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250 MATHEMATICAL MODELING IN PETROLEUM GEOLOGY

the geologic system under study (Buryakovsky et al., 1982; Buryakovsky and
Chilingarian, 1991b).
The necessity to take geologic time into account meets with significant difficulties.
One of the reasons for this difficulty is the use of absolute and relative geologic time
scales. The difference between them is substantial: the absolute time scale has the
beginning common for the entire Earth, which is not an attribute of the relative time
scale based on paleontology and stratigraphy. Another reason is the lack of repro-
ducibility of the geologic time in physical and chemical experiments.
Two methods in constructing the dynamic geologic models may be offered:
analytical and statistical. The better approach in modeling such systems is the
combination of mathematical analysis (i.e., differential equations) with the statistical-
probabilistic assignment of numerical values for the parameters, causing the change in
dynamic geologic systems. This approach allows the deterministic description of main
features of the dynamics of the geologic systems. At the same time, it allows to account
for the statistical-probabilistic nature of various geologic parameters, which cause the
evolution of the systems. The implementation of analytical solution is accomplished
using the statistical sampling technique or the so-called Monte Carlo method.

11.3.1. Analytical approach

Two important issues must be addressed before constructing analytical models:
1. The key properties of the system under study, as well as those of the surrounding
rocks, should be defined. These properties should be described by strictly defined
quantitative constraints.
2. The limitations assumed in describing these properties should be clearly delin-
eated and should reflect the substance of a particular geologic system.
It is natural to choose as the main parameters those properties of the system
and of the surrounding rocks, which would stimulate or restrain the course of the
geologic processes.
In the following discussion, the writers use as synonyms the properties of the
geologic system and their respective parameters. They may have a dual nature, i.e.,
they may be either deterministic or stochastic, depending on the formalization ap-
proach at each stage of simulation of a geologic system.
Two significant assumptions ought to be made while developing the differential
equations of geologic processes.
1. The rate of change of the geologic system, or the speed of the geologic process, is
proportional to the state of the system.
2. Influence of various natural factors is proportional to the product of the number
(or quantitative estimates) of the events accelerating the process by the number
(or quantitative estimates) of the events retarding the process.
The first assumption leads to the differential equation similar to

dx=dt ¼ ðtÞ f ðxÞ (11.76)
where x is a variable (quantitatively measured natural factor) describing the
evolution of geologic system, ðtÞ is a coefficient of proportionality (generally