Porous Medium Equation Research Articles

On state‐constrained porous‐media systems with gradient‐type multiplicative noise

AbstractThe aim of the present paper is to provide necessary and sufficient conditions to maintain a stochastic coupled system with porous media components and gradient‐type noise in a prescribed set of constraints by using internal controls. This work is a complementary contribution to the results obtained by the same authors, also on the viability problem associated to the porous media equation, but with Lipschitz noise. Second, the present paper provides a different framework in which the quasi‐tangency condition can be obtained with optimal speed. In comparison with the aforementioned result, and from a technical point of view, here, we transform the stochastic system into a random‐PDE one, via the rescaling approach, and then we study the viability of random sets. As an application, (stronger) conditions for the stabilization of the stochastic porous media equations are obtained. These are illustrated on a simple example.

The fractional porous medium equation on manifolds with conical singularities II

AbstractThis is the second of a series of two papers that studies the fractional porous medium equation, with and , posed on a Riemannian manifold with isolated conical singularities. The first aim of the article is to derive some useful properties for the Mellin–Sobolev spaces including the Rellich–Kondrachov theorem and Sobolev–Poincaré, Nash and Super Poincaré type inequalities. The second part of the article is devoted to the study the Markovian extensions of the conical Laplacian operator and its fractional powers. Then based on the obtained results, we establish existence and uniqueness of a global strong solution for initial data and all . We further investigate a number of properties of the solutions, including comparison principle, contraction and conservation of mass. Our approach is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.

The porous medium equation as a singular limit of the thin film Muskat problem

The singular limit of the thin film Muskat problem is performed when the density (and possibly the viscosity) of the lighter fluid vanishes and the porous medium equation is identified as the limit problem. In particular, the height of the denser fluid is shown to converge towards the solution to the porous medium equation and an explicit rate for this convergence is provided in space dimension d ⩽ 4. Moreover, the limit of the height of the lighter fluid is determined in a certain regime and is given by the corresponding initial condition.

Physics-informed machine learning for solving partial differential equations in porous media

Physical phenomenon in nature is generally simulated by partial differential equations. Among different sorts of partial differential equations, the problem of two-phase flow in porous media has been paid intense attention. As a promising direction, physics-informed neural networks shed new light on the solution of partial differential equations. However, current physics-informed neural networks’ ability to learn partial differential equations relies on adding artificial diffusion or using prior knowledge to increase the number of training points along the shock trajectory, or adaptive activation functions. To address these issues, this study proposes a physics-informed neural network with long short-term memory and attention mechanism, an ingenious method to solve the Buckley-Leverett partial differential equations representing two-phase flow in porous media. The designed network structure overcomes the dependency on artificial diffusion terms and enhances the importance of shallow features. The experimental results show that the proposed method is in good agreement with analytical solutions. Accurate approximations are shown even when encountering shock points in saturated fields of porous media. Furthermore, experiments show our innovative method outperforms existing traditional physics-informed machine learning approaches. Cited as: Shan, L., Liu, C., Liu, Y., Tu, Y., Dong, L., Hei, X. Physics-informed machine learning for solving partial differential equations in porous media. Advances in Geo-Energy Research, 2023, 8(1): 37-44. https://doi.org/10.46690/ager.2023.04.04

Aronson–Bénilan estimates for weighted porous medium equations under the geometric flow

In this paper, we study Aronson–Bénilan gradient estimates for positive solutions of weighted porous medium equations coupled with the geometric flow on a complete measure space . As an application, by integrating the gradient estimates, we derive the corresponding Harnack inequalities.

Numerical analysis of fractional heat transfer and porous media equations within Caputo-Fabrizio operator

<abstract><p>This paper presents a comparative study of two popular analytical methods, namely the Homotopy Perturbation Transform Method (HPTM) and the Adomian Decomposition Transform Method (ADTM), to solve two important fractional partial differential equations, namely the fractional heat transfer and porous media equations. The HPTM uses a perturbation approach to construct an approximate solution, while the ADTM decomposes the solution into a series of functions using the Adomian polynomials. The results obtained by the HPTM and ADTM are compared with the exact solutions, and the performance of both methods is evaluated in terms of accuracy and convergence rate. The numerical results show that both methods are efficient in solving the fractional heat transfer and porous media equations, and the HPTM exhibits slightly better accuracy and convergence rate than the ADTM. Overall, the study provides a valuable insight into the application of the HPTM and ADTM in solving fractional differential equations and highlights their potential for solving complex mathematical models in physics and engineering.</p></abstract>

Numerical Approximation to Porous Medium Equation Using a Quarter-Sweep Based Finite Difference and Explicit Four-Points Group

In this study, numerical approximation is used to tackle the one-dimensional nonlinear porous media problem. The emphasis is on using a quarter-sweep finite difference technique to apply the approximation. The quarter-sweep computation approach is highly helpful in terms of reducing complexity while solving big and sparse linear equations. This is done in order to produce a finite difference scheme that works well for the porous medium equation. Internal calculations are carried out using an explicit four-points group methodology, and the resulting nonlinear system is solved using the Newton method. To demonstrate the efficiency of the suggested finite difference approximation and the explicit four-points group technique together, several examples involving the porous medium equation are described. Analyzing the maximum number enables an assessment of the computational time and iteration efficiency of the numerical approach. The method's accuracy is further evaluated by looking at the largest absolute errors the grid points can produce. The suggested method is compared with a half-sweep finite difference method that uses the Newton explicit group method and a conventional quarter-sweep finite difference approximation.

Meshfree numerical approach for some time-space dependent order partial differential equations in porous media

<abstract><p>In this article, the meshfree radial basis function method based on the Gaussian function is proposed for some time-space dependent fractional order partial differential equation (PDE) models. These PDE models have significant applications in chemical engineering and physical science. Some main advantages of the proposed method are that it is easy to implement, and the output response is quick and highly accurate, especially in the higher dimension. In this method, the time-dependent derivative terms are treated by Caputo fractional derivative while space-dependent derivative terms are treated by Riesz, Riemann-Liouville, and Grünwald-Letnikov derivatives. The proposed method is tested on some numerical examples and the accuracy is analyzed by $ \|L\|_\infty $.</p></abstract>

Analytical and computational results for boundary-layer equations in porous medium

The aim of this paper is to determine analytic solutions to the steady/unsteady 2-D flows of a Newtonian fluid passing through a porous media. Using the stream function, the analytical general solutions of the non-linear equations corresponding to boundary-layer flows are determined by employing the extended approach of variable separation. Some specific flow problems, described by given initial and boundary conditions, are investigated using the obtained general analytic solutions.

Porous medium equation with Elzaki transform homotopy perturbation

We have shown how to solve 1-D fourth order parabolic linear PDE with varable coefficients in this article. We have applied the Elzaki transform homotopy pertur?bation method. It is used in conjunction with the homotopy perturbation approach in this method. To further showcase the Elzaki transform-homotopy perturbation technique?s competence and reliability, we have provided solution the parabolic linear PDE. It was found that both HPM and Laplace decomposition method were similar when compared to resulting analytical solutions. Results suggest that the Elzaki transform in conjunction with the homotopy perturbation technique is capa?ble and practicable for use in these cases.

Cauchy-Dirichlet problems for the porous medium equation

We consider the porous medium equation subject to zero-Dirichlet conditions on a variety of two-dimensional domains, namely strips, slender domains and sectors, allowing us to capture a number of different classes of behaviours. Our focus is on intermediate-asymptotic descriptions, derived by formal arguments and validated against numerical computations. While our emphasis is on non-negative solutions to the slow-diffusion case, we also derive a number of results for sign-change solutions and for fast diffusion. Self-similar solutions of various kinds play a central role, alongside the identification of suitable conserved quantities. The characterisation of domains exhibiting infinite-time hole closure is a particular upshot and we highlight a number of open problems.

Global existence and blow-up of solutions to the porous medium equation with reaction and singular coefficients

We study global in time existence versus blow-up in finite time of solutions to the Cauchy problem for the porous medium equation with a variable density $ \rho(x) $ and a power-like reaction term posed in the one dimensional interval $ (-R,R) $, $ R>0 $. Here the weight function is singular at the boundary of the domain $ (-R,R) $, indeed it is such that $ \rho(x)\sim (R-|x|)^{-q} $ as $ |x|\to R $, with $ q\ge0 $. We show a different behavior of solutions depending on the three cases when $ q>2 $, $ q = 2 $ and $ q<2 $.

The porous medium equation with large data on Cartan-Hadamard manifolds under general curvature bounds

<p style='text-indent:20px;'>We consider very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds, that are assumed to satisfy general curvature bounds and to be stochastically complete. We identify a class of initial data that can grow at infinity at a prescribed rate, which depends on the assumed curvature bounds through an integral function, such that the corresponding solution exists at least on <inline-formula><tex-math id="M1">\begin{document}$ [0,T] $\end{document}</tex-math></inline-formula> for a suitable <inline-formula><tex-math id="M2">\begin{document}$ T>0 $\end{document}</tex-math></inline-formula>. The maximal existence time <inline-formula><tex-math id="M3">\begin{document}$ T $\end{document}</tex-math></inline-formula> is estimated in terms of a suitable weighted norm of the initial datum. Our results are sharp, in the sense that slower growth rates yield global existence, whereas one can construct data with critical growth for which the corresponding solutions blow up in finite time. Under further assumptions, uniqueness of very weak solutions is also proved, in the same growth class.</p>

Wasserstein model reduction approach for parametrized flow problems in porous media

The aim of this work is to build a reduced order model for parametrized porous media equations. The main challenge of this type of problems is that the Kolmogorov width of the solution manifold typically decays quite slowly and thus makes usual linear model order reduction methods inappropriate. In this work, we investigate an adaptation of the methodology proposed in [Ehrlacher et al., Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces, ESAIM: Mathematical Modelling and Numerical Analysis (2020)], based on the use of Wasserstein barycenters [Agueh & Carlier, Barycenters in the Wasserstein Space, SIAM Journal on Mathematical Analysis (2011)], to the case of non-conservative problems. Numerical examples in one-dimensional test cases illustrate the advantages and limitations of this approach and suggest further research directions that we intend to explore in the future.

Homogenization of nonlinear nonlocal diffusion equation with periodic and stationary structure

<abstract><p>This paper is devoted to the homogenization of a class of nonlinear nonlocal parabolic equations with time dependent coefficients in a periodic and stationary structure. In the first part, we consider the homogenization problem with a periodic structure. Inspired by the idea of Akagi and Oka for local nonlinear homogenization, by a change of unknown function, we transform the nonlinear nonlocal term in space into a linear nonlocal scaled diffusive term, while the corresponding linear time derivative term becomes a nonlinear one. By constructing some corrector functions, for different time scales $ r $ and the nonlinear parameter $ p $, we obtain that the limit equation is a local nonlinear diffusion equation with coefficients depending on $ r $ and $ p $. In addition, we also consider the homogenization of the nonlocal porous medium equation with non negative initial values and get similar homogenization results. In the second part, we consider the previous problem in a stationary environment and get some similar homogenization results. The novelty of this paper is two folds. First, for the determination equation with a periodic structure, our study complements the results in literature for $ r = 2 $ and $ p = 1 $. Second, we consider the corresponding equation with a stationary structure.</p></abstract>

Complexity of asymptotic behavior of solutions for the fractional porous medium equation

In this paper, we study the large time behavior of solutions to the fractional porous medium equation $ u_{t} = \nabla\cdot (u\nabla^{\alpha-1}u) $ in $ \mathbb{R}^{N} $ with $ 0<\alpha<2 $. More precisely, we reveal that for any given $ 0<\mu<\frac{2N}{N+\alpha} $ and $ \beta>\frac{2-\mu}{2\alpha} $, there exists an initial-value $ u_{0}(x) $ such that the complexity of asymptotic behavior for the rescaled solutions $ t^{\frac{\mu}{2}}u(t^{\beta}\cdot,t) $ occurs in $ C_{0}^{+}(\mathbb{R}^{N}) $. For this purpose, we apply the $ L^{1} $-$ L^{\infty} $ smoothing effect and establish the propagation estimates for the solutions.

Small Scale Formations in the Incompressible Porous Media Equation

We construct examples of solutions to the incompressible porous media (IPM) equation that must exhibit infinite in time growth of derivatives provided they remain smooth. As an application, this allows us to obtain nonlinear instability for a class of stratified steady states of IPM.

Exact analytical solution of fluid flow equation in heterogeneous porous media

Exact analytical solution of fluid flow equation in heterogeneous porous media

Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities

We establish boundedness estimates for solutions of generalized porous medium equations of the form∂tu+(−L)[um]=0in RN×(0,T), where m≥1 and −L is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, Lévy operators. Our quantitative estimates take the form of precise L1–L∞-smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of −L and I−L.In the linear case m=1, it is well-known that the L1–L∞-smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques.We establish a similar scenario in the nonlinear setting m>1. First, we can show that operators for which ultracontractivity holds, also provide L1–L∞-smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by 0-order Lévy operators like −L=I−J⁎. They do not regularize when m=1, but we show that surprisingly enough they do so when m>1, due to the convex nonlinearity. This reveals a striking property of nonlinear equations: the nonlinearity allows for better regularizing properties, almost independently of the linear operator.Finally, we show that smoothing effects, both linear and nonlinear, imply families of inequalities of Gagliardo-Nirenberg-Sobolev type, and we explore equivalences both in the linear and nonlinear settings through the application of the Moser iteration.

Calculation of effective properties of geocomposites based on computed tomography images

Biot’s parameter is included in the formula for calculating effective stresses and should be taken into account when assessing the stress-strain state of a water-saturated rock mass. A method for calculating Biot’s tensor parameter based on asymptotic averaging of the equilibrium equation for a fluid-saturated porous medium is proposed. Calculations of elastic properties and Biot’s coefficient were carried out on various types of rocks - limestone, dolomite, hyaloclastite, basalt. The calculations were carried out using 3D models of geocomposites built from X-ray computed tomography images. The results of 3D calculations of Young's modulus and Biot’s coefficient coincided with the results of experimental determinations of these properties by the ultrasonic method. This fact shows the expediency of using a computational approach that implements asymptotic averaging to estimate the effective properties using 3D models of the real rock structure. The results of 3D and 2D simulation of effective properties of geocomposites are compared. 2D models were built using photographs of rock sections. It was found that the values of Young's modulus and Biot’s parameter for 2D models differ from the corresponding experimental and 3D calculation results by 20-30 %. Therefore, 2D modeling is not suitable for evaluating the effective properties of porous geomaterials. Based on the results of calculations and experiments, the dependences of Young's modulus and Biot’s coefficient on porosity were obtained and studied. These dependencies are used in non-linear numerical simulation of the consolidation of water-saturated soils. The results of the study showed that Biot’s coefficient does not depend on Young's modulus of the rock skeleton material. The influence of the pore shape on Young's modulus and Biot’s coefficient was studied using 2D calculations. A method for predicting the shape of pores from the values of porosity and Young's modulus using neural networks is proposed. Using this method, a specific algorithm for predicting the shape of pores in hyaloclastites has been implemented and studied.