Conductivity/Diffusivity

Determining the bulk conductivity associated with the wetting or non-wetting fluid is equivalent to treating the conducting pore space as a digitized resistor network and solving the set of linear equations for current when a known electrical potential is applied. To construct the correct set of network equations, the continuity of current across the voxel boundary (a voxel is a cubic element designated as either fluid or solid) was first imposed. To first order, this implies that the component of the current density J ^x_i,j,k entering a voxel surface in the x direction is

$$J^x_{i,j,k}=\frac{2(V_{i,j,k}-V_{i+1,j,k})}{\triangle x}\frac... ...k}}{\sigma_{i,j,k}+\sigma_{i+1,j,k}} (\hat{x} \cdot {\hat{n}})$$ (14)

where V_i,j,k is the electrical potential specified on each node labeled i,j,k, $\triangle x$is the lattice spacing, the _{i, j,k,} is the conductivity assigned to each node and n is unit normal pointing out of the voxel surface. To describe steady state current, the net current flux through the entire voxel surface is set equal to zero. i.e

0=J ^x_{i+1, j,k}+J ^x_i,j,k+J ^y_i,j+1,k +J ^y_i,j,k+J ^z_i,j,k+1+J ^z_i,j+1,k (15)

where J ^x_i+1,j,k and J ^x_i,j,k, and so forth, are evaluated at opposite faces of the the voxel. The resulting set of linear equations for V_i,j,k was solved using a conjugate-gradient relaxation algorithm [9]. The bulk conductivity is then determined by calculating the average current < J > for a given applied potential difference and using Ohm's Law $<J>=\sigma_{b} \triangle V$. The relative diffusivity is then obtained by next calculating $\sigma_{bi}$_bi for each separate fluid, i, and using D_ri =  _bi / _b.