| Abstract: | In a paper presented at last year's Amsterdam meeting (viz. Geophysical Prospecting 14 (1966), 3, 301–341), J. R. Schopper derived formulas relating formation factor, permeability and porosity, by means of a statistical-network approach and by treating the electric and hydraulic resistance analogously. The model of the porous medium consists of a network of branch resistors, their values being statistically distributed about a mean Ro with a relative standard deviation (variation coefficient) s. A properly defined total resistance R of the network can be expressed by the relationship:  ((1′)) Here α is a geometrical factor dependent only on the shape of the network (i.e. the number of meshes in the longitudinal and transversal direction), ε is a characteristical constant dependent only on the individual mesh shape (i.e. the number of nodes and branches within a mesh). This network constant ε enters the equations relating formation factor, permeability and porosity, ε had been found to be in the range zero to one by calculating algebraically two special limiting network cases. However, for a better understanding of which value exactly this constant will have in actual porous media, networks with various mesh shapes have to be treated generally. Because of the basically statistical approach, the networks have to be large so that a general algebraic treatment is precluded. Hence numerical methods using digital computers must be applied. The determination of the total resistance R of any resistance network leads to the problem of solving a system of linear, inhomogeneous equations; i.e. Ohm's law written in matrix form:  ((2′)) - (R) is the matrix of the coefficients, composed of the individual branch resistances.
- (I) is the column vector, its components being fictitious circular mesh currents.
- (U) is the inhomogeneity column, its components being source voltages within the individual meshes.
The matrix (R) has characteristic properties that depend on the mesh shape on the one hand and on the number and arrangement of the meshes on the other hand. With the regular arrangement of identical meshes investigated here, the matrix always has a banded structure and is symmetrical with respect to the main diagonal, positive definite, and non-singular. For the numerical determination of the wanted constant ε the coefficients matrix is provided with values having a known distribution. Here, in particular, a computer-generated pseudorandom homogeneous distribution is used. The system, of equations is solved for R by a modified Cholesky method. Equation (1′) can then be solved fore. The main features of an ALGOL program written for this purpose and optimized with respect to storage space requirement and computer time are discussed. Networks of triangular, square and hexagonal meshes have been investigated. The results are discussed. |
