Model For Flow In Porous Media Research Articles

Two-phase Porous Media Flow Model Based on the Incompressible Navier-Stokes Equation.

Two-phase porous media flow is important in many applications from drug delivery to groundwater diffusion and oil recovery and is of particular interest to biomedical diagnostic test developers using cellulose and nitrocellulose membranes with limited fluid sample volumes. This work presents a new two-phase porous media flow model based on the incompressible Navier-Stokes equation. The model aims to address the limitations of existing methods by incorporating a partial saturation distribution in porous media to account for limited fluid volumes. The basic parameters of the model are the pore size distribution and the contact angle. To validate the model, we solved five analytical solutions and compared them to corresponding experimental data. The experimentally measured penetration length data agreed with the model predictions, demonstrating model accuracy. Our findings suggest that this new two-phase porous media flow model can provide a valuable tool for researchers developing fluidic assays in paper and other porous media.

Perfusion estimation using synthetic MRI-based measurements and a porous media flow model.

The measurement of perfusion and filtration of blood in biological tissue give rise to important clinical parameters used in diagnosis, follow-up, and therapy. In this paper, we address techniques for perfusion analysis using processed contrast agent concentration data from dynamic MRI acquisitions. A new methodology for analysis is evaluated and verified using synthetic data generated on a tissue geometry.

A Hybrid Model for Cardiac Perfusion: Coupling a Discrete Coronary Arterial Tree Model with a Continuous Porous-Media Flow Model of the Myocardium.

This paper presents a novel hybrid approach for the computational modeling of cardiac perfusion, combining a discrete model of the coronary arterial tree with a continuous porous-media flow model of the myocardium. The constructive constrained optimization (CCO) algorithm captures the detailed topology and geometry of the coronary arterial tree network, while Poiseuille's law governs blood flow within this network. Contrast agent dynamics, crucial for cardiac MRI perfusion assessment, are modeled using reaction-advection-diffusion equations within the porous-media framework. The model incorporates fibrosis-contrast agent interactions and considers contrast agent recirculation to simulate myocardial infarction and Gadolinium-based late-enhancement MRI findings. Numerical experiments simulate various scenarios, including normal perfusion, endocardial ischemia resulting from stenosis, and myocardial infarction. The results demonstrate the model's efficacy in establishing the relationship between blood flow and stenosis in the coronary arterial tree and contrast agent dynamics and perfusion in the myocardial tissue. The hybrid model enables the integration of information from two different exams: computational fractional flow reserve (cFFR) measurements of the heart coronaries obtained from CT scans and heart perfusion and anatomy derived from MRI scans. The cFFR data can be integrated with the discrete arterial tree, while cardiac perfusion MRI data can be incorporated into the continuum part of the model. This integration enhances clinical understanding and treatment strategies for managing cardiovascular disease.

Shape design of an artificial pump-lung using high-resolution hemodynamic simulation with high-performance computing

Accurate and fast prediction of the hemodynamics of the artificial pump-lung is critical in the design process. In this study, a comprehensive computational framework, including a sliding mesh method, a coupled free flow and porous media flow model, a hemolysis prediction method, a k−ω shear stress transport turbulence model, and solution algorithms, is introduced to accurately predict the velocity field, pressure heads, and hemolysis. The framework is used to do the shape design of an artificial pump-lung on a supercomputer. High-resolution hemodynamics simulation results are obtained and analyzed, and the parallel performance of the algorithm is studied. The numerical results indicate that the proposed framework is capable of accurately predicting the velocity field, pressure heads, and hemolysis, and the performance of the designed artificial pump-lung meets the biocompatibility requirements. Additionally, the parallel performance results demonstrate the potential of the framework to efficiently perform the design of artificial pump-lungs using a large number of processors.

Homogenization of ferrofluid flow models in porous media with Langevin magnetization law

The paper is concerned with the homogenization of the equations describing the flow of a ferrofluid through a heterogeneous porous medium Ω in the presence of an applied magnetic field. We discuss two models where the magnetization M is parallel to the magnetic field H. In the first one M and H satisfy the relation M=λ01ΩfH in Ω, where λ0 is a positive constant and 1Ωf is the characteristic function of Ωf (the pore space). In the second model, M and H satisfy the Langevin magnetization law M=MsL(b1|H|)|H|1ΩfH, where L is the Langevin function given by L(x)=1tanh⁡x−1x, Ms is the saturation magnetization and b1 is a positive physical constant. The velocity and the pressure satisfy the Stokes equation with a Kelvin magnetic force. We perform the homogenization of the equations of each of the two models. Using the two-scale convergence method, we rigorously derive the homogenized equation for the magnetic potential and determine the asymptotic limit of the magnetization. Then we rigorously derive a Darcy law.

Simulation of Mechanical Test in Digital Rock Affected by Rock Heterogeneity and Flow in Porous Media

As the main reservoir of oil and gas resources, the physical and mechanical properties of the rock are complicated. The strong heterogeneity of rock and the flow in porous media further influences the nature of stress propagation. Therefore, considering the heterogeneous particle model and fluid-structure coupling, it is very important to study the variation law of rock mechanical properties under a stress state, which lays a foundation for formation fracturing and mining. In this paper, the mechanical properties of the rock are studied by the ideal, heterogeneous, and flow models in porous media. The results show that the stress of the contact surface between particles of the same size is large. Still, the stress of the contact surface between different particles is larger, which is more likely to become the weak mechanical link of failure. When the displacement of the z-axis is the same, the x and y-axis increase the axial stress of the z-axis. The peak stress is high and more sensitive when the particle size difference is small. When water is in the pores, the rock strength obtained under consolidated undrained is greater than that under consolidated drained.

Numerical Study on the Migration of Leaked Oil in Submarine Sediment Caused by Pipeline Leakage Using Two-phase Flow Model

Oil leakage from submarine pipelines can cause severe environmental problems and economic losses. In this study, a two-phase immiscible flow model in porous media is proposed to simulate the migration of leaked oil caused by the leakage of a buried submarine pipeline in sandy sediment. The relationship between saturation, pore pressure, and relative permeability of different fluids in porous media is considered in present model, which is distinguished from previous studies. The spilled oil spreads radially from the leakage point into surrounding sediment driven by the leakage velocity. Moreover, it continues to migrate in sediment under the capillary pressure and gravity. Interestingly, the effect of wave-induced transient response on the ultimate migration of spilled oil in a sandy sediment is negligible. However, increasing sediment porosity can greatly enhance the upward migration of leaked oil based on systematically parametric analysis. Meanwhile, the migration rate of leaked oil decreases with increasing oil density or viscosity. The oil-contaminated area significantly expands with the increase of leakage rate. Fitting formulas between dimensionless time required for the leaked oil to reach the overlying seawater and non-dimensional influencing parameters are well established, which can provide useful guidelines for similar disasters.

Mechanically robust and rapidly degradable hydrogels for temporary water plugging in oilfields

AbstractHydrogels have been widely used for in‐depth water control and water plugging in most mature oilfields; however, the poor mechanical properties as well as non‐degradable characteristics severely limit their applications. Herein, a multifunctional macromolecular cross‐linker (GA) based on a natural macromolecule, gelatin, is fabricated and used to prepare polyacrylamide hydrogels. Cross‐linking with GA leads to a special network involving physical entanglements that dissipate energy and topological cross‐links that distribute stress. Therefore, the hydrogels manifest remarkable mechanical strength, toughness and tear‐resistance. At the same time, GA can be readily degraded at high temperature, leading to disintegration of the hydrogels. The flow model in porous media reveals that the hydrogels present a significant ability of water plugging and rapid degradation, where the plugging ratio and degradation ratio reach 99% and 100%, respectively. Such mechanical properties and degradation performance enable the hydrogels to serve as a novel temporary water plugging agent for heterogeneous reservoirs of oilfield.

Probing the generation mechanism of emulsion based on shearing actions between multiphase fluids in porous media

The generation, migration, and coalescence processes of emulsions in porous media of the actual formations are relatively complex. However, few studies have been reported about the generation mechanism of emulsions under shearing action, especially under shearing action between multiphase fluids. In this paper, the concept of emulsification caused by fluid shearing was proposed for the first time. An oil-water two phases flow model in porous media was established to study the generation process of emulsion caused by shearing actions between oil and water. Fluid flow velocity, pressure and distribution characteristics of oil and water during this process were recorded. The effects of flow velocity and oil-water interface properties on the generation process of emulsion were investigated. Combined with the microscopic visualization oil displacement model, the generation characteristics of emulsions in porous media were further explored. The results indicated that the complex flow between oil and water in porous media would cause shear emulsification of crude oil. With the increase of flow velocity of oil-water two phases, the shear emulsification capacity was gradually enhanced. The generation capacity of emulsion in the micro shearing unit was mainly affected by the flow velocity of the displacing phase, and the particle size of the emulsion was mainly affected by the flow velocity of the shearing phase. When the interfacial tension between oil and water was lower than 10−1 mN/m, the shear emulsification rate did not change significantly. These results are of great significance for the supplement and improvement of the generation mechanism of emulsions in porous media, providing theoretical guidance for further enhancing oil recovery by regulating emulsification.

Petroleum reservoir parameters estimation using non-isothermal transient model and optimization methods

Traditionally, oil well tests aim to characterize the reservoir permeability field from pressure transient analysis (PTA) of drawdown and build up based on isothermal flow models in porous media. With the advancement of well test instrumentation, more accurate temperature records became available and have encouraged analyses based on non-isothermal models that made possible the temperature transient analysis (TTA). In addition to the characterization of reservoir parameters, such as permeability and porosity by TTA, taking non-isothermal effect into account on PTA represents better physical phenomena, especially in high transmissibility reservoirs and in cases where the pressure sensor is placed away from the sandface. This work consists in the development and numerical implementation of a non-isothermal model of well test considering a unidimensional radial reservoir coupled to a production well and in the use of this simulator, associated with multivariable optimization methods, to solve the inverse problem of reservoir parameters characterization. Nelder-Mead Simplex, Newton-Raphson and BGFS methods were used to evaluate reservoir parameters for one of the reservoir configurations analyzed and the first method presented better computational efficiency. It was then used to solve the inverse problem of radial composite reservoirs.

On convergence of explicit finite volume scheme for one-dimensional three-component two-phase flow model in porous media

Abstract In this work, we develop and analyze an explicit finite volume scheme for a one-dimensional nonlinear, degenerate, convection–diffusion equation having application in petroleum reservoir. The main difficulty is that the solution typically lacks regularity due to the degenerate nonlinear diffusion term. We analyze a numerical scheme corresponding to explicit discretization of the diffusion term and a Godunov scheme for the advection term. L ∞ {L}^{\infty } stability under appropriate CFL conditions and BV {\rm{BV}} estimates are obtained. It is shown that the scheme satisfies a discrete maximum principle. Then we prove convergence of the approximate solution to the weak solution of the problem, and we mount convergence results to a weak solution of the problem in L 1 {L}^{1} . Results of numerical experiments are presented to validate the theoretical analysis.

Adjoint Numerical Method for a Multiphysical Inverse Problem of Two-Phase Well Testing in Petroleum Reservoirs

Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

Numerical Simulation of Flow within a storage area of HDPE modular pavement

The development of stromwater management usually promote to provide the safe passage of stromwater. However physical modelling need expensive laboratory experiments. Due to that this numerical study is performed to study the flow within storage area of HDPE modular pavement. This paper studied and compared the infiltration rate of diagonal modular and conventional pavement (control), determined the velocity magnitude and pressure of modular pavement at various rainfall intensities. FLOW-3D was used to run a simulation on a porous media flow model using the Navier-Stokes equation. Real rainfall data of Malaysia was used as the model inputs to get better analysis of pavement design. The present findings showed that storage area modular pavement has lower fraction of fluid than control, which means that it has greater holding capacity and capable to capture all the rainfall volume from 5mm/h to 85mm/h. Besides, rainfall intensity has a strong influence on velocity magnitude and pressure. The HDPE diagonal modular pavement strong enough to sustain with an increasing of velocity magnitude and pressure during extreme rainfall. Therefore, HDPE modular pavement indicates a better water interception capacity than conventional pavement. FLOW-3D helps the critical analysis of pavement design process and useful as supplementary tool.

About Identifiability of Oil and Water Relative Permeability Curves and Reservoir Heterogeneity through Integrated Well Test Study

Relative permeabilities (RP) play an important role in reservoir engineering. RP functions together with the permeability anisotropy coefficient predetermine waterflooding efficiency and oil/water ratio in production forecast. Traditionally multiphase flow parameters are estimated from core analysis data. But such measurements suffer from small representative elementary volume (REV) and limited characterization of reservoir properties. Therefore using core data in 3D reservoir modelling could lead to distorted description of actual flow conditions. Despite those functions (RP) could be history matched with assimilated production data, such a procedure would require long history of waterflooding. So the authors’ idea is to apply integrated well test study to estimate the displacement efficiency and/or relative permeability functions at downhole conditions. Well testing is traditionally applied in reservoir engineering for single-phase reservoir parameters estimation. Our approach is extended to multi-phase flow and is based on data collection at well bottomholes and subsequent data assimilation in flow model.In this paper we summarize both features of mathematical problem statement and identifiability of multiphase parameters through inverse problem solution, but also discuss 10+ -year experience in interpretation of two-phase well testing using our technologies. To estimate multiphase parameters, we have to jointly consider complex data of well logging and dynamic data of well testing. Both data sets are used in the inverse problem quality criteria, where the least squares method has been applied. Forward problem corresponds to a 3D multiphase fluid flow model in porous media. The latter consists of continuity equation with multiphase Darcy equation instead of moment balances. For data assimilation modern methods of optimal control theory were successfully applied. For all synthetic test cases real reservoir parameters were accurately recovered through the use of forward simulation data.In order to estimate reservoir heterogeneity, we applied a single-phase model to get the ratio of vertical to horizontal permeability. Data of well self-interference testing were used for vertical permeability estimation. Depending on the depth of reservoir pressure perturbation, reservoir properties could be properly inferred from observations. Two other options of 3D interference testing using vertical and horizontal wells are also presented.In other words, all the problems considered are identifiable, and the level of their correctness is completely predetermined by the amount and quality of observed data. And transition to digital (intelligent) oil-and-gas production and closed-loop reservoir management [1] provides a possibility to remove discussed restrictions and all inverse problems considered should be accounted as fully identifiable.

A Class of Lagrangian–Eulerian Shock-Capturing Schemes for First-Order Hyperbolic Problems with Forcing Terms

In this work, we develop an improved shock-capturing and high-resolution Lagrangian–Eulerian method for hyperbolic systems and balance laws. This is a new method to deal with discontinuous flux and complicated source terms having concentrations for a wide range of applications science and engineering, namely, 1D shallow-water equations, sedimentation processes, Geophysical flows in 2D, N-Wave models, and Riccati-type problems with forcing terms. We also include numerical simulations of a 1D two-phase flow model in porous media, with gravity and a nontrivial singular $$\delta $$ -source term representing an injection point. Moreover, we present approximate solutions for 2D nonlinear systems (Compressible Euler Flows and Shallow-Water Equations) for distinct benchmark configurations available in the literature aiming to present convincing and robust numerical results. In addition, for the linear advection model in 1D and for a smooth solution of the nonlinear Burgers’ problem, second order approximations were obtained. We also present a high-resolution approximation of the nonlinear non-convex Buckley–Leverett problem. Based on the work of A. Harten, we derive a convergent Lagrangian–Eulerian scheme that is total variation diminishing and second-order accurate, away from local extrema and discontinuous data. Additionally, using a suitable Kružkov’s entropy definition, introduced by K. H. Karlsen and J. D. Towers, we can verify that our improved Lagrangian–Eulerian scheme converges to the unique entropy solution for conservation laws with a discontinuous space-time dependent flux. A key hallmark of our method is the dynamic tracking forward of the no-flow curves, which are locally conservative and preserve the natural setting of weak entropic solutions related to hyperbolic problems that are not reversible systems in general. In the end, we have a general procedure to construct a class of Lagrangian–Eulerian schemes to deal with hyperbolic problems with or without forcing terms. The proposed scheme is free of Riemann problem solutions and no adaptive space-time discretizations are needed. The numerical experiments verify the efficiency and accuracy of our new Lagrangian–Eulerian method.

A hybrid porous model for full reactor core scale CFD investigation of a prismatic HTGR

Three-dimensional (3-D) Computational Fluid Dynamics (CFD) analysis of a whole nuclear reactor core is a great challenge due to the large geometric volume and complex structures. This research presents a hybrid porous (HP) model to simplify a prismatic High Temperature Gas-cooled Reactor (HTGR) core, so 3-D CFD investigation can be performed on a full reactor core scale. In the HP model, the prototypic small coolant channels in the nuclear fuel blocks are lumped together to form multiple equivalent large coolant channels, and then the porous medium flow model is applied to each of them. Therefore, heat transfer in fuel blocks is computed by a hybrid combination of solid energy and porous flow energy equations. The similarity between the HP model and prototypic model is achieved by deriving the porous flow permeability, inertial resistance factor, and artificial thermophysical properties. Compared with the widely used whole porous (WP) flow model, the HP model preserves more realistic geometric structures, and therefore more accurate physical processes. The General Atomics’ Modular High Temperature Gas-cooled Reactor (MHTGR) design was chosen as a prototype to demonstrate the methodology. Simulations were performed using the prototypic CFD model and HP model at steady-state forced circulation, steady-state natural circulation, and transient conditions that correspond to normal operation, extended period of pressurized cool down, and short-term transients after reactor shutdown, respectively. The comparison shows good agreement between the HP model and prototypic model in the maximum fuel temperature, average solid temperature, and helium flow rate, which demonstrates the potential applicability of the HP model for a full reactor core scale simulation in the future. As a benefit, the HP model reduces the mesh quantity by a factor of 50 from a prototypic model. Correspondingly, the computation time was reduced by a factor of at least 30.

Demonstration and application of NMM‐based fractured porous medium flow model

AbstractIn natural rock masses, the shapes of three‐dimensional (3‐D) blocks cut by arbitrary fracture networks may be very complex. Owing to the geometric complexity and difficulty of mesh discretization of 3‐D blocks and fracture facets, explicit consideration of fracture networks in flow analysis of fractured porous medium (FPM) is very challenging. Using the numerical manifold method based on independent covers (NMMIC), an FPM flow model was proposed that can deal with very complex 3‐D fracture networks. In this paper, the convergence of NMMIC was first demonstrated. The theoretical basis of the arbitrary refinement of computational meshes was proven. Moreover, three peculiarities of NMMIC meshes, that is, arbitrary shape, arbitrary connection, and arbitrary refinement of independent covers, were concluded. Finally, some two‐dimensional (2‐D) tunnel flow examples were analyzed and the numerical results were compared with the analytical results. 3‐D examples with complex fracture distributions were also analyzed. In addition, the computational scale of the developed program was tested by increasing the number of computational elements. The results show that our model can accurately analyze the groundwater flow of rocks surrounding tunnels with complex fracture distributions.

Non-Darcy flow models in porous media via Atangana-Baleanu derivative

Abstract Two fractional Swartzendruber models applying the Atangana-Baleanu (A-B) derivative are proposed to describe non-Darcy flow in porous media. The analytical solutions of models are obtained by using Laplace transform. Fitting curves with the experimental data display the proposed models are suitable to describe non-Darcy flow problems in low and high permeability media. In addition, sensitivity analysis is performed to clarify the influence of relevant parameters on the A-B Swartzendruber models. In the light of the two-scale thermodynamics, the physical explanations of A-B Swartzendruber model I and II are revealed. The proposed A-B Swartzendruber models connect fluid flow of different scales, and provide a unified description of non-Darcy flow in porous media with low and high permeability.

Resistance Analysis of Low Reynolds number flow in porous media

At present, some researchers believe that Darcy’s law may not be valid, which may be the root cause of complex porous media problems. Porous media flow model is generally regarded as pipe flow, but in theory, it is necessary to consider it as bypass flow. However, there are few studies in this area. According to the seepage characteristics, we established the model of flowing around a single sphere in porous media and derived the resistance correlation of Darcy velocity (Re < 1). However, the Darcy experimental data analysis shows that the predicted value of the flow around the sphere is far lower than the experimental value, which proves that the assumption of the model of flowing around a single sphere is unreasonable. Even for the seepage of the extremely low Reynolds number, the flow interference around the sphere should be taken into account. Therefore, when the interference coefficient is 45, our formulas are superior to the classical Kozeny-Carman equation and Ergun equation in the prediction of all Darcy’s experimental data and Charles Ritter’s experimental data. Ergun equation has the largest error.

Linearized domain decomposition methods for two-phase porous media flow models involving dynamic capillarity and hysteresis

We discuss two linearization and domain decomposition methods for mathematical models for two-phase flow in a porous medium. The medium consists of two adjacent regions with possibly different parameterizations. The model accounts for non-equilibrium effects like dynamic capillarity and hysteresis. The θ-scheme is adopted for the temporal discretization of the equations yielding nonlinear time-discrete equations. For these, we propose and analyze two iterative schemes, which combine a stabilized linearization iteration of fixed-point type, the L-scheme, and a non-overlapping domain decomposition method into one iteration. First, we prove the existence of unique solutions to the problems defining the linear iterations. Then, we give the rigorous convergence proof for both iterative schemes towards the solution of the time-discrete equations.The developed schemes are independent of the spatial discretization or the mesh and avoid the use of derivatives as in Newton based iterations. Their convergence holds independently of the initial guess, and under mild constraints on the time step. The numerical examples confirm the theoretical results and demonstrate the robustness of the schemes. In particular, the second scheme is well suited for models incorporating hysteresis. The schemes can be easily implemented for realistic applications.