that for the solution to converge to effective permittivity, a large 
number of grid blocks is needed. Furthermore, the convergence rate depends on the 
numerical method we use. 
To analyze the validity of the results obtained, we make use of the Hashin-Shtrikman 
bounds that show the range of effective values for different spatial distributions, 
for the given volume ratio of constituents. In agreement to the analytical solution, 
the geometric mean is found to be the value for the effective permittivity in the 2-
D case. However, a different choice of internodal dielectric coefficients can 
enhance the computation for the finite difference simulation. 
For the 3-D case no analytical expression could be found and from the computations 
we can conclude that the effective permittivity is close to the Bruggeman mixing law 
and the third power law average. 
Conclusions and significance:
1.For the 2-D and 3-D problem the convergence rate for the Finite Element (FE) model 
is faster than for the Finite Difference (FD) model.
2.For the isotropic 2-D case, a geometric mean for the internodal dielectric 
permittivity leads to the fastest convergence rate, which corresponds to the 
effective parameter value.  
3.The solution for the 3 –D problem is close to the Bruggeman mixing rule and the 
third power law average.
4.Coarse gridded FD computations, show results that do not satisfy the Hashin-
Shtrikman bounds.
5.Fine gridded numerical computations can be used to analyze the measured effective 
dielectric permittivity in terms of the spatial distribution of the components. 
6.The results suggest that it is possible to obtain unambiguous mixing formula if 
both the fraction of constituents and a geometric factor are specified.
7.Spatial distribution coefficients of the constituents (e.g. water, sand and rock) 
are important to interpret and to analyse electric resistivity measurements, e.g. 
borehole measurements. 
8.This research shows that the conventional FD scheme used in reservoir simulators 
can be improved by an optimal choice of internodal permeabilities.