An approximate solution of some two-dimensional problems of nonstationary filtration

A method of approximate solution of the two-dimensional problems of nonstationary filtration of a gas is proposed. It is based on the application of the method of conformai mapping and the method of successive change of stationary states. The nonstationary influx of a gas into the well is computed by way of example.     

A solution to the hydraulic fracture problem in the traveling wave form

Solutions to the system of equations describing the propagation of hydraulic fracture cracks in a porous medium are obtained in the traveling wave form. The only sought solution is the separatrix of integral curves on the “penetration depth-crack width” plane. Some necessary dependencies that should be given at the crack inlet are found for the fluid flow rate and the fluid pressure. The crack width and the fluid penetration depth are related by power laws in the limiting cases when the crack propagation processes or the fluid penetration processes are dominant.     

Numerical solution of one problem in the spherical-radial filtration of a gas

The article discusses the solution of the problem of the influx of gas into the closed cylinder of a stratum probe on a well-logged cable, for a centrally symmetrical model of the filtration flow. The solution is obtained by the method of finite differences.     

Numerical investigation of the temperature field in a reservoir with a hydraulic fracture

The temperature distribution in a reservoir with a hydraulic fracture is studied by numerical modeling of transient temperature fields taking into account the Joule–Thomson effect and the adiabatic effect. It is shown that the presence of a hydraulic fracture in the reservoir leads to a nonmonotonic change in the reservoir temperature: as the well pressure decreases, the temperature first decreases due to the adiabatic expansion of the fluid and then increases due to the Joule–Thomson effect. As the water–oil displacement front approaches the wellbore, the temperature decreases slightly due to heat exchange in the fracture–reservoir system.     

Model of stepwise propagation of the tip of a hydraulic fracture in the absence of filtration

Quasi-static stepwise propagation of a hydraulic fracture in rock with a regular structure in the absence of filtration is considered. It is proposed to use a brittle fracture diagram taking into account the hydraulic fracturing fluid pressure and the confining pressure. Fracture curves describing the brittle rock fracture where the hydraulic fracturing fluid partially fills the fracture are constructed and used to predicted the possibility of stepwise propagation of hydraulic fracturing in the case where the fluid gradually flows into the fracturing crack. The regularity of the structure of the brittle rocks fracture is estimated from the results of two full-scale experiments: the critical stress intensity factor and the tensile strength limit of the rock. Experiments on pulsed loading of polymethylmethacrylate samples with stepwise crack propagation along concentric circular arcs were performed. The results of the experiments are consistent with theoretical predictions.     

Propagation of a plane-strain hydraulic fracture with a fluid lag: Early-time solution

This paper studies the propagation of a plane-strain fluid-driven fracture with a fluid lag in an elastic solid. The fracture is driven by a constant rate of injection of an incompressible viscous fluid at the fracture inlet. The leak-off of the fracturing fluid into the host solid is considered negligible. The viscous fluid flow is lagging behind an advancing fracture tip, and the resulting tip cavity is assumed to be filled at some specified low pressure with either fluid vapor (impermeable host solid) or pore-fluids infiltrating from the permeable host solid. The scaling analysis allows to reduce problem parametric space to two lumped dimensionless parameters with the meaning of the solid toughness and of the tip underpressure (difference between the specified pressure in the tip cavity and the far field confining stress). A constant lumped toughness parameter uniquely defines solution trajectory in the parametric space, while time-varying lumped tip underpressure parameter describes evolution along the trajectory. Further analysis identifies the early and large time asymptotic states of the fracture evolution as corresponding to the small and large tip underpressure solutions, respectively. The former solution is obtained numerically herein and is characterized by a maximum fluid lag (as a fraction of the crack length), while the latter corresponds to the zero-lag solution of Spence and Sharp [Spence, D.A., Sharp, P.W., 1985. Self-similar solution for elastohydrodynamic cavity flow. Proc. Roy. Soc. London, Ser. A (400), 289–313]. The self-similarity at small/large tip underpressure implies that the solution for crack length, crack opening and net fluid pressure in the fluid-filled part of the crack is a given power-law of time, while the fluid lag is a constant fraction of the increasing fracture length. Evolution of a fluid-driven fracture between the two limit states corresponds to gradual expansion of the fluid-filled region and disappearance of the fluid lag. For small solid toughness and small tip underpressure, the fracture is practically devoid of fluid, which is localized into a narrow region near the fracture inlet. Corresponding asymptotic solution on the fracture lengthscale corresponds to that of a crack loaded by a pair of point forces which magnitude is determined from the coupled hydromechanical solution in the fluid-filled region near the crack inlet. For large solid toughness, the fluid lag is vanishingly small at any underpressure and the solution is adequately approximated by the zero-lag self-similar large toughness solution at any stage of fracture evolution. The small underpressure asymptotic solutions obtained in this work are sought to provide initial condition for the propagation of fractures which are initially under plane-strain conditions.     

Application of equation regulation method to the solution of inverse problems in multiphase materials

In this paper, we propose a new method of determining local material properties of multiphase composites given the experimentally measured displacements and known traction boundary conditions. In the proposed method, an “observation” term is added to the original differential equation, and the modified equation is solved in terms of a regulation parameter. We call this approach the equation regulation (ER) method. By appropriately adjusting the value of the regulation parameter based on the noise level in the input data, we get faster convergence and improved stability than prevailing methods of solving the inverse problem in elliptic ordinary differential equations. Several numerical examples to the solution of this non-linear problem with continuous and discontinuous coefficient functions are given to show the accuracy and reliability of the proposed method.     

Homotopy solution of the inverse generalized eigenvalue problems in structural dynamics

The structural dynamics problems, such as structural design, parameter identification and model correction, are considered as a kind of the inverse generalized eigenvalue problems mathematically. The inverse eigenvalue problems are nonlinear. In general, they could be transformed into nonlinear equations to solve. The structural dynamics inverse problems were treated as quasi multiplicative inverse eigenalue problems which were solved by homotopy method for nonlinear equations. This method had no requirements for initial value essentially because of the homotopy path to solution. Numerical examples were presented to illustrate the homotopy method.     

Numerical solution of the two-dimensional nonstationary problem of the motion of a shell under the action of detonation products

The problem of the motion of an incompressible cylindrical shell with an explosive charge is solved numerically for the propagation of a plane detonation wave from the end of the charge. The strength of the shell is not taken into account. A three-term equation of state [1] is assumed for the detonation products. A comparison is made with the one-dimensional case.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 76–79, July–August, 1972.The authors thank G. S. Roslyakov and V. M. Paskonov for assistance in the work and for helpful advice.     

Numerical-analytical solution of problems of nonlinear filtration,mapped on the two-sheet region of the plane of a hodograph

A numerical solution is obtained to plane problems of nonlinear filtration, reduced using the linearizing transformation of a hodograph, to the associated boundary-value problems on two sheets of the plane of the hodograph. A study is made of flows set up by a source-source system and by a five-point area system. The article discusses the law of filtration with a limiting gradient and a piecewise-linear filtration law. The range of problems which can be reduced to linear boundary-value problems after the transformation of the hodograph is considerably broadened if mapping on non-single-sheet regions is admitted [1]. Specifically, we can consider in this manner a flow set up by two sources of differing intensity, a flow in a rectangular element of the symmetry of a grid of wells, etc. Actually, it is simplest of all to construct a flow by direct numerical solution of the problem in the two-sheet region of the plane of the hodograph, and then to return to the physical plane using known inversion formulas. Under these circumstances, it is possible to make complete use of known asymptotic solutions, which considerably reduces the volume of the calculations. Precisely this approach is used in the present work. Another scheme of a numerical solution is proposed in [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 19–28 January–February, 1975.The authors are indebted to L. A. Chudov for his advice and his evaluation.     

Numerical simulation of the dynamics of a reinforced shell subject to nonstationary load

A numerical algorithm to determine the stress-strain state of a reinforced cylindrical shell subject to impulsive loading is elaborated. Results from a numerical analysis of the dynamic behavior of a reinforced elastoplastic shell under explosive loading are presented __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 83–90, July 2008.     

Self-similar solution of the hydraulic fracture problem for two closely spaced beams

The fracture of two closely spaced elastic beams by an incompressible viscous fluid is considered. The beam bending is described in the framework of the Kirchhoff-Love model. The self-similar solutions are sought by the group methods. The numerical results obtained when solving the corresponding system of differential equations with various boundary conditions are graphically illustrated in the form of pressure and velocity distributions within the fluid and in the form of the distance between the beams in the fracture zone.     

Numerical solution of an inverse problem of nozzle theory for a two-phase gas-particle mixture

In the present paper gas flows with monodisperse and polydisperse particles in plane and axisymmetric nozzles are calculated by the inverse method [1, 2]. The gas velocity distribution is specified on the axis of symmetry of the nozzle, while the gas and particle parameters are specified in the entrance section. As a result of the numerical integration of a system of equations describing a flow of gas with condensate particles in it we determine the gas and particle parameters, the gas streamlines, and the particle trajectories with allowance for the mutual influence of the gas and particles. One of the gas streamlines is taken as the nozzle contour and the limiting trajectories and pure gas zone are found. A difference method is described which makes it possible to calculate the subsonic, transonic, and supersonic flow regions using a single algorithm, its features are noted, and the results of the calculation for monodisperse mixtures with particle diameters 1 and 5 m and fractions by weight 0.3 are given. A comparison is made with the results of calculations by other methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 106–114, July–August, 1986.The authors express their gratitude to N. B. Ponomarev and G. E. Dumnov for their useful discussions and help in carrying out the calculations.     

Biquaternion solution of the kinematic control problem for the motion of a rigid body and its application to the solution of inverse problems of robot-manipulator kinematics

The problem of reducing the body-attached coordinate system to the reference (programmed) coordinate system moving relative to the fixed coordinate system with a given instantaneous velocity screw along a given trajectory is considered in the kinematic statement. The biquaternion kinematic equations of motion of a rigid body in normalized and unnormalized finite displacement biquaternions are used as the mathematical model of motion, and the dual orthogonal projections of the instantaneous velocity screw of the body motion onto the body coordinate axes are used as the control. Various types of correction (stabilization), which are biquaternion analogs of position and integral corrections, are proposed. It is shown that the linear (obtained without linearization) and stationary biquaternion error equations that are invariant under any chosen programmed motion of the reference coordinate system can be obtained for the proposed types of correction and the use of unnormalized finite displacement biquaternions and four-dimensional dual controls allows one to construct globally regular control laws. The general solution of the error equation is constructed, and conditions for asymptotic stability of the programmed motion are obtained. The constructed theory of kinematic control of motion is used to solve inverse problems of robot-manipulator kinematics. The control problem under study is a generalization of the kinematic problem [1, 2] of reducing the body-attached coordinate system to the reference coordinate system rotating at a given (programmed) absolute angular velocity, and the presentedmethod for solving inverse problems of robotmanipulator kinematics is a development of the method proposed in [3–5].