Modeling of multiphase flow in low permeability porous media:Effect of wettability and pore structure properties

Multiphase flow in low permeability porous media is involved in numerous energy and environmental applications.However,a complete description of this process is challenging due to the limited modeling scale and the effects of complex pore structures and wettability.To address this issue,based on the digital rock of low permeability sandstone,a direct numerical simulation is performed considering the interphase drag and boundary slip to clarify the microscopic water-oil displacement process.In addition,a dual-porosity pore network model(PNM)is constructed to obtain the water-oil relative permeability of the sample.The displacement efficiency as a recovery process is assessed under different wetting and pore structure properties.Results show that microscopic displacement mechanisms explain the corresponding macroscopic relative permeability.The injected water breaks through the outlet earlier with a large mass flow,while thick oil films exist in rough hydrophobic surfaces and poorly connected pores.The variation of water-oil relative permeability is significant,and residual oil saturation is high in the oil-wet system.The flooding is extensive,and the residual oil is trapped in complex pore networks for hydrophilic pore surfaces;thus,water relative permeability is lower in the water-wet system.While the displacement efficiency is the worst in mixed-wetting systems for poor water connectivity.Microporosity negatively correlates with invading oil volume fraction due to strong capillary resistance,and a large microporosity corresponds to low residual oil saturation.This work provides insights into the water-oil flow from different modeling perspectives and helps to optimize the development plan for enhanced recovery.

Prediction of Porous Media Fluid Flow with Spatial Heterogeneity Using Criss-Cross Physics-Informed Convolutional Neural Networks

Recent advances in deep neural networks have shed new light on physics,engineering,and scientific computing.Reconciling the data-centered viewpoint with physical simulation is one of the research hotspots.The physicsinformedneural network(PINN)is currently the most general framework,which is more popular due to theconvenience of constructing NNs and excellent generalization ability.The automatic differentiation(AD)-basedPINN model is suitable for the homogeneous scientific problem;however,it is unclear how AD can enforce fluxcontinuity across boundaries between cells of different properties where spatial heterogeneity is represented bygrid cells with different physical properties.In this work,we propose a criss-cross physics-informed convolutionalneural network(CC-PINN)learning architecture,aiming to learn the solution of parametric PDEs with spatialheterogeneity of physical properties.To achieve the seamless enforcement of flux continuity and integration ofphysicalmeaning into CNN,a predefined 2D convolutional layer is proposed to accurately express transmissibilitybetween adjacent cells.The efficacy of the proposedmethodwas evaluated through predictions of several petroleumreservoir problems with spatial heterogeneity and compared against state-of-the-art(PINN)through numericalanalysis as a benchmark,which demonstrated the superiority of the proposed method over the PINN.

Anisotropic dynamic permeability model for porous media

Based on the tortuous capillary network model,the relationship between anisotropic permeability and rock normal strain,namely the anisotropic dynamic permeability model(ADPM),was derived and established.The model was verified using pore-scale flow simulation.The uniaxial strain process was calculated and the main factors affecting permeability changes in different directions in the deformation process were analyzed.In the process of uniaxial strain during the exploitation of layered oil and gas reservoirs,the effect of effective surface porosity on the permeability in all directions is consistent.With the decrease of effective surface porosity,the sensitivity of permeability to strain increases.The sensitivity of the permeability perpendicular to the direction of compression to the strain decreases with the increase of the tortuosity,while the sensitivity of the permeability in the direction of compression to the strain increases with the increase of the tortuosity.For layered reservoirs with the same initial tortuosity in all directions,the tortuosity plays a decisive role in the relative relationship between the variations of permeability in all directions during pressure drop.When the tortuosity is less than 1.6,the decrease rate of horizontal permeability is higher than that of vertical permeability,while the opposite is true when the tortuosity is greater than 1.6.This phenomenon cannot be represented by traditional dynamic permeability model.After the verification by experimental data of pore-scale simulation,the new model has high fitting accuracy and can effectively characterize the effects of deformation in different directions on the permeability in all directions.

Flow characteristics and regime transition of aqueous foams in porous media over a wide range of quality,velocity,and surfactant concentration

Aqueous foam is broadly applicable to enhanced oil recovery(EOR).The rheology of foam as a function of foam quality,gas and liquid velocities,and surfactant concentration constitute the foundation of its application.The great variations of the above factors can affect the effectiveness of N2 foam in EOR continuously in complex formations,which is rarely involved in previous relevant studies.This paper presents an experimental study of foam flow in porous media by injecting pre-generated N2 foam into a sand pack under the conditions of considering a wide range of gas and liquid velocities and surfactant concentrations.The results show that in a wide range of gas and liquid velocities,the pressure gradient contours are L-shaped near the coordinate axes,but V-shaped in other regions.And the surfactant concentration is a strong factor influencing the trend of pressure gradient contours.Foam flow resistance is very sensitive to the surfactant concentration in both the high-and low-foam quality regime,especially when the surfactant concentration is less than CMC.The foam quality is an important variable to the flow resistance obtained.There exists a transition point from low-to high-quality regime in a particular flow system,where has the maximum flow resistance,the corresponding foam quality is called transition foam quality,which increases as the surfactant concentration increases.The results can add to our knowledge base of foam rheology in porous media,and can provide a strong basis for the field application of foams.

Numerical Approach of a Coupled Pressure-Saturation Model Describing Oil-Water Flow in Porous Media

Two-phase flow in porous media is a very active field of research,due to its important applications in groundwater pollution,CO_(2)sequestration,or oil and gas production from petroleum reservoirs,just to name a few of them.Fractional flow equations,which make use of Darcy's law,for describing the movement of two immiscible fluids in a porous medium,are among the most relevant mathematical models in reservoir simulation.This work aims to solve a fractional flow model formed by an elliptic equation,representing the spatial distribution of the pressure,and a hyperbolic equation describing the space-time evolution of water saturation.The numerical solution of the elliptic part is obtained using a finite-element(FE)scheme,while the hyperbolic equation is solved by means of two dif-ferent numerical approaches,both in the finite-volume(FV)framework.One is based on a monotonic upstream-centered scheme for conservation laws(MUSCL)-Hancock scheme,whereas the other makes use of a weighted essentially non-oscillatory(ENO)reconstruc-tion.In both cases,a first-order centered(FORCE)-αnumerical scheme is applied for inter-cell flux reconstruction,which constitutes a new contribution in the field of fractional flow models describing oil-water movement.A relevant feature of this work is the study of the effect of the parameterαon the numerical solution of the models considered.We also show that,in the FORCE-αmethod,when the parameterαincreases,the errors diminish and the order of accuracy is more properly attained,as verified using a manufactured solution technique.

Evaluation of gas wettability and its effects on fluid distribution and fluid flow in porous media

The special gas wettability phenomenon of reservoir rocks has been recognized by more and more researchers.It has a significant effect on efficient development of unconventional reservoirs.First,based on the preferentially gas-covered ability and surface free energy changes,definition and evaluation methods have been established.Second,a method for altering rock wettability and its mechanisms have been studied,surface oriented phenomena of functional groups with low surface energy are the fundamental reason for gas wettability alteration of rock.Third,the effect of gas wettability on the surface energy,electrical properties and dilatability are investigated.Last,the effects of gas wettability on capillary pressure,oil/gas/water distribution and flow are investigated with capillary tubes and etchedglass network models.The gas wettability theory of reservoir rocks has been initially established,which provides theoretical support for the efficient production of unconventional reservoirs and has great significance.

Numerical simulation of two-phase flow in fractured porous media using streamline simulation and IMPES methods and comparing results with a commercial software

Streamline simulation is developed to simulate waterflooding in fractured reservoirs. Conventional reservoir simulation methods for fluid flow simulation in large and complex reservoirs are very costly and time consuming. In streamline method, transport equations are solved on one-dimensional streamlines to reduce the computation time with less memory for simulation. First, pressure equation is solved on an Eulerian grid and streamlines are traced. Defining the "time of flight", saturation equations are mapped and solved on streamlines. Finally, the results are mapped back on Eulerian grid and the process is repeated until the simulation end time. The waterflooding process is considered in a fractured reservoir using the dual porosity model. Afterwards, a computational code is developed to solve the same problem by the IMPES method and the results of streamline simulation are compared to those of the IMPES and a commercial software. Finally, the accuracy and efficiency of streamline simulator for simulation of two-phase flow in fractured reservoirs has been proved.

An overview on nonlinear porous flow in low permeability porous media

This paper gives an overview on nonlinear porous flow in low permeability porous media, reveals the microscopic mechanisms of flows, and clarifies properties of porous flow fluids. It shows that, deviating from Darcy＇s linear law, the porous flow characteristics obey a nonlinear law in a low-permeability porous medium, and the viscosity of the porous flow fluid and the permeability values of water and oil are not constants. Based on these characters, a new porous flow model, which can better describe low permeability reservoir~ is established. This model can describe various patterns of porous flow, as Darcy＇s linear law does. All the parameters involved in the model, having definite physical meanings, can be obtained directly from the experiments.

Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability

Based on Huang＇s accurate tri-sectional nonlin- ear kinematic equation （1997）, a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary. This model contains double moving boundaries, including an internal moving boundary and an external mov- ing boundary, which are different from the classical Stefan problem in heat conduction： The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary. Through a similarity transfor- mation, the nonlinear partial differential equation （PDE） sys- tem is transformed into a linear PDE system. Then an ana- lytical solution is obtained for the dimensionless simplified mathematical model. This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications. Finally, through plotted comparison curves from the exact an- alytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensi- tive effects of the dimensionless variable on the dimension- less pressure distribution and dimensionless pressure gradi- ent distribution become more serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensi- tive effects of the dimensionless variable become more serious; the dimensionless threshold pressure gradient （TPG） has a great effect on the external moving boundary but has little effect on the internal moving boundary.

A control volume based finite element method for simulating incompressible two-phase flow in heterogeneous porous media and its application to reservoir engineering

Applying the standard Galerkin finite element method for solving flow problems in porous media encounters some difficulties such as numerical oscillation at the shock front and discontinuity of the velocity field on element faces.Discontinuity of velocity field leads this method not to conserve mass locally.Moreover,the accuracy and stability of a solution is highly affected by a non-conservative method.In this paper,a three dimensional control volume finite element method is developed for twophase fluid flow simulation which overcomes the deficiency of the standard finite element method,and attains high-orders of accuracy at a reasonable computational cost.Moreover,this method is capable of handling heterogeneity in a very rational way.A fully implicit scheme is applied to temporal discretization of the governing equations to achieve an unconditionally stable solution.The accuracy and efficiency of the method are verified by simulating some waterflooding experiments.Some representative examples are presented to illustrate the capability of the method to simulate two-phase fluid flow in heterogeneous porous media.

Volumetric lattice Boltzmann method for pore-scale mass diffusionadvection process in geopolymer porous structures

Porous materials present significant advantages for absorbing radioactive isotopes in nuclear waste streams.To improve absorption efficiency in nuclear waste treatment,a thorough understanding of the diffusion-advection process within porous structures is essential for material design.In this study,we present advancements in the volumetric lattice Boltzmann method(VLBM)for modeling and simulating pore-scale diffusion-advection of radioactive isotopes within geopolymer porous structures.These structures are created using the phase field method(PFM)to precisely control pore architectures.In our VLBM approach,we introduce a concentration field of an isotope seamlessly coupled with the velocity field and solve it by the time evolution of its particle population function.To address the computational intensity inherent in the coupled lattice Boltzmann equations for velocity and concentration fields,we implement graphics processing unit(GPU)parallelization.Validation of the developed model involves examining the flow and diffusion fields in porous structures.Remarkably,good agreement is observed for both the velocity field from VLBM and multiphysics object-oriented simulation environment(MOOSE),and the concentration field from VLBM and the finite difference method(FDM).Furthermore,we investigate the effects of background flow,species diffusivity,and porosity on the diffusion-advection behavior by varying the background flow velocity,diffusion coefficient,and pore volume fraction,respectively.Notably,all three parameters exert an influence on the diffusion-advection process.Increased background flow and diffusivity markedly accelerate the process due to increased advection intensity and enhanced diffusion capability,respectively.Conversely,increasing the porosity has a less significant effect,causing a slight slowdown of the diffusion-advection process due to the expanded pore volume.This comprehensive parametric study provides valuable insights into the kinetics of isotope uptake in porous structures,facilitating the development of porous materials for nuclear waste treatment applications.

GLOBAL EXISTENCE AND BLOW-UP PHENOMENA OF CLASSICAL SOLUTIONS FOR THE SYSTEM OF COMPRESSIBLE ADIABATIC FLOW THROUGH POROUS MEDIA

By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result on the global existence and the blow-up phenomena of classical solutions of these systems. These results show that the dissipation is strong enough to preserve the smoothness of ‘small ’ solution.

Interactions between gas flow and reversible chemical reaction in porous media

Taking into consideration the gas compressibility and chemical reaction reversibility, a model was developed to study the interactions between gas flow and chemical reaction in porous media and resolved by the finite volume method on the basis of the gas-solid reaction aA(g)+bB(s)cC(g)+dD(s).The numerical analysis shows that the equilibrium constant is an important factor influencing the process of gas-solid reaction. The stoichiometric coefficients, molar masses of reactant gas, product gas and inert gas are the main factors influencing the density of gas mixture. The equilibrium constant influences the gas flow in porous media obviously when the stoichiometric coefficients satisfy a/c≠1.

Non-Darcy flows in layered porous media(LPMs)with contrasting pore space structures

Compared to single layer porous media,fluid flow through layered porous media(LPMs)with contrasting pore space structures is more complex.This study constructed three-dimensional(3-D)pore-scale LPMs with different grain size ratios of 1.20,1.47,and 1.76.The flow behavior in the constructed LPMs and single layer porous media was numerically investigated.A total of 178 numerical experimental data were collected in LPMs and single layer porous media.In all cases,two different flow regimes(i.e.,Darcy and Non-Darcy)were observed.The influence of the interface of layers on Non-Darcy flow behavior in LPMs was analyzed based pore-scale flow data.It was found that the available correlations based on single layer porous media fail to predict the flow behavior in LPMs,especially for LPM with large grain size ratio.The effective permeability,which incorporated the influence of the interface is more accurate than the Kozeny-Carman equation for estimating the Darcy permeability of LPMs.The inertial pressure loss in LPMs,which determines the onset of the Non-Darcy flow,was underestimated when using a power law expression of mean grain size.The constant B,an empirical value in the classical Ergun equation,typically equals 1.75.The inertial pressure loss in LPMs can be significantly different from it in single lager porous media.For Non-Darcy flow in LPMs,it is necessary to consider a modified larger constant B to improve the accuracy of the Ergun empirical equation.

TWO PHASE COMPRESSIBLE FLOW IN POROUS MEDIA

We study the mathematical model of two phase compressible flows through porous media. Under the condition that the compressibility of rock, oil, and water is small, we prove that the initial-boundary value problem of the nonlinear system of equations admits a weak solution.

Quantitative criteria for identifying main flow channels in complex porous media

To identify the type of main flow channels of complex porous media in oil and gas reservoirs,the"main flow channel index"is defined as the ratio of comprehensive permeability obtained from well test to matrix permeability obtained from core analysis or well logging.Meanwhile,a mathematical model is established based on equivalent flow assumption,the classification method for main flow channels is put forward,and quantitative characterization of main flow channels is realized.The method has been verified by analysis of typical gas reservoirs.The study results show that the"main flow channel index"can quantitatively classify types of flow channels.If the index is less than 3,the matrix pore is the main flow channel;if the index is between 3 and 20,the fracture is the main flow channel and the matrix pore acts as the supplement one;if the index is more than 20,the fracture is the only seepage channel.The dynamic analysis of typical gas reservoirs shows that the"main flow channel index"can be used to identify the type of flow channel in complex porous media,guiding the classified development of gas reservoirs,and avoiding development risk.

Empirical mode decomposition of multiphase flows in porous media:characteristic scales and speed of convergence

We apply a proper orthogonal decomposition(POD)to data stemming from numerical simulations of a fingering instability in a multiphase flow passing through obstacles in a porous medium,to study water injection processes in the production of hydrocarbon reservoirs.We show that the time evolution of a properly defined flow correlation length can be used to identify the onset of the fingering instability.Computation of characteristic lengths for each of the modes resulting from the POD provides further information on the dynamics of the system.Finally,using numerical simulations with different viscosity ratios,we show that the convergence of the POD depends non-trivially on whether the fingering instability develops or not.This result has implications on proposed methods to decrease the dimensionality of the problem by deriving reduced dynamical systems after truncating the system’s governing equations to a few POD modes.

Enduring effect of permeability texture for enhancing accuracy and reducing uncertainty of reservoir fluid flow through porous media

Modeling reservoir permeability is one of the crucial tasks in reservoir simulation studies.Traditionally,it is done by kriging-based methods.More rigorous modeling of the permeability results in more reliable outputs of the reservoir models.Recently,a new category of geostatistical methods has been used for this purpose,namely multiple point statistics(MPS).By this new category of permeability modeling methods,one is able to predict the heterogeneity of the reservoir permeability as a continuous variable.These methods consider the direction of property variation in addition to the distances of known locations of the property.In this study,the reservoir performance of a modified version of the SPE 10 solution project as a pioneer case is used for investigating the efficiency of these methods and paralleling them with the kriging-based one.In this way,the permeability texture concept is introduced by applying some MPS methods.This study is accomplished in the conditions of real reservoir dimensions and velocities for the whole reservoir life.A continuous training image is used as the input of calculation for the permeability modeling.The results show that the detailed permeability of the reservoir as a continuous variable makes the reservoir simulation show the same fluid front movement and flooding behavior of the reservoir similar to the reference case with the same permeability heterogeneity.Some MPS methods enable the reservoir simulation to reproduce the fluid flow complexities such as bypassing and oil trapping during water flooding similar to the reference case.Accordingly,total oil production is predicted with higher accuracy and lower uncertainty.All studied cases are identical except for the permeability texture.Even histograms and variograms of permeabilities for the studied reservoir are quite similar,but the performance of the reservoir shows that kriging-based method results have slightly less accuracy than some MPS methods.Meanwhile,it results in lower uncertainty in outputs for this water flooding case performance.

Measurement of Fluid Flow in Pipe and Porous Media by High-Resolution Magnetic Resonance Imaging

The objective of this study is to understand the process of fluid flow in pipe and porous media with different pore structures. High-resolution Magnetic Resonance Imaging （MRI） technique was used to visualize the pore structure and measure fluid flow. The porous media was formed by packed bed of glass beads. Flow measurement was carried out by a modified spin echo sequence. The results show that the velocity distribution in pipe is annular and the linear relation between MRI velocity and actual velocity is found in pipe flow measurement. The flow distribution in porous media is rather heterogeneous, and it is consistent with heterogeneous pore structure. The flow through pores with the high volume flow rate is determined largely by geometrical effects such as pore size and cross-sectional area.

THE TRANSIENT TWO-DIMENSIONAL FLOW THROUGH DOUBLE POROUS MEDIA

This paper presents the analytical solutions in Laplace domain for two-dimensional nonsteady flow of slightly compressible liquid in porous media with double porosity by using the methods of integral transforms and variables separation. The effects of the ratio of storativities to , interporosity flow parameter on the pressure behaviors for a vertically fractured well with infinite conductivity are investigated by using the method of numerical inversion. The new log-log diagnosis graph of the pressures is given and analysed.