Fundamentals of Fluid Flow in Porous Media


Chapter 2

Relative Permeability

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,
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:
Ohm's Law with Current Density (J) and Charge Density (F)

Thermal conductivity is also another well-known conductivity and it is defined by Fourier’s law,
Thermal Conductivity 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
Kirchhoff's Law

at any node i, and for any closed loop of resistors:
Equation 2-142

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
Equation 2-143

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:
Equation 2-144


[G] = The conductivity matrix,

[V] = The voltage drop vector,

[I] = The current vector.

This way of approaching the problem is different from Dodds’ approach (1968) that used the resistances instead of the conductivities resulting in the inverse set of equations:
Equation 2-145

where R is the resistance matrix.

For the optimum numerical solution, eq. (2‑145) is preferred because of less memory space and less execution time requirements (George and Liu (1981)). So, for a network that consists of n nodes, a set of n linear equations is created in which all the elements of the current vector have the value of zero. The linear set however, is not positively definite. Therefore, it is not that easy to get a solution for eq. (2‑145), and the solution most probably will not be unique. Also, this solution will not provide any information for the overall conductivity of the network, which is the main aim of this work. So, it is necessary to have an additional independent equation that relates the overall conductivity of the network, Gt, the overall voltage drop, Vt, across the network and the overall current, It. This set up provides the extra information that relates It, Vt and Gt.
Equation 2-146

where e denotes the resistors that connect the network to the external node.

If the value of the overall current ( It ) is defined, then a linear and positive definite set of equations is obtained which can be used for obtaining the solution for the voltage vector. If, in addition to the above, It = 1, then the overall conductivity of the network can be automatically calculated, since
Equation 2-147

Therefore, if the conductivity of each resistor is known, the overall conductivity of the network can be calculated. This calculation was performed by Kantzas (1985) and the result was a set of relative permeability curves that provided a good agreement with experimental results in Berea Sandstone.




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