by Oleg KUZNETSOV, Dr. Sc. (Technol.), Andrey KARAKIN, Dr. Sc. (Phys. & Math.), Tatiana LEVCHENKO, Cand. Sc. (Technol.); All-Russia Research Institute of Geological, Geophysical and Geochemical Systems
Twenty years ago one of the authors of the present article published a material in the Russian-language magazine Nauka i zhizn (Science and Life) on nonlinear effects observed in studying the earth's interior; he stressed that such effects should be incorporated in mathematical models of geologic media, too. A new discipline, nonlinear geophysics, was thus born.
This publication touched off a spate of comments in the press and elsewhere. Research teams, set up here and there, got busy with studying these odd processes in the earth's crust. A few years before that, in 1980, the All-Union Research Institute of Nuclear Physics and Geochemistry (today the All-Russia Research Institute of Geological, Geophysical and Geochemical Systems) founded a specialized laboratory, the world's first.
Now let's take a simple example. Filling a cylindrical glass with water, we shall see at first glance that the dependence between the water's volume and level is linear. But when the glass is full, just one drop will cause the water to run over the brim. A drop too much! But why so? Because in this case two nonlinear effects are at work: the first - when the addition of more water to the glass will not change its level, and the second (the trigger effect) - when a weak impact entails major consequences.
A NEW TREND IN EARTH SCIENCES
Geophysics, concerned with the exploration and development of mineral deposits, also studies the technogenic effect on the geological medium. Its object is the earth and its bowels. The structure and elements of this object of research are incompatible in scope with the size of a laboratory, while the ongoing geologic processes take millions and billions of years.
Consequently, we cannot observe most of such processes in laboratory conditions in full. Held experiments are very costly and laborious, they involve large expanses, whereas the available research methods are not accurate enough. Small wonder that in some respects humankind knows far less about the in-depth life of its "cradle" than it does about stellar systems lying at colossal distances away. And if now we can attempt describing processes, nonlinear for the most part, this is due to contemporary computer hard- and software that makes it possible to include novel parameters into the arsenal of mathematical models and research methods.
The earth's upper crust, which is composed mainly of solid and brittle
Nonlinear processes in the upper crust: 1 - plastic deformations, earthquakes, seismoelectric effect, electromagnetic radiation; 2 - hydrodynamic, vibratory and acoustic effects on a deposit; 3 - slow migration of liquids and gases, chemical and physicochemical conversions.
water-saturated rock and which is of the utmost interest to man, concentrates all useful deposits accessible to mining as well as sources of earthquakes-not lying deep but awful in their action. These nonlinear phenomena are accompanied by emission of elastic waves (the main destructive factor), cross thermodynamic effects (mechanical-electrical and mechanical-chemical), along with kinetic impacts (which occur in the "beaks" of fractures). That is why one of the most important aspects of this particular domain of geophysics involves the dynamics of the upper strata of the crust and their elements.
A separate line of the earth sciences deals with the planet's liquid media (central core, ocean), in particular, with the motion of relevant mixtures playing an important part in the vital activity of man and in the interaction of heterodox phenomena and structural changes in the earth's shell. For instance, soil and the bottoms of water pools contain liquescent (diluted) rocks: sols (colloidal solutions where tiniest particles ravel freely), water suspensions (where the solid phase either settles or comes up readily) as well as emulsions (drops within a liquid). The dynamics of like substances (aerosols and fumes in the atmosphere, too) is well known by and large.
But it is not as simple when we come to mixtures of the much concentrated solid (or quasisolid) fraction. This is a huge class of processes (sedimentation in the ocean, bottomset beds) that should be considered in technogenic activities. For instance, the mining of minerals often goes along with the dilution of rocks. Also, pipe laying over thousands of kilometers across swamps, drift sands, lakes and rivers is hampered by bottom mud, slime and noncompacted sediments. But when disturbed and put in motion, the quasisolid structures that impede work are destroyed. Only knowing their dynamics can we control and optimize the physicochemical reactions proceeding there.
As a matter of fact, any geophysical system is multifactorial (what with different temperatures, pressure, density and the like). It comprises many elements and stages, and is active under all aggregated conditions (solid, liquid and gaseous states, plasma), which are interchangeable and which make up an integral whole. Such interconversions and interchanges of process stages as well as any qualitative modifications of underlying processes are nonlinear as a rule. The trigger effects (earthquakes, changes of the climate and oceanic currents, etc.), which play a major role in nature, are but a particular instance of this phenomenon.
TRADITIONAL NONLINEAR MODELS IN GEOPHYSICS
The group theory method that passed from abstract mathematics to theoretical physics in the early 20th century is a potent tool in studying corresponding relationships. It allows us to go into the general characteristics of equations without their solution. Besides, constructions on the basis of Lagrangian and Hamiltonian functions* have attained fundamental importance now. They are used in deriving formulae of motion for complex physical fields in conformity with the laws of invariance** and conservation (of matter, momentum, energy).
Nonlinear wave mechanics is an essential part of this trend in geophysics. The point is that we obtain the lion's share of experimental information on plutonic matter with the aid of seismic surveys (over long distances) and acoustics (within a well, or hole). We should be able to tell apart dynamic and kinematic processes. Inertial forces play a significant role in dynamic processes (acoustic and shock effects propagating in water, air and solid bodies), while viscous, elastic and other effects triggered by changes in the conditions of matter are implicated in kinematic processes. Solitions (solitary waves) are the most graphic and widely known example of nonlinear oscillations. They move like elastic particles and disperse when colliding and, though changing their velocity of motion, preserve the same form. Similar effects accompany the tsunami, too. An equation deduced for them is also valid for describing other
* With reference to Joseph Louis Lagrange and William Rowan Hamilton. J. L. Lagrange (1736 - 1813) - French mathematician and mechanic, honorary member of the St. Petersburg Academy of Sciences (1776); W.R. Hamilton (1805 - 1865)-(Irish mathematician, honorary corresponding member of the St. Petersburg Academy of Sciences (1837). - Ed.
** Invariance - what is not changing or capable of being changed, e.g. a value remaining constant with the change of physical conditions or during certain transformations. - Ed.
phenomena. Meanwhile its application domain keeps expanding, something to excite a mathematician's interest. The most amazing characteristic of this relationship is that it can be transformed into a linear dependence amenable to solution through state-of-the-art methods.
NEW RESEARCH METHODS IN GEOPHYSICS
Study of complex processes calls for unorthodox approaches like, say, the pattern recognition method (identification of objects preassigned to a definite class) or the synergetic, statistical and stochastic (probabilistic) method relying on a body of mathematics. We resort to such approaches when fundamental natural laws are little known or when their use poses difficulties, e.g. in map making, prospecting for mineral deposits or ecological problem solving. In this case we proceed from a dependence (usually a nonlinear one) among values characterizing a given phenomenon, which are often not explicit either. The earth sciences make use of what we call the geoinformation (documentation) science relative multifactorial objects.
The equations of hydrodynamics deduced more than two hundred years ago (their variables include such macrocharacteristics as pressure, temperature, density, etc.) must be one of the first nonlinear models. Using local laws of conservation of matter, momentum and energy, these equations have provided a basis for contemporary geophysical theories for subterranean hydrodynamics and hydrodynamics of mixtures. Yet the problem of formulating essential characteristics for specific media is still outstanding in some cases, including a mathematical recording of the condition of matter. This is one of the priorities of nonlinear geophysics.
Of special importance here is a body of mathematics for describing open systems. Unlike closed (isolated) systems these comprise a large number of elements (atoms and molecules in chemistry and physics; cells and microorganisms in biology; planets and stars in astronomy; such elements exchange energy, matter and even information with ambient bodies, they are capable of gradual self-organization thus giving rise to more complex and odd structures.* Two tendencies are proper to any geologic open system, and these are toward irreversible changes as a whole, on one hand, and toward equilibrium of each constituent element, on the other; hence an open system like that is an object of synergetics. In keeping with the well-known definition formulated by Dr. H. Haken of Germany in the mid-1980s, it studies systems composed of many subsystems whose interaction results in the appearance of spatial, temporal (or mixed) structures on a macroscopic scale. However, it is often hard to tell whether this occurs in consequence of self-organization: every new stage of a corresponding process "erases" the information on the previous one. And adequate description and modeling is just as well difficult, too.
Here a new, kinetic method of the synergetics of geologic systems (worked out by Drs G. Zmievskaya and A. Bondareva - M. V. Keldysh Institute of Applied Mathematics, RAS; and by Dr. T. Levchenko, Research Institute of Geosystems) may be helpful: it allows to address an important and still obscure area of the earth sciences - that of nonequilibrated fluctuating transformations (i.e. characterized by random deviations from mean physical values). These include the incipient phases of matter as part of a complex model taking in the hydrodynamic and kinetic description of a medium. Such kind of description covers the role of different physical mechanisms of phase transitions in the earth's interior that lead, e.g., to condensation of water vapor and metal in cracks and fissures at various depths, to defects inside minerals and on their surface, and cavitation (formation of cavities in liquids filled with gas and vapor or their mixture).
What concerns condensation, it has been studied well for gas-vapor media responsible for the formation of aerosols, clouds and fogs. But we know all too little yet about the behavior of supersaturated vapors in plutonic fissures, faults and open pores (apertures) in a wide range of thermodynamic parameters which are of great practical significance for the exploration of oil and gas deposits. The same is true of physical processes occurring deep within the earth and connected with the appearance of free electric and space (bulk) charges, atmospherics-type discharges and the transition of mechanical stresses to thermal effects. All that produces strong electric fields that cause ionization of gas and formation of gas-discharge plasma. Today technologies are being developed for exploiting it for the extraction of metal from ore.
Clusters of defects arising in a solid body's lattice at high temperatures (a process similar to condensation of liquids) cause microcracks and other changes in rock, which should be taken into account in evaluating the rock structure. The mechanism of cavitation, too, is of no less practical significance (when a new phase "germinates" from smallish cavitation bubbles of different dimensions in a viscous liquid): it is this mechanism that is at work in the acoustic action on productive oil-and-gas strata, and in the movement of oil and gas in pipelines.
Likewise important are nonlinear effects in the atmosphere, especially nonsteady-state effects in the magnetosphere and ionosphere, since they are associated with transformations inside our planet.
All these phenomena, so much different in their physical essence, can fit into one mathematical model, for they are particular cases of diffusion and, consequently, a good object for what we call the stochastic (probabilistic) analog method. Developed for solving quasilinear and nonlinear problems on a spatial-temporal scale compatible, among other things, with the microstructure of matter, this method is applied for the description of water and metal vapor condensation. Its application domain can be
* See: A. Volynsky, "Self-Organization of Matter: Universal Phenomenon", Science in Russia, No. 3, 2003. - Ed.
extended via complexification of physical conditions for a given process, introduction of new macroparameters and thermodynamic characteristics of the medium, and so on.
This method may be used for synthesizing models describing the formation of cavities and clusters of defects and furthermore, for developing comprehensive models with respect to multifactorial physical phenomena. It would be pertinent to recall here that mountain rocks of inorganic origin have a crystalline structure. When deformed or shifted, their "grains" give rise to linear and point defects that cause a release of excessive free energy. Since the linear defects contribute most of all to the pool of this energy, they are considered oftener in modeling a solid body's destruction. However, the point defects may be taken into account as well: at definite temperatures and stresses they appear in the shape of clusters and, if present in subsurface layers, come to be implicated in pore formation. Besides, during natural seismic phenomena (earthquakes, volcanic eruptions) and during technogenic, particularly, acoustic, pulse-initiated effects on productive horizons (techniques practiced for stimulating the influx of hydrocarbons), the gas phase in the form of tiny disperse bubbles is formed within the gas phase. It is here that nonlinear effects - induced by pulsations and accompanied by cavitation - are born.
A body of mathematics with stochastic differential equations has
already been used for assessing residual reserves at various stages of oil pool development. We deem it advisable to apply this array for direct problem solving, say, at the preparatory stage prior to mining. Using it in combination with geological exploration (seismic surveying) data, field workers can estimate oil reserves in wells. This information is likewise used for planning a network of such wells, for determining the actual volume of water injection as well as for selecting appropriate methods of bed stimulation... Relying on such information, we can forecast oil-extraction evolution in time, consider different scenarios of field work, and optimize some of their parameters.
As to indirect problem solving in course of mining, such indicators as production rate (yield, output), oil withdrawal from liquid, the number of running and injection (input) wells and bed stimulation characteristics make it possible to refine data on commercial deposits of particular raws computed by other methods and determine how investments affect the output rate. Finally, all this information may underwrite a direct task of forecasting further mining prospects for a particular deposit.
NONLINEAR EFFECTS SERVING MAN
All these processes are studied and widely used for the exploration of mineral deposits as well as optimization and intensification of their output, of oil in particular. More often than not it becomes necessary to spot oil-rich strata by using geophysical methods, for instance, by bed stimulation with a strong acoustic field. It modifies the medium's physical characteristics, including the phase state of oil and gases dissolved in it and, as a consequence, the correlation of heavy and light fractions. This results in the destruction of pseudocrystals of paraffin wax and other high-molecular hydrocarbons and in the modification of acoustic parameters of the field. Measuring them we can tell whether oil or water is present there. An acoustic field thus modifies the medium and at the same time evolves as an indicator of such modification-we come to deal with a set of nonlinear phenomena.
Proceeding from these principles, Dr. V. Dryagin (Dr. O. Kuznetsov's pupil) worked out an effective technology for determining a degree of saturation of oil-bearing formations, now widely applied by geophysicists in practice. Furthermore, a high-frequency acoustic field is employed for other purposes as well - e.g. for acting upon productive strata so as to recover their permeability. A response to a low-frequency acoustic field (a few Hz units) invited the idea of direct prospecting for hydrocarbons. All that gave birth to an innovative set of methods, ANCHAR, developed in this country and allowing to explore for promising territories both on land and offshore (continental shelves of seas and oceans).
The use of a quasistationary electric field likewise changes the characteristics of seamy (fissured) rocks and causes a variety of electrochemical and electrokinetic processes. The technology of well casing, which makes use of electrolysis, electrophoresis and other techniques, is based on this phenomenon.
A number of nonlinear effects is caused by a mechanical and hydrodynamic action on beds for oil displacement. The mechanical technique employs powerful vibrators generating low-frequency oscillations in rock. The other, hydrodynamic technique provides for the injection of water or alkali, acids or surfactants. However, lately we have been trying to avoid chemical reagents for ecological safety reasons.
A wealth of practical experience combined with software data gives a powerful impulse toward devising new theoretical constructions for exploration of useful minerals, including methods of their prospecting and surveying, along with prognostication models for technogenic effects on the earth crust. Even though in nonlinear geophysics we are making our first strides toward an in-depth understanding of the world we live in, we shall certainly accomplish this purpose and learn more about our planet, its origins, evolution and future.
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