Research

Jump to: Review

27 pages, 10875 KiB

Open AccessArticle

Multiscale Model Reduction with Local Online Correction for Polymer Flooding Process in Heterogeneous Porous Media

by Maria Vasilyeva and Denis Spiridonov

Mathematics 2023, 11(14), 3104; https://0-doi-org.brum.beds.ac.uk/10.3390/math11143104 - 13 Jul 2023

Viewed by 817

In this work, we consider a polymer flooding process in heterogeneous media. A system of equations for pressure, water saturation, and polymer concentration describes a mathematical model. For the construction of the fine grid approximation, we use a finite volume method with an [...] Read more.

In this work, we consider a polymer flooding process in heterogeneous media. A system of equations for pressure, water saturation, and polymer concentration describes a mathematical model. For the construction of the fine grid approximation, we use a finite volume method with an explicit time approximation for the transports and implicit time approximation for the flow processes. We employ a loose coupling approach where we first perform an implicit pressure solve using a coarser time step. Subsequently, we execute the transport solution with a minor time step, taking into consideration the constraints imposed by the stability of the explicit approximation. We propose a coupled and splitted multiscale method with an online local correction step to construct a coarse grid approximation of the flow equation. We construct multiscale basis functions on the offline stage for a given heterogeneous field; then, we use it to define the projection/prolongation matrix and construct a coarse grid approximation. For an accurate approximation of the nonlinear pressure equation, we propose an online step with calculations of the local corrections based on the current residual. The splitted multiscale approach is presented to decoupled equations into two parts related to the first basis and all other basis functions. The presented technique provides an accurate solution for the nonlinear velocity field, leading to accurate, explicit calculations of the saturation and concentration equations. Numerical results are presented for two-dimensional model problems with different polymer injection regimes for two heterogeneity fields. Full article

(This article belongs to the Special Issue Mathematical Models of Multiphase Flows in Porous Media)

► Show Figures

Figure 1

13 pages, 2410 KiB

Open AccessArticle

Approach to the Numerical Study of Wave Processes in a Layered and Fractured Porous Media in a Two-Dimensional Formulation

by Amir A. Gubaidullin, Olga Yu. Boldyreva and Dina N. Dudko

Mathematics 2023, 11(1), 227; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010227 - 02 Jan 2023

Cited by 2 | Viewed by 847

Abstract

A new approach to the numerical study of arbitrary waveform impulses in a layered porous and fractured-porous medium in a two-dimensional formulation has been developed. Layers can have different characteristics and contain fractures. A computer implementation of the mathematical model based on the [...] Read more.

A new approach to the numerical study of arbitrary waveform impulses in a layered porous and fractured-porous medium in a two-dimensional formulation has been developed. Layers can have different characteristics and contain fractures. A computer implementation of the mathematical model based on the finite-difference MacCormack method has been completed. A number of test calculations have been carried out confirming the reliability of the numerical solutions obtained. The possibility of using the proposed approach to solve problems of wave dynamics is shown. Full article

(This article belongs to the Special Issue Mathematical Models of Multiphase Flows in Porous Media)

► Show Figures

Figure 1

17 pages, 6041 KiB

Open AccessArticle

Evolution of Filtration Pressure Waves in a Hydraulic Fracture during Transient-Well-Operation Modes

by Vladislav S. Shagapov, Rustem A. Bashmakov, Nina O. Fokeeva and Anastasia A. Shammatova

Mathematics 2023, 11(1), 98; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010098 - 26 Dec 2022

Cited by 1 | Viewed by 1825

Abstract

At present, a significant part of oil is extracted from difficult-to-develop reservoirs with low permeability. Hydraulic fracturing is one of the most important methods of production stimulation. Scientific articles do not describe the connections between the flow-rate in the well and the change [...] Read more.

At present, a significant part of oil is extracted from difficult-to-develop reservoirs with low permeability. Hydraulic fracturing is one of the most important methods of production stimulation. Scientific articles do not describe the connections between the flow-rate in the well and the change in pressure in the hydraulic fracture or between the changing pressure in the well and the pressure in the hydraulic fracture, except in some cases of constant fluid-flow in the well and constant production. We obtained both the exact analytical solutions and the simple approximate solutions which describe the connection between the well-fluid flow-rate and the pressure evolution in a fracture. The work has also solved the inverse problem: how to determine the parameters of a hydraulic fracture, knowing the change in well flow-rate and the change in fluid-flow. The results show good comparison with practical data. Full article

(This article belongs to the Special Issue Mathematical Models of Multiphase Flows in Porous Media)

► Show Figures

Figure 1

15 pages, 768 KiB

Open AccessArticle

Acoustic Sounding of Hydraulic Fractures in a Low-Permeability Reservoir

by Vladislav Sh. Shagapov, Emiliya Galiakbarova and Zulfiya Khakimova

Mathematics 2023, 11(1), 97; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010097 - 26 Dec 2022

Viewed by 902

Abstract

Theoretical models indicate the possibility of diagnosing the presence and the conductivity of the hydraulic fractures in reservoirs of the permeability of order milli Darcy by means of an acoustic TV set, which is a cylindrical probe of the length of several meters [...] Read more.

Theoretical models indicate the possibility of diagnosing the presence and the conductivity of the hydraulic fractures in reservoirs of the permeability of order milli Darcy by means of an acoustic TV set, which is a cylindrical probe of the length of several meters with a generator of impulse signals and pressure sensors. It is suggested to generate an impulse signal in a fluid, filling the gap between the probe body and the outer wall of the borehole. It is supposed that the impulse signal is generated in a fluid located in the gap between the probe body and an outer wall of the borehole. The signal evolution recorded by means of the pressure sensors as a damping of its amplitude and the appearance of the reflected burst of pressure allow one to estimate the presence and the conductivity of the fractures in the bottomhole zone. We consider fractures, which are radial or longitudinal to an open part of the borehole. The length of the impulse signal is less than the length of the probe, but exceeds the width of the gap between the probe body and the outer wall of the borehole. We take into consideration the damping of the impulse signal due to the viscosity effects in a boundary layer near the borehole walls. The width of a fracture is much less than the wavelength, and this is why the radial fracture is admitted by the reflecting surface. We provide the results of the dispersion analysis and numerical experiments on the influence of the filtration characteristics of the fractures, the width of the gap and the type of fluid on the evolution of the impulse signals in the channel. Full article

(This article belongs to the Special Issue Mathematical Models of Multiphase Flows in Porous Media)

► Show Figures

Figure 1

15 pages, 2600 KiB

Open AccessArticle

Reservoir Permeability Identification under Three-Phase Filtration Using a Priori Information on Wells

by Andrey V. Elesin, Alfiya Sh. Kadyrova, Anatoliy I. Nikiforov and Aleksey V. Tsepaev

Mathematics 2022, 10(23), 4558; https://0-doi-org.brum.beds.ac.uk/10.3390/math10234558 - 01 Dec 2022

Viewed by 920

Abstract

A method is proposed to solve the identification problem of the permeability field of a three-dimensional reservoir under conditions of three-phase fluid filtration. The peculiarity of the method lies in the fact that the wells retain the proportionality of the values of the [...] Read more.

A method is proposed to solve the identification problem of the permeability field of a three-dimensional reservoir under conditions of three-phase fluid filtration. The peculiarity of the method lies in the fact that the wells retain the proportionality of the values of the permeability coefficients of the layers obtained a priori from the results of geophysical studies. The approximation of the permeability field was carried out in layers using surface splines. The problem of identifying the permeability field belongs to the class of inverse coefficient problems for a system of nonlinear partial differential equations describing the process of fluid flow in a porous medium. The solution of the problem was reduced to minimizing the residual function constructed on the total liquid production rate at the wells. Minimization of the residual function was carried out by the Levenberg–Marquardt method. The black oil model was used to describe the fluid filtration process. The filtration equations were solved numerically by the method of the simultaneous solution, and the control volume method was used to approximate the equations by spatial variables. The proposed approach was used to solve a model problem. The influence of various types of errors on the results of identification was investigated. Full article

(This article belongs to the Special Issue Mathematical Models of Multiphase Flows in Porous Media)

► Show Figures

Figure 1

14 pages, 1760 KiB

Open AccessFeature PaperArticle

Mathematical Model of the Process of Non-Equilibrium Hydrate Formation in a Porous Reservoir during Gas Injection

by Marat K. Khasanov, Svetlana R. Kildibaeva, Maxim V. Stolpovsky and Nail G. Musakaev

Mathematics 2022, 10(21), 4054; https://0-doi-org.brum.beds.ac.uk/10.3390/math10214054 - 01 Nov 2022

Cited by 4 | Viewed by 1150

Abstract

Increasing the efficiency of natural gas storage in geological formations is possible by transferring gas from a free state to a gas hydrate state, since gas hydrates have a number of unique properties. For example, 1 m³ of methane hydrate contains 164 [...] Read more.

Increasing the efficiency of natural gas storage in geological formations is possible by transferring gas from a free state to a gas hydrate state, since gas hydrates have a number of unique properties. For example, 1 m³ of methane hydrate contains 164 m³ of gas under normal conditions. It is possible to store a sufficiently large amount of gas in a small volume at a relatively low pressure. To study the regularities of the process of formation of underground gas hydrate gas storage, this article presents a mathematical model of the process of methane injection into a natural reservoir saturated with methane and water, accompanied by the formation of gas hydrate. Unlike previous works, the constructed mathematical model additionally takes into account a number of factors: the filtration flow of water, the real gas properties, the Joule–Thomson effects and adiabatic compression. The process of gas hydrate formation is considered as a non-equilibrium phase transition. Numerical solutions of the problem are constructed that describe the distributions of parameters (temperature, pressure, phase saturations) in a reservoir. Dependences are obtained that reveal the regularities of the process of non-equilibrium formation of gas hydrate in a natural reservoir during gas injection. The influence of gas injection pressure and temperature, as well as reservoir porosity and permeability, on the distributions of pressure, temperature, water saturation and hydrate saturation in the reservoir, as well as on the dynamics of changes in these parameters and the mass of gas hydrate formed in the reservoir over time, are analyzed. Full article

(This article belongs to the Special Issue Mathematical Models of Multiphase Flows in Porous Media)

► Show Figures

Graphical abstract

14 pages, 2445 KiB

Open AccessArticle

A Theoretical Analysis of Profile Conformance Improvement Due to Suspension Injection

by Konstantin Mikhailovich Fedorov, Alexander Yanovich Gilmanov, Alexander Pavlovich Shevelev, Alexander Vyacheslavovich Kobyashev and Denis Alekseevich Anuriev

Mathematics 2021, 9(15), 1727; https://0-doi-org.brum.beds.ac.uk/10.3390/math9151727 - 22 Jul 2021

Cited by 6 | Viewed by 1322

Abstract

This study is focused on a solution for the problem of suspension penetration in a porous formation. Such a solution forms the basis of injection profile diversion technology for oil reservoir sweep improvement. A conventional model of deep-bed suspension flow was used to [...] Read more.

This study is focused on a solution for the problem of suspension penetration in a porous formation. Such a solution forms the basis of injection profile diversion technology for oil reservoir sweep improvement. A conventional model of deep-bed suspension flow was used to describe the suspension injection process. The suspension slug was followed by water injection, and the inflow injection profile before and after treatment was investigated. For the first time, the criteria that determine the effectiveness of the inflow profile improvement process are introduced. The effect of the suspension filtration coefficient on the particle penetration depth was studied. A specific filtration coefficient value for the maximum penetration depth was achieved. The obtained analytical solution was generalized on multi-reservoir strata with poor interlayer crosslinking. The efficiency of profile conformance improvement was described by the differences in the root-mean-square deviations of the inflow velocities in interlayers from mean values before and after the treatment. It was shown that the complex criterion of suspension treatment efficiency should include a reduction in total injectivity. An increase in suspension slug volume improves the injectivity profile but decreases the total injectivity of an injector. Full article

(This article belongs to the Special Issue Mathematical Models of Multiphase Flows in Porous Media)

► Show Figures

Figure 1

Review

Jump to: Research

17 pages, 3895 KiB

Open AccessReview

Mathematical Modeling of a Non-Isothermal Flow in a Porous Medium Considering Gas Hydrate Decomposition: A Review

by Stanislav L. Borodin, Nail G. Musakaev and Denis S. Belskikh

Mathematics 2022, 10(24), 4674; https://0-doi-org.brum.beds.ac.uk/10.3390/math10244674 - 09 Dec 2022

Cited by 4 | Viewed by 1079

Abstract

Deposits of natural gas hydrates are some of the most promising sources of hydrocarbons. According to studies, at the current level of natural gas consumption, the traditional reserves will last for about 50 years, and the gas hydrate deposits will last for at [...] Read more.

Deposits of natural gas hydrates are some of the most promising sources of hydrocarbons. According to studies, at the current level of natural gas consumption, the traditional reserves will last for about 50 years, and the gas hydrate deposits will last for at least 250 years. Therefore, interest in the study of gas hydrates is associated first of all with gas production from gas hydrate deposits. Additionally, gas hydrates are widely studied for solving practical problems, such as transportation and storage of natural gas, utilization of industrial gases and environmental and technological disasters associated with gas hydrates. When solving practical problems related to gas hydrates, in addition to laboratory and field studies, mathematical modeling is also widely used. This article presents the mathematical models of non-isothermal flow in a porous medium considering the decomposition of gas hydrate. The general forms of the mass conservation equations, Darcy’s law and the energy conservation equation are given. The article also presents derivations of the equations for taking into account the latent heat of phase transitions and non-isothermal filtration parameters for the energy conservation equation. This may be useful for researchers to better understand the construction of the model. For the parameters included in the basic equations, various dependencies are used in different works. In all the articles found, most often there was an emphasis on one or two of the parameters. The main feature of this article is summarizing various dependencies for a large number of parameters. Additionally, graphs of these dependencies are presented so that the reader can independently evaluate the differences between them. The most preferred dependencies for calculations are noted and explained. Full article

(This article belongs to the Special Issue Mathematical Models of Multiphase Flows in Porous Media)

► Show Figures