Page 245 - Geology and Geochemistry of Oil and Gas

212 MATHEMATICAL MODELING IN PETROLEUM GEOLOGY

where H r is the relative entropy, H max is the maximum possible entropy, and p is the
i
fraction (probability) of the ith component of the N-component system.
The H max is determined as follows:
N
X
H max ¼ ð1=NÞ ½logð1=Nފ ¼ log N (11.6)
i¼1

The H r (even as H entropy) becomes maximum when all the component fractions
are equal to each other, and becomes minimum when one of the components is
certain and the others are impossible.
The relative entropy, H r , is convenient in quantitative studies and graphic rep-
resentations of relative heterogeneity of multicomponent systems. It may be em-
ployed in investigations of heterogeneity of sediments and rocks, laterally or
vertically, as well as in the studies of ternary mixtures encountered in lithology,
geochemistry, and hydrochemistry, for example. A triangular diagram of a three-
component mixture with relative entropy contours is presented in Fig. 11.5. A point
within the triangle will be represented by a number expressing the degree of het-
erogeneity of the system. The maximum relative entropy (H r ¼ 1) lies in the center of
the triangle and the minima (H r ¼ 0) at the apexes.
A quantitative rating of the heterogeneity of different types of clastic rocks can be
made using this triangle. The classification diagram of clastic rocks includes eight
major types (Fig. 11.6): (1) sand, (2) silty–clayey sand, (3) sandy–clayey silt, (4) silt,
(5) sandy–silty–clayey rock, (6) clayey–sandy silt, (7) silty–sandy shale, and (8) shale.
The most homogeneous ones (sand, silt, or shale) have the relative entropy of 0.45.
The least homogeneous rocks (silty–clayey sand, sandy–clayey silt, clayey–sandy silt,
and silty–sandy shale) have the relative entropy of 0.78. Unsorted rocks have the
relative entropy of 0.83–0.85 depending on the clay content; loam, 0.90; and a
mixture of equal proportions (by wt) of sand, silt, and clay, about 1.

Fig. 11.5. Relative entropy of ternary mixture.