Fundamentals of Fluid Flow in Porous Media
Two Phase Relative Permeability Literature Survey: Conductivity and Permeability, the Main Algorithm
The term conductivity is one of the most commonly used terms that appear in all the texts and studies that deal with flow problems. It describes the difficulty of flow through a certain medium, where the flow refers to momentum, energy, mass, electricity, etc. The most well known conductivity is the electrical conductivity, which is described by Ohm’s law,
I is the electric current (in Amperes),
V is the voltage drop (in Volts),
G is the electrical conductivity of a resistor (in I/Ohm).
Eq. (2‑138) can be rewritten in terms of current density J and charge density F as follows:
Thermal conductivity is also another well-known conductivity and it is defined by Fourier’s law,
qx = The heat flux,
k = The thermal conductivity,
( dT / dx ) = The temperature gradient along the x direction.
In porous media, the conductivity to flow (or permeability) is given by Darcy’s law. Darcy’s law and equations (2‑139) and (2‑140)are obviously similar. Many researchers who have seen this similarity tried to give solutions for the permeability simulation problems using algorithms originally designed for the electrical conductivity analogs. This strategy is very reasonable. The same strategy is used in Kantzas (1985).
First, let us consider the case of calculating the overall conductivity of a network of resistors. This can be done by applying Kirchhoffs’ laws
at any node i, and for any closed loop of resistors:
where m is the number of bonds that form a closed loop.
The second law applies in such cases where we have many sources of electricity within a loop. If we consider a network that contains only resistors then application of the first law can give
This says that we have one equation for every node in the network. A group of equations can be formed that describes the whole network, such that: