Ratner S.V., Southern Scientific Center
Mathematical modeling of the processes of transportation and sedimentation of the products of gas and mud volcano activity in a sea basin
Ratner S.V., Southern Scientific Center
Zaretskaya M.V., Kuban State University
Abstract
The processes of transportation of substance – gas and mud volcano production – in the water, taking into account diffusion, absorption, convection and near-bottom flows is very complicated. In order to investigate this process some mathematical models are necessary. A problem of substance transportation are solved in this paper with the use of numerical-analytical decision of transport equation.
The result of last years investigations revealed a wide-spread expansion of mud volcanism in the area of Azov and Black Seas. This unique natural phenomenon attracts attention of scientists from different areas of knowledge due to its multiple-factor influence on environment. For example, investigation of the problem what portion of carbohydrates in the Azov and Black Sea basin has natural origin as a product of mud volcano activity, helps to get precise estimation of the level of negative anthropogenic influence on the water media conditions in this region.
Mud volcanoes are the part of more common natural phenomenon –gas volcanism. The unified mechanism of gas volcanism can appear either as mud volcanism with eruptions of breccia, water, gases and solid fragments or through gas jets, which are present in the Black Sea in abundance (fig.1). Gas jets are the exits of gas streams without eruption of substantial mass of breccia, that usually form in conditions of lack of heavy layer of clay sediments. The important point in characterization of the structure of gas emanations during the eruptions of mud volcanoes in the Black Sea is the fact that the main geochemical peculiarity of describing basin is the presence of significant accumulations of reduced gases - sulphuretted hydrogen and methane in the deep-water part of sea and bottom deposits.
The series of theoretical and experimental research in the framework of international project financing by EU in 2003-2005 years were devoted to the problem of investigation and estimation of the contribution of highly intensive gas jets, located in the Black Sea, in total methane emission to the atmosphere (http://www.crimea-info.org/project3).
Figure 1. The typical seismic recording of a gas jet.
In result of this project the large quantity of new experimental data about physical and chemical processes coherent with methane’s eruption and raising was obtained. All findings are integrated in a database with gridding and available through Internet (http://www.giscenter.ru/crimea/), that helps to use them as input- data set for the purpose of mathematical modeling and simulation of mud volcano activity.
In spite of the large amount of performed works in this project there are many problems connected with mud volcano activity in the Black Sea area which are not yet studied. Russian scientists are well informed about the potential danger of methane and sulphuretted hydrogen for ecological situation not only in this region, but also in a global scale. The transport of fluids in the water under the influence of different surface and near-bottom streams is a complicated process demanding for investigation mathematical modeling and computer simulations.
In the present paper the problem of transportation of mud volcano activity products is considered. We will use the transport equation with turbulent diffusion, gravitational settling and natural resolution as a basic equation, describing a process of substance propagation:
,
here (x,y,z) – concentration of a substance; – velocity vector components in the directions x, y, and z; – a parameter of interaction between a substance and environment, a coefficient of absorption (reciprocal to the time period sufficient for diminution of concentration in times from initial value ); – coefficients of diffusion in horizontal and vertical directions; – a function of inner sources.
In case then the source of blowout is centred in a point function can be express by - function as:
,
here C – constant, which characterize a power of a source of blowout.
Will consider all processes in mesoscale approximation. The models of complex physicochemical interaction of substances and turbulent displacements are not included. But diffusion and absorption of substances, convection and the influence of surface displacements of media, which can lead to different effects for zones with different properties are taking into consideration.
The results of experimental study of sea and ocean water areas, obtained with fast probing devices revealed that vertical profiles of temperature, salinity and sonic speed have thin-layer stratification with quasi-homogeneous layers up to several dozens meters depth. Vertical profiles of flow velocity and water density also have graduated structure [4]. All this facts allows to consider the media as multi-layerd. The media movement in each layer is considered as steady and available for active measuring.
For multi-layerd media the transport equation has a following form:
(1)
Here (x,y,z) – function of concentration of substances in n-th layer; – components of velocity vector in directions x, y, z for n-th layer; – absolute value of vertical velocity due to gravity in n-th layer; – coefficients of diffusion in a vertical and horizontal directions for n-th layer; shows that point source located in i-th layer; n – number of layer.
On the interfaces between layers will also use the connection conditions in the form
i = 2, 3,…, N-1; z =. (2)
Different substances, which are ejecting in sea water can have different behavior: they can be conservative, variable, disintegrating after some reactions with substances of sea water, tailing due to gravity, they also can be gases rising up to the surface.
On the upper border will specify ether the conditions of absence tailing fractions or the condition of interception of the interface water-air for lightweight fraction, that is
or
(3)
Sea bottom in the suburbs of substances blow out has some zones with different capacity of accumulation of precipitable fractions. We will call them polytypic underlying surface. They properties will be described by parameters, characterizing the ability of selected area to keep a part of precipitable fractions in case of surface media movement.
The boundary conditions for each type of underlying surface have a following form:
, , (4)
The intensity of the process of interaction with the bottom can be described with empiric coefficient .
The problem statement (1)–(4) is quite common. By data varying it is possible to get specific problem statements for the case of settling-out of heavy admixture from the point-source, some types of transport problems for lightweight substances, in particular the problem of settling-out of fractions on the surface during the process of secondary propagation of substances taking into account different near-bottom and near-surface streams.
Hence we will get mixed boundary problem for a transport equation. Zones cover the plane хОу completely and in general case they can be unconnected. For the purpose of simplicity will consider them as simply connected.
Following [5], it is possible to get a problem solution for the case of homogeneous underlying surface and then summarize it for a case of different types of bottom surface [6].
A software implementation and verification of presented model is included as a stage in a new international project (Russia-Turkey-Ukraine), targeting the risk-map of Black-Azov Seas basin construction. Using this map it will be possible to specify zones of high-risk for navigation and trunk pipeline due to high probability of mad and gas volcano activity.
The work is partly financed by RFBR (projects 08-08-00447-а and 07-05-00858) and The Program 16 of the Presidium of Russian Academy of Science.
References
1. Klerkx J., 2002. Contribution of high-intensity gas seeps in the Black Sea to methane emission to the atmosphere. The CRIMEA project. NewsLetter & Information Service of the E.G.S., Issue №01, 10 November 2002.
2. Seismoactive fluid- abyssal systems of Northern Caucasia. М.: Institute of Physics of the Earth RAS, 2005, 225 p (in Russian).
3. Marchuk G.I. Mathematical modeling in environmental problems. Nauka, М., 1982 (in Russian).
4. Monin А.S., Fedorov К.N., Shevcov V.P. About vertical mezo- and microstructure of ocean streams // Doklady Akademii Nauk USSR. 1973. V.208. No. 4. pp. 833–836 (in Russian).
5. Zaretskaya M.V. Modelling the process of macro transportation in medium of complicated structure of parameters distribution // Computational technologies, V. 8, № 5, 2003, 58–62 (in Russian).
6. Zaretskaya M.V. Investigation of the influence of heterogeneity of bottom zones on the process of transportation in layered medium // Ecological bulletin of scientific centers of BSEC, 2003. No.1. С.42–45. (in Russian)