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 ...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.展开更多
In-situ stress is a common stress in the exploration and development of oil reservoirs. Therefore, it is of great significance to study the propagation characteristics of borehole acoustic waves in fluid-saturated por...In-situ stress is a common stress in the exploration and development of oil reservoirs. Therefore, it is of great significance to study the propagation characteristics of borehole acoustic waves in fluid-saturated porous media under stress.Based on the acoustoelastic theory of fluid-saturated porous media, the field equation of fluid-saturated porous media under the conditions of confining pressure and pore pressure and the acoustic field formula of multipole source excitation in open hole are given. The influences of pore pressure and confining pressure on guided waves of multipole borehole acoustic field in fluid-saturated porous media are investigated. The numerical results show that the phase velocity and excitation intensity of guided wave increase significantly under the confining pressure. For a given confining pressure, the phase velocity of the guided wave decreases with pore pressure increasing. The excitation intensity of guided wave increases at low frequency and then decreases at high frequency with pore pressure increasing, except for that of Stoneley wave which decreases in the whole frequency range. These results will help us get an insight into the influences of confining pressure and pore pressure on the acoustic field of multipole source in borehole around fluid-saturated porous media.展开更多
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 ef...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.展开更多
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 physicsi...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.展开更多
The measurement of nuclear magnetic resonance(NMR)porosity is affected by temperature.Without considering the impact of NMR logging tools,this phenomenon is mainly caused by variations in magnetization intensity of th...The measurement of nuclear magnetic resonance(NMR)porosity is affected by temperature.Without considering the impact of NMR logging tools,this phenomenon is mainly caused by variations in magnetization intensity of the measured system due to temperature fluctuations and difference between the temperature of the porous medium and calibration sample.In this study,the effect of temperature was explained based on the thermodynamic theory,and the rules of NMR porosity responses to temperature changes were identified through core physics experiments.In addition,a method for correcting the influence of temperature on NMR porosity measurement was proposed,and the possible factors that may affect its application were also discussed.展开更多
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.T...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.展开更多
Extensive improvements in small-scale thermal systems in electronic circuits,automotive industries,and microcomputers conduct the study of microsystems as essential.Flow and thermic field characteristics of the cohere...Extensive improvements in small-scale thermal systems in electronic circuits,automotive industries,and microcomputers conduct the study of microsystems as essential.Flow and thermic field characteristics of the coherent nanofluid-guided microchannel heat sink are described in this perusal.The porous media approximate was used to search the heat distribution in the expanded sheet and Cu:γ-AlOOH/water.A hybrid blend of Boehme copper and aluminum nanoparticles is evaluated to have a cooling effect on the microchannel heat sink.By using Akbari Ganji and finite element methods,linear and non-linear differential equations as well as simple dimensionless equations have been analyzed.The purpose of this study is to investigate the fluid and thermal parameters of copper hybrid solution added to water,such as Nusselt number and Darcy number so that we can reach the best cooling of the fluid.Also,by installing a piece of fin on the wall of the heat sink,the coefficient of conductive heat transfer and displacement heat transfer with the surrounding air fluid increases,and the efficiency of the system increases.The overall results show that expanding values on the NP(series heat transfer fluid system maximizes performance with temperatures)volume division of copper,as well as boehmite alumina particles,lead to a decrease within the stream velocity of the Cu:AlOOH/water.Increasing the volume fraction of nanoparticles in the hybrid mixture decreases the temperature of the solid surface and the hybrid nanofluid.The Brownian movement improves as the volume percentage of nanoparticles in the hybrid mixture grows,spreading the heat across the environment.As a result,heat transmission rates rise.As the Darcy number increases,the thermal field for solid sections and Cu:AlOOH/water improves.展开更多
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 o...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.展开更多
Since its inception in the 1970s,multi-dimensional magnetic resonance(MR)has emerged as a powerful tool for non-invasive investigations of structures and molecular interactions.MR spectroscopy beyond one dimension all...Since its inception in the 1970s,multi-dimensional magnetic resonance(MR)has emerged as a powerful tool for non-invasive investigations of structures and molecular interactions.MR spectroscopy beyond one dimension allows the study of the correlation,exchange processes,and separation of overlapping spectral information.The multi-dimensional concept has been re-implemented over the last two decades to explore molecular motion and spin dynamics in porous media.Apart from Fourier transform,methods have been developed for processing the multi-dimensional time-domain data,identifying the fluid components,and estimating pore surface permeability via joint relaxation and diffusion spectra.Through the resolution of spectroscopic signals with spatial encoding gradients,multi-dimensional MR imaging has been widely used to investigate the microscopic environment of living tissues and distinguish diseases.Signals in each voxel are usually expressed as multi-exponential decay,representing microstructures or environments along multiple pore scales.The separation of contributions from different environments is a common ill-posed problem,which can be resolved numerically.Moreover,the inversion methods and experimental parameters determine the resolution of multi-dimensional spectra.This paper reviews the algorithms that have been proposed to process multidimensional MR datasets in different scenarios.Detailed information at the microscopic level,such as tissue components,fluid types and food structures in multi-disciplinary sciences,could be revealed through multi-dimensional MR.展开更多
The seepage characteristics of multiscale porous media is of considerable significance in many scientific and engineering fields.The Darcy permeability is one of the key macroscopic physical properties to characterize...The seepage characteristics of multiscale porous media is of considerable significance in many scientific and engineering fields.The Darcy permeability is one of the key macroscopic physical properties to characterize the seepage capacity of porous media.Therefore,based on the statistically fractal scaling law of porous media,fractal geometry is applied to model the multiscale pore structures.And a two-dimensional fractal orifice-throat model with multiscale and tortuous characteristics is proposed for the seepage flow through porous media.The analytical expression for Darcy permeability of porous media is derived,which is validated by comparing with available experimental data.The results show that the Darcy permeability is significantly influenced by porosity,orifice-throat fractal dimension,minimum to maximum diameter ratio,orifice-throat ratio and tortuosity fractal dimension.The present results are helpful for understanding the seepage mechanism of multiscale porous media,and may provide theoretical basis for unconventional oil and gas exploration and development,porous phase transition energy storage composites,CO2 geological sequestration,environmental protection and nuclear waste treatment,etc.展开更多
Chen's technique of computing synthetic seismograms,which decomposes every vector with a set of basis of orthogonality and completeness before applying the Luco-Apsel-Chen(LAC)generalized reflection and transmissi...Chen's technique of computing synthetic seismograms,which decomposes every vector with a set of basis of orthogonality and completeness before applying the Luco-Apsel-Chen(LAC)generalized reflection and transmission coefficients method,is confirmed to be efficient in dealing with elastic waves in multi-layered media and accurate in any frequency range.In this article,we extend Chen's technique to the computation of coupled seismic and electromagnetic(EM)waves in layered porous media.Expanding the involved mechanical and electromagnetic fields by a set of scalar and vector wave-function basis,we obtain the fundamental equations which are subsequently solved by using a recently developed version of the LAC generalized reflection and transmission coefficients method.Our approach and corresponding program is validated by reciprocity tests.We also show a numerical example of a two-layer model with an explosion source.The P-to-EM conversion waves radiated from the interface may have potential application.展开更多
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 consum...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.展开更多
Magnetohydrodynamic(MHD)bioconvection of an incompressible electrically conducting nanofluid near a vertical wavy surface saturated porous medium containing both nanoparticle and gyrotactic microorganisms is investiga...Magnetohydrodynamic(MHD)bioconvection of an incompressible electrically conducting nanofluid near a vertical wavy surface saturated porous medium containing both nanoparticle and gyrotactic microorganisms is investigated.The nanofluid is represented by a model that includes both Brownian motion and thermophoresis effects.A suitable set of non-dimensional variables are used to transform the governing boundary layer equations into a dimensionless form.The resulting nonlinear system is mapped to the vertical flat plate domain,and a non-similar solution is used to the obtained equations.The obtained non-similar system is then solved numerically using the fourth-order Runge-Kutta method.The influence of various physical parameters on the local Nusselt number,the local Sherwood number,the local density number of the motile microorganisms,the dimensionless velocity,the dimensionless temperature,and the rescaled density of motile microorganisms is studied.It is found that the local Nusselt number,the local Sherwood number,and the local density number of the motile microorganisms decrease by increasing either the Grashof number or the magnetic field parameter.展开更多
The fractal Brownian motion is utilized to describe pore structures in porous media. A numerical model of laminar flow in porous media is developed, and the flow characteristics are comprehensively analyzed and compar...The fractal Brownian motion is utilized to describe pore structures in porous media. A numerical model of laminar flow in porous media is developed, and the flow characteristics are comprehensively analyzed and compared with those of homogeneous porous media. Moreover, the roles of the fractal dimension and porosity in permeability are quantitatively described. The results indicate that the pore structures of porous media significantly affect their seepage behaviors. The distributions of pressure and velocity in fractal porous media are both non-uniform;the streamline is no longer straight but tortuous. When Reynolds number Re < 1, the dimensionless permeability is independent of Reynolds number, but its further increase will lead to a smaller permeability. Moreover, due to the higher connectivity and enlarged equivalent aperture of internal channel network, the augment in porosity leads to the permeability enhancement, while it is small and insensitive to porosity variation when ε < 0.6. Fractal dimension also plays a significant role in the permeability of porous media. The increase in fractal dimension leads to the enhancement in pore connectivity and a decrease in channel tortuosity,which reduces the flow resistance and improves the transport capacity of porous media.展开更多
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 gra...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.展开更多
Fines migration induced by injection of low-salinity water(LSW) into porous media can lead to severe pore plugging and consequent permeability reduction. The deepbed filtration(DBF) theory is used to model the aforeme...Fines migration induced by injection of low-salinity water(LSW) into porous media can lead to severe pore plugging and consequent permeability reduction. The deepbed filtration(DBF) theory is used to model the aforementioned phenomenon, which allows us to predict the effluent concentration history and the distribution profile of entrapped particles. However, the previous models fail to consider the movement of the waterflood front. In this study, we derive a stochastic model for fines migration during LSW flooding, in which the Rankine-Hugoniot condition is used to calculate the concentration of detached particles behind and ahead of the moving water front. A downscaling procedure is developed to determine the evolution of pore-size distribution from the exact solution of a large-scale equation system. To validate the proposed model,the obtained exact solutions are used to treat the laboratory data of LSW flooding in artificial soil-packed columns. The tuning results show that the proposed model yields a considerably higher value of the coefficient of determination, compared with the previous models, indicating that the new model can successfully capture the effect of the moving water front on fines migration and precisely match the effluent history of the detached particles.展开更多
In this paper,an efcient multigrid-DEIM semi-reduced-order model is developed to accelerate the simulation of unsteady single-phase compressible fow in porous media.The cornerstone of the proposed model is that the fu...In this paper,an efcient multigrid-DEIM semi-reduced-order model is developed to accelerate the simulation of unsteady single-phase compressible fow in porous media.The cornerstone of the proposed model is that the full approximate storage multigrid method is used to accelerate the solution of fow equation in original full-order space,and the discrete empirical interpolation method(DEIM)is applied to speed up the solution of Peng-Robinson equation of state in reduced-order subspace.The multigrid-DEIM semi-reduced-order model combines the computation both in full-order space and in reducedorder subspace,which not only preserves good prediction accuracy of full-order model,but also gains dramatic computational acceleration by multigrid and DEIM.Numerical performances including accuracy and acceleration of the proposed model are carefully evaluated by comparing with that of the standard semi-implicit method.In addition,the selection of interpolation points for constructing the low-dimensional subspace for solving the Peng-Robinson equation of state is demonstrated and carried out in detail.Comparison results indicate that the multigrid-DEIM semi-reduced-order model can speed up the simulation substantially at the same time preserve good computational accuracy with negligible errors.The general acceleration is up to 50-60 times faster than that of standard semi-implicit method in two-dimensional simulations,but the average relative errors of numerical results between these two methods only have the order of magnitude 10^(−4)-10^(−6)%.展开更多
Natural gas hydrate(NGH)has recently received more attention as a cleaner alternative energy source that not only reduces carbon emissions caused by the use of conventional fossil fuels but also plays a key role in gl...Natural gas hydrate(NGH)has recently received more attention as a cleaner alternative energy source that not only reduces carbon emissions caused by the use of conventional fossil fuels but also plays a key role in global climate change.Furthermore,hydrate-based technologies,particularly hydrate-based carbon capture and storage,have enormous promise for decreasing global carbon emissions,and porous media play an important role in all hydrate-based technologies.Accordingly,this paper reviews the recent applications of porous media in the field of methane hydrate(MH)formation and analyzes the influence of porous media systems on MH phase equilibria and formation kinetics.This is because the efficiency of hydrate-based technologies is determined mainly by the phase equilibrium and formation kinetics of hydrates.The influence of the nature of the media on MH formation in porous media systems is comprehensively summarized to understand how porous media can efficiently enhance the kinetics of hydrate formation.Promoters are necessary for rapid hydrate formation,and the effect of various promoters on MH formation was also evaluated.Based on the aforementioned overview and understanding,the mechanisms for MH formation in various porous media systems are proposed.Finally,the future perspectives and challenges of hydrate-based technologies in tackling global climate change were discussed.This review provides a fundamental understanding of the application and development of porous media in rapid hydrate formation,a fair evaluation of the performance of various porous media systems,and critical insights into major research foci.展开更多
Water-based fire extinguishing agent is the main means to deal with smoldering fires.However,due to the hydrophobic properties of the particle surface,the porous medium channel provide resistance and slow down the ext...Water-based fire extinguishing agent is the main means to deal with smoldering fires.However,due to the hydrophobic properties of the particle surface,the porous medium channel provide resistance and slow down the extinguishing agent flow during the downward permeation process.To promote the liquid permeation process in such porous media,this work studied liquid imbibition process and analyzed the oscillating and attenuating process of liquid level in capillary channel by theoretical,experimental,and numerical methods.An empirical mathematical equation was proposed to describe the oscillating process,and the effects of the capillary diameter and contact angle parameters on the transportation process were analyzed.Based on this,the“relay-mode”was proposed to promote the liquid transportation forward.Finally,the transient simulation results of liquid permeation in coal stacks showed when the liquid flowed through the channel with changed diameter from large to small ones,the transportation distance was several times longer than that through the unidiameter ones.The trend of liquid“relay-mode”in capillaries can be used to promote the permeation in granular materials porous media stacks.The relevant results also provide new thoughts to develop the water-based fire extinguishing agents and then improve the firefighting efficiency of deep-seated fire in porous media stacks.展开更多
While experimental designs developed in recent decades have contributed to research on dynamic nonequilibrium effects in transient two-phase flow in porous media,this problem has been seldom investigated using direct ...While experimental designs developed in recent decades have contributed to research on dynamic nonequilibrium effects in transient two-phase flow in porous media,this problem has been seldom investigated using direct numerical simulation(DNS).Only a few studies have sought to numerically solve Navier—Stokes equations with level-set(LS)or volume-of-fluid(VoF)methods,each of which has constraints in terms of meniscus dynamics for various flow velocities in the control volume(CV)domain.The Shan—Chen multiphase multicomponent lattice Boltzmann method(SC-LBM)has a fundamental mechanism to separate immiscible fluid phases in the density domain without these limitations.Therefore,this study applied it to explore two-phase displacement in a single representative elementary volume(REV)of two-dimensional(2D)porous media.As a continuation of a previous investigation into one-step inflow/outflow in 2D porous media,this work seeks to identify dynamic nonequilibrium effects on capillary pressure—saturation relationship(P_(c)—S)for quasi-steady-state flow and multistep inflow/outflow under various pressure boundary conditions.The simulation outcomes show that P_(c),S and specific interfacial area(a_(nw))had multistep-wise dynamic effects corresponding to the multistep-wise pressure boundary conditions.With finer adjustments to the increase in pressure over more steps,dynamic nonequilibrium effects were significantly alleviated and even finally disappeared to achieve quasisteady-state inflow/outflow conditions.Furthermore,triangular wave-formed pressure boundary conditions were applied in different periods to investigate dynamic nonequilibrium effects for hysteretical Pc—S.The results showed overshoot and undershoot of P_(c)to S in loops of the nonequilibrium hysteresis.In addition,the flow regimes of multistep-wise dynamic effects were analyzed in terms of Reynolds and capillary numbers(Re and Ca).The analysis of REV-scale flow regimes showed higher Re(1<Re<10)for more significant dynamic nonequilibrium effects.This indicates that inertia is critical for transient twophase flow in porous media under dynamic nonequilibrium conditions.展开更多
基金Supported by the National Natural Science Foundation of China(52274048)Beijing Natural Science Foundation Project of China(3222037)Shaanxi Provincial Technical Innovation Project of China(2023-YD-CGZH-02).
文摘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.
基金Project supported by the National Natural Science Foundation of China (Grant No.42074139)the Natural Science Foundation of Jilin Province,China (Grant No.20210101140JC)。
文摘In-situ stress is a common stress in the exploration and development of oil reservoirs. Therefore, it is of great significance to study the propagation characteristics of borehole acoustic waves in fluid-saturated porous media under stress.Based on the acoustoelastic theory of fluid-saturated porous media, the field equation of fluid-saturated porous media under the conditions of confining pressure and pore pressure and the acoustic field formula of multipole source excitation in open hole are given. The influences of pore pressure and confining pressure on guided waves of multipole borehole acoustic field in fluid-saturated porous media are investigated. The numerical results show that the phase velocity and excitation intensity of guided wave increase significantly under the confining pressure. For a given confining pressure, the phase velocity of the guided wave decreases with pore pressure increasing. The excitation intensity of guided wave increases at low frequency and then decreases at high frequency with pore pressure increasing, except for that of Stoneley wave which decreases in the whole frequency range. These results will help us get an insight into the influences of confining pressure and pore pressure on the acoustic field of multipole source in borehole around fluid-saturated porous media.
基金supported by National Natural Science Foundation of China(Grant No.42172159)Science Foundation of China University of Petroleum,Beijing(Grant No.2462023XKBH002).
文摘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.
基金the National Natural Science Foundation of China(No.52274048)Beijing Natural Science Foundation(No.3222037)+1 种基金the CNPC 14th Five-Year Perspective Fundamental Research Project(No.2021DJ2104)the Science Foundation of China University of Petroleum,Beijing(No.2462021YXZZ010).
文摘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.
基金This paper is supported by“National Natural Science Foundation of China(Grant No.42204106)”.
文摘The measurement of nuclear magnetic resonance(NMR)porosity is affected by temperature.Without considering the impact of NMR logging tools,this phenomenon is mainly caused by variations in magnetization intensity of the measured system due to temperature fluctuations and difference between the temperature of the porous medium and calibration sample.In this study,the effect of temperature was explained based on the thermodynamic theory,and the rules of NMR porosity responses to temperature changes were identified through core physics experiments.In addition,a method for correcting the influence of temperature on NMR porosity measurement was proposed,and the possible factors that may affect its application were also discussed.
基金financially supported by National Natural Science Foundation of China(No.U20B6003).
文摘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.
文摘Extensive improvements in small-scale thermal systems in electronic circuits,automotive industries,and microcomputers conduct the study of microsystems as essential.Flow and thermic field characteristics of the coherent nanofluid-guided microchannel heat sink are described in this perusal.The porous media approximate was used to search the heat distribution in the expanded sheet and Cu:γ-AlOOH/water.A hybrid blend of Boehme copper and aluminum nanoparticles is evaluated to have a cooling effect on the microchannel heat sink.By using Akbari Ganji and finite element methods,linear and non-linear differential equations as well as simple dimensionless equations have been analyzed.The purpose of this study is to investigate the fluid and thermal parameters of copper hybrid solution added to water,such as Nusselt number and Darcy number so that we can reach the best cooling of the fluid.Also,by installing a piece of fin on the wall of the heat sink,the coefficient of conductive heat transfer and displacement heat transfer with the surrounding air fluid increases,and the efficiency of the system increases.The overall results show that expanding values on the NP(series heat transfer fluid system maximizes performance with temperatures)volume division of copper,as well as boehmite alumina particles,lead to a decrease within the stream velocity of the Cu:AlOOH/water.Increasing the volume fraction of nanoparticles in the hybrid mixture decreases the temperature of the solid surface and the hybrid nanofluid.The Brownian movement improves as the volume percentage of nanoparticles in the hybrid mixture grows,spreading the heat across the environment.As a result,heat transmission rates rise.As the Darcy number increases,the thermal field for solid sections and Cu:AlOOH/water improves.
文摘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.
基金supported by the National Natural Science Foundation of China(No.61901465,82222032,82172050).
文摘Since its inception in the 1970s,multi-dimensional magnetic resonance(MR)has emerged as a powerful tool for non-invasive investigations of structures and molecular interactions.MR spectroscopy beyond one dimension allows the study of the correlation,exchange processes,and separation of overlapping spectral information.The multi-dimensional concept has been re-implemented over the last two decades to explore molecular motion and spin dynamics in porous media.Apart from Fourier transform,methods have been developed for processing the multi-dimensional time-domain data,identifying the fluid components,and estimating pore surface permeability via joint relaxation and diffusion spectra.Through the resolution of spectroscopic signals with spatial encoding gradients,multi-dimensional MR imaging has been widely used to investigate the microscopic environment of living tissues and distinguish diseases.Signals in each voxel are usually expressed as multi-exponential decay,representing microstructures or environments along multiple pore scales.The separation of contributions from different environments is a common ill-posed problem,which can be resolved numerically.Moreover,the inversion methods and experimental parameters determine the resolution of multi-dimensional spectra.This paper reviews the algorithms that have been proposed to process multidimensional MR datasets in different scenarios.Detailed information at the microscopic level,such as tissue components,fluid types and food structures in multi-disciplinary sciences,could be revealed through multi-dimensional MR.
文摘The seepage characteristics of multiscale porous media is of considerable significance in many scientific and engineering fields.The Darcy permeability is one of the key macroscopic physical properties to characterize the seepage capacity of porous media.Therefore,based on the statistically fractal scaling law of porous media,fractal geometry is applied to model the multiscale pore structures.And a two-dimensional fractal orifice-throat model with multiscale and tortuous characteristics is proposed for the seepage flow through porous media.The analytical expression for Darcy permeability of porous media is derived,which is validated by comparing with available experimental data.The results show that the Darcy permeability is significantly influenced by porosity,orifice-throat fractal dimension,minimum to maximum diameter ratio,orifice-throat ratio and tortuosity fractal dimension.The present results are helpful for understanding the seepage mechanism of multiscale porous media,and may provide theoretical basis for unconventional oil and gas exploration and development,porous phase transition energy storage composites,CO2 geological sequestration,environmental protection and nuclear waste treatment,etc.
基金supported by the Natural R&D Special Fund for Public Welfare Industry(No.200808069)National Natural Science Foundation of China(Nos.40974038,40774028 and 40821062)
文摘Chen's technique of computing synthetic seismograms,which decomposes every vector with a set of basis of orthogonality and completeness before applying the Luco-Apsel-Chen(LAC)generalized reflection and transmission coefficients method,is confirmed to be efficient in dealing with elastic waves in multi-layered media and accurate in any frequency range.In this article,we extend Chen's technique to the computation of coupled seismic and electromagnetic(EM)waves in layered porous media.Expanding the involved mechanical and electromagnetic fields by a set of scalar and vector wave-function basis,we obtain the fundamental equations which are subsequently solved by using a recently developed version of the LAC generalized reflection and transmission coefficients method.Our approach and corresponding program is validated by reciprocity tests.We also show a numerical example of a two-layer model with an explosion source.The P-to-EM conversion waves radiated from the interface may have potential application.
文摘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.
文摘Magnetohydrodynamic(MHD)bioconvection of an incompressible electrically conducting nanofluid near a vertical wavy surface saturated porous medium containing both nanoparticle and gyrotactic microorganisms is investigated.The nanofluid is represented by a model that includes both Brownian motion and thermophoresis effects.A suitable set of non-dimensional variables are used to transform the governing boundary layer equations into a dimensionless form.The resulting nonlinear system is mapped to the vertical flat plate domain,and a non-similar solution is used to the obtained equations.The obtained non-similar system is then solved numerically using the fourth-order Runge-Kutta method.The influence of various physical parameters on the local Nusselt number,the local Sherwood number,the local density number of the motile microorganisms,the dimensionless velocity,the dimensionless temperature,and the rescaled density of motile microorganisms is studied.It is found that the local Nusselt number,the local Sherwood number,and the local density number of the motile microorganisms decrease by increasing either the Grashof number or the magnetic field parameter.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.51776037 and 51806147)Natural Science Foundation of Jiangsu Province,China(Grant No.BK20170082).
文摘The fractal Brownian motion is utilized to describe pore structures in porous media. A numerical model of laminar flow in porous media is developed, and the flow characteristics are comprehensively analyzed and compared with those of homogeneous porous media. Moreover, the roles of the fractal dimension and porosity in permeability are quantitatively described. The results indicate that the pore structures of porous media significantly affect their seepage behaviors. The distributions of pressure and velocity in fractal porous media are both non-uniform;the streamline is no longer straight but tortuous. When Reynolds number Re < 1, the dimensionless permeability is independent of Reynolds number, but its further increase will lead to a smaller permeability. Moreover, due to the higher connectivity and enlarged equivalent aperture of internal channel network, the augment in porosity leads to the permeability enhancement, while it is small and insensitive to porosity variation when ε < 0.6. Fractal dimension also plays a significant role in the permeability of porous media. The increase in fractal dimension leads to the enhancement in pore connectivity and a decrease in channel tortuosity,which reduces the flow resistance and improves the transport capacity of porous media.
基金financially supported by the National Key Research and Development Program of China(No.2019YFC1804303)the National Natural Science Foundation of China(Grant Nos.41877171 and 41831289)。
文摘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.
基金the National Natural Science Foundation of China(Nos.51804316,51734010,and U1762211)the National Science and Technology Major Project of China(No.2017ZX05009)the Science Foundation of China University of Petroleum,Beijing(No.2462017YJRC037)。
文摘Fines migration induced by injection of low-salinity water(LSW) into porous media can lead to severe pore plugging and consequent permeability reduction. The deepbed filtration(DBF) theory is used to model the aforementioned phenomenon, which allows us to predict the effluent concentration history and the distribution profile of entrapped particles. However, the previous models fail to consider the movement of the waterflood front. In this study, we derive a stochastic model for fines migration during LSW flooding, in which the Rankine-Hugoniot condition is used to calculate the concentration of detached particles behind and ahead of the moving water front. A downscaling procedure is developed to determine the evolution of pore-size distribution from the exact solution of a large-scale equation system. To validate the proposed model,the obtained exact solutions are used to treat the laboratory data of LSW flooding in artificial soil-packed columns. The tuning results show that the proposed model yields a considerably higher value of the coefficient of determination, compared with the previous models, indicating that the new model can successfully capture the effect of the moving water front on fines migration and precisely match the effluent history of the detached particles.
基金This study is supported by the National Natural Science Foundation of China(Nos.51904031,51936001)the Beijing Natural Science Foundation(No.3204038)the Jointly Projects of Beijing Natural Science Foundation and Beijing Municipal Education Commission(No.KZ201810017023).
文摘In this paper,an efcient multigrid-DEIM semi-reduced-order model is developed to accelerate the simulation of unsteady single-phase compressible fow in porous media.The cornerstone of the proposed model is that the full approximate storage multigrid method is used to accelerate the solution of fow equation in original full-order space,and the discrete empirical interpolation method(DEIM)is applied to speed up the solution of Peng-Robinson equation of state in reduced-order subspace.The multigrid-DEIM semi-reduced-order model combines the computation both in full-order space and in reducedorder subspace,which not only preserves good prediction accuracy of full-order model,but also gains dramatic computational acceleration by multigrid and DEIM.Numerical performances including accuracy and acceleration of the proposed model are carefully evaluated by comparing with that of the standard semi-implicit method.In addition,the selection of interpolation points for constructing the low-dimensional subspace for solving the Peng-Robinson equation of state is demonstrated and carried out in detail.Comparison results indicate that the multigrid-DEIM semi-reduced-order model can speed up the simulation substantially at the same time preserve good computational accuracy with negligible errors.The general acceleration is up to 50-60 times faster than that of standard semi-implicit method in two-dimensional simulations,but the average relative errors of numerical results between these two methods only have the order of magnitude 10^(−4)-10^(−6)%.
基金supported by the Doctoral Start-up Foundation of Liaoning Province(2019-BS-159)the Scientific Research Fund of Liaoning Provincial Education Department(LJKZ0381)the Key Scientific Research Project of Liaoning Provincial Department of Education(L2020002)。
文摘Natural gas hydrate(NGH)has recently received more attention as a cleaner alternative energy source that not only reduces carbon emissions caused by the use of conventional fossil fuels but also plays a key role in global climate change.Furthermore,hydrate-based technologies,particularly hydrate-based carbon capture and storage,have enormous promise for decreasing global carbon emissions,and porous media play an important role in all hydrate-based technologies.Accordingly,this paper reviews the recent applications of porous media in the field of methane hydrate(MH)formation and analyzes the influence of porous media systems on MH phase equilibria and formation kinetics.This is because the efficiency of hydrate-based technologies is determined mainly by the phase equilibrium and formation kinetics of hydrates.The influence of the nature of the media on MH formation in porous media systems is comprehensively summarized to understand how porous media can efficiently enhance the kinetics of hydrate formation.Promoters are necessary for rapid hydrate formation,and the effect of various promoters on MH formation was also evaluated.Based on the aforementioned overview and understanding,the mechanisms for MH formation in various porous media systems are proposed.Finally,the future perspectives and challenges of hydrate-based technologies in tackling global climate change were discussed.This review provides a fundamental understanding of the application and development of porous media in rapid hydrate formation,a fair evaluation of the performance of various porous media systems,and critical insights into major research foci.
基金the funding support of National Natural Science Foundation of China (21978204)
文摘Water-based fire extinguishing agent is the main means to deal with smoldering fires.However,due to the hydrophobic properties of the particle surface,the porous medium channel provide resistance and slow down the extinguishing agent flow during the downward permeation process.To promote the liquid permeation process in such porous media,this work studied liquid imbibition process and analyzed the oscillating and attenuating process of liquid level in capillary channel by theoretical,experimental,and numerical methods.An empirical mathematical equation was proposed to describe the oscillating process,and the effects of the capillary diameter and contact angle parameters on the transportation process were analyzed.Based on this,the“relay-mode”was proposed to promote the liquid transportation forward.Finally,the transient simulation results of liquid permeation in coal stacks showed when the liquid flowed through the channel with changed diameter from large to small ones,the transportation distance was several times longer than that through the unidiameter ones.The trend of liquid“relay-mode”in capillaries can be used to promote the permeation in granular materials porous media stacks.The relevant results also provide new thoughts to develop the water-based fire extinguishing agents and then improve the firefighting efficiency of deep-seated fire in porous media stacks.
基金University of Queensland International Scholarship(UQI)for its support(Grant No.42719692)。
文摘While experimental designs developed in recent decades have contributed to research on dynamic nonequilibrium effects in transient two-phase flow in porous media,this problem has been seldom investigated using direct numerical simulation(DNS).Only a few studies have sought to numerically solve Navier—Stokes equations with level-set(LS)or volume-of-fluid(VoF)methods,each of which has constraints in terms of meniscus dynamics for various flow velocities in the control volume(CV)domain.The Shan—Chen multiphase multicomponent lattice Boltzmann method(SC-LBM)has a fundamental mechanism to separate immiscible fluid phases in the density domain without these limitations.Therefore,this study applied it to explore two-phase displacement in a single representative elementary volume(REV)of two-dimensional(2D)porous media.As a continuation of a previous investigation into one-step inflow/outflow in 2D porous media,this work seeks to identify dynamic nonequilibrium effects on capillary pressure—saturation relationship(P_(c)—S)for quasi-steady-state flow and multistep inflow/outflow under various pressure boundary conditions.The simulation outcomes show that P_(c),S and specific interfacial area(a_(nw))had multistep-wise dynamic effects corresponding to the multistep-wise pressure boundary conditions.With finer adjustments to the increase in pressure over more steps,dynamic nonequilibrium effects were significantly alleviated and even finally disappeared to achieve quasisteady-state inflow/outflow conditions.Furthermore,triangular wave-formed pressure boundary conditions were applied in different periods to investigate dynamic nonequilibrium effects for hysteretical Pc—S.The results showed overshoot and undershoot of P_(c)to S in loops of the nonequilibrium hysteresis.In addition,the flow regimes of multistep-wise dynamic effects were analyzed in terms of Reynolds and capillary numbers(Re and Ca).The analysis of REV-scale flow regimes showed higher Re(1<Re<10)for more significant dynamic nonequilibrium effects.This indicates that inertia is critical for transient twophase flow in porous media under dynamic nonequilibrium conditions.
