Diffusion In Porous Media Research Articles

Hydrodynamic couplings of colloidal ellipsoids diffusing in channels

The hydrodynamic interactions (HIs) of two colloidal spheres characterized by the translation–translation (T–T) couplings have been studied under various confinements, but little is known regarding the HIs of anisotropic particles and rotational motions, which are common in nature and industry. Here, we study the T–T, rotation–rotation (R–R) and translation–rotation (T–R) hydrodynamic couplings of two colloidal ellipsoids sediment on the bottoms of channels in experiment, theory and simulation. We find that the degree of confinement and the particle shape anisotropy are critical tuning factors resulting in anomalous hydrodynamic and diffusive behaviours. The negative R–R coupling reflects the tendency of opposite rotations of two neighbouring ellipsoids. The positive T–R coupling reflects that an ellipsoid rotates away from the channel axis as another ellipsoid approaches. As the channel width increases, the positive T–T coupling changes to an abnormal negative coupling, indicating that the single-file diffusion can exist even in wide channels. By contrast, only positive T–T couplings were observed for spheres in channels. The T–T coupling increases with the aspect ratiop. The R–R coupling is the maximum at a moderatep ~ 2.8. The T–R coupling is the maximum at a moderate degree of confinement. The spatial range of HIs is longer than that of spheres and increases withp. We propose a simple model which reproduces some coupling phenomena between two ellipsoids, and it is further confirmed by low-Reynolds-number hydrodynamic simulation. These findings shed new light on anisotropic particle diffusion in porous media, transport through membranes, microfluidics and microrheology.

Diffusive Mass Transfer and Gaussian Pressure Transient Solutions for Porous Media

This study revisits the mathematical equations for diffusive mass transport in 1D, 2D and 3D space and highlights a widespread misconception about the meaning of the regular and cumulative probability of random-walk solutions for diffusive mass transport. Next, the regular probability solution for molecular diffusion is applied to pressure diffusion in porous media. The pressure drop (by fluid extraction) or increase (by fluid injection) due to the production system may start with a simple pressure step function. The pressure perturbation imposed by the step function (representing the engineering intervention) will instantaneously diffuse into the reservoir at a rate that is controlled by the hydraulic diffusivity. Traditionally, the advance of the pressure transient in porous media such as geological reservoirs is modeled by two distinct approaches: (1) scalar equations for well performance testing that do not attempt to solve for the spatial change or the position of the pressure transient without reference to a well rate; (2) advanced reservoir models based on numerical solution methods. The Gaussian pressure transient solution method presented in this study can compute the spatial pressure depletion in the reservoir at arbitrary times and is based on analytical expressions that give spatial resolution without gridding-meaning solutions that have infinite resolution. The Gaussian solution is efficient for quantifying the advance of the pressure transient and associated pressure depletion around single wells, multiple wells and hydraulic fractures. This work lays the basis for the development of advanced reservoir simulations based on the superposition of analytical pressure transient solutions.

Modeling Vapor Migration for Estimating the Time to Reach Steady State Conditions

AbstractThe design and planning of soil vapor sampling for vapor intrusion assessment require an estimate of the time for vapor migration from the contamination source to reach steady state prior to vapor sampling and analysis for volatile organic compounds (VOCs). This study presents the model derivation, analytical solutions, as well as the assumptions and limitations of a one‐dimensional VOC vapor transport model based on diffusion in porous media and equilibrium partitioning of VOCs in solid, aqueous, and vapor phases. The model assumes a finite domain with boundary conditions that represent the scenarios of vapor migration in the real environment. The derivation of the conceptual model is presented along with its practical use and implications as illustrated through case examples. Consideration of the upper (or exit) boundary condition along with the distance between the source and the applicable boundary, rather than the distance from the source to the measurement point, are shown to be critical in the time estimates as compared to an expression typically used and cited in guidance documents. The study reveals the importance of defining a conceptual model and relevant boundaries in assessing near steady state conditions, and suggests a tiered approach in refining the estimate with increasing level of effort for practical applications in vapor assessment.

Generalized finite difference method with irregular mesh for a class of three-dimensional variable-order time-fractional advection-diffusion equations

Fractional advection-diffusion equation, as a generalization of classical advection-diffusion equation, has been always mentioned to simulate anomalous diffusion in porous media. This work introduces a meshless generalized finite difference method (GFDM) to solve a class of three-dimensional variable-order time fractional advection-diffusion equation (TFADE) in finite domains. Three examples with known analytic solutions in different domains are given to demonstrate that the method is accurate and stable. To reduce computational and storage cost, we discretize the time derivative terms of TFADE by a fast finite difference method (FFDM) based on sum-of-exponentials (SOE) approximation. Meanwhile, discretizing space derivative terms, GFDM generates a linear equation set including function values of neighboring nodes with various weight coefficients. Then the partial derivatives of TFADE are indicated as the linear system above. Also, this paper investigates the irregular mesh in the finite spatial domain, which is more closely meets the description of practice problems. Numerical results indicate that models with irregular mesh can also be simulated by GFDM which maintains high accuracy. Furthermore, the method is stable and accurate in solving three-dimensional irregular domain problems, where the relative errors can be less than 0.01%. This paper shows that FFDM based on SOE approximation can improve computational efficiency, and GFDM can flexibly and efficiently solve three-dimensional variable-order and variable-coefficient TFADE.

CO2 huff-n-puff experimentation and numerical simulation for 3D printed rock samples

3D printed rock samples advance the experimental research of fluid flow physics in the petroleum industry. 3D printed replicas can be flexibly tailored to different experimental setups, in addition to the low cost of creation with uniform materials. The conducted experiments, on 3D printed samples, have less geometrical uncertainties, which is related to the accurately described structure of the pore network. The difficulties of describing fluid-rock interaction are diminished because of the compositional uniformity of the 3D-printing material. Therefore, 3D printed replicas were used, in this paper, to demonstrate the CO2 effects on oil recovery through a huff and puff experiment. The CT scan of a Berea core sample was reconstructed using image processing to prepare a 3D printable object. The two samples, i.e., the 3D printed gypsum replica and its original Berea core sample, were inserted simultaneously in a newly designed experimental setup to conduct the huff and puff experiment under the same conditions. The oil recovery was reported at three different CO2 injection pressures and temperatures. After running the experiment, the synthetic core sample was broken laterally and longitudinally to examine the CO2 action in the sample visually. As a pioneer study, the CO2 effect on a 3D-printed core sample was pictured, which can be used to explain the physics of CO2 diffusion in porous media. Finally, a supporting simulation model was created based on the CT scan to simulate the experiment. The simulation results matched the recovery results for the 3D printed replica and generated oil composition crosssections similar to the pictured ones.

A reducing mechanism on wave speed for chemotaxis systems with degenerate diffusion

This paper is concerned with traveling wave solutions for a chemotaxis model with degenerate diffusion of porous medium type. We establish the existence of semi-finite traveling waves, including the sharp type and \(C^1\) type semi-finite waves. Our results indicate that chemotaxis slows down the wave speed of semi-finite traveling wave, that is, the traveling wave speed for chemotaxis with porous medium (degenerate) diffusion is smaller than that for the porous medium equation without chemotaxis. As we know, this is a new result not shown in the existing literature. The result appears to be a little surprising since chemotaxis is a convective force. We prove our results by the Schauder’s fixed point theorem and estimate the wave speed by a variational approach.

Large time behavior in a chemotaxis-Stokes system modeling coral fertilization with arbitrarily slow porous medium diffusion

This paper concerns with a kind of chemotaxis-Stokes systems generalizing the prototype{nt+u⋅∇n=∇⋅(nm−1∇n)−∇⋅(n∇c)−nv,ct+u⋅∇c=Δc−c+v,vt+u⋅∇v=Δv−nv,ut=Δu+∇P+(n+v)∇Φ,∇⋅u=0 which characterizes the process of coral fertilization in ocean. By virtue of a novel approach on the basis of some conditional estimates for signal gradient and fluid velocity, it is proved that when m>1 an associated initial-boundary problem possesses a globally bounded weak solution in spatially three-dimensional setting, which extends the corresponding results obtained in [15]. Moreover, the obtained solutions stabilize to a certain constant equilibrium (n∞,v∞,v∞,0) with n∞:=1|Ω|{∫Ωn0−∫Ωv0}+ and v∞:=1|Ω|{∫Ωv0−∫Ωn0}+ as t→∞.

Experimental and microscopic simulation study on CO2 diffusion in an oil-water liquid system

CO2 diffusion is important in CO2 enhanced oil recovery and CO2 capture and storage. However, most reported studies only focused on diffusion in single phase. In this study, a new model considering CO2 diffusion in multi-phase was established, and a new mathematical model is proposed based on the diffusion stage division. Diffusion experiments were conducted to determine the diffusion coefficients. Then, microscopic simulations were conducted to investigate CO2 diffusion in porous media using the determined coefficients. According to the results, the coupling between the experimental pressure and model indicated the accuracy of the model. The pressure and temperature were key parameters for controlling the coefficients. Water distribution had a critical impact on CO2 diffusion in porous media. CO2 diffusion shows worse performance in models with severe water block effect. The findings of this study can help guide the design of CO2 EOR and CCS projects.

Strong dissipativity of generalized time-fractional derivatives and quasi-linear (stochastic) partial differential equations

In this paper strong dissipativity of generalized time-fractional derivatives on Gelfand triples of properly in time weighted Lp-path spaces is proved. In particular, as special cases the classical Caputo derivative and other fractional derivatives appearing in applications are included. As a consequence one obtains the existence and uniqueness of solutions to evolution equations on Gelfand triples with generalized time-fractional derivatives. These equations are of typeddt(k⁎u)(t)+A(t,u(t))=f(t),0<t<T, with (in general nonlinear) operators A(t,⋅) satisfying general weak monotonicity conditions. Here k is a non-increasing locally Lebesgue-integrable nonnegative function on [0,∞) with lims→∞k(s)=0. Analogous results for the case, where f is replaced by a time-fractional additive noise, are obtained as well. Applications include generalized time-fractional quasi-linear (stochastic) partial differential equations. In particular, time-fractional (stochastic) porous medium and fast diffusion equations with ordinary or fractional Laplace operators and the time-fractional (stochastic) p-Laplace equation are covered.

Application of ultrasound in regeneration of silica gel for industrial gas drying processes

Industrial gas dehumidification contributes to a considerable portion of the total industrial energy consumption. Depending on the desired level of dryness, solid desiccants can be a more suitable dehumidification method compared to the conventional dehumidification processes. In this study, the ultrasound-enhanced regeneration of silica gel as a substitute to conventional heating methods is investigated. To analyze the energy-savings effect of applying ultrasound a method of constant total power levels of 20 and 25 (W) corresponding to total specific power levels of 327 and 409 was adopted and for each Watt of applied ultrasonic power, the same amount was deducted from thermal power. The moisture content and regeneration temperature were measured and compared in ultrasound-assisted and non-ultrasound (heat-only) regeneration processes. Experimental results showed that applying ultrasound along with thermal power improves the regeneration process and reduces the energy required to regenerate silica gel by as much as 26%. Various transport modes contributing to mass diffusion in porous media are analyzed and an apparent diffusion coefficient for porous media that includes ultrasonication effects is proposed. Regarding regeneration temperature, with application of ultrasound, regeneration at lower temperatures by as much as ∼11% was achieved.

Global bounded weak solutions and asymptotic behavior to a chemotaxis-Stokes model with non-Newtonian filtration slow diffusion

In this paper, we deal with the following chemotaxis-Stokes model with non-Newtonian filtration slow diffusion (namely, p>2){nt+u⋅∇n=∇⋅(|∇n|p−2∇n)−χ∇⋅(n∇c),ct+u⋅∇c−Δc=−cn,ut+∇π=Δu+n∇φ,divu=0 in a bounded domain Ω of R3 with zero-flux boundary conditions and no-slip boundary condition. Similar to the study for the chemotaxis-Stokes system with porous medium diffusion, it is also a challenging problem to find an optimal p-value (p≥2) which ensures that the solution is global bounded. In particular, the closer the value of p is to 2, the more difficult the study becomes. In the present paper, we prove that global bounded weak solutions exist wheneverp>p⁎(≈2.012). It improved the result of [21,22], in which, the authors established the global bounded solutions for p>2311. Moreover, we also consider the large time behavior of solutions, and show that the weak solutions will converge to the spatially homogeneous steady state (n‾0,0,0). Comparing with the chemotaxis-fluid system with porous medium diffusion, the present convergence of n is proved in the sense of L∞-norm, not only in Lp-norm or weak-* topology.

Systemic States of Spreading Activation in Describing Associative Knowledge Networks II: Generalisations with Fractional Graph Laplacians and q-Adjacency Kernels

Associative knowledge networks are often explored by using the so-called spreading activation model to find their key items and their rankings. The spreading activation model is based on the idea of diffusion- or random walk -like spreading of activation in the network. Here, we propose a generalisation, which relaxes an assumption of simple Brownian-like random walk (or equally, ordinary diffusion process) and takes into account nonlocal jump processes, typical for superdiffusive processes, by using fractional graph Laplacian. In addition, the model allows a nonlinearity of the diffusion process. These generalizations provide a dynamic equation that is analogous to fractional porous medium diffusion equation in a continuum case. A solution of the generalized equation is obtained in the form of a recently proposed q-generalized matrix transformation, the so-called q-adjacency kernel, which can be adopted as a systemic state describing spreading activation. Based on the systemic state, a new centrality measure called activity centrality is introduced for ranking the importance of items (nodes) in spreading activation. To demonstrate the viability of analysis based on systemic states, we use empirical data from a recently reported case of a university students’ associative knowledge network about the history of science. It is shown that, while a choice of model does not alter rankings of the items with the highest rank, rankings of nodes with lower ranks depend essentially on the diffusion model.

A FRACTAL MODEL FOR PREDICTING OXYGEN EFFECTIVE DIFFUSIVITY OF POROUS MEDIA WITH ROUGH SURFACES UNDER DRY AND WET CONDITIONS

Oxygen diffusion in porous media (ODPM) with rough surfaces (RS) under dry and wet conditions is of great interest. In this work, a novel fractal model for the oxygen effective diffusivity of porous media with RS under dry and wet conditions is proposed. The proposed fractal model is expressed in terms of relative roughness, the water saturation, fractal dimension for tortuosity of tortuous capillaries, fractal dimension for pores, and porosity. It is observed that the normalized oxygen diffusivity decreases with increasing relative roughness and fractal dimension for capillary tortuosity. It is found that the normalized oxygen diffusivity increases with porosity and fractal dimension for pore area. Besides, it is seen that that the normalized oxygen diffusivity under wet condition decreases with increasing water saturation. The determined normalized oxygen diffusivity is in good agreement with experimental data and existing models reported in the literature. With the proposed analytical fractal model, the physical mechanisms of oxygen diffusion through porous media with RS under dry and wet conditions are better elucidated. Every parameter in the proposed fractal model has clear physical meaning, with no empirical constant.

Transport Characteristics of the Near-Surface Layer of the Nucleus of Comet 67P/Churyumov–Gerasimenko

The paper considers the free molecular flow of gas through the dusty porous surface layer of a comet nucleus. The study is based on computer models of generation of a porous medium and Knudsen gas diffusion. We consider various types of homogeneous and heterogeneous layers constructed from nonintersecting spheres, including layers that contain microcracks or inner cavities. Using the test-particle method, we quantitatively estimate the free path distribution function, layer permeability, and effective kinetic characteristics of sublimation products passing through a nonisothermal porous layer. In addition, in this approach, we consider the volumetric absorption of visible solar radiation in the near-surface absorbing layer. Simple approximation expressions are obtained for all the transport characteristics under study, which makes it possible to estimate the characteristics with sufficient accuracy for practical applications in the physics of comets. The results will be used to construct new consistent models of energy transfer in the near-surface layer of a comet nucleus and, first of all, to analyze the results of the observations of comet 67P/Churyumov‒Gerasimenko.

Addressing nonlinear transient diffusion in porous media through transformations

The nonlinear differential equation describing flow of a constant compressibility liquid in a porous medium is examined in terms of the Kirchhoff and Cole-Hopf transformations. A quantitative measure of the applicability of representing flow by a slightly compressible liquid – which leads to a linear differential equation, the Theis equation – is identified. The classical Theis problem and the finite-well-radius problem in a system that is infinite in its areal extent are used as prototypes to address concepts discussed. This choice is dictated by the ubiquity of solutions that depend on these archetypal examples for examining transient diffusion. Notwithstanding that the Kirchhoff and Cole-Hopf transformations arrive at a linear differential equation, for the specific purposes of this work – the estimation of the hydraulic properties of rocks, the Kirchhoff transformation is much more advantageous in a number of ways; these are documented. Insights into the structure of the nonlinear solution are provided. The results of this work should prove useful in many contexts of mathematical physics though developed in the framework of applications pertaining to the earth sciences.

Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion

&lt;p style='text-indent:20px;'&gt;We are interested in the existence and stability of traveling waves of arbitrary amplitudes to a chemotaxis model with porous medium diffusion. We first make a complete classification of traveling waves under specific relations among the biological parameters. Then we show all these traveling waves are asymptotically stable under appropriate perturbations. The proof is based on a Cole-Hopf transformation and the energy method.&lt;/p&gt;

Prediction of effective diffusivity of porous media using deep learning method based on sample structure information self-amplification

Effective diffusivity is one of the basic transport coefficients used to describe the mass transport capability of a porous medium. In this study, a deep learning method based on a convolutional neural network (CNN) with sample structure information self-amplification is proposed to predict the effective diffusivity of a porous medium, which is considerably influenced by the morphological and topological parameters of the porous medium. In this method, the geometric structures of three-dimensional (3D) porous media are reproduced via a stochastic reconstruction method. Datasets of the effective diffusivities of the reconstructed porous media were first established by the pore-scale lattice Boltzmann method (LBM) simulation. A large number of geometric structures of 3D porous media are obtained using the proposed sample structure information self-amplification approach. The 3D geometric structure information and corresponding effective diffusivities are directionally applied to a CNN for training and prediction. The effective diffusivities of media with porosities ranging from 0.48 to 0.58 are employed as training datasets, and the effective diffusivities of media with a broader porosity range of 0.39 to 0.79 are predicted by CNN. The CNN model can achieve a fast and accurate prediction of the effective diffusivity. The relative error between the CNN and LBM is 0.026%–8.95% with porosities ranging from 0.39 to 0.79. For a typical case with a porosity of 0.5, the computation time required by the CNN model is only 3 × 10−4 h, while the computation time for the same case is 16.96 h using the LBM. These findings indicate that the proposed deep learning method has a powerful learning ability; it is time-saving, provides accurate predictions, and can serve as a promising and powerful tool to predict the transport coefficients of complex porous media.

Existence of global weak solutions for a two-dimensional Keller-Segel-Navier-Stokes system with porous medium diffusion and rotational flux

This article concerns a two-dimensional Keller-Segel-Navier-Stokes system with porous medium diffusion and rotational flux describing the coral fertilization. Based on the Gagliardo-Nerenberg inequality and an energy-type argument, we show that, in the context of the nonlinear diffusions of sperm and eggs with index m&gt;1 and l&gt;0, the corresponding initial-boundary value problem possesses at least one global bounded weak solution. For more information: https://ejde.math.txstate.edu/

A hybrid model for the combined impact of non-Darcy flow, stimulated matrix permeability, and anomalous diffusion flow in the unconventional reservoirs

Abstract A hybrid model for unconventional gas reservoirs that couples three different parameters is presented in this paper. The first is the anomalous diffusion in a fractal porous media. The second is the stimulated or induced matrix permeability in the stimulated reservoir volume (SRV). The third is the non-Darcy flow permeability in the hydraulic fractures. This model is generated from the multi-linear flow model for fractal reservoirs controlled by diffusive flow mechanism with adjustment for fluid flux through hydraulic fracture face considering minimum fracture relative permeability (kmr) and non-Darcy flow Number (FND). Pressure distributions, flow regimes, and reservoir performances have been investigated for three types of unconventional reservoirs. The first is formations with homogenous matrix permeability where petrophysical properties of stimulated and un-stimulated reservoir volume are the same. The second is fractal reservoirs with different petrophysical properties in the two volumes without considering normal or classic diffusion mechanism. The third is the fractal reservoirs where anomalous diffusive flow mechanism dominates fluid flow. A set of comparisons has been generated between the three types for better understand the impact of non-Darcy flow permeability and stimulated matrix permeability on reservoir performance under normal and anomalous diffusion flow mechanisms. The outcomes of this study are: (1) Generating a new analytical model that describes pressure distribution in fractal unconventional reservoirs and couples non-Darcy flow, stimulated or induced matrix permeability with the anomalous diffusion in porous media. (2) Understanding the impact of these different parameters on reservoir performance. (3) Developing different models for all types of flow regimes that are expected to be observed during the entire production life. (4) Comparing the productivity index of reservoirs: having homogeneous matrix permeability, fractal with normal diffusion, and fractal with anomalous diffusion. The most interesting points in this study are: (1) Minimum fracture relative permeability (kmr) significantly enhances the productivity index by eliminating the impact of non-Darcy flow while non-Darcy flow number (FND) works conversely (2) Increasing stimulated matrix permeability enhances reservoir performance (3) The impact of non-Darcy flow permeability of hydraulic fractures is seen at early and intermediate production time where transient flow is dominant. (4) Two different trends are recognized for the impact of anomalous diffusion flow: the first is a positive impact on the transient flow period and the second is negative in the pseudo-steady state flow.

Extraction of density-layered fluid from a porous medium

We consider axisymmetric flow towards a point sink from a stratified fluid in a vertically confined aquifer. We present two approaches to solve the equations of flow for the linear density gradient case. Firstly, a series method results in an eigenfunction expansion in Whittaker functions. The second method is a simple finite difference method. Comparison of the two methods verifies the finite difference method is accurate, so that more complicated nonlinear, density stratification can be considered. Such nonlinear profiles cannot be considered with the eigenfunction approach. Interesting results for the case where the density stratification changes from linear to almost two-layer are presented, showing that in the nonlinear case there are certain values of flow rate for which a steady solution does not occur.