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.

MODFLOW-CFPv2 模型在岩溶隧道突涌水及对地下水环境影响中的应用: 以云南鹤庆锰矿沟岩溶水系统为例

滇中地区构造复杂、岩溶发育,隧洞突涌水及泉流量衰减是隧洞施工过程中最棘手的问题之一。锰矿沟岩溶水系统岩溶管道化程度高,岩溶裂隙与岩溶管道2种含水介质差异显著。采用MODFLOW-CFPv2双重介质数值模型对锰矿沟岩溶水系统展开数值模拟研究,精细刻画岩溶管道与引水隧洞,进而掌握隧洞施工对地下水流场影响以及泉流量变化的规律。结果表明:(1)MODFLOW-CFPv2模型能够刻画岩溶地区复杂的地质结构,较好地模拟研究区地下水位的动态特征和岩溶泉流量响应特征。(2)在强排工况下隧洞单位长度最大涌水量为164 m^(3)/(d·m),单位长度稳定涌水量为69 m^(3)/(d·m),锰矿沟岩溶泉流量也出现显著下降的趋势,在模拟期内平均泉流量从天然条件下1578 L/s下降至1098 L/s,总体减少了30.4%;峰值泉流量从2133 L/s下降至1375 L/s,减少了35.5%,强排工况施工会对隧洞工程施工和地下水环境造成显著影响;限排工况下隧洞单位长度最大涌水量为39 m^(3)/(d·m),单位长度稳定涌水量为24 m^(3)/(d·m),隧洞单位长度涌水量显著降低,锰矿沟岩溶泉流量的下降趋势也得到了一定程度的改善,模拟期内平均泉流量降低至1284 L/s,减少了18.6%,峰值泉流量降低至1617 L/s,减少了22.1%。采用的MODFLOW-CFPv2双重介质模型具有较精确刻画岩溶区管道、溶洞、裂隙共存的高度非均质岩溶水系统的能力,能够定量评价香炉山隧洞施工对锰矿沟岩溶水系统地下水流场及泉流量的影响,为香炉山隧洞工程的突涌水灾害防治提供参考依据,也为岩溶地区复杂地质条件下地下水研究提供借鉴经验。

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.

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.

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.

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.

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.

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.

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.

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.

A New Numerical Solution of Fluid Flow in Stratigraphic Porous Media

A new numerical technique based on a lattice-Boltzmann method is presented for analyzing the fluid flow in stratigraphic porous media near the earth's surface. The results obtained for the relations between porosity, pressure,and velocity satisfy well the requirements of stratigraphic statistics and hence are helpful for a further study of the evolution of fluid flow in stratigraphic media.

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.

Determining factors for carotid mean/max intima-media thickness and brachial flow-mediated dilation in healthy young women

Background: Many factors can contribute to atherosclerotic-type vascular changes in older individuals or men. Thus, confining the investigation to young women with no clinical evidence of the condition could enhance understanding of the early stages of cardiovascular disease. The aim of this study was to determine whether carotid mean/max intima-media thickness (IMT) and brachial flow-mediated dilation (FMD) values, which are well-known event-related indices, are associated with laboratory data and the other vascular indices of atherosclerosis in healthy young women. Methods: Carotid mean/max IMT and brachial FMD were measured in young women with no clinical evidence of atherosclerosis (n = 110;mean age, 39 years) who were instructed not to eat, drink or smoke after 9 PM the evening before testing. All participants also underwent laboratory assessment, including simultaneous measurements of arterial stiffness such as augmentation index (AI), cardioankle vascular index (CAVI) and brachial-ankle pulse wave velocity (baPWV). Results: Mean IMT was signifi-cantly and positively associated with age (p = 0.002), CAVI (p = 0.044), low-density lipoprotein-cholesterol (LDL-C, p = 0.047) and high-sensitive C-reactive protein (hs-CRP, p = 0.002) values but was not related to FMD, AI, baPWV or triglycerides (TG) in the multivariate regression analysis. Similarly, max IMT was positively associated with age (p p = 0.003) and hs-CRP (p = 0.005) values but was not related to FMD, AI, CAVI, baPWV, TG or blood pressure level in the multivariate regression analysis. The association between LDL-C and max IMT was much stronger than that between LDL-C and mean IMT. Brachial FMD was positively associated only with heart rate in the multivariate regression analysis. Conclusions: These results suggest that mean IMT more closely represents the sclerotic aspect of vascular change, whereas max IMT represents the atherotic aspect in healthy young women. Although the relationship between the autonomic nervous system and heart rate is well-known, there may be a complex interaction between the autonomic nervous system and endothelial function.

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.

Plane Transverse MHD Flow through Porous Media

Plane, transverse MHD flow through a porous structure is considered in this work. Solution to the governing equations is obtained using an inverse method in which the streamfunction of the flow is considered linear in one of the space variables. Expressions for flow quantities are obtained for finitely conducting and infinitely conducting fluids.

Numerical Calculation of Transient Flow of Polymer Foam in Porous Media

Based on the mathematical model of one dimension transient flow of the polymer foam in porous media, the numerical calculation method of the flow mentioned above by using the finite difference method is given. Through the experiments of one dimension transient flow of HPAM (Hydrolytic Polyacrylamide) foam in the artificial sandstone core, the HPAM foam generation and coalescence coefficient of the mathematical model mentioned above are determined. The profiles of the liquid phase saturation, the pressure drop and the number density of one dimension transient flow of HPAM foam with the dimensionless time in artificial sandstone core are numerically calculated and analyzed by using the numerical calculation method.

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.

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.

THE TRANSIENT ELLIPTIC FLOW OF POWER-LAW FLUID IN FRACTAL POROUS MEDIA

The steady oil production and pressure distribution formulae of vertically fractured well for power-law non-Newtonian fluid were derived on the basis of the elliptic flow model in fractal reservoirs. The corresponding transient flow in fractal reservoirs was studied by numerical differentiation method: the influence of fractal index to transient pressure of vertically fractured well was analyzed. Finally the approximate analytical solution of transient flow was given by average mass conservation law. The study shows that using elliptic flow method to analyze the flow of vertically fractured well is a simple method.