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\title{Application of Numerical Methods for New Estimate of Rheology Constants in the 2D Computer Model of the Mantle Wedge Thermal Convection as a Possible Physical Mechanism of Hydrocarbons Transport}
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             \author[1]{  S.V.Gavrilov}

             \author[2]{A. L.  Kharitonov}

             \affil[1]{  Schmidt Institute of Physics of the Earth}

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\date{\small \em Received: 6 December 2019 Accepted: 31 December 2019 Published: 15 January 2020}

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\begin{abstract}
        


For both Newtonian and non-Newtonian mantle rheology laws, the numerical model of the 2D dissipationdriven mantle wedge thermal convection is constructed for the case of subduction of the Black sea micro-plate under the Crimea peninsula with the account taken of the phase transitions in the mantle. The horizontal extent of the positive 2D heat flux anomaly zone localized in the rear of the Crimea mountains is shown to correspond to the model subduction velocity ?10 mm per year for the water content of one weight %. For Newtonian rheology upwelling convective flow transporting heat to the Earth?s surface is formed at the subduction velocity of ~102 mm per year, which appears too excessive and probably evidence of that the non-Newtonian rheology dominates in the mantle wedge. In the case of non- Newtonian rheology, the velocity in convective vortices in the mantle wedge exceeds ?10 m per year. The subduction velocity may be less than 10 mm a year for the water content in the mantle wedge over ~1 weight %. The upwelling convective flow is shown to transport mantle hydrocarbons to the Earth?s surface since the zone of oil and gas accumulation coincides with the 2D one of heat flux anomaly.

\end{abstract}


\keywords{2D thermal convection, Newtonian and non-newtonian rheology constants, phase transitions, hydrocarbons transport.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
nteraction of the lithospheric plates in the Crimea-Caucasus region leads to the thrusting of the Black Sea micro-plate under the Crimea peninsula (under the Scythian plate)  {\ref [Nimetulayeva, 2004]}. As a consequence, the seismic focal plane is formed along which the Crimea ascends as the result of seismic jerks. The velocities of vertical uplift of the Crimea mountains and sinking of the near-Crimean area of the Black Sea micro-plate equal to \textasciitilde 4 mm per year and \textasciitilde 10 mm per year, respectively. Mountainous Crimea is a folded fault region being a part of the Alps-Himalaya-Indonesia belt \hyperref[b14]{[Yudin, 2001]}.\par
In  {\ref [Ushakov et al., 1977]} the subduction velocity of the Black Sea micro-plate under the Crimea peninsula is estimated of \textasciitilde 1 mm per year as the best fit to the observed sedimentary layer distribution. Other estimations are unknown to the knowledge of the authors. However the obtained estimate of \textasciitilde 1 mm per year appears to be an underestimate, being not correspondent to the vertical velocities of \textasciitilde 4 and \textasciitilde 10 mm per year of Mountainous Crimea and the Black Sea micro-plate.\par
According to \hyperref[b1]{[Gavrilov, 2014;} {\ref Gerya, 2011;} {\ref Gerya et al., 2006}] two types of dissipation driven smallscale thermal convection in the mantle wedge are possible, viz. 3D finger-like convective jets, raising to volcanic chain, and 2D transversal Karig vortices, aligned perpendicularly to subduction. These two types of convection are shown to be spatially separated due to the pressure and temperature dependence of mantle effective viscosity, the Karig vortices, if any of them formed, being located behind the volcanic arc \hyperref[b1]{[Gavrilov, 2014]}. Despite the firmly established localization of the seismic focal plane there is just a single definite conclusion concerning the velocity of subduction of the Black Sea micro-plate  {\ref [Ushakov et al., 1977]}. It is not completely clear if volcanism played a substantial role in forming Mountainous Crimea, or the mountains are of a purely thrust-and-fold origin.  {\ref [Nimetulayeva, 2004]} indicates the contradictory statements on the Crimean volcanism to have been published, however in Fig. 2.4 in  {\ref [Nimetulayeva, 2004]}, the volcanic eruption in the Mountainous Crimea is depicted. The abovementioned picture is reproduced here in Fig. \hyperref[fig_6]{1} with the convective vortices drawn additionally. It is worth assuming the two heat flux anomaly maxima observed in the south of the Crimea peninsula  {\ref [Smirnov, 1980;} {\ref Nimetulayaeva, 2004, Fig.2.4]} owe their origin to respectively 3D and 2D upward convective heat transfer from the mantle wedge to the Earth's surface (see Fig. \hyperref[fig_6]{1} of this paper). The latter 2D maximum located in the rear of the Mountainous Crimea is much greater as compared to the former 3D maximum located in the Mountainous Crimea. The 2D I © 2020 Global Journals lobal Journal of Researches in Engineering ( ) Volume Xx X Issue I Ve rsion I heat flux anomaly maximum is associated with the 2D upward convective flow in the mantle wedge. Numerical modeling of 2D mantle wedge thermal convection occurring in the form of the Karig vortices and presumably transporting heat to the Earth's surface in the rear of the Mountainous Crimea allows judging about the mean velocity of subduction of the Black Sea micro-plate under the Crimea peninsula as well as about the rheological mantle parameters. The horizontal extent of the 2D heat flux anomaly in the rear of the Mountainous Crimea is shown to correspond to the mean subduction velocity >10 mm per year for the observed subduction angle 15°. Numerical convection models accounting for the effects of phase transitions as well as the pressure, temperature, and viscous stresses viscosity dependence fit in well with the heat flux observational data in the case of non-Newtonian mantle rheology at the mean concentration of water in the mantle wedge of \textasciitilde 1 wt. \%. 
\section[{II.}]{II.} 
\section[{Algorithm and Computation Complexity}]{Algorithm and Computation Complexity}\par
Thermo-mechanical model of the mantle wedge between the base of the overlying Scythian plate and the upper surface of the Black Sea micro-plate subducting under the Scythian one with a velocity V at an angle ? is obtained for the infinite Prandtl number fluid as the solution of non-dimensional 2D hydrodynamic equations in the Boussinesq approximation for the stream-function ? and temperature T.) 660 ( ) 660 ( ) 410 ( ) 410 ( 2 2 2 2 2 2 ? ? 4 ? ) ( ? ) ( x x x xz xz xx zz xx zz Ra Ra RaT ? ? ? ? ? ? ? ? ? ? ? ? ? ? , (\textbf{1})Q Ra Di T T T T ik z x x z t ? ? ? ? ? ? ? ? ? 2 ? ? ? 2 , (\textbf{2})\par
Here ? is dynamic viscosity, ? and indices denote partial derivatives with respect to coordinates x (horizontal), z (vertical) and time t , ? is the Laplace   z x V ? ? , x z V ? ? ? ,\textbf{(3)} 
\section[{RT pV E}]{RT pV E}b h A m * * * exp 2 ? ? ? ? ? ? ? ? ? ? ? ? , (\textbf{5}) T z) 1 ( 72 . 6 8 . 14 exp 7 - 10 5.0 ? ? ? ? ? , (\textbf{6}) RT pV E m b h n r w AC * * exp * 1 ? 2 1 ? ? ? ? ? ? ? ? ? ? ? ? , (\textbf{7}) ] ? 2 2 / ) ? ? [( ? 4 ? 2 2 2 2 xz xx zz ik ? ? ? (8) non-dimensional viscosity is T z) 1 ( 0 . 5 0 . 10 exp ] 2? /2 ) ? - [(? .00 1 ? 1/3 2 xz 2 xx zz ? ? ? ? ? ? . (\textbf{9})\par
Following  {\ref [Trubitsyn \& Trubitsyn, 2014]} we assume the phase functions Where for "wet" olivine A=5.3×10 15 s -1 , m=2.5, the grain size h =10 -1 -10 mm, b * =5×10 -8 cm is the Burgers vector \hyperref[b16]{[Zharkov, 2003]}, E * =240 kJ . mol -1 is activation energy, V * =5×10 3 mm 3. mol -1 is activation volume, ? =300 GPa is the shear modulus normalizing factor, R is the gas constant. At the chosen constants and the grain size h=1.6 mm, non-dimensional viscosity also denoted To check as to how the estimate of the velocity of subduction of the Black Sea micro-plate is sensitive to the accepted linear rheological law here we make extra computations for non-Newtonian rheology, in which case the viscosity formulae (5)-Where according to  {\ref [Trubitsyn, 2012]} for "wet" olivine n=3, r=1.2, m=0,) (l ? as ? ? ? ? ? ? ? ? ? ? ? ? ) ( ) ( ) ( ) ( 1 2 1 l l l w T z z th , ) ( ? ? ) ( ) ( 0 ) ( ) ( 0 ) ( l l l l T T g z T z ? ? ? ,\textbf{(10)}2 / 1 2 ) ? ( ? ik = , E * =480 kJ . mol -1 ,\par
V * =11×10 3 mm 3. mol -1 , A=10 2 ? -1 ×(MPa) -n , C w >10 -3 for "wet" olivine is the weight water concentration (in \%\%).\par
It should be noted the constants in \hyperref[b6]{(7)} vary considerably in the papers referred to by  {\ref [Trubitsyn, 2012]} and heretofore, we gave averaged values of constants. At C w =10 -3 on accounting for where the signs are changed as z-axis is pointing upwards, ) ( ) (    
\section[{T z l}]{T z l}? ? ? ? ? ? ? ? ) ( ) ( 0 ) ( ) ( 0 2 ) ( ) ( ) ( ? / ) ( ? ? 2 ? , (\textbf{11}) x l l l l l l l l l l l x T w T T Ra z z ch w Ra ? ? ? ? ? ? ? ? ) ( ) ( 0 ) ( ) ( ) ( ) ( 0 2 ) ( ) ( ) ( ) ( ) ( ) ( ? ?? ? 2 ? ? ?? . (\textbf{12})\par
Equations ( \hyperref[formula_0]{1})-( \hyperref[formula_2]{2}) are solved for the isothermal horizontal and insolated vertical boundaries regarded no-slip impenetrable ones except for the "windows" for in-and outgoing subducting plate, where the plate velocity is specified. Vertical boundary distant from subduction zone is assumed penetrable at right angle, the latter boundary condition appears not too imposing in the case of very flat subduction. Q in ( \hyperref[formula_2]{2}) is non-zero in the continental and oceanic crust 40 and 7 km thick. Initial vertical boundaries temperature is calculated for the half-space cooling model for 10 9 yr and 10 8 yr for Scythian (continental) and Black Sea (oceanic) plates respectively. 
\section[{III.}]{III.} 
\section[{Results and Discussion}]{Results and Discussion}\par
Assuming the second (more remote from the trench) heat flux q maximum in Fig. \hyperref[fig_6]{1} appears above the convective flow, ascending to C 2 point in Fig. \hyperref[fig_6]{1}, and the convection cell dimension is equal to the two adjacent q minima separation (i.e. the q minima are located above the descending convective flows) we can estimate the convection cell dimension as \textasciitilde 250 km. To preliminarily access the mean velocity of subduction of the Black Sea micro-plate the coordinate\par
x dependence of the growth rate ? ? ( x ) of transversal convective rolls for the constant viscosity fluid model can be allowed for. In such the model the averaged temperature and pressure viscosity dependence is accounted for in an averaged manner, the factor describing the temperature-and pressure viscosity dependence being equal to its mean value \hyperref[b1]{[Gavrilov, 2014]}.\par
Analytical formulae in \hyperref[b1]{[Gavrilov, 2014]}   It should be noted the growth rates ? ? ( x ) are viscosity independent as convection is driven by viscous heat release (which is directly proportional to viscosity), while, on the other hand, the greater is the viscosity the more difficult is to arouse the convection. Fig. 2 clearly demonstrates the convective zone with ?? ( x )>0 amounts to ? ? 1 2 x x\par
250 km (i.e. the single convective cell of ?250 size is actually aroused) at V=40.5 mm per year, the latter value being a preliminary estimate of the mean subduction velocity. The ? ? ( x ) maximum is ?320 km distant from the trench which is very close to the distance from the trench to the observed 2D heat flux anomaly (?400 km, see Fig. \hyperref[fig_6]{1}).\par
To compute more accurate consistent model of small-scale convection in the mantle wedge between the overriding Scythian plate and subducting Black Sea micro-plate it is necessary from the computational point of view first to specify vanishing non-dimensional numbers Ra ?0, Di =0 in ( \hyperref[formula_0]{1})-( \hyperref[formula_2]{2}), i.e. to ignore convection and viscous dissipation. This approach is applied as convection with Ra and Di (4) passes through very vigorous stages, and the time steps in integrating ( \hyperref[formula_0]{1})-( \hyperref[formula_2]{2}) become too small thus making it difficult to model the thermal structure of the plates. Solving (1)-( \hyperref[formula_2]{2}) by the finite element method in space on the grid 104×104 and the 3-rd order Runge-Kutta method in time one obtains for Ra ?0, Di =0 and V=45 mm a year non-dimensional quasi steady-state ? and T shown in Figs. \hyperref[b2]{3}, where the streamlines are depicted with step 0.25 and the isotherms with an interval of 0.05. Subducting plate was considered rigid, while the viscosity at the zone of plates friction (at temperatures below 1200 K) was reduced by 2 orders of magnitude as compared to \hyperref[b4]{(5)}. The latter viscosity reduction at the plates contact zone accounts for lubrication effected by deposits partially entrained by the subducting plate. Such a lubrication prevents the overriding Scythian plate from gluing to the subducting one  {\ref [Gerya, 2011]}. It is worth noting the isotherm T=0.15 in Fig. 3a,c approximately corresponding to the Earth's surface is depressed at subduction zone by\textasciitilde 7 km which is of the order of a typical trench depth. Fig. \hyperref[fig_5]{3} shows the results of computation for formulae ( \hyperref[formula_7]{7}) -(9) for non-Newtonian rheology case for the water content C w =10 -3 weight \%\% (Fig. 3a, b) and C w =3×10 -1 weight \%\% (Fig. 3c, d). The velocity V= 45 mm per year is chosen as resulting in the best convective zone size fitting in with the observed heat flux (positive and negative) anomaly size at the point C 2 in Fig. \hyperref[fig_6]{1}, i.e. in the rear of the Mountainous Crimea. The Black Sea microplate subducting with a given velocity V is considered rigid and is shown in Fig. 3b,d by the equidistant diagonal streamlines. The induced mantle wedge flow above the subducting plate is seen to occur in the form of a single vortex at C w =10 -3 weight \%\% (Fig. 3b These convective vortices are seen actually to correspond to a single convection cell aroused at subduction velocity V=45 mm per year. The latter convection cell dimension is of the order of \textasciitilde 300 km, i.e., is very close to the observed minima q separation under the C 2 point in Fig. \hyperref[fig_6]{1}.\par
Thus the for the non-Newtonian mantle wedge rheology case with the viscosity reduced by 3 orders of magnitude as compared to (7)-9) the computation shows the convection in the mantle wedge to occur at C w =3×10 -1 weight \%\% in the form of two micro vortices at V=45 mm per year. Convection of this type can provide abnormal 2D heat flux q observed in the rear of the Mountainous Crimea and the upwelling of the mantle hydrocarbons to the Earth's surface along the arrow "c" \hyperref[b15]{[Yudin, 2003]}. Considerable velocity in convective vortices in Fig. 4 is due to the local viscous stresses increase resulting in the drop in viscosity in convective zone. In the case of Newtonian rheology the convection is aroused at the subduction velocity of over 10 2 mm×a -1 , which appears unrealistic.\par
According to  {\ref [Zharkov, 2019, p.143}], the water content in the mantle transition zone in the mantle wedge may amount to \textasciitilde 3 wt. \%. To investigate the role of water infused into the mantle wedge from the subducting slab the above computations were carried out for the mean water content of 1 wt. \% and subduction velocity of 30, 20, and 10 mm per year. The results of the convection computation are shown in Figs. 5a and 5b for V=30 and 20 mm per year respectively, where the streamlines corresponding to subducting Black Sea micro-plate are shown with the interval of 10, and the streamlines, corresponding to convective vortices with the interval of 10 6 . The mean non-dimensional velocity in the left micro-vortex are \textasciitilde 15.2×10 7 , \textasciitilde 7.1×10 7 and \textasciitilde 0.05×10 7 for the velocity of subduction of V = 30, 20, and 10 mm per year respectively. Thus, the convection may be considered to arise at the subduction velocity over \textasciitilde 10 mm per year for the mean water content C w \textasciitilde 1 wt.\%. Since the meant water content in the mantle wedge could hardly exceed \textasciitilde 1 wt.\% even at the water content in the mantle transition zone of 3 wt\%, the obtained subduction velocity of \textasciitilde 10 mm per year may be regarded the minimum estimate of that of subduction of the Black Sea micro-plate.\par
It is worth noting, that in the case of Newtonian rheology, the mantle wedge dissipation-driven convection in the form of transversal rolls, as in Fig. 4, is characteristic of very small subduction angles, the convection of this type being absent already at subduction angle ?=30° \hyperref[b2]{[Gavrilov \& Abbott, 1999]}. At the subduction angle under consideration here, ?=15°, the convective transversal rolls do not appear at V<10 cm×yr -1 for the Newtonian rheology case. Arrow (c) above the boundaries of the oppositely revolving convective vortices in Figs.4, 5 indicate a possible direction of transport of non-organic mantle hydrocarbons to the Earth's surface. Computations for Newtonian mantle rheology with the viscosity ( \hyperref[formula_5]{5})-( \hyperref[formula_6]{6}) shows the transversal rolls to be aroused at far greater distance from the trench than the observed 2D heat flux anomaly. the model constructed here favors the non-Newtonian mantle wedge rheology as better fitting in with the observed heat flux anomaly localization. It should be noted that numerous thermo-mechanical mantle models in the zones of subduction (see, e.g.  {\ref [Gerya et al., 2006;} {\ref Gerya, 2011]} and the vast number of references there) showed convection in the form of transversal rolls never to occur as the models with extremely small subduction angle and sufficiently great subduction velocity were not investigated.\par
IV. 
\section[{Conclusions}]{Conclusions}\par
The size of the cell of 2D mantle wedge dissipation-driven convection in the case of the realistic non-Newtonian rheology equals \textasciitilde 300 km at the subduction velocity 10 mm×yr \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}}\end{figure}
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\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-5.png}
\caption{\label{fig_3}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{fig_4}T}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-7.png}
\caption{\label{fig_5}3 T}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-8.png}
\caption{\label{fig_6}Fig. 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-9.png}
\caption{\label{fig_7}}\end{figure}
      			\footnote{Application of Numerical Methods for New Estimate of Rheology Constants in the 2D Computer Model of the Mantle Wedge Thermal Convection as A Possible Physical Mechanism of Hydrocarbons Transport} 			\footnote{© 2020 Global Journals} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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