Exact PDF equations and closure approximations for advective-reactive transport

Mathematical models of advection–reaction phenomena rely on advective flow velocity and (bio) chemical reaction rates that are notoriously random. By using functional integral methods, we derive exact evolution equations for the probability density function (PDF) of the state variables of the advection–reaction system in the presence of random transport velocity and random reaction rates with rather arbitrary distributions. These PDF equations are solved analytically for transport with deterministic flow velocity and a linear reaction rate represented mathematically by a heterog eneous and strongly-correlated random field. Our analytical solution is then used to investigate the accuracy and robustness of the recently proposed large-eddy diffusivity (LED) closure approximation [1]. We find that the solution to the LED-based PDF equation, which is exact for uncorrelated reaction rates, is accurate even in the presence of strong correlations and it provides an upper bound of predictive uncertainty. 2013 Elsevier Inc. All rights reserved. 1. Introductio n The mathematical modeling of realistic advection–reaction phenomena involve many random parameters. For example, the dynamics of solute transport in an heterogeneous porous medium is significantly influenced by the epistemic uncertainty in the porosity distribution , advective (Darcy) flow velocity and (bio) chemical reaction rates [2]. These random fields render the governing equations of the advection–reaction system stochastic. A common approach to quantifyi ng the statistical propertie s of the state variables of the system, e.g., the solute concentratio n, relies in modeling their probability density functions (PDFs) directly via determinist ic equations. This has advantag es over other uncertainty quantification methods, such as polynomial chaos [3–6], probabilistic collocation [7,8], perturba tion methods [35–38], and generalized spectral decompositi ons [9–13]. In particular , PDF methods allow to directly ascertain the tails of probabilistic distribut ions thus facilitating the assessment of rare events and associated risks. This is in contrast to many other stochasti c approach es that use the variance (or the standard deviation) of a system variable as a measure of predictive uncertainty . Another key advantage of PDF methods is that they do not suffer from the curse of dimensionality. Moreover, they can be used to tackle several open problems in stochasti c dynamics such as discontinui ties in parametric space [6] and long-term integration [14,15]. 324 D. Venturi et al. / Journal of Computational Physics 243 (2013) 323–343 As is well known, in general it is not possible to determine an exact one-point (in space and time) PDF equation for the solution to an arbitrary partial different ial equation. This is due to the fact that the solution to the PDE can be nonlocal in space and time. Howeve r, a functional differential equation for the Hopf characte ristic function al can always be determined [17–19], although effective analytic al or numerical methods to solve it are still lacking. Moreover, the Hopf functional equation is equivalent to an infinite hierarchy of equations involving multi-poi nt PDFs of increasing order [20–23]. First-order stochastic partial different ial equations (SPDEs), such as the advection–reaction equation, are often amenable to exact treatment with PDF methods [1,24–26]. The main reason is that these equations can be reduced to a finite set of ordinary differential equations along the characterist ic curves. This enables one to obtain an exact evolution equation for the joint PDF of the state variables and the external random fields appearing in the SPDE. In general, these random fields are high-dimens ional, e.g., represented in terms of many random variables in a Karhunen –Loève expansion, and therefore the corresponding exact equation for the joint PDF is high-dim ensional as well [27,26]. From a computational viewpoint this could be an issue despite the recent advances in numerical methods for high-dimens ional problems such as proper generalized decompositi on [28–30], sparse grid collocation [31,7] or functional ANOVA techniques [32–34]. A closure approximat ion can significantly reduce the number of parameters appearing in the PDF equation and therefore it can provide an effective computational tool that allows for an efficient integration. In particular, the large-eddy diffusivity (LED) closure for advection–reaction equations [1] has been shown to be effective for uncorrelated and weakly correlated random reaction rates. However, the performance of the LED-based PDF equation s for strongly correlated reaction rates remains unexplored. Its investigation is the main objective of the present study. To this end, we consider the prototype problem proposed in [1] and obtain analytical solutions to the PDF equation for two different random reaction models. The analytical solutions will be employed as useful benchma rk to test the accuracy and effectivenes s of the LED closure. This paper is organized as follows. In Section 2 we formulat e the governing equation s of advectivereactive transport in porous media and obtain an evolution law for the correspondi ng indicator function ([39, Ch. 3] ). Section 2.2 presents the LED-based PDF equation for random advection–reaction transport. The exact evolution equation for the joint responseexcitation PDF of the advection–reaction system is derived and discussed in Section 2.3. In Section 3 and Section 4 we compare the LED approximat ion with exact analytica l results for a prototype advection–reaction problem involving linear reactions and strongly correlated random reaction rates. Finally, the main findings and their implication s are summarized in Section 5. We also include two brief appendices, where we obtain analytica l solutions to the equation for the joint responseexcitation PDF and the advection–reaction equation in physical space. 2. Problem formulation Let us consider the dimensio nless form of the advection–reaction equation for a scalar concentr ation field cðx; tÞ