A viscosity solution of the Hamilton–Jacobi equation with exponential dependence of Hamiltonian on the momentum

The initial – boundary value problem is considered for the Hamilton-Jacobi of evolutionary type in the case when the state space is one-dimensional. The Hamiltonian depends on the state and momentum variables, and the dependence on the momentum variable is exponential. The problem is considered on fixed bounded time interval, and the state variable changes from a given fixed value to infinity. The initial and boundary functions are subdifferentiable. It is proved that such a problem has a continuous generalized viscosity) solution. The representative formula is given for this solution. Sufficient conditions are indicated under which the generalized solution is unique. Hamilton-Jacobi equations with an exponential dependence on the momentum variable are atypical for theory, but such equations arise in practical problems, for example, in molecular genetics.

FINITE ELEMENT ANALYSIS OF THE DIFFUSION MODEL OF THE BIOCLOGGING OF THE GEOBARRIER

The distribution of an organic chemical and the filtration process in the soil which contains a thin geochemical barrier are considered. Microorganism colonies develop in the presence of organic chemicals in the soil which leads to the so-called phenomenon of bioclogging of the pore space. As a result, the conductivity characteristics of both the soil as a whole and the geochemical barrier change. Conjugation conditions as a component of the mathematical model of chemical filtration in the case of inhomogeneity of porous media and the presence of fine inclusions were modified for the case of bioclogging. The numerical solution of the corresponding nonlinear boundary value problem with modified conjugation conditions was found by the finite element method. The conditions of the existence of a generalized solution of the corresponding boundary value problem are indicated. The results on the theoretical accuracy of finite element solutions are presented. Differences in the value of pressure jumps at a thin geochemical barrier were analyzed for the case considered in the article and the classical case on a model example of filtration consolidation of the soil in the base of solid waste storage. The excess pressure in 600 days after the start of the process reaches 25 % of the initial value when taking into account the effect of bioclogging, while is only 6 % for the test case disregarding the specified effect.

Stability estimation of the generalized solution to the direct problem for the acoustic equation

The initial-boundary value problem for the acoustic equation with data on a timelike surface is considered in this paper. Such a problem arises, for example, if it is required to determine the acoustic pressure inside the region from a fixed response to part of the boundary from the source involved at the same boundary. It is assumed that the medium is at rest up to a certain instant of time and the parameters of the medium, for example, acoustic density, are known. The problem is considered in a triangular domain. The advisability of this was shown in the second half of the last century in the works of Romanov V.G. (for example, [1]), where it was proved that the solution to the direct problem of acoustic is representable as the sum of a singular and a continuous terms. The author has written out the form of the singular part, investigated the problem in an integral statement, and also proved conditional well-posedness theorems for three cases: for a small parameter of the domain, for small data, and for the source representability of the sought solution. It is known that the initial-boundary value problem for the acoustic equation with data on a timelike surface is ill-posed. In this paper, the original ill-posed problem is reduced to an inverse problem with respect to some direct (well-posed) problem. The theorem is proved and a stability estimate of the generalized solution to the direct problem is obtained.

Generalized solvability and optimal control for an integro-differential equation of a hyperbolic type

We consider an integro-differential operator with Volterra type integral term. We provide a priory inequalities in negative norms for certain spaces. Further, using obtained inequalities we prove well-posedness (existence and uniqueness of the (weak) generalized solution) of the corresponding boundary value problem as well as a theorem on optimal control existence.

On the solvability of a boundary value problem for a quasilinear equation of mixed type with two degeneration lines

The article investigates the existence of a generalized solution to one boundary value problem for an equation of mixed type with two lines of degeneration in the weighted space of S.L. Sobolev. In proving the existence of a generalized solution, the spaces of functions U(Ω) and V (Ω) are introduced, the spaces H1(Ω) and H 1 * (Ω) are defined as the completion of these spaces of functions, respectively, with respect to the weighted norms, including the functions K(y) and N(x). Using an auxiliary boundary value problem for a first order partial differential equation, Kondrashov’s theorem on the compactness of the embedding of W 2 1 (Ω) in L2(Ω) and Vishik’s lemma, the existence of a solution to the boundary value problem is proved.

The Effect of Compositional Gradient in Field Development

The existence of fluid’s compositional gradient in a reservoir drives convective flow which brings significant impacts to the operations, e.g., in formulation of injected fluid for well stimulation and enhanced oil Recovery (EOR). However, fluid compositional gradient is not always included in modeling reservoir performance due to PVT sampling limitation and simulation constraint. This work aims to show the significance of compositional convection in oil/gas reservoir and provides our experiences in dealing with this issue in Indonesian’s fields. PHE ONWJ as one of the most prolific producers of oil and gas in Indonesia currently operates an offshore block that has been producing for almost 40 years. Operating in a relatively mature well, PHE ONWJ often encounters significant fluid property change namely oil viscosity and specific gravity that changes overtime as depletion process occur. Data from X field, operated by PHE ONWJ, shows that compositional convection impacts workover and tertiary operations, by deviating from simulation results. We present the evidence of compositional convection using mechanistic models. We firstly adopt field data for setting the initial composition stratification. The stratification is identified through DST or fluid sampling. We secondly perform similarity simulation to analyze the effect of compositional gradient towards oil production. Similarity simulation is performed in the simplified domain for providing generalized solution. This solution is then scaled for the real domain. Finally, we show our approach to encounter the problems. Based on the similarity study inspired by the case of X Field, it shows that the compositional stratification affects geochemistry and near-wellbore flow behavior. The compositional convection develops multiple fluid properties at different depth, which create cross flow among layers. It also causes scale deposition in near wellbore which reduces the permeability and alters rock-fluid interactions, such as wettability and relative permeability. The alteration of near-wellbore geochemistry creates severe flow assurance issues in the wellbore. The mixing of multiple fluids from different layers cause paraffin and scale deposition. In some fields, the mixing triggers severe corrosions which could impact on wellbore integrity. The compositional stratification forces us to develop multiple treatments for different layers in single wellbore. Since the fluid’s properties are different for each layer, the compatibility between injected fluid and reservoir fluids varies.

Оn the Analysis of Certain Flotation Processes with Velocities Depending on Time and Height of the Column

— The present paper is an extension of the previous paper of the author where the flotation column dynamics has been investigated. Here we consider the case when particle sedimentation rate and bubble lifting speed depend on time and position in the column. We use the methods for examining the transmission lines set out in the papers mentioned in the References. We formulate a mixed problem for the system describing the processes in the column and present it in a suitable operator form. Then we prove an existence - uniqueness of generalized solution by the fixed point method. We show an explicit approximated solution as a step in the sequence of successive approximations.