Abstract—This paper proposes a technique for determining the effective specific electrical conductivity of rock samples when their digital models are used. A modified algorithm for reconstructing the internal structure of the sample from the core’s nondestructive imaging data can be used to construct a relevant discrete model that approximates the pore space with a high degree of accuracy. Unlike existing approaches, the reconstructed discrete geometric model of a heterogeneous medium sample is hierarchical and oriented to the application of parallel computational schemes of multiscale finite element methods for a forward mathematical simulation of electromagnetic processes. The paper presents the results of solving the problem of determining the effective specific electrical conductivity of fluid–saturated rock samples and compares them with the data from laboratory experiments.

In real-life applications, it is not always possible to decouple multiple physical processes occurring simultaneously in the medium, thus indirect methods based on, for example, electromagnetic measurements may be used to adequately describe the macroscopic behavior of the studied objects. We consider a multiscale problem of modelling the electromagnetic field in the hydrocarbon-bearing rock under the thermal and mechanical effects. The goal of this study is to develop a unified approach to modelling the multi-physical processes in geological media, such as oil reservoirs and carbon deposits. Geological media are characterized by the presence of multiple geometrical scales and highly heterogeneous physical properties. We use upscaling techniques and the effective characteristics of the media that adequately reflect its macroscopic behavior. We carry out mathematical modelling of the processes of heat and mass transfer, elastic deformation and mesoscale electromagnetic interactions with regard to the complex internal structure of the media. Our algorithm for the macroscopic description of the behavior of the geological media is based on the methods of the effective medium theory and numerical homogenization techniques. We present the results of the mathematical modelling of heat and mass transfer, elastic deformation and electromagnetic field in a heterogeneous medium, obtained using the computational schemes based on the multiscale finite element methods.

Предложены два метода аппроксимации тороидального источника электромагнитного поля для решения системы уравнений Максвелла в естественных переменных векторным методом конечных элементов: электрическая постановка (плотность электрического тока вдоль обмотки катушки); магнитная постановка (эффективная плотность магнитного тока в сердечнике катушки). Выполнен сравнительный анализ методов в зависимости от электрофизических и геометрических характеристик источника и свойств среды. Численные эксперименты демонстрируют преимущество аппроксимации тороидальной генераторной катушки плотностью магнитного тока. Objective. In well-logging applications, the electromagnetic field is generated by the toroidal or solenoidal coils. The physically correct and efficient approximation of these coils is of utmost importance. In this paper, we propose two techniques to approximate the toroidal coil: the electric source and the magnetic source. Methods. In the frequency domain the electromagnetic field is governed by the second-order partial differential equation for the electric or magnetic fields. The computational scheme is based on the vector finite element method with a hierarchical basis of the complete second order. The tetrahedral discretization of the computational domain is non-uniform. When approximating the toroidal coil as an electric current density in the coil winding, the finite element mesh must capture the geometry of the winding. For a coil with multiple winding turns, it results in an extremely fine mesh, which leads to large linear systems, hence, poor solver performance. To overcome this, we develop a new technique for approximating a toroidal coil as a magnetic current density in the core of the coil. This approach does not require including the winding into the finite element mesh. Therefore, the size of the resulting linear system is independent of the number of turns in the coil winding. Results. The article shows that the magnetic current density in the generator coil does not depend on the host medium, but is determined only by the geometric (core size and number of turns) and electrophysical characteristics of the toroidal source core. The paper presents the distribution of the electromagnetic field strength for both formulations in a homogeneous medium. Conclusion. The behavior of the electromagnetic field is the same for both methods of source approximation, but the magnetic formulation does not depend on the geometry of the toroidal coil therefore the computational mesh does no.

The paper considers the problem of calculating effective anisotropic thermal, electric and mechanical properties of composite media. We propose numerical algorithms for the homogenization of heterogeneous media based on the upscaling technique and effective medium theory. The algorithms rely on the direct mathematical modelling of thermal conductivity, elastic deformation, and electromagnetic field. We discretize mathematical models using multiscale non-conforming finite element methods. We investigate the effect of the physical properties and the arrangement of the microinclusions on the effective tensors for various types of field excitation.

Рассматриваются особенности применения модифицированной вариационной постановки векторного метода конечных элементов (ВМКЭ), основанной на замене тонких сильнопроводящих объектов токонесущими поверхностями, для моделирования гармонического электрического поля в областях с криволинейными экранами при различном типе возбуждения поля. Исследуется применимость модифицированной вариационной постановки в широком диапазоне частот Purpose. The paper addresses applicability of the modified variational formulation of vector FEM for the harmonic electric field to the media with cylindrical shields. Thin highly conductive objects are treated as surfaces with the equivalent surface current density. We consider the excitation of the field by a local source (current loop) located either inside or outside the cylindrical shield. Methodology. The simulations are carried out on unstructured tetrahedral meshes. Since the modified variational formulation treats thin highly conductive objects as surfaces, only the surface of a cylinder is discretized. The results yielded by the modified variational formulation are compared with the results of the classic vector FEM. Findings. For the frequency range between 100 KHz and 100 MHz, the modified variational formulation provides correct results when the field source is located outside the cylindrical shield. The modified variational formulation reduces computational cost, since the volume of the thin shield is not discretized. When the field source is located inside the shield, the modified variational formulation gives valid results only in the proximity of the source. Originality/value. The limitations for the application of the reduced variational formulation for the modelling of harmonic electric field in the media with hollow cylindrical shields are investigated

In the paper, we present a computational scheme for mathematical simulation of heat transfer processes in phase-changing multiscale media. In this problem, the correct approximation of heat flux jumps on phase transition boundaries is the main difficulty. Our approach is based on using a nonconforming multiscale finite element method to solve Stefan’s problem. We propose to divide a solution of Stefan’s problem into two components. A discontinuous component is determined in phase transition zones (fine level). The discontinuous component is approximated by a discontinuous Galerkin method. A continuous component is determined everywhere (coarse level). The continuous component is approximated by the classic finite element method. In this approach, a discrete analogue of Stefan’s problem can be solved in parallel. For the correct approximation of the heat flux jump on the phase transition boundary, we introduce a special lifting operator in the variational formulation of the multiscale discontinuous Galerkin method. Results of verification procedure for the developed computational scheme are shown using Stefan’s problem with the analytical solution. A validation procedure is performed using a comparative analysis of mathematical simulation results with physical experimental data. In the physical experiment, a phase change material sample was heated. At discrete time moments, the temperature was recorded using a sensor. We performed the mathematical simulation of the heat transfer process in the phase change material sample using experiment conditions. The difference between the calculated temperature field and physical experiment data was less than 5%.

Статья посвящена математическому моделированию процесса фильтрации жидкости в пористой геологической среде под действием давления. Разрабатывается вычислительная схема решения задачи Дарси на основе смешанной конечноэлементной постановки с использованием разрывного метода Галёркина. Выполнено построение специализированных базисных систем для скорости в пространстве H и давления в пространстве L . Проведены вычислительные эксперименты на классе модельных задач. The paper addresses mathematical modelling modelling for the process of fluid filtration in porous geological media under pressure. A computational scheme based on a mixed non-conformal variational formulation for solving the Darcy problem with a tensor coefficient of permeability of the medium was developed, implemented and verified. Moreover, the discontinuous Galerkin method for obtaining the finite element approximation was used. A specialized hierarchical basis system for velocity in the H space and a basis system with discontinuous functions on boundaries of finite elements for pressure in the L space were constructed. Computational experiments on a class of problems close to real ones have shown that for a computational domain with multidirectional rectangular inclusions of arbitrary size and concentration, the numerical fields of pressure and velocity are determined with a relative error of 1e-2 even on a coarse grid. An increase in the contrast of the permeability coefficient of the medium with respect to the permeability coefficient of inclusions does not change the relative error in determining the numerical fields. To this end, we can conclude that the constructed computational scheme is stable to significant variation of the coefficient. The authors have developed and verified a software package that is able to export a ready-made solution to the problem in the “.dat” format file for further graphical display and analysis in the Tecplot software package.

We present mathematical simulating results of a gas-liquid mixture flow in a cylindrical pipe. The gas-liquid mixture consists of the mineralized water and propane gas. To analyze the mineral composition of the fluid, a sensor is mounted in the pipe. The sensor affects the topology of the flow velocity vector field. The gas-liquid mixture flow is described by the unsteady Navier-Stokes equations. When the flow rate of gas-liquid mixtures exceeds 20 m/s, eddy flows occur in the pipe with obstacles. A method of discretization should take into account the problem specifics: rapidly changing gradients, the prevalence of the convective term in the Navier-Stokes equations. A computational scheme of discontinuous Galerkin method has the local conservative property and is best suited for solving such singularly perturbed problems. To perform a spatial discretization, a computational scheme of the discontinuous Galerkin method in the function spaces H(div) and L 2 is used. Application of the multiscale approach allows breaking down the solution of the simulation problem into several smaller ones that can be solved using parallel computations. Mathematical modeling results of the gas-liquid mixture flow in the pipe with different options for the sensor location are presented.

A mixed variational formulation based on a discontinuous Galerkin method for solving the Darcy problem with a tensor permeability coefficient is considered. Two special hierarchical basis systems in H-div space for velocity and in H-1 space for pressure are constructed. The influence of these basis on the properties of the matrix of the discrete analogue is investigated.

We present results of mathematical modeling of the thermal conductivity process with phase transitions in heterogeneous media. For demonstrating the mathematical modeling results, we use a uniformly porous medium as an experimental sample. We suppose the sample matrix consists of sandstone and the pores are completely filled with a solid paraffin. When the sample is heated, the paraffin, in the pores, goes into the liquid phase. To solve the Stefan problem in a three-dimensional formulation, a computational scheme of a multi-scale discontinuous Galerkin method on tetrahedral finite elements is used. The algorithm for calculating the effective thermal conductivity is based on the solution of the thermal conductivity direct problem with phase transitions and Fourier's law. The effect of the molten paraffin volume concentration in the pores on the effective thermal conductivity is shown.

We present an iterative algorithm for mathematical modeling of an elastic deformation process in a fluid-saturated fractured-porous medium. A three-dimensional multi-physics problem describes the coupled isothermal processes of the solid elastic deformation and slightly compressible fluid flow under external pressure. Mathematical models of these processes are connected via interface conditions for the pressure and density fields on the surface of a fractured-porous medium. For solving the multi-physics problem, a special multiscale procedure was developed. We use a heterogeneous multiscale finite element discretization on coarse polyhedral grids for the solid elastic deformation problem. Multiscale shape functions are constructed using special interface conditions for a hydrodynamic pressure on the surface of pores. We apply a discontinuous Galerkin method and a stabilized finite element discretization on fine tetrahedral grids for solving the hydrodynamics problem in fluid-saturated pores. In this case, we can realize an effective parallel procedure for solving the multi-physics problem. In each pore, hydrodynamics problems can be solved in parallel and independently. Verifications of the computational schemes are presented. We consider three-dimensional media with a different volume concentration of cracks and pores. Computational modeling results are presented. A time of solving the multi-physics problem using fine and coarse grids is shown.

При решении задач электромагнетизма в широком частотном диапазоне в областях с тонкими пластинами, оболочками и экранами численными методами возникает проблема резкого роста сеточной дискретизации вблизи внутренних структур с разномасштабными габаритными размерами. В работе предложена модификация вариационной постановки векторного метода конечных элементов, основанная на снижении размерности модели в окрестности тонких включений, которая позволяет преодолеть эту проблему за счет специфического учета таких структур на уровне вариационной постановки. Так как редуцирование модели обычно приводит к появлению ограничений на область ее применимости, выполнено исследование диапазона допустимых частот, контрастности электрофизических характеристик матрицы и включений, геометрических особенностей внутренней структуры, для которых предложенная модель позволяет получить корректные с точки зрения физики результаты. Purpose. In this paper, we propose a reduced variational formulation for the Helmholtz equation for the electric field, in which thin highly conductive objects are approximated by surfaces with the equivalent surface current density. We conduct a study aimed at defining the range of application for the reduced variational formulation, focusing on highly contrasting thin objects of various geometrical shape and arrangement in a wide frequency range. Methodology. The modelling is performed on unstructured tetrahedral meshes. Since the reduced variational formulation treats thin highly conductive objects as surfaces, no volume mesh is constructed inside of them.We compare the results obtained by the vector FEM using the proposed variational formulation with the results obtained using standard formulation. Findings. Due to the fact that the proposed variational formulation does not require volume meshing of the thin objects, its computational cost is significantly lower. However, the reduced formulation yields correct results in a restricted frequency range. It also imposes some limitations on the minimal contrast and maximal thickness of the thin highly conductive objects. Originality/value. The proposed reduced variational formulation can be applied to simulate the time-harmonic electric field in the media with thin highly conductive inclusions of either regular or chaotic arrangement, as well as thin shielding plates or casings of various geometrical forms.

In this paper, we propose a numerical method to obtain an effective electrical resistivity of heterogeneous media under the influence of a direct current. The heterogeneous multiscale finite element method is used to solve the direct problem of simulation of an electrostatic field. The computational experiments using the developed software complex showed that even the small inclusion concentrations define the effective resistivity of the media. In addition, the change in the localization, orientation, and geometrical shape of inclusions also leads to a significant change of the effective properties of the media.

A method for three-dimensional modeling of pulsed soundings using a fast Fourier transform is developed: instead of the time-domain problem, it is proposed to solve a set of frequency-domain problems obtained by processing the original excitation pulse by a fast Fourier transform. The error of the proposed method compared to the method of solving the time-domain problem is numerically estimated for a model problem with a sinusoidal signal.

A numerical method for computing effective electric characteristics of heterogeneous media is proposed in this paper. We analyze relation between effective characteristics and interior structure of complex medium like form and location of the inclusions. The computed results of the direct problem with inclusions are compared with the results of the direct problem without inclusions. The problem without inclusions is considered as a problem for anisotropic medium with computed tensor characteristics.