Porous-DeepONet:Learning the Solution Operators of Parametric Reactive Transport Equations in Porous Media

Reactive transport equations in porous media are critical in various scientific and engineering disciplines,but solving these equations can be computationally expensive when exploring different scenarios,such as varying porous structures and initial or boundary conditions.The deep operator network(DeepONet)has emerged as a popular deep learning framework for solving parametric partial differential equations.However,applying the DeepONet to porous media presents significant challenges due to its limited capability to extract representative features from intricate structures.To address this issue,we propose the Porous-DeepONet,a simple yet highly effective extension of the DeepONet framework that leverages convolutional neural networks(CNNs)to learn the solution operators of parametric reactive transport equations in porous media.By incorporating CNNs,we can effectively capture the intricate features of porous media,enabling accurate and efficient learning of the solution operators.We demonstrate the effectiveness of the Porous-DeepONet in accurately and rapidly learning the solution operators of parametric reactive transport equations with various boundary conditions,multiple phases,and multiphysical fields through five examples.This approach offers significant computational savings,potentially reducing the computation time by 50–1000 times compared with the finite-element method.Our work may provide a robust alternative for solving parametric reactive transport equations in porous media,paving the way for exploring complex phenomena in porous media.

Periodic Solutions of Porous Medium Equations with Weakly Nonlinear Sources

In this paper, we show the existence of the time periodic solutions to the porous medium equations of the formut= Δ (|u| m-1 u)+B(x,t,u)+f(x,t) in Ω×Rwith the Dirichlet boundary value condition, where m>1, Ω is a bounded domain in R N with smooth boundary Ω , the continuous function f and the Hlder continuous function B(x,t,u) are periodic in t with period ω and the nonlinear sources are assumed to be weaker, i.e., B(x,t,u) u≤b 0|u| α+1 with constants b 0≥0 and 0≤α<m.

Rheological behavior of hydrolyzed polyacrylamide solution flowing through a molecular weight adjusting device with porous medium

The separate-layer injection in different interlayers and the injection of the same-molecular-weight polymer so- lution in a layer are necessary in the polymer flooding process because of heterogeneous multilayer sandstone reservoirs in EOR projects. To alleviate the matching problems between the layer permeability and the injected polymer molecular weight, a molecular weight adjusting device with porous medium was designed on the basis of mechanical degradation principle. In terms of four variables （polymer concentration, pore diameter, length of shear component and flow rate ）, the theological behavior of hydrolyzed polyacrylamide （HPAM） solu- tion flowing through the device was investigated in detail. The change of these variables is able to control the shear rate of HPAM solutions through ceramic foam, and achieve the desired degree of shear degradation and the final theological parameters-viscosity loss, viscoelasticity and pressure drop. Therefore, a linear relationship between viscosity loss and shearing rate was established so as to obtain the targeted viscosity easily. Field tests in the Daqing Oil Field showed that the polymer molecular weight could drop 20% to 50%. In a word, the results could guide the industrial application of the novel device and the further study of polymer degradation flowing through the porous medium.

Analytic Solution for Fluid Flow over an Exponentially Stretching Porous Sheet with Surface Heat Flux in Porous Medium by Means of Homotopy Analysis Method

In this paper, the analytical solution of a viscous and incompressible fluid towards an exponentially stretching porous sheet with surface heat flux in porous medium, for the boundary layer and heat transfer flow, is presented. The equations of continuity, momentum and the energy are transformed into non-linear ordinary differential by using similarity transformation. The solutions of these highly non-linear ordinary differential equations are found analytically by means of Homotopy Analysis Method (HAM). The result obtained by HAM is compared with numerical results presented in the literature. The accuracy of the HAM is indicated by close agreement of the two sets of results. By this method, an expression is obtained which is admissible for all values of effective parameters. This method has the ability to control the convergence of the solution.

Utilization of Maxwell-Cattaneo law for MHD swirling flow through oscillatory disk subject to porous medium

The present study aims to investigate the salient features of incompressible, hydromagnetic, three-dimensional flow of viscous fluid subject to the oscillatory motion of a disk. The rotating disk is contained in a porous medium. Furthermore, a time-invariant version of the Maxwell-Cattaneo law is implemented in the energy equation. The flow problem is normalized by obtaining similarity variables. The resulting nonlinear system is solved numerically using the successive over-relaxation method. The main results are discussed through graphical representations and tables. It is perceived that the thermal relaxation time parameter decreases the temperature curves and increases the heat trans- fer rate. The oscillatory curves for the velocity field demonstrate a decreasing tendency with the increasing porosity parameter values. Two- and three-dimensional flow phenom- ena are also shown through graphical results.

Simulation of solute transportation within porous particles during the bioleaching process

A mathematical model, accounting for the sulfuric acid and ferric ions diffusion and the copper sulfide mineral leaching process, was developed for an ore particle by considering its porous structure. It was simulated with the simulation tool COMSOL Multiphysics. The simulation results show that the highest acid and ferric concentrations near the particle surface are apparent, while the concentrations in the central particle increase slightly as the less-porous ore core with low permeability prevents the oxidation from penetrating. The extraction of the mineral near the particle surface is the maximum, mainly because of ample sulfuric acid, ferric ions, bacteria, and oxygen available for the leaching process. Because of low oxidation concentration in the central part of the particle, the reaction rate and copper sulphide conversion are small. The simulation shows good agreement with the experimental results.

Linear stability of triple-diffusive convection in micropolar ferromagnetic fluid saturating porous medium

The triple-diffusive convection in a micropolar ferromagnetic fluid layer heated and soluted from below is considered in the presence of a transverse uniform magnetic field. An exact solution is obtained for a flat fluid layer contained between two free boundaries. A linear stability analysis and a normal mode analysis method are carried out to study the onset convection. For stationary convection, various parameters such as the medium permeability, the solute gradients, the non-buoyancy magnetization, and the micropolar parameters （i.e., the coupling parameter, the spin diffusion parameter, and the micropolar heat conduction parameter） are analyzed. The critical magnetic thermal Rayleigh number for the onset of instability is determined numerically for a sufficiently large value of the buoyancy magnetization parameter M1. The principle of exchange of stabilities is found to be true for the micropolar fluid heated from below in the absence of the micropolar viscous effect, the microinertia, and the solute gradients. The micropolar viscous effect, the microinertia, and the solute gradient introduce oscillatory modes, which are non-existent in their absence. Sufficient conditions for the non-existence of overstability are also obtained.

Exact solutions to one-dimensional transient response of incompressible fluid-saturated single-layer porous media

Based on the Biot theory of porous media, the exact solutions to one- dimensional transient response of incompressible saturated single-layer porous media un- der four types of boundary conditions are developed. In the procedure, a relation between the solid displacement u and the relative displacement ＇w is derived, and the well-posed initial conditions and boundary conditions are proposed. The derivation of the solution for one type of boundary condition is then illustrated in detail. The exact solutions for the other three types of boundary conditions are given directly. The propagation of the compressional wave is investigated through numerical examples. It is verified that only one type of compressional wave exists in the incompressible saturated porous media.

Experimental study on the velocity-dependent dispersion of the solute transport in different porous media

The hydrodynamic dispersion is an important factor influencing the reactive solute transport in the porous media, and many previous studies assumed that it linearly varied with the average velocity of the groundwater flow. Actually, such linear relationship has been challenged by more and more experimental observations, even in homogeneous media. In this study, we aim to investigate the relationship between hydrodynamics dispersion and the flow velocity in different types of porous media through a laboratory-controlled experiment. The results indicate that (1) the dispersion coefficient should not be a linear function of the flow velocity when the relationship between the flow velocity and the hydraulic gradient can be described by Darcy's law satisfactorily;(2) Power function works well in describing the dispersion coefficient changing with the flow velocity for different types of porous media, and the power value is between 1.0-2.0 for different particle sizes.

Modelling Simultaneous Transport of Bioreactive Solutes and Microorganisms in Porous Media

Recent years have seen the development of a number of mathematical models for the description of the simultaneous transport of microorganisms and bioreactive solutes in porous media. Most models are based on the advection dispersion equation, with terms added to account for interactions with the surfaces of the solid matrix, transformations and microbial activities. Those models based on the advection dispersion equation have all been shown to represent laboratory experimental data adequately although various assumptions have been made concerning the pore scale distribution of bacteria. This paper provides an overview of the recent work on modelling the transport and fate of microorganisms and bioreactive solutes in porous media and examines the different assumptions regarding the pore scale distribution of microorganisms.

FEM analyses for influences of pressure solution on thermo-hydro-mechanical coupling in porous rock mass

The model of pressure solution for granular aggregate was introduced into the FEM code for analysis of thermo-hydro- mechanical （T-H-M） coupling in porous medium. Aiming at a hypothetical nuclear waste repository in an unsaturated quartz rock mass, two computation conditions were designed： 1） the porosity and the permeability of rock mass are fimctions of pressure solution; 2） the porosity and the permeability are constants. Then the corresponding numerical simulations for a disposal period of 4 a were carried out, and the states of temperatures, porosities and permeabilities, pore pressures, flow velocities and stresses in the rock mass were investigated. The results show that at the end of the calculation in Case 1, pressure solution makes the porosities and the permeabilities decrease to 10%-45% and 0.05%-1.4% of their initial values, respectively. Under the action of the release heat of nuclear waste, the negative pore pressures both in Case 1 and Case 2 are 1.2-1.4 and 1.01-l.06 times of the initial values, respectively. So, the former represents an obvious effect of pressure solution. The magnitudes and distributions of stresses within the rock mass in the two calculation cases are the same.

Numerical investigation of Dufour and Soret effects on unsteadyMHD natural convection flow past vertical plate embedded innon-Darcy porous medium

The Dufour and Soret effects on the unsteady twodimensional magnetonyaro dynamics （MHD） doublediffusive free convective flow of an electrically conducting fluid past a vertical plate embedded in a nonDarcy porous medium are investigated numeri cally. The governing nonlinear dimensionless equations are solved by an implicit finite difference scheme of the CrankNicolson type with a tridiagonal matrix manipulation. The effects of various parameters entering into the problem on the unsteady dimension less velocity, temperature, and concentration profiles are studied in detail. Furthermore, the time variation of the skin friction coefficient, the Nusselt number, and the Sherwood number is presented and analyzed. The results show that the unsteady velocity, tem perature, and concentration profiles are substantially influenced by the Dufour and Soret effects. When the Dufour number increases or the Soret number decreases, both the skin friction and the Sherwood number decrease, while the Nusselt number increases. It is found that, when the magnetic parameter increases, the velocity and the temperature decrease in the boundary layer.

Similarity Solutions of Unsteady Convective Boundary Layer Flow along Isothermal Vertical Plate with Porous Medium

Similarity solution of unsteady convective boundary layer flow along isothermal vertical plate with porous medium is analyzed. The plate surface is reactive with the fluid and generates inert specie which diffuses inside the boundary. The flux of the specie at the plate is proportional to specie concentration at the plate. The governing equations of continuity, momentum, energy and specie diffusion are transformed into ordinary differential equation by using the similarity transformation and solved numerically by using free parameter method along with shooting technique. The dimensionless velocity, temperature and concentration profiles are obtained and presented through figures for different parameters entering into the problem. The local Skin-friction co-efficient, Nusselt number and Sherwood number at the plate for physical interest are also discussed through tables.

GAS BUBBLE-FACILITATED TRANSPORT OF METALS IN LITHOSPHERE

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Axisymmetrical analytical solution for vertical vibration of end-bearing pile in saturated viscoelastic soil layer

Based on elasticity and the theory of saturated porous media, and regarding the pile and the soil as a single phase elastic and a saturated viscoelastic media, respectively, the dynamical behavior of vertical vibration of an end-bearing pile in a saturated viscoelastic soil layer is investigated in the frequency domain using the Helmholtz decomposition and variable separation method. The axisymmetrical analytical solutions for vertical vibrations of the pile in a saturated viscoelastic soil layer are obtained, and the analytical expression of the dynamical complex stiffness of the pile top is presented. Responses of dynamic stiffness factor and equivalent damping of pile top with respect to the frequency are shown in figures using a numerical method. Effects of the saturated soil parameters, modulus ratio of the pile to soil, slenderness ratio of pile and pile＇s Poisson ratio, etc. on the stiffness factor and damping are examined. It is shown that, due to the effect of the transversal deformation of the pile and the radial force of the saturated viscoelastic soil acting on the pile, the dynamic stiffness factor and the damping derived from the axisymmetrical solution are greatly different from those derived from the classical Euler-Bernoulli rod model, especially at some specific excitation frequencies. Therefore, there are limitations on applicability of the Euler-Bernoulli rod model in analyzing verticai vibration of the pile. More accurate analysis should be based on a three-dimensional model.

Experimental investigation on rainfall infiltration and solute transport in layered porous and fractured media

Layered structures with upper porous and lower fractured media are widely distributed in the world. An experimen- tal investigation on rainfall infiltration and solute transport in such layered structures can provide the necessary foundation for effectively preventing and forecasting water bursting in mines, controlling contamination of mine water, and accomplishing ecological restoration of mining areas. A typical physical model of the layered structures with porous and fractured media was created in this study. Then rainfall infiltration experiments were conducted after salt solution was sprayed on the surface of the layered structure. The volumetric water content and concentration of chlorine ions at different specified positions along the profile of the experiment system were measured in real-time. The experimental results showed that the lower fractured media, with a considerably higher permeability than that of the upper porous media, had significant effects on preventing water infil- tration. Moreover, although the porous media were homogeneous statistically in the whole domain, spatial variations in the features of effluent concentrations with regards to time, or so called breakthrough curves, at various sampling points located at the horizontal plane in the porous media near the porous-fractured interface were observed, indicating the diversity of solute transport at small scales. Furthermore, the breakthrough curves of the outflow at the bottom, located beneath the underlying fractured rock, were able to capture and integrate features of the breakthrough curves of both the upper porous and fractured media, which exhibited multiple peaks, while the peak values were reduced one by one with time.

Numerical simulation of effect of convection-diffusion on oxygen transport in microcirculation

The entire process of oxygen transport in microcirculation by developing a3 D porous media model is calculated numerically with coupled solid deformation-fluid seepage-convection and diffusion. The principal novelty of the model is that it takes into account volumetric deformation of both capillary and tissues resulting from capillary fluctuation. How solid deformation, fluid seepage, and convection-diffusion combine to affect oxygen transport is examined quantitatively：（1） Solid deformation is more significant in the middle of capillary, where the maximum value of volumetric deformation reaches about 0.5%.（2） Solid deformation has positive influence on the tissue fluid so that it flows more uniformly and causes oxygen to be transported more uniformly, and eventually impacts oxygen concentration by 0.1%–0.5%.（3） Convection-diffusion coupled deformation and seepage has a maximum（16%） and average（3%） increase in oxygen concentration,compared with pure molecular diffusion. Its more significant role is to allow oxygen to be transported more evenly.

Thermal Analysis of MHD Non-Newtonian Nanofluids over a Porous Media

In the present research,Tiwari and Das model are used for the impact of a magnetic field on non-Newtonian nanofluid flow in the presence of injection and suction.The PDEs are converted into ordinary differential equations(ODEs)using the similarity method.The obtained ordinary differential equations are solved numerically using shooting method along with RK-4.Part of the present study uses nanoparticles(NPs)like TiO_(2) andAl_(2)O_(3) and sodium carboxymethyl cellulose(CMC/water)is considered as a base fluid(BF).This study is conducted to find the influence of nanoparticles,Prandtl number,and magnetic field on velocity and temperature profile,however,the Nusselt number and coefficient of skin friction parameters are also presented in detail with the variation of nanoparticles and parameters.The obtained results of the present study are presented usingMATLAB.In addition to these,some simulations of partial differential equations are also shown using software for graphing surface plots of velocity profile and streamlines along with surface plots and isothermal contours of the temperature profile.

Exact solutions of MHD second Stokes flow of generalized Burgers fluid

This work concerns with the exact solutions of magnetohydrodynamic（MHD）flow of generalized Burgers fluid describing the second Stokes problem. The modified Darcy law is taken into account. The related velocity distribution and shear stress are expressed as a combination of steady-state and transient solutions computed by means of integral transformations. The effects of various parameters on the flow field are investigated. The MHD flow results in reduction of velocity distribution and associated thickness of the boundary layer.

Size Exclusion Mechanism, Suspension Flow through Porous Medium

A lot of investigations have been done in order to understand the mechanisms of the transport of particulate suspension flow through porous medium. In general, Deep Bed Filtration studies have been conducted to analyse the mechanism involved in the processes of capturing and retaining particles occurs throughout the entire depth of the filter and not just on the filter surface. In this study, the deep bed filtration mechanism and the several mechanisms for the capture of suspended particles are explained then the size exclusion mechanism has been focused (particle capture from the suspension by the rock by the size exclusion). The effects of particle flux reduction and pore space inaccessibility due to selective flow of different size particles will be included in the model for deep bed filtration. The equations for particle and pore size distributions have been derived. The model proposed is a generalization of stochastic Sharma-Yortsos equations. Analytical solution for low concentration is obtained for any particle and pore size distributions. As we will see, the averaged macro scale solutions significantly differ from the classical deep bed filtration model.