Parvin I. Rahimly

The work considers the problem of gas hydrate dissociation in a porous medium using the two-term Forchheimer law, corresponding to high flow rates of reservoir fluids. Such rates can arise during the decomposition of gas hydrates, since a large amount of gas is released. Intensive emissions of gases from the earth’s interior are observed on the ocean floor. They are also associated with a large number of subvertical geological structures under the ocean floor, coming to the surface in the form of local ring funnels (pockmarks). Many similar objects have also been found on land. Particular interest in this problem is caused by climate threats associated with the release of greenhouse gases. The movement of gas released from the hydrate to the breakthrough channel is similar to the gas inflow to the well (without hydrate), which is usually described by a two-term filtration law. In this work, a mathematical model of gas hydrate dissociation with a nonlinear Forchheimer-type law of motion is developed. The system is split in two blocks by physical processes, taking into account the quadratic correction to the velocity in the filtration law. The first block is responsible for the convective transfer of saturation parameters in the model, water, gas and hydrate saturations are taken into account. The second block corresponds to the equation of dissipative piezoconductivity with a different number of thermodynamic degrees of freedom, taking into account heat and mass transfer in a porous medium. The performed splitting allows using explicit-implicit difference schemes when solving problems and avoiding strong refinement of the step in time and space. For numerical modeling, the support operator method is used, which makes it possible to discretize partial differential equations on irregular grids, which allows taking into account the complex geometry and lithology of the reservoir. A difference scheme based on the support operator method is developed, which, due to the mutually consistent approximation of vector analysis operations (divergence and gradient), allows to take into account the various flux laws between adjacent grid cells, including quadratic corrections to the velocity. Based on the developed numerical algorithms and their program implementations, calculations of gas hydrate dissociation are performed both in a reservoir of simple geometric structure and in a heterogeneous reservoir of complex configuration. The results obtained correspond to the physics of the processes under consideration.

This work considers the isothermal process of incompressible viscous fluid filtration in an oil-saturated, fractured-porous reservoir. A study of the pressure and water saturation distribution process is carried out for a case in which a production well is put into operation. For this problem, i.e., a mathematical model in a two-dimensional formulation, a numerical method and a parallel algorithm are proposed. The mathematical model of two-phase filtration is written in accordance with the classical laws of continuum mechanics and Darcy’s law and also includes a function of fluid exchange between low-permeability pores and high-permeability natural fractures within the framework of the Warren–Root model. The numerical solution is based on the finite-difference method and a splitting scheme of physical processes and spatial coordinates. For a split system with respect to piezoconductivity, an implicit finite-difference scheme with fixed saturations is constructed, and with respect to saturation transfer, explicit and implicit difference schemes are constructed. For parallel implementation of the developed numerical approach, a method based on geometric parallelism is selected. Testing of the developed method is performed using the example of calculating liquid mass transfer for a wide range of parameters. To verify the model, the obtained calculated pressure curves are compared with field data recorded by a deep-well measuring device. The results allow for estimation of the distribution of reservoir pressure and water saturation depending on the permeability of the fracture set and the pore part. The obtained results allow for monitoring of well operations, reducing unexpected accident risks and optimizing the development system in order to increase oil production in fractured-porous reservoirs. Computational experiments confirm the efficiency of the developed numerical algorithm and its parallel implementation.

This work presents a mathematical model for solving three-dimensional radiation problems of magnetohydrodynamics. An implicit fully conservative difference scheme is used to solve the system of differential equations. Two methods are used to solve the system of difference equations: the method of separate and the method of combined solution of equations, which are split by physical processes. A software implementation of the developed numerical algorithms is carried out, and calculations are performed modeling the compression of plasma by a magnetic field. The time dynamics of the parameters of matter and the magnetic field are studied. During the calculation process, at its various stages, both numerical methods used in the program are involved. The results obtained correspond to the physics of the process. В данной работе представлена математическая модель для решения трехмерных радиационных задач магнитной гидродинамики. Для решения системы дифференциальных уравнений применена неявная полностью консервативная разностная схема. Используется два метода решения системы разностных уравнений: метод раздельного и метод комбинированного решения уравнений, которые расщеплены по физическим процессам. Произведена программная реализация разработанных численных алгоритмов, выполнены расчеты, моделирующие сжатие плазмы магнитным полем. Изучалась динамика по времени параметров вещества и магнитного поля. В процессе расчета на его различных стадиях были задействованы оба используемых в программе численных метода. Полученные результаты соответствуют физике процесса.

Предлагается пространственная двухкомпонентная (H2O, CH4) трехфазная (гидрат, свободные вода и газ) модель фильтрации с учетом диссоциации газовых гидратов, основанная на расщеплении по физическим процессам, с использованием неклассического закона движения (с учетом его нелинейности). Представленная математическая модель позволяет рассчитывать двумерные течения в областях с нерегулярной структурой пластов. С ее помощью можно проводить как профильные, так и площадные расчеты, учитывая при этом сложную геометрию осадочных бассейнов. При ее апробации для решения задач теории фильтрации в осадочных бассейнах применен и программно реализован метод опорных операторов. Этот метод позволяет рассчитывать процессы фильтрации в средах с разрывными физическими свойствами, что достигается использованием нерегулярных сеток. В результате появляется возможность моделировать зоны сдвига и получать численное решение в условиях разномасштабности задачи. При этом на сетках с крупными ячейками, где присутствуют разрывы материальных свойств, сохраняется качественная аппроксимация переноса насыщенностей и градиентов термодинамических величин. Построенная сеточная модель также аппроксимирует тождества метода опорных операторов на различных слоях по времени. На основе разработанной вычислительной технологии создан программный комплекс, инструментальные средства которого способны на сетках нерегулярной структуры решать двумерные задачи многофазного и многокомпонентного моделирования процессов диссоциации газовых гидратов в пористой среде осадочных бассейнов литологически сложной структуры. Для тестирования программного комплекса проведены модельные расчеты пьезопроводных процессов в трехфазной среде с гидратизированными твердофазными включениями в двумерном случае на нерегулярных сетках. Расчеты показали уменьшение значения депрессии внутри пространственных областей при использовании нелинейных фильтрационных законов движения в среде по сравнению с классическим законом Дарси, что позволяет корректно описывать физику низкопроницаемых коллекторов.

The work is devoted to the application of irregular computational grids to problems of underground fluid dynamics in the presence of gas hydrates taking into account gravity. Discretization of filtration equations is carried out using the support operator method. The obtained algorithms are tested on the problem of gravitational differentiation of gas and water in a porous medium taking into account gas hydrates. One of the criteria used in testing is the horizontality of the gas-water contact in the stationary case. The calculations are divided into two stages. At the first stage, a regular rectangular grid is used, which makes it possible to check compliance with this criterion. At the second stage, calculations are carried out on an irregular grid for a system of heterogeneous layers of complex geometric structure corresponding to real geological structures. The results obtained correspond to the physics of the process. The approach can be used to study the migration of deep gas released during the dissociation of gas hydrates to the earth's surface.

The work provides a description of the spatial filtration problem in a three-phase hydrate equilibrium zone. A mathematical model is presented for studying two-dimensional fluid flows taking into account the solid hydrate phase and the irregular structure of formations. A non-classical form of the motion law is used, applicable at low permeability and low pressure drops. Efficient computational algorithms based on the support operator method are proposed that make it possible to separate the hyperbolic and dissipative subsystems of the problem. The algorithms are implemented on meshes of irregular structure to model the two-dimensional multiphase processes of gas hydrate dissociation. Testing is carried out on model piezoconductive processes with saturation transfer, where it is shown that depression processes are less expressed when using a nonlinear law of motion.

The work is devoted to numerical modeling of the processes of heat and mass transfer of a two-phase fluid in the environment of a production well, which is necessary for monitoring the development of fractured-porous reservoirs. This work proposes an efficient approach to constructing a solution to the problem. To solve the problem, a model of the “double medium” type is used, where the pore part of the reservoir is considered as the first medium, and the system of natural fractures is considered as the second medium. For the resulting mathematical model, the difference schemes with time weights are constructed based on the algorithm of splitting by physical processes, which ensure the correctness and consistency of fluxes in the fracture system and the pore reservoir. In the numerical solution, the approximations of differential operators obtained in the framework of the finite difference method are used. For the parallel implementation of the developed numerical approach, the domain decomposition method and the matrix sweep algorithm are chosen. The program implementation is made using the MPI standard. Computational experiments are carried out, the results of which confirm the effectiveness of the developed numerical algorithm and its parallel implementation. In numerical experiments, the distributions of pressure and temperature near an operating production well are obtained, on the basis of which it is possible to adjust the operation of wells in order to increase production.