Mathematical model of water injection into a geothermal steam-saturated reservoir

A mathematical model of the process of water injection into a geothermal high-temperature steam-saturated reservoir is proposed. The steam and water zones are separated by a moving interface (phase transition front) whose location is determined in the process of solving the problem. It is shown that for low permeabilities the solution contains a thermodynamic contradiction expressed as supercooling of the steam ahead of the interface. Critical curves bounding the regions of existence of the frontal solution are constructed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 92–98, November–December, 1996.     

Emulsion flow through a porous medium with allowance for interphase mass transfer between the components

A new model of the flow of two miscible, mutually-insoluble fluids in a porous medium with the formation of an emulsion and adsorption of the fluid components on the skeleton is proposed. The model takes into account the effect of interphase mass transfer on the emulsion dynamics and the active porosity. A continuous general solution of the one-dimensional model and the problem of breakdown of a discontinuity is constructed. The flow regimes generated in displacement problems which depend on the shape of the adsorption isotherms and the densities of the fluid components are considered. The time dependence of the production rate is constructed for frontal displacement regimes and for displacement regimes with the formation of a zone of mixing (Riemann wave) of the initial reservoir and injected fluids. These functions coincide, at least qualitatively, with the experimental data [1] indicating an initial increase in production rate even against a background of falling reservoir pressure, transition through a maximum, and subsequent decline. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 77–88, January–February, 1997. The work was carried out with financial support from the Russian Foundation for Fundamental Research (project No. 96-01-00991).     

Flow of fluids with substantially different mobilities through a porous medium in the presence of phase transitions: Boundary layer phenomena, spatial phase structures, and instability of the flow

A model of flow through a porous medium with phase transitions which permits an efficient qualitative investigation is proposed for two fluids with sharply different (high-contrast) mobilities. It is shown that the model problem of flow toward a unit sink is singularly perturbed and can be solved using analytic asymptotic matching methods. The nature of the singularity is associated with violation of the condition of the flow contrast in certain zones. The solution can be unstable depending on the direction of interphase mass transfer and the zone in which the process takes place. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 124–135, March–April, 2000. The work was carried out with support from the European Foundation INTAS (grant No. 94-4367) and the Russian Foundation for Basic Research (project No. 95-01-01179a).     

Diffusive dispersion of natural gas hydrates in ocean sediments

A mathematical model of the process of dissolving and dispersion of natural methane hydrates in ocean sediments is proposed. A condition which must be satisfied on the solution boundary of the hydrates in the region of their stable thermodynamic state is derived. The calculated solution rates are several centimeters per year. It is shown that the temperature perturbations initiated by the transition of a gas from the hydrate into the dissolved state are small and the process can be considered isothermal. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 170–173, January–February, 1999. The work was carried out with financial support from the Russian Foundation for Fundamental Research (project No. 96-01-00521).     

Estimating the total effect on a formation during borehole drilling

A stationary mathematical model describing the time-integrated effect on an oil-saturated reservoir during drilling is considered. Calculated results are compared with the solution of the problem in an exact nonstationary formulation. The formation of an invaded zone in straight borehole drilling in water-and oil-saturated reservoirs is studied by numerical modeling. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 113–122, March–April, 2008.     

Radial angular temperature distributions for nonisothermal two-phase filtration of oil and water

The distributions of phase saturations, pressure, and temperature in a porous medium of nonuniform permeability are studied by numerical modeling of nonisothermal two-phase filtration of oil and water with the Joule-Thomson effect and adiabatic effect taken into account. It is shown that the presence of nonuniformity in the near-well zone of the reservoir results in nonmonotonic angular and radial distributions of temperature and phase saturations. During oil and water filtration, there is transition from negative to positive temperature anomalies or vice versa, depending on the ratio of the reservoir permeabilities and the presence of a segment on which the angular temperature distribution in the well is nonuniform. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 124–130, November–December, 2008.     

Formation of a gas hydrate due to injection of a cold gas into a porous reservoir partly saturated by water

Specific features of formation of gas hydrates due to injection of a gas into a porous medium initially filled by a gas and water are considered. Self-similar solutions of an axisymmetric problem, which describe the distributions of the basic parameters in the reservoir, are constructed. The existence of solutions is demonstrated, which predict gas hydrate formation both on the frontal surface and in the volume zone. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 137–150, May–June, 2008.     

Decomposition of gas hydrates in low-temperature reservoirs

The process of dissociation of gas hydrates coexisting with gas and ice in low-temperature reservoirs is considered. A qualitative analysis of the phase transitions which enables possible configurations of the solutions to be predicted is carried out on the basis of the phase diagram for methane hydrate. Mathematical models of hydrate decomposition in reservoirs which take into account the formation of an extended dissociation zone and the presence of two phase transition fronts are proposed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 101–111, January–February, 1998. The work was carried out with financial support from the Russian Foundation for Fundamental Research (project No. 96-01-00521).     

A Condensation Heating Model for Evaluating Early-Period SAGD Performance

In SAGD, it is important to obtain steam chamber conformance along the horizontal wellbore to shorten the ramp-up time and maximize economics. However, reservoir heterogeneity, wellbore undulation, and operation conditions make it hard to achieve this objective. This paper proposes a new method using a condensation temperature model to evaluate the steam chamber conformance of early-period SAGD by interpreting temperature falloff data. The initial temperature distribution of the model is categorized into three zones: a hot-zone of steam temperature, a cold-zone of reservoir temperature, and a transition-zone from steam temperature to reservoir temperature. All three zones are assumed to have circular shapes. This assumption is valid since the model focuses on the early stages of SAGD. The model takes into account the steam condensation on the chamber edge during shut-in, thereby being able to calculate the condensation-front (chamber edge) movement. A semi-analytical approach is developed to solve this model. Sensitivity analysis indicates that the sizes of the steam zone and transition zone, the observing location, and the thermal diffusivity of the formation directly affect the temperature behavior. The synthetic case study shows that the temperature falloffs from the condensation model and from simulation are in good agreement and suggests that the condensation model has the potential to estimate the chamber size during the early stages of SAGD, if calibrated to the field data. Because of the ready-to-use temperature data and the semi-analytic solution, the condensation model proposed in this paper can provide a quick and reliable estimation of the steam chamber size to help the engineers monitor and optimize the chamber development thereafter.     

Rarefaction waves in nonequilibrium-boiling fluid flows

The depressurization of a high-pressure vessel initially filled with water heated to below the saturation point is investigated. After depressurization, the pressure in the vessel drops and the boiling fluid flows out into the atmosphere. The experiments [1–3] showed that, when the first rarefaction wave passes through the fluid and the pressure falls below the saturation point, a two-phase mixture with a small steam volume fraction (below 20%) is formed in the vessel. Intense boiling starts only after the arrival of a rarefaction wave traveling at a speed ∼ 10 m/s called the "boiling shock" in [4]. To explain the specific features of this process we developed a mathematical model which takes the difference in the phase velocities into account. Although in bubbly flows this difference is only ∼ 1 m/s, it is enough to cause bubble fragmentation. The calculation showed that the fragmentation proceeds like a chain reaction, i. e. one fragmentation event creates the conditions for the succeeding events. The avalanche-like bubble number growth leads to sharp boiling intensification and the rapid transition of the non-equilibrium mixture to the equilibrium state. This process occurs in a narrow region, namely, in a slow boiling wave which, in the numerical calculations, looks like a shock. The model developed has made it possible to obtain numerical solutions which agree well with the experimental data, to study the wave structure, and to explain the wave mechanism. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 20–33, July–August, 2000. The work received financial support in part from the Russian Foundation for Basic Research (project 99-03-32042) and from INTAS (grant OPEN 97-2027).     

Mechanism of deep gel plugging emplacement in a nonhomogeneous reservoir

The process of determining the mechanism of deep gel plugging emplacement in the high-permeability zones of a reservoir is analyzed numerically for small characteristic gelling times. It is shown that the efficiency of the process is determined by the sequence and size of the injected reactant plugs, which when they react in the reservoir form the high-viscosity gel. The configuration and site of the plug are determined by the size of the water slug separating the reactant solutions. Tyumen'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 98–103, January–February, 1994.     

Possibility of gas washout from a gas-hydrate massif by circulation of warm water

A possibility of gas extraction from a gas-hydrate massif by means of warm water circulation through a system of wells is demonstrated. A technological scheme and a theoretical model of this process are proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 100–111, July–August, 2009.     

Separationless flow past a body of revolution with simultaneous suction and ejection of fluid

The problem of the axisymmetric potential flow past a body of revolution with a channel along the axis in the presence of a recirculation flow zone near the body, first proposed by G. Yu. Stepanov, is solved. Kazan’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 14–21, January–February, 2000. The study was carried out with the support of the Russian Foundation for Basic Research (project No. 96-01-00123).     

One problem of the theory of dimensional electrochemical machining of metals

A method is proposed for determining the shape of the anode-article boundary for a given shape of the cathode-tool in plane problems of the theory of dimensional electrochemical machining of metals. Under the assumptions used, the boundary of the anode-article is divided into the working zone, where metal dissolution occurs, and an adjacent zone, where the treatment (dissolution) is terminated. The initial problem is reduced to a problem of a fictitious plane-parallel potential flow of an ideal fluid with a nonlinear condition on the free surface. The point of separation of the fictitious flow from the solid boundary corresponds to the point separating these two zones of the anode boundary. The Brillouin-Will condition of smooth separation is imposed at the separation point to construct a closed system of equations determining the problem solution. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 214–220, May–June, 2009.     

Physicomathematical modeling of the processes of capillary impregnation of porous materials

The process of capillary impregnation of porous materials is studied numerically. A physicomathematical model of liquid diffusion in a porous sample is proposed. The model involves an analytical presentation of the diffusion coefficient, which describes available experimental data. A method of solving one-dimensional unsteady problems of impregnation is developed and tested on a self-similar solution of the corresponding boundary-value problem of impregnation. If the impregnation process is sufficiently long, the motion of the liquid in the sample is described by a stable self-similar solution. A classification of moisture diffusion on the basis of the initial humidity on the sample boundary is proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 42–51, January–February, 2009.     

Critical phenomena in particle dissolution in the melt during electron-beam surfacing

A model for the electron-beam surfacing process is proposed that takes into account the dissolution of the modifying particles in the melt. Critical conditions are determined for various modes of surfacing resulting in nearly homogeneous or composite coatings. A detailed parametric study of the one-dimensional version of the model is performed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 131–142, January–February, 2007.     

Oil Displacement for One-Dimensional Three-Phase Flow with Phase Change in Fractured Media

In this article, the numerical simulations for one-dimensional three-phase flows in fractured porous media are implemented. The simulation results show that oil displacement in matrix is dominated by oil–water capillary pressure only under certain conditions. When conditions are changed to decrease the amount of water entering into the fractured media from the boundary of the flow field, water in fracture may be vaporized to superheated steam. In these cases, the appearance of superheated steam in fracture rather than in matrix will decrease the fracture pressure and generate the pressure difference between matrix and fracture, which results in oil flowing from matrix to fracture. Assuming that oil is wetting to steam, the matrix steam–oil capillary pressure will decrease the matrix oil-phase pressure as the matrix steam saturation increases. After the steam–oil capillary pressure finally exceeds the pressure difference due to the appearance of superheated steam in fracture, the oil displacement in matrix will stop. It is also shown that variations of the water relative permeability curve in matrix do not result in different mechanisms for oil displacement in matrix. The simulation results suggest that the amount of liquid water supply from the boundary of flow field fundamentally influence the mechanisms for oil displacement in matrix.     

An inverse elastoplastic problem for plates

We study an inverse elastoplastic problem of determining the residual stresses, the plasticity zone, and the external loads for a plate for known residual deflections which occur after removal of these loads and elastic unloading. Assuming that the deformation theory of plasticity is valid at the active stage of deformation, we prove the theorem of unique solution. An iterative method of solution is proposed and a variational formulation of the problem is given. Some simple examples are considered. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 186–194, July–August, 1999.     

Hydraulic fracturing in an inhomogeneous reservoir

A method of estimating the change in well productivity resulting from the hydraulic fracturing of a finite reservoir, piecewise-homogeneous in the horizontal and vertical directions, with an arbitrary number of cracks is proposed. A correction coefficient that enables these estimates to take into account the effect of capillary trapping of mud and fracturing fluid filtrate is derived [1, 2]. Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 109–114, September–October, 1988.     

Calculation of the effect of a constant electric field on the gas dynamics and nitrogen oxide emission in a laminar diffusion flame

A laminar methane diffusion flame in a constant electric field whose direction is opposite to that of the velocity of the gas flow from the burner nozzle is considered. The mathematical model used includes the complete system of Navier-Stokes equations for the velocity and passive admixture concentration fields, the charge transport equation, together with the Poisson equation for the self-consistent electric field, and the chemical kinetic equations for a thin combustion zone. The calculations show that the flame shortens and its thickness increases in the root zone when an electric field is imposed. This effect is accompanied by a reduction in the nitrogen oxide emission, while is consistent with the experimental data. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 48–53, January–February, 2000. The work was supported by the Russian Foundation for Basic Research (project No. 96-01-014-50).