Application of kinetic approach to porous medium flow simulation in environmental hydrology problems on high-performance computing systems

Abstract

A kinetically-based system of equations for three-phase porous media ow simulation is considered. A simple case with the following assumptions is discussed: phase transitions are absent, phases do not dissolve and do not mix, the rock compressibility is negligible. Such systems are under consideration in applied problems when the pressure changes slightly and thermal processes are absent, for example, in environmental problems. The continuity equation is modi ed via introduction of the regularizing term and the second-order time derivative. Due to conversion to the hyperbolic type the corresponding di erence equation stability is improved. An explicit algorithm is developed and adapted to high-performance computing systems. High parallelization e ciency is achieved on a classical cluster as well as on a hybrid cluster with graphics accelerators.Keywords: Multiphase ow in a porous medium, quasigasdynamic system of equations, explicit nitedi erence schemes, parallel implementation.Mathematical modelling of multiphase uid ows in porous media is necessary for solving many practically important problems, for example, for the simulation of processes of contaminant in ltration into the soil [1, 9, 10, 17]. It is well known that numerical simulation of these processes is very time-consuming and impossible without the employment of high-performance computing systems. Nowadays the rapid growth in the computer performance is mainly achieved due to the use of hybrid architectures including multicore CPUs and di erent accelerators like graphics processing units (GPU). Use of GPU for general-purpose computations is a perspective modern trend to solve large scale applied problems with high accuracy for the reasonable time. However, such architectures cause serious di culties in the software development. For example, wellknown IMPES-method for modelling porous media ows [8] assumes the solution of elliptic equation. But the Laplace operator inversion requires high computational costs and leads to decrease of parallelization efciency. Computational algorithms with logically simple structure like explicit nite-di erence schemes can be adapted easily to hybrid supercomputers and allow to exploit them more e ciently [5]. Such algorithms show perfect scaling on parallel architectures therefore they are widely used in numerical investigations of subsurface ows [7, 12].In the present paper a new algorithm for porous medium ow simulation is discussed. Currently kinetic approaches such as Lattice Boltzmann schemes [2, 7, 11] and kinetically-consistent nite di erence (KCFD) [3] schemes seem to be the most promising methods in computational uid dynamics. The new model is constructed by the analogy with KCFD schemes and the related quasigasdynamic (QGD) system of equations [13].

Non-Isothermal Compositional Model for Simulation of Multiphase Porous Media Flows

“…The hyperbolic QGD model of porous medium flow proposed and substantiated by the authors in previous works [3,7,8] has been elaborated. If to pass from conservation laws for phases to conservation laws for components and to determine lacking closure relations one can formulate the next compositional model:…”

“…In cases when calculations with the critical accuracy are really needed, the computational grid steps on space as well as on time are strongly restricted and explicit methods can gain over implicit ones [1]. Following this topic the authors proposed an approach to porous medium flow modeling based on the analogy with the quasigasdynamic (QGD) system of equations [2]: the mass balance equation for each fluid phase was modified to get a regularizing term and to change the equation type from parabolic to hyperbolic [3]. Note that hyperbolization is a modern trend in CFD [4][5][6] to increase the stability of employed explicit schemes.…”

Comparative analysis of explicit type algorithms for solving problems of filtration with the use of hyperbolized equations

“…Рассмотренный подход был многократно протестирован и применялся для решения задач нефтедобычи и задач просачивания на суперкомпьютерах, в том числе с гибридной архитектурой [15,16]. Были достигнуты чрезвычайно высокие ускорения вычислений.…”

An algorithm of the explicit type to solve problems of non-isothermal multiphase flows in porous media

“…Присутствие регуляризатора дало новые вычислительные возможности, в частности позволило использовать явную схему с центральными разностями для аппроксимации конвективных членов этого уравнения. Во-вторых, была проведена гиперболизация полученного уравненияпоявилась вторая производная по времени с малым параметром τ, за счет чего повысилась устойчивость соответствующей явной разностной схемы [2,9,10].…”

“…Предложенные модель и алгоритм с успехом использовались для решения тестовых и модельных задач фильтрации, в том числе на гибридных суперкомпьютерах. В случае изотермического течения одной жидкости в пористой среде верификация проводилась с помощью задачи о плоскорадиальном притоке к скважине [8], численное решение сравнивалось с точным аналитическим [1,10]. Для оценки адекватности разработанного подхода в многофазном случае проводилось сравнение численных результатов, полученных авторами с помощью различных методов [9].…”