Computer solution of the problem of the depletion of a stratum with a gasified liquid in the case of the work of a round battery of boreholes

The article discusses the problem of the depletion of a round stratum with a gasified liquid, in the case of the work of a round battery of boreholes with constant end-face pressures. The nonlinear differential equations describing the filtration process are replaced by finite-difference equations, which are solved numerically using a computer program. The condition for the convergence of the finite-difference scheme used is given. The proposed algorithm is used to solve a concrete example. By making analogous calculations for different variants of a given number of boreholes, of the end-face pressures, of the radius of the battery, valuable information can be obtained with respect to their optimal values, the time of the start of secondary methods of exploitation, the volume of gas dissolved in the petroleum, as well as the output of petroleum, which is easily calculated from the known values of the pressure and the petroleum saturation at any given moment of time.     

Investigation of non-steady-state centeally symmetric filtration of a liquid in closed heterogeneous media

This paper is the final stage in a study of the properties of the non-steady-state filtration of a homogeneous liquid towards a central well with a hemispherical end face in heterogeneous media with a dual porosity, consisting of hemispherical regions set one inside the other with different values of the medium parameters [1, 2]. Exact solutions are found for the problems of the decrease in seam pressure as a function of time and distance, as well as of the time change in output of a well with a hemispherical end face operating at a fixed flow rate, or at a fixed end face pressure, respectively, in closed heterogeneous media. The effect of the magnitudes of the heterogeneous media parameters on the change in their production process indicators is established on the basis of numerical calculations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 98–104, May–June, 1976.     

Centrally symmetrical filtration of liquid in cracked porous media with a hemispherical supply contour

The motion of a homogeneous liquid in a well with a hemispherical face is studied for the case of transient, spherically radial filtration in cracked porous media comprising mutually superposed hemispherical regions with different crack permeabilities, having a supply contour in the outer hemispherical region. Using a Laplace integral transformation with respect to the time variable, the systems of differential equations describing the filtration of liquid in these media are solved for zero initial and corresponding boundary conditions. Exact solutions are obtained for the reduction in stratal pressure with time and distance, and also for the changes taking place in the output of a well operating under conditions of specified face pressure. On the basis of corresponding numerical calculations, the influence of the parameters of the cracked porous strata and the radius of the surface containing the supply contour on the indices of the production process is established.     

Unsteady-state filtration of a liquid to a rectilinear gallery in a two-layer fractured porous stratum

In connection with investigations of the filtration of a liquid in oil-bearing strata with fractured and fractured-porous types of collectors, the article discusses the unsteady-state filtration of a homogeneous liquid in a two-layer stratum of great extension, with different values of the permeabilities of the system of fractures of the intercalations.     

Nonlinear waves in short-term injection of a non-Newtonian fluid into a stratum

The one-dimensional nonlinear pressure waves excited in short-term (instantaneous) injection of finite mass of power-law non-Newtonian fluid into a stratum (or instantaneous extraction from the stratum) are analyzed. The propagation laws for plane, cylindrical, and spherical waves in closed (pressurized) and open (unpressurized) strata are obtained.Baku. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 89–96, May–June, 1996.     

Unsteady-state flow induced by a well in a deformable stratum interacting with the adjacent rocks

Unsteady-state plane radial flow induced by a well in a thin deformable stratum is studied taking into account the stratum interaction with the adjacent rocks. The stratum permeability is assumed to depend on the lateral deformation. The behavior of the well productivity properties for harmonic reservoir pressure oscillations and after pressure drawdown in a bilayered stratum is analyzed. Steady-state well behavior under these conditions was studied in [1]. A qualitative estimate of the well productivity variation due to a stepwise pressure change in an adjacent stratum was previously derived in [2].Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 85–90, January–February, 1996.     

Analysis of the transition from constant output to fixed well-bottom pressure for elastic regime of oil extraction

The elastic flow regime of stratal fluids has been well studied in the literature [1, 2] and the results have been widely applied in practice in the development of oilfields [3, 4]. Consideration is given below to the new problem of the reduction in the output of a well at fixed well-bottom pressure after it has been operating for some time at constant output. There is a practical aspect to this sort of problem. For instance, the degassing of the reservoir in the region of the well is considered an undesirable feature in the exploitation process. If degassing begins, the resistance to flow grows sharply and the well outputs fall considerably. Once the well-bottom pressure has fallen to the saturation pressure, it must not be allowed to fall further, or gas will be given off in the well region of the reservoir. It is desirable to keep the well-bottom pressure higher than the saturation pressure.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 82–87, March–April, 1985.     

Numerical modeling of the process of heating a petroleum stratum by means of high-frequency electromagnetic radiation

Conclusions We have completed a numerical study on a two-dimensional model in which consideration has been given to the nonlinear heat-capacity function c(T) of the heating process for a petroleum stratum. It has been demonstrated that the efficiency of the heating depends significantly on correct selection of the parameters, and we have particular reference here to the absorption factor a defined by the radiation frequency. We have determined the optimum values of , as well as the optimum heating times. We have carried out calculations to confirm the possibility of utilizing high-frequency electromagnetic heating of petroleum strata in order to intensify the production yield of high-viscosity petroleum and to combat other well problems, and these can also be utilized for the development of practical recommendations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 59–64, July–August, 1990.     

A basis for a theory of nonlocal elastic filtration using the equations of elasticity

The proposed [1–3] nonlocal formulation of the hypothesis that the ground pressure is constant in nonstationary pressure filtration in a deep elastic stratum is derived from the equilibrium equations for the stratum-roof system. The roof is considered to be a flat plate [4] and the floor of the stratum is assumed to be rigid. An equation is established for the scale of the region of influence on the stress and pressure distributions at a point.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 82–86, July–August, 1971.The author wishes to thank V. N. Nikolaevskii for a number of useful observations.     

Depeletion of the elastic reserve of a sealed and fractured formation with anomalously high formation pressure

Analysis of the exploitation of deposits with anomalously high formation pressures has shown [1] that when the weighted-mean formation pressure drops below a certain critical value (close to the hydrostatic pressure) a rather sharp sudden fall in output, together with a change in the rate of decline of pressure, is observed. The fall in output is attributable to the closing of the joints and the resulting catastrophic deterioration in the permeability of the reservoir [2]. In this paper an attempt is made to develop a joint closing hypothesis, to calculate the motion of the joint closing front from the bottom of the well to the edge of a homogeneous circular formation, and to derive expressions for predicting the fall in the output of the wells and the pressure in the formation. In order to obtain solutions it is assumed that the reservoir depletion regime is quasisteady, so that the results should be regarded as approximate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 73–83, November–December, 1985.     

Construction of a one-dimensional stochastic model of an inhomogeneity in a stratum

The role of nonrandom (determinate) and random components of the percolation resistance is shown for the example of a one-dimensional stochastic model of a stratum for a determinate flow. The problem reduces to the analysis of the Fokker-Planck equation and the determination based on it of the first, second, and higher moments of the distribution density of the magnitude of the resistance. On the basis of this investigation a simple equation is derived for determining the fluctuation as a measure of the random coexistence, in a complex process, of both indicated components of resistance. A model is also constructed for the case in which the pressure differential is a determinate quantity and the flow is a random quantity. Based on the inequalities, from an analysis we compare both models. It is necessary that we take into account the inhomogeneity of the strata for planning and analyzing the development of petroleum deposits.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 59–65, July–August, 1973.     

Linear analysis of convective instability of a fluid in a horizontal annular cavity occupied by a porous medium

The problem of convective instability in an infinite horizontal annular porous stratum located in an impermeable rock mass is considered. Using the Bubnov-Galerkin method, the values of the first seven critical Rayleigh numbers are found. The forms of the corresponding critical motions are established. The change in the modes of instability of the critical motions when the thickness of the porous stratum is varied is analyzed.Makhachkala. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 19–25, May–June, 1996.     

Calculation of a stratum saturated with liquid under pressure as a hydraulic seismograph

The reaction of the pore pressure of a stratum completely saturated with liquid to periodic long-wavelength variations in the stressed state of a rock mass is a alyzed. The amplitude-frequency characteristic of the stratum used as a hydraulic sensor is found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 81–86, July–August, 1984.     

Some problems of centrally symmetrical percolation of a gas

A solution is considered for the centrally symmetrical problem of unsteady inflow of gas to a stratum sampler on a logging cable in a thick stratum with constant pressure at the discharge and variable pressure when filling the closed cylinder of the sampler with gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 117–121, November–December, 1971.     

Unsteady dissipative structures in non-Newtonian fluid flow through a porous medium

The nonuniform space-time pressure and velocity distributions in an initially nonempty stratum with constant initial pressure created by pumping a non-Newtonian fluid through the boundary of the stratum are investigated. The injected fluid and the fluid present in the stratum before injection have identical physical properties. The conditions of formation of traveling fronts and localized structures are analyzed as functions of the nonlinearity of the rheological law of the fluid and the injection regime.Baku. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 106–112, November–December, 1994.     

The spherical problem of the filtration of a liquid and a gas with a change in the permeability of rocks in accordance with an exponential law

The article considers the problem of the filtration of liquids (or gases), pumped through a borehole at a constant rate with elastic filtration conditions. The permeability of the stratum is assumed to be an exponential function of the coordinates. The viscosities of the injected and displaced liquids are assumed to be different. To increase the capacity of strata, i.e., of collectors used for the burial of industrial waste flows and gases, various methods are employed to increase the fracturing and the permeability of the rocks (hydro-pulse techniques, explosions, and other methods). As a result of this, a spherical region is formed in the rocks, in which the permeability varies along the radius. The character of this change is well described by an exponential function. The pumping of waste flows or industrial gases into such a cavity leads to the displacement of the stratum liquid (or gas). The problem of the displacement of one liquid by another liquid not miscible with it under rigid filtration conditions was first discussed in [1–5]. Here a study was made of a region of finite dimensions, bounded by two boundaries, with given pressures or mass flow rates (the linear and axisymmetric flow problems). The permeability of the stratum was assumed to be independent of the coordinates. A special characteristic of these problems is the fact that it is impossible to consider unbounded or semi-bounded filtration conditions in them since, under rigid filtration conditions, the condition of bounded character of the pressure (the head) is not satisfied at infinity. Elastic filtration conditions for two immiscible liquids were first discussed in [6], and later in [7, 8] and other reports. Here an investigation was made of the linear and axisymmetric problems for an unbounded region. In [9, 10] solutions are given to some problems with spherical symmetry for an unbounded region, with rigid filtration conditions and a jumpwise change of the permeability along the radius. In the problems of [6–10] the condition of the bounded character of the pressure is satisfied. In [11] the case of a hyperbolic change in the permeability of the rocks is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 42–51, November–December, 1974.     

Numerical investigation of micellar-polymer displacement of oil from layered strata

A mathematical model developed earlier [8] is used to make a numerical investigation of the mechanism and wave dynamics of a process in layered strata consisting of hydrodynamically connected layers with different flow parameters. A study is made of a micellar solution with an exterior hydrocarbon phase, i.e., a solution that on contact with the stratal fluids forms a two-phase structure: a hydrocarbon fluid (micellar solution and stratal oil) and water with polymer admixture. The concentration of salts in the stratum is assumed to be constant.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 87–93, March–April, 1984.     

Elasticoplastic conditions of the filtration of a liquid in porous media

The article discusses questions in the theory of filtration in porous media, taking account of elastic, elasticoplastic, and plastic deformations. Parameters are introduced to evaluate irreversible effects in petroleum- and water-bearing strata, i.e., coefficients of the change in the porosity and the permeability. Equations are derived for filtration under unsteady-state and steady-state working conditions of wells and galleries. Two limiting cases, which allow analytical solutions, are separated out. In the general case, the equations of elasticoplastic filtration conditions are solved on an electronic computer. The numerial calculations show that the predominating effect results from taking account of the irreversible change in the permeability, depending on the change of the pressure in the stratum.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 84–90, September–October, 1973.The author is grateful to V. N. Nikolaevskii for his evaluation of the work.     

Investigation of heat flux at the forward critical point of bodies placed in a high-temperature subsonic stream

We present the results of measurements of the total heat flux at the forward critical point of axially symmetric models placed in a high-temperature subsonic stream of air and nitrogen in an apparatus which uses a high-frequency inductive discharge [1] to heat the gas. The heat fluxes were measured for cylindrical models whose forward part had one of three possible shapes: a hemisphere, a hemisphere with a blunt nose, or a flat circular end-face. A water-cooled calorimeter sensor was set up at the forward critical point of the model; the calorimeter sensor was made of different materials, so that it was possible to estimate the radiant and convective components of the total heat flux and determine the effect of the sensor material on the heat flux measured. The convective component of the heat flux was compared to a calculated value obtained by Fay and Riddell's formula. The heat-flux values found for two shapes of models were used in determining the effective radius of streamline flow for a model with a plane end-face.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 133–141, September–October, 1973.The authors are grateful to N. I. Nesterov for his help with the work.     

Water filtration to partially penetrating wells in a two-layer confined stratum

Water filtration to partially penetrating wells in a uniform confined stratum has been extensively studied recently. Considerably less study has been made of filtration to partially penetrating wells in layered strata, which are frequently encountered in practice. Some particular cases of this problem were considered in [1–4], and its most complete solution was given in [4]. However, this solution is presented in a general form which is difficult to apply in practice.In the following we present the solution of the water filtration problem to partially penetrating wells in a two-layer confined stratum for the cases of the operating portion of the well located in both the upper and lower stratum layers. The problem is solved by the method developed in [5, 6], where the potential of a point sink is first found, and then the potential of a line sink of intensity q is found, which is then used as the operating portion of the well.Applying to the resulting solutions the known method of filtration resistances, approximate relations are presented for the final calculations.