This lecture will present stochastic PDE's (Partial Differential Equations) to model various “transport” phenomena like water flow, solute transport and wave propagation, in heterogeneous geologic porous media. The material properties are represented by random functions of space F(x) (random fields). The resulting transport PDE’s contain random field coefficients, and their solutions are stochastic (randomly heterogeneous).

In this framework, the objectives are to solve the stochastic PDE’s, to analyze the behavior of the solutions (fluxes, velocities, pressures, stresses and strains), and to propose an upscaled version of the governing equations with effective or equivalent “macro-scale” coefficients that incorporate the effects of heterogeneity (e.g., anisotropic macro-permeability tensor Kij).  

The methods used for obtaining the random solutions involve the so-called “sigma-expansion” method, where “sigma” stands for the standard deviation of the random parameters… or of their logarithms (it will be shown that this choice can make quite a difference regarding the robustness of the solution). The perturbation equations are then solved with Green's functions in space, and/or with spectral representation in the space of wave-vectors k (Fourier-Wiener-Khinchin representation with Stieltjes integrals).

Once the random fluctuations or their moments are known, the next task is to implement “relevant” averaging operations, and to study the behavior of averaged macro-scale quantities (mean flux, mean pressure gradient, mean displacement, etc.). Macro-scale governing PDE’s can emerge from this upscaling step, e.g., one obtains a mass conservation equation div(Q) = 0 combined with a macro-scale Darcy equation Q =  K Grad(P) where macro-permeability K is a 2nd rank tensor embodying the geometric anisotropy of the geologic medium. However, due to nonlinear or stochastic interactions, the macro-scale PDE does not necessarily resemble the original “local scale” stochastic PDE…as will be shown.

Through this lecture, several types of phenomena of practical importance will be used as examples: hydrodynamic dispersion of a tracer; single phase flow (Darcy); two-phase flow (Darcy-Muskat); seismic elastic wave dispersion and attenuation....

KEY-WORDS: Stochastic PDE; Random fields; Random media; Geologic porous media; Perturbation methods; Fourier-Wiener-Khinchin; Homogenization; Upscaling; Darcy-Muskat; Elastic waves.

Keywords : Stochastic PDE • first-order hyperbolic equation • Itô integral • multiplicative noise • finite volume method • monotone scheme • Godunov scheme • Young measures • Kruzhkov smooth entropy.

We are interested in the Cauchy problem for a nonlinear hyperbolic scalar conservation law in d space dimensions with a multiplicative stochastic perturbation of type: 

(1):    du + div[v(x,t)f(u)] dt = g(u) dW in Ω×Rd×(0,T),

u(ω,x,0) = u0(x), ω ∈ Ω, x ∈ Rd,

where div is the divergence operator with respect to the space variable, d is a positive integer, T > 0 and W = {Wt,Ft;0 ≤ t ≤ T} is a standard adapted one-dimensional continuous Brownian motion defined on the classical Wiener space (Ω,F,P). I will present in this talk the discretization of Problem (1) by monotone finite volume schemes.

Firstly, I will introduce the well-posedness theory for solutions of such kind of stochastic problems. Then, the main part of the talk will be devoted to the study of the monotone numerical scheme used to approximate the solution of (1). I will show that under a stability condition on the time step, one is able to show the convergence of the finite volume approximation towards the unique stochastic entropy solution of the equation.

A large number of physical phenomenon could be mathematically described with the help of hyperbolic conservation laws. The inherent complexities make it only natural to able to account for possible random- ness/noise in the descriptions. In this talk, we will try to make case for Lévy (jump-diffusion) type of noise and contend that it brings us a step closer to reality. We will describe some of the recent advances related existence, uniqueness, stability for stochastic conservation laws that are driven by Lévy noise.

The presentation will discuss a model for viscoelastic flows where PDEs for the complex fluid flow are coupled with SDEs for the microstructure.

Precisely, a system coupling the Navier-Stokes equations and stochastic dumbbells is studied numerically after simplification in an asymptotic "hydrostatic" regime with free-surface boundary conditions.

This talk will start with the presentation of a conductance based neuron model, describing the generation and propagation of a nerve impulse along a neural cell. Such a model consists in a partial differential equation whose parameters have some markovian behavior. Often, a natural deterministic limit may be derived and adding the corresponding fluctuations, leads to a model consisting in a stochastic partial differential equation.

One of the main questions in considering such models is the reliability of signal transmission or propagation failure. We will tackle this question by different means, mainly numerically.

We formulate a criterion on initial data which guarantees that solutions to stochastic porous medium equations with linear multiplicative noise exhibit a waiting time phenomenon almost surely. Up to a logarithmic factor, it coincides with the optimal criterion known from the deterministic setting. A novel iteration technique and stochastic counterparts of weighted integral estimates used in the deterministic setting are  key ingredients of our approach, which may be modified to prove basic results on finite speed of propagation, too.

Some numerical experiments on the size of waiting times will be presented as well.

This is joint work with Julian Fischer (Leipzig) and Hubertus Grillmeier (Erlangen).

We examine new numerical methods to approximate SPDEs and discuss both convergence and efficiency. We are particularly interested in the time discretisation of multiplicative noise. Our techniques are primarily based on approximating the mild solution of the SPDE where we can try and exploit exact solutions in the numerics. Proofs of convergence are for globally Lipschitz nonlinearities. In the case where this condition does not hold, rather than tamed methods, we examine instead using an adaptive timestep for the SPDE. We take as applications SPDEs aising from models of neural and also from models of reactive single phase flow in a porous media.

In this talk, we analyze a semi-discrete finite volume scheme for a stochastic balance laws driven by multiplicative Lévy noise. Using BV estimates of approximate solutions, generated by finite volume scheme, we show that approximate solutions converges to the unique BV entropy solution of the underlying problem.

Moreover, we show that expected value of the L1-difference between approximate solution and the unique entropy solution converges at rate O(∆x1/2), where ∆x being a spatial mesh size.

Stochastic methods are frequently used to model flow in porous media: At small-scales through Lattice Boltzmann methods of probabilistic inspiration, and also through methods of random walks used to estimate laboratory-scale dispersivity properties or porous medium conductivity, knowing its microstructure provided for example by micro tomo X. On a larger scale, homogenization methods have been introduced, both to up-scale fractured media or heterogeneous porous media or subsurface reservoirs by periodic homogenization or by perturbation development of PDE solutions of the problem and a suitable averaging procedure. Finally, calibration methods, parameter estimation using data and quantifying uncertainties also use this type of concepts.

The presentation will focus on reviewing some practical applications of stochastic approaches and some open problems will be discussed.

The numerical solution of flow problems in geophysical applications provide  typical situations  that require techniques for uncertainty quantification.  For  low-parametric problems  polynomial chaos expansion (PCE) techniques are an appropriate tool.   The approach leads to  deterministic systems for the stochastic moments. However,  the systems can become intricate with increasing complexity of the underlying system. This fact increases in turn  the computational effort of PCE and significantly reduces the scalability in parallelization. We present a hybrid and adaptive stochastic Galerkin method which extends the classical polynomial chaos expansion by a multi-element discretization in the probability space of the parameters. It leads to a deterministic system that is coupled to a lesser degree than in element-free PCE versions, respectively, fully decoupled in stochastic elements. Therefore, the HSG-FV method allows for more efficient computation. We present numerical examples in two spatial dimensions  for a range of nonlinear flow problems including sedimenting flows and fluid  flow in heterogeneous porous media. Finally we consider  mixed-type two-phase porous media flow which combines hyperbolic transport equations  with an elliptic equation for the pressure.

The convergence of a Finite Volume method for a stochastic scalar conservation law with a multiplicative noise has been obtained by Bauzet, Charrier, Gallouët. With slightly different assumptions, one proves the same result by using the kinetic formulation of the Finite Volume scheme.

This is a conjoint work with Sylvain Dotti.