The Dual Porosity Model

Classical theory for systems of the first order partial differential equations for a scalar function can be rephrased as the submanifold theory of contact manifolds (geometric first order jet spaces). In the same spirit, we will develop the geometric theory of systems of partial differential equations of second order for a scalar function as the Contact Geometry of Second Order, following E. Cartan.We will formulate the submanifold theory of second order jet spaces as the geometry of PD manifolds (R;D 1 ,D 2 ) of second order. Moreover we will establish the First Reduction Theorem for (R;D 1,D 2) admitting non-trivial Cauchy characteristic systems. By utilizing Parabolic Geometry, we will give, directly or combined with reduction theorems, several classes of systems of partial differential equations of second order, for which the model equation of each class admits the Lie algebra of infinitesimal contact transformations, which is finite dimensional and simple.

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).

Exact solutions of some nonlinear systems of partial differential equations by using the first integral method

Comparison of systems of stochastic partial differential equations

Flow simulation of fractured reservoirs usually is performed using a dual porosity model. The dual porosity system is modeled by using two coupled grids: one for matrix and one for fracture. These two continua communicate by transfer functions. Until now, there were no mathematical models of dual porosity, three-phase, compressible flow for streamline simulators. To realize this model, it was necessary to reformulate the matrix and fracture pressure equations The conventional transfer function has been incorporated as a source/sink term, not only in the streamline saturation equations (as it was in incompressible case), but also in the pressure equation. The dual porosity model has been implemented into a streamline simulator. This tool has its main application in the geological modeling domain for analyzing uncertainty, model ranking and screening of geologically detailed models, including fractures. This paper describes the mathematical model for a three-phase compressible dual porosity streamline simulator and compares the results and run times of the streamline-based approach with a conventional dual porosity grid-based commercial simulator. The results from the streamline simulator for dual porosity show good agreement with those produced by a commercial finite difference simulator with order of magnitude improvement in simulation time. Streamline methods as a reservoir simulation tool have generated much interest in petroleum engineering because of the capability to calculate fluid flow in multi-million cell geological models with reasonable CPU times. However, important physical properties of geo-scale fluid flow models are still not properly modeled by streamline methods. Enhancing the range of physical properties that can be simulated accurately in a timely manner will enable improved workflows in the geological modeling domain.

This study developed new analytical models for predicting the temperature distribution within a geothermal reservoir following reinjection of water having a temperature different from that of the reservoir. The study consisted of two parts: developing new analytical models for the heat conduction rate into multi-dimensional, parallelepiped matrix blocks and developing new analytical models for the advance of the thermal front through the geothermal reservoir. In the first part of the study, a number of semi-empirical models for the multi-dimensional heat conduction were developed to overcome the limitations to the exact solutions. The exact solution based on a similarity solution to the heat diffusion equation is the best model for the early-time period, but fails when thermal conduction fronts from opposing sides of the matrix block merge. The exact solution based on an infinite series solution was found not to be useful because it required tens of thousands of terms to be include d for accuracy. The best overall model for the entire conduction time was a semi-empirical model based on an exponential conduction rate. In the second part of the study, the early-time period exact solution based on similarity methods and the semi-empirical exponential model were used to develop new analytical models for the location of the thermal front within the reservoir during injection. These equations were based on an energy balance on the water in the fractured network. These convective models allowed for both dual and triple porosity reservoirs, i.e., one or two independent matrix domains. A method for incorporating measured fracture spacing distributions into these convective models was developed. It was found that there were only minor differences in the predicted areal extent of the heated zone between the dual and triple porosity models. Because of its simplicity, the dual porosity model is recommended. These new models can be used for preliminary reservoir studies. Although they are not as accurate as numerical simulators, they are simple, easy and inexpensive to use. These new models can be used to get general information about reservoir behavior before committing to the considerable greater expense of numerical simulation.

For a class of nonlinear systems of partial differential equations of high order in a cylindrical domain the following boundary value problem is considered: Cauchy-type conditions are prescribed on the upper and lower bases of the cylinder, and a Robin-type condition is prescribed on the lateral part of the boundary. This boundary value problem is reduced in an equivalent way to a nonlinear functional equation in a certain subspace of the Sobolev space. Under certain assumptions about the nonlinear terms an a priori estimate is obtained for the solution of the problem in question and the existence of the solution is established, while in the case when these conditions fail, the absence of a solution is established. The question of uniqueness is also discussed for the solution. Bibliography: 18 titles.

A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear system of two-dimensional partial differential equations with corresponding boundary and initial conditions. Using the classical Lie scheme, we establish that the base system of partial differential equations (under some restrictions on coefficients) is invariant under the infinite-dimensional Lie algebra, which enables us to construct families of exact solutions. Moreover, exact solutions with a more general structure are found using another (non-Lie) technique. Finally, it is shown that some of the solutions obtained describe the hydrostatic pressure and the glucose concentration in peritoneal dialysis.

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.

A new family of exact solutions to the unsteady Navier–Stokes equations using canonical transformation with complex coefficients

Singular perturbation problems for systems of partial differential equations of elliptic type

Existence of series solutions for certain nonlinear systems of time fractional partial differential equations

Kranc: a Mathematica package to generate numerical codes for tensorial evolution equations

Boundary value problems are considered for a system of stationary partial differential equations that stems from the models describing the implantation of impurities into semiconductor devices and is similar to the equations of carrier transport. The existence of solutions by functional analytic means (energetic method) is proved. Besides, an upper estimate for the set of solutions is obtained.

Degeneration of the solutions of certain well-posed systems of partial differential equations depending on a small parameter