Fresh water injection into a geothermal reservoir which contains superheated vapor and solid phase salt

A mathematical model is proposed for the process with two unknown phase transition interfaces. At the leading interface, evaporation occurs and a salt precipitates. The interface that moves at a lower velocity is a dissolution surface. In the isothermal approximation the self-similar solution is obtained. It is shown that fresh water injection may lead to solid precipitate transfer from one region to another, substantially increasing the precipitate content behind the evaporation front. This significantly decreases permeability and, hence, the flow velocity through permeable rocks. With approaching the critical parameter region the branches of the obtained two-valued self-similar solution also approach and at critical values merge, which corresponds to the disappearance of the solution. The non-existence of the self-similar solution can be interpreted as the filling of the pores with salt precipitate and the flow arrest in the geothermal reservoir.     

Front Regime of Heat and Mass Transfer in a Gas Hydrate Reservoir under the Negative Temperature Conditions

The analytical self-similar solution to the nonlinear problem of the front regime of heatand- mass transfer in a gas hydrate reservoir under the negative temperature conditions is obtained. In the initial state the reservoir is assumed to be saturated with a heterogeneous gas hydrate–ice–gas mixture. In particular cases there may be no ice or/and gas. The ice and gas are formed behind the gas hydrate dissociation front. The calculations are presented for a stable hydrate–gas system. The critical curves are constructed in the well-pressure–reservoir-permeability plane. These curves separate the domains of the front regime and the regime of volume gas hydrate dissociation ahead of the front. The velocity of the gas hydrate dissociation front is investigated as a function of various problem parameters. The characteristic temperature and pressure distributions corresponding to various regimes on the diagram are investigated.     

Numerical simulation of convective flows in a soil during the evaporation of water containing a dissolved admixture

The concentration profile is investigated for a solution that saturates a low-permeability soil. The simulation results showed the presence of three flow regimes. The salt accumulated near the phase transition boundary increases the solution density and may lead to the development of natural concentration-induced convection which interacts with the rising flow (forced convection). The stability threshold of the forced flow and the effect on it of natural convection that arises are determined. It is shown that at intense flow to the evaporation surface the admixture concentration increases at this boundary rapidly and reaches the saturation concentration. In this case, the admixture precipitates. In the slow evaporation regime the admixture diffuses from the high-concentration region, which prevents the development of convective flow.     

Numerical solution for spherical laser-driven shock waves

The history of the flow behind a laser-driven shock is investigated in the context of variable energy blast waves. Thereby the total laser energy absorbed by the blast is assumed to vary proportionally to some power of time. Due to the high temperatures and pressures occurring in the initial phase of the flow a real gas model has been employed. It accounts for vibration, dissociation, electronic excitation, ionization and intermolecular forces. Radiative and conductive heat transfer are considered as well. The numerical computations were carried out using the method of characteristics. A self-similar strong shock solution serves as initial condition. It turns out that the exponent which determines the time-dependent addition of energy at the shock front is limited for physical reasons. The computed far-field solutions expand the temporal scope of the self-similar solution domain, which has been the main subject of the classical literature, into the non-self-similar domain at late time. The differences between the solutions obtained for real gas and perfect gas are less significant than in the case of the classical point explosion. Received 30 July 1996 / Accepted 21 August 1997     

Motion of a strong shock front in a two-dimensional gas layer with moving internal boundary

We examine the problem of planar one-dimensional motion of a strong shock wave with moving internal boundary in which the initial position of the front, its intensity, the mass of the gas involved in the motion, and the energy contained in this gas are known. The problem is not self-similar and its exact solution, which involves working with partial differential equations, presents serious difficulties. In the following we determine the law of shock-front motion in this problem via the method of [1], which makes it possible to find a system of ordinary differential equations for the problem. The method is based on an initial specification of the power-law coupling between the dimensionless Lagrangian and Eulerian variables and replacement of the energy equation by this coupling and the energy integral. The solution is sought in the first approximation.     

Contribution to the theory of laminar momentumless wakes

The self-similar laws of decay of the velocity field in a plane momentumless viscous incompressible wake of a hydrodynamic propulsion system were first analyzed in [1]. Turbulent wakes in the near and far flow regions were investigated in [2] on the basis of an integral calculation method. In [3] the asymptotic laws of degeneration of the passive admixture concentration, temperature and velocity fields were obtained, making it possible to estimate the effect of purely molecular diffusion and convective transfer o the passive impurity concentration distribution in the wake. In the present paper the limiting self-similar solutions of the problem of a swirled momentumless viscous incompressible wake are obtained, together with the self-similar solutions of the problem of the development of a plane momentumless wake in a medium nonuniform with respect to temperature (passive admixture concentration).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 35–39, March–April, 1986.     

An Analytic Solution for the Frontal Flow Period in 1D Counter-Current Spontaneous Imbibition into Fractured Porous Media Including Gravity and Wettability Effects

Including gravity and wettability effects, a full analytical solution for the frontal flow period for 1D counter-current spontaneous imbibition of a wetting phase into a porous medium saturated initially with non-wetting phase at initial wetting phase saturation is presented. The analytical solution applicable for liquid–liquid and liquid–gas systems is essentially valid for the cases when the gravity forces are relatively large and before the wetting phase front hits the no-flow boundary in the capillary-dominated regime. The new analytical solution free of any arbitrary parameters can also be utilized for predicting non-wetting phase recovery by spontaneous imbibition. In addition, a new dimensionless time equation for predicting dimensionless distances travelled by the wetting phase front versus dimensionless time is presented. Dimensionless distance travelled by the waterfront versus time was calculated varying the non-wetting phase viscosity between 1 and 100 mPas. The new dimensionless time expression was able to perfectly scale all these calculated dimensionless distance versus time responses into one single curve confirming the ability for the new scaling equation to properly account for variations in non-wetting phase viscosities. The dimensionless stabilization time, defined as the time at which the capillary forces are balanced by the gravity forces, was calculated to be approximately 0.6. The full analytical solution was finally used to derive a new transfer function with application to dual-porosity simulation.     

Self-similar singular solution of fast diffusion equation with gradient absorption terms

The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation (ODE). Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.     

Self-similar solution of the one-dimensional problem of thermocapillary motion of an emulsion

It is proved that the problem of one-dimensional motion of an emulsion under the action of thermocapillary forces has a self-similar solution in a semi-infinite interval. The behavior of the solution is illustrated by numerical examples for aluminum-lead emulsions, in which the carrier phase is lead or aluminum. The solution is compared with the solution of the self-similar problem linearized in the low impurity concentration. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 61–70, September–October, 2007.     

Three-dimensional isothermal lava flows over a non-axisymmetric conical surface

Within the thin-layer approximation for a highly-viscous heavy incompressible fluid, a hydrodynamicmodel of a 3D isothermal lava flow over a non-axisymmetric conical surface is constructed. Using analytical methods, a self-similar solution for the law of leading-edge propagation is obtained in the case of a flow from a non-axisymmetric source located at the apex of a conical surface with smoothly varying properties. In the case of a flow over a substantially non-axisymmetric surface, it is shown that there exists a self-similar solution for the free-surface shape and the law of leading-edge motion. This solution is studied numerically for particular examples of the substrate surface and the source. In the general case of a non-self-similar flow over a substantially non-axisymmetric conical surface, a local analytical solution is obtained for the free-surface shape and the velocity field near the leading flow front.     

Phase discontinuities in water flows through a porous medium

Flow of a fluid through a porous medium is considered with allowance for heat conduction processes and phase transitions. Discontinuities in flows between both single-phase zones saturated with water and steam and single-and two-phase zones saturated with an equilibrium steam-water mixture are studied. It is shown that only the evaporation fronts are evolutionary for a convex-downward shock adiabat of the discontinuity inside the steam-water mixture. The structure of these fronts is considered and a condition supplementary to the conservation laws and necessary for the well-posed formulation of problems whose solution contains this front is found from the condition of existence of a discontinuity structure between the water (steam) and the steam-water mixture.     

Large time asymptotics for solutions to a nonhomogeneous Burgers equation

In this article, the solutions to a nonhomogeneous Burgers equation subject to bounded and compactly supported initial profiles are constructed. In an interesting study, Kloosterziel （Journal of Engineering Mathematics 24, 213-236 （1990）） represented a solution to an initial value problem （IVP） for the heat equation, with an initial data in a class of rapidly decaying functions, as a series of self-similar solutions to the heat equation. This approach quickly revealed the large time behaviour for the solution to the IVP. Inspired by Kloosterziel＇s approach, the solution to the nonhomogeneous Burgers equation is expressed in terms of the self-similar solutions to the heat equation. The large time behaviour of the solutions to the nonhomogeneous Burgers equation is obtained.     

Salt precipitation during solution evaporation in low-permeability rocks

The effect of admixture diffusion on the process of precipitation during solution evaporation in high-temperature low-permeability rocks is investigated. It is found that at permeabilities lower than 0.01 mD the two-valued solution is restructured becoming single-valued. As a result, during salt precipitation in low-permeability rocks where admixture diffusion is important, a solution exists for any flow regime and the channels in the porous medium are not blocked by solid-phase salt.     

Convergence of Anisotropically Decaying Solutions of a Supercritical Semilinear Heat Equation

We consider the Cauchy problem for a semilinear heat equation with a supercritical power nonlinearity. It is known that the asymptotic behavior of solutions in time is determined by the decay rate of their initial values in space. In particular, if an initial value decays like a radial steady state, then the corresponding solution converges to that steady state. In this paper we consider solutions whose initial values decay in an anisotropic way. We show that each such solution converges to a steady state which is explicitly determined by an average formula. For a proof, we first consider the linearized equation around a singular steady state, and find a self-similar solution with a specific asymptotic behavior. Then we construct suitable comparison functions by using the self-similar solution, and apply our previous results on global stability and quasi-convergence of solutions.     

Accumulation and Precipitation of Salts during Groundwater Evaporation and Flow

A mathematical model for the precipitation of dissolved salts during groundwater evaporation is proposed. An asymptotic solution of the problem is obtained for a specified flow law. It is shown that there are two different regimes of solute precipitation which are determined by the evaporation front velocity and the groundwater flow rate. The dependence of the precipitated salt mass on the soil surface temperature, the atmospheric humidity, the initial solute concentration and the filtration rate is illustrated.     

The propagation of a passive admixture from a local instantaneous source in a turbulent mixing zone

The propagation of a passive admixture from a local instantaneous source in a turbulent mixing zone in a stable stratified fluid (two-dimensional problem) is simulated numerically. The location of the source does not coincide with the center of the turbulent zone. The calculation results indicate that the average admixture concentration distribution depends significantly on the initial data. The location of the maximum concentration in homogeneous and linearly stratified fluid is fairly slowly displaced toward the center of the zone. Calculations in a pycnocline show that situations are possible when the propagation of a passive admixture is largely determined by a convective flow generated by the turbulent mixing zone.     

Two-Valued Solutions in the Problem of Salt Precipitation during Groundwater Evaporation

The problem of the evaporation of groundwater containing a dissolved admixture is studied. It is shown that in the salt precipitation regime the solution is nonunique. At critical parameter values the solutions merge and in a certain region the solution of the problem disappears. The nonexistence of a solution corresponds to clogging of the pore space with precipitated salt.     

Self-similar collapse of a circular cavity of a power-law liquid

Using the lubrication approximation we investigate the self-similar axisymmetric flow of a power-law liquid towards a central circular cavity. It is shown that this problem has a self-similar solution of the second kind. The self-similarity exponent is found by solving a non-linear eigenvalue problem arising from the requirement that the integral curve that represents the solution must join the appropriate singular points in the phase plane of the governing equation. The eigenvalues for different values of the rheological index are computed. Numerical integration of the equations allows us to determine the shape of the solution in terms of the physical variables. We make a detailed analysis of the influence of the rheology on the properties of the solutions.     

定态火焰在可燃预混气中产生的压力波

火焰在可燃预混气中传播时，在火焰面前方产生一道压力波。忽略点火及火焰的初期加速，仅考虑火焰达到稳定传播速度的情况。用Ｏｐｅｎｈｅｉｍ自相似解分析流场，得到相应的控制方程及定解条体；用自适应步长的四阶Ｒｕｎｇｅ－Ｋｕｔｔａ法对方程积分，讨论了流场压力波结构及弱激波近似声波解；认为火焰为间断面，能量释放在火焰面后瞬时完成。利用火焰面两侧的能量关系，得到了火焰位置、燃速及对应Ｃ－Ｊ条件的火焰位置、Ｃ－Ｊ燃速。