MODELS OF DINAMIC GEOLOGICAL SYSTEMS
As a rule, geologic systems, with subsequent technologic impact change either in ‘‘geologic’’ or ‘‘technologic’’ time scale. Thus, in order to develop adequate dynamic models of geologic and technologic processes, it is necessary to introduce time factor. The time factor is of a special importance for the problems of any forecasting. Such problems indeed call for the creation and application of the mathematical models. The successful forecast may depend on the historical evaluation of he geologic system under study.
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 is 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.
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 delineated 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 approach 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.
Statistical approach, based on the empirical data, is simpler than the analytical one and is justified from the viewpoint of lithosphere evolution. It is based on the inference of interconnections through generalization, analysis, and comparison of the structural–functional features of geologic systems at certain discrete moments of the geologic time. Approximation of the discrete (discontinuous) data by a continuous function allows to obtain an empirical equation for a parameter (or a set of parameters) of the geologic system under study as a function of time.
The relationship between porosity of shales and depth of burial was studied by numerous investigators, this is due to the fact that porosity of argillaceous sediments is a complex function of numerous natural factors, often superimposed on each other. These factors include:
Effective stress (total overburden stress minus the pore pressure);
Speed of sediment deposition;
Thickness of sedimentary formations;
Shape and sorting of grains;
Amount and type of cementing material;
Chemistry of interstitial fluids.
This multitude of variables complicates the quantitative evaluation of the influence of individual factors on the porosity of argillaceous sediments.