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.

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.

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.

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.

Effects of physical parameter range on dimensionless variable sensitivity in water flooding reservoirs

The similarity criterion for water flooding reservoir flows is concerned with in the present paper. When finding out all the dimensionless variables governing this kind of flow, their physical meanings are subsequently elucidated. Then, a numerical approach of sensitivity analysis is adopted to quantify their corresponding dominance degree among the similarity parameters. In this way, we may finally identify major scaling law in different parameter range and demonstrate the respective effects of viscosity, permeability and injection rate.

Bifurcation of Nonlinear Problems Modeling Flows Through Porous Media

This paper deals with a class of nonlinear boundary value problems which appears in the study of models of flows through porous media. Existence results of asymptotic bifurcation and continua are reported both for operator equations and for boundary value problems.

DYNAMIC PRODUCTION PREDICTION AND PARAMETER IDENTIFICATION FOR GAS WELL WITH VERTICAL FRACTURE

In order to devoid the hard work and factitious error in selecting charts while analyzing and interpreting hydraulic fracturing fracture parameters, on the basis of the non-Darcy flow factor, this paper put out the non-Darcy flow mathematical model of real gas in the formation and fracture, established the production history automatic matching model to identify fracture parameters, and offered the numerical solutions of those models, which took the variation of fracture conductivity in production process. These results offered a precise and reliable method to understand formation, analyze and evaluate the fracturing treatment quality of gas well.

Interstitial fluid flow:simulation of mechanical environment of cells in the interosseous membrane

In vitro experiments have shown that subtle fluid flow environment plays a significant role in living biological tissues, while there is no in vivo practical dynamical measurement of the interstitial fluid flow velocity. On the basis of a new finding that capillaries and collagen fibrils in the interosseous membrane form a parallel array, we set up a porous media model simulating the flow field with FLUENT software, studied the shear stress on interstitial cells＇ surface due to the interstitial fluid flow, and analyzed the effect of flow on protein space distribution around the ceils. The numerical simulation results show that the parallel nature of capillaries could lead to directional interstitial fluid flow in the direction of capillaries. Interstitial fluid flow would induce shear stress on the membrane of interstitial cells, up to 30 Pa or so, which reaches or exceeds the threshold values of cells＇ biological response observed in vitro. Interstitial fluid flow would induce nonuniform spacial distribution of secretion protein of mast cells. Shear tress on cells could be affected by capillary parameters such as the distance between the adjacent capillaries, blood pressure and the permeability coefficient of capillary＇s wall. The interstitial pressure and the interstitial porosity could also affect the shear stress on cells. In conclusion, numerical simulation provides an effective way for in vivo dynamic interstitial velocity research, helps to set up the vivid subtle interstitial flow environment of cells, and is beneficial to understanding the physiological functions of interstitial fluid flow.

Prediction of Low-Permeability Reservoirs Performances Using Long and Short-Term Memory Machine Learning

In order to overcome the typical limitations of numerical simulation methods used to estimate the production of low-permeability reservoirs,in this study,a new data-driven approach is proposed for the case of water-driven hypo-permeable reservoirs.In particular,given the bottlenecks of traditional recurrent neural networks in handling time series data,a neural network with long and short-term memory is used for such a purpose.This method can reduce the time required to solve a large number of partial differential equations.As such,it can therefore significantly improve the efficiency in predicting the needed production performances.Practical examples about water-driven hypotonic reservoirs are provided to demonstrate the correctness of the method and its ability to meet the requirements for practical reservoir applications.

SENSITIVITY COEFFICIENTS OF SINGLE-PHASE FLOW IN LOW-PERMEABILITY HETEROGENEOUS RESERVOIRS

Theoretical equations for computing sensitivity coefficients of wellbore pressures to estimate the reservoir parameters in low-permeability reservoirs conditioning to non-Darcy flow data at low velocity were obtained. It is shown by a lot of numerical calculations that the wellbore pressures are much more sensitive to permeability very near the well than to permeability a few gridblocks away from the well. When an initial pressure gradient existent sensitivity coefficients in the region are closer to the active well than to the observation well. Sensitivity coefficients of observation well at the line between the active well and the observation well are influenced greatly by the initial pressure gradient.

HORIZONTAL WELL PRESSURE ANALYSIS IN BOX-BOUNDED RESERVOIRS

In this paper, solutions to the 3D transient flow mathematical model for horizontal wells in box-rounded reservoirs are presented. The solutions are derived in Laplace transform domain by employing integral transform and point-source superposition. Both efficient computation of pressure responses and practical technology of oil field application mentioned here may be used to interpret the data from unsteady-state horizontal well testing.

Simulation of flow through porous anode in MFC at higher power density

Microbial fuel cell （MFC） is a novel environmental friendly energy device which has received great attention due to its technology for producing electricity directly fi-om organic or inorganic matter by using bacteria as catalyst. To date, many experiments have been carried out to achieve the maximum power output with advective flow through porous anode to the cathode in the MFC. However, the precise mechanical mechanism of flow through anode and the quantified relationship between electrode spacing and MFC performance are not yet clearly understood. It hasbeen found experimentally that the power output can be increased apparently at certain electrode spacing configuration. Based on these available experimental data, this paper investigates the effect of spacing between electrodes, the Darcy number of porous anode and the Reynolds number on the power production performance of MFC by using lattice Boltzmann method. The numerical simulation results present that the distance between electrodes significantly influences the flow velocity and residence time of the organic matter attached to the anode in the MFC. Moreover, it is found that the Darey number of porous anode and the Reynolds number can regulate the output efficiency of MFC. These results perform better understanding of the complex phenomena of MFC and will be helpful to optimize MFC design.

Some Advances in Applications of Lattice Boltzmann Method for Complex Thermal Flows

Over the past two decades,there have been enormous advances in lattice Boltzmann(LB)numerical simulation and modelling.The lattice Boltzmann method has become a practical and promising tool for many fluid problems.A majority of recent studies have relied on numerical computations of isothermal flows.However,much less efforts have been devoted to complex thermal flows,such as flows in porous media subjected to external magnetic force,flows with temperature-dependent properties.In this paper,an overview is made based on some accomplishments in these numerical endeavours.Along with the paper’s sections,the state-of-the-art trend and the LBM advances in modelling and in computational aspects for specific classes of problems of major interest will be fully touched on.Concluding remarks are given and the axis of our future studies will be traced.

RADICAL FLOW IN POROUS MEDIA WITH DISPERSION AND ADSORPTION

The radical transport of chemical concentration in porous media withdispersion and adsorption was studied in this paper. Using Langmuir''s adsorption model, thenumerical equation of concentration transport was derived. The flows with and without adsorptionwere simulated and analyzed. Comparison of the obtained solution with the known analytical solutionfor flow without adsorption shows the presented numerical method is correct and effective, which canbe used in reservoir engineering.

A STUDY ON TRANSIENT FLUID FLOW OF HORIZONTAL WELLS IN DUAL-PERMEABILITY MEDIA

To adopt horizontal wells in dual media reservoirs, a good understanding of the related fluid flows is necessary. Most of the recent studies focus on dual porosity media instead of dual permeability media. In this article, through both integral transformation and sink-source superposition, a new Laplace-domain solution is obtained for the slightly-compressible fluid flow in the 3-D dual-permeability media in which the horizontal well is operating in a constant rate of production. Major asymptotic characteristics of diagnosis curves of dimensionless downhole pressure are analyzed by the limited analysis. Effects of parameters of dual-permeability media including mobility ratio k, storativity ratio ω and inter-porosity flow parameter k on the downhote pressure are studied by using the Laplace numerical inversion. The new solution obtained in this article includes and improves the previous results and then can be used as a basis for either pressure transient analysis or formation behavior evaluation for the typical reservoir with horizontal wells.

Polymer flow through water- and oil-wet porous media

The effect of wettability on polymer behavior in porous media is investigated through a series of synthetic polymer floods conducted in water- and oil-wet Bentheim and Berea sandstone cores. A new experimental setup is used in which the core effluent polymer concentrations are not measured during polymer flooding, but instead are passed through a capillary tube that is connected to the core outlet. Using the notion of intrinsic viscosity, an approximate expression is presented that can be used to determine the breakthrough time of the injected polymer solution. In addition, polymer adsorption, inaccessible pore volume, and apparent viscosities are evaluated for all cores and wetting conditions. In general, in the presence of residual oil, the oil-wet cores display the lowest degree of polymer retention（Berea） and the lowest inaccessible pore volume-IPV（Bentheim）. The largest wettability impact on the polymer behavior in porous media appears to be for the Berea formation; polymer retention in oil-wet cores decreases 90% and IPV 52% compared to the corresponding amounts for the water-wet ones. On the other hand, the estimated polymer retention and IPV values appear to be similar for the oil- and water-wet Bentheim cores, this is attributed to questionable wettability alteration.

AN ELLIPTIC-PARABOLIC SYSTEM ARISING FROM THE FLUID-SOLITE-HEAT FLOW THROUGH SATURATED POROUS MEDIA

In this paper an elliptic-parabolic coupled system arising from the fluid-solute-heat flowthrough a saturated porous medium is considerd.The uniqueness and the existence of classicalsolutions are proved.The asymptotic behavior of solutions for large time is shown,too.

NUMERICAL SIMULATION FOR A PROCESS ANALYSIS OF A COKE OVEN

A computational fluid dynamic model is established for a coking process analysis of a coke oven using PHOENICS CFD package. The model simultaneously calculates the transient composition, temperatures of the gas and the solid phases, velocity of the gas phase and porosity and density of the semi-coke phase. Numerical simulation is illustrated in predicting the evolution of volatile gases, gas flow paths, profiles of density, porosity of the coke oven charge, profiles of temperatures of the coke oven gas and the semi-coke bed. On the basis of above modeling, the flow of coke oven gas （COG） blown from the bottom of the coke oven into the porous semi-coke bed is simulated to reveal whether or not and when the blown COG can uniformly flow through the porous semi-coke bed for the purpose of desulfurizing the semi-coke by recycling the COG. The simulation results show that the blown COG can uniformly flow through the semi-coke bed only after the temDerature at the center of the semi-coke bed has risen to above 900℃.

THEORY OF FLUID-SOLID COUPLED FLOW THROUGH FRACTURED LOW-PERMEABILITY OIL RESERVOIR AND ITS APPLICATIONS

During the development of low permeability reservoirs. the interaction between fluid flow and rock mass deformation is obvious. On the basis of fluid mechanics in porous media and elasto plastic theory. the author presents an equivalent continuum model to simulate fluid flow in fractured low permeability oil reservoir coupled with geo stress. The model not only reflects the porosity change of matrix, but also the permeability change due to the opening and closing of fracture. By analyzing of simulation results, the changes in porosity and permeability and their effect on oil development are studied.

An h-Adaptive Finite Element Solution of the Relaxation Non-Equilibrium Model for Gravity-Driven Fingers

The study on the fingering phenomenon has been playing an important role in understanding the mechanism of the fluid flow through the porous media.In this paper,a numerical method consisting of the Crank-Nicolson scheme for the temporal discretization and the finite element method for the spatial discretization is proposed for the relaxation non-equilibrium Richards equation in simulating the fingering phenomenon.Towards the efficiency and accuracy of the numerical simulations,a predictor-corrector process is used for resolving the nonlinearity of the equation,and an h-adaptive mesh method is introduced for accurately resolving the solution around the wetting front region,in which a heuristic a posteriori error indicator is designed for the purpose.In numerical simulations,a traveling wave solution of the governing equation is derived for checking the numerical convergence of the proposed method.The effectiveness of the h-adaptive method is also successfully demonstrated by numerical experiments.Finally the mechanism on generating fingers is discussed by numerically studying several examples.