Page 289 - Geology and Geochemistry of Oil and Gas

MODELS OF DYNAMIC GEOLOGIC SYSTEMS 251
time-dependent), and f ðxÞ is any function of the variable x. In the case of a mul-
tiphase process, a system of Equations 11.76 type is written jointly.
The second assumption allows to put together a system of differential equations,
which takes into account the effects of interrelationships among variables:

dx 1 =dt ¼ 1 ðtÞ f ðx 1 Þ þ g ðtÞ f ðx 1 Þ f ðx 2 Þ
1 12 1 2
dx 2 =dt ¼ 2 ðtÞ f ðx 2 Þ þ g ðtÞ f ðx 1 Þ f ðx 2 Þ ð11:77Þ
2 21 1 2
where x 1 and x 2 are variables (natural factors) accelerating and retarding the proc-
ess, respectively; g ðtÞ and g ðtÞ are interdependency quotients of these variables (or
12 21
natural factors), which are generally time-dependent.
In some particular cases, the factors and g may not be time-dependent, i.e., they
are constant. In those cases, Eq. 11.76 forms the so-called ‘‘model of proportional
effects’’, or an ‘‘organism growth model’’ (Volterra, 1976).
Various functions of the affecting parameters can be used in Eqs. 11.76 and 11.77.
This creates the necessary diversity in mathematical descriptions of the dynamics of
the geologic systems. For example, when f ðxÞ ¼ x, the process in Eq. 11.76 is de-
scribed by the exponential curve; when f ðxÞ ¼ xða xÞ, where a is a constant, proc-
ess is described by the logistical curve (S-like or the Gompertz curve), etc.
The signs of and g quotients in Eqs. 11.77 may vary. If the first equation has
positive 1 and negative g , then the two sign combinations are possible for the 2
12
and g 21 in the second equation. In the case of negative 2 and positive g , the
21
processes of system construction and destruction are antagonistic. In the case of
positive 2 and negative g , the processes merge into a single process controlled by
21
the same natural factors, and the prevalence of the constructive component over the
destructive one depends on the relation between these factors.
Depending on the signs of and g, the geologic processes can be stable or unstable
in time (Fig. 11.24). The former case is characterized by a point (center) or a con-
vergent spiral on the phase plane in the (x 1 , x 2 ) coordinates. The latter case is
characterized by a saddle or a divergent spiral.
Taking into account the specifics of geologic time–space continuum, the math-
ematical models (Eqs. 11.76 and 11.77) allow to forecast the status, structure, and
behavior of a geologic system at depths not yet studied through geologic techniques,
assuming a normal stratigraphic succession of consecutive time intervals.

11.3.2. Statistical approach
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 contin-
uous 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.