Stochastic Moment Equations Research Articles

Stochastic kinetic theory applied to coarse-grained polymer model.

A stochastic field theory approach is applied to a coarse-grained polymer model that will enable studies of polymer behavior under non-equilibrium conditions. This article is focused on the validation of the new model in comparison with explicit Langevin equation simulations under conditions with analytical solutions. The polymers are modeled as Hookean dumbbells in one dimension, without including hydrodynamic interactions and polymer-polymer interactions. Stochastic moment equations are derived from full field theory. The accuracy of the field theory and moment equations is quantified using autocorrelation functions. The full field theory is only accurate for a large number of polymers due to keeping track of rare occurrences of polymers with a large stretch. The moment equations do not have this error because they do not explicitly track these configurations. The accuracy of both methods depends on the spatial degree of discretization. The timescale of decorrelation over length scales bigger than the spatial discretization is accurate, while there is an error over the scale of single mesh points.

Data assimilation with multiple types of observation boreholes via the ensemble Kalman filter embedded within stochastic moment equations

Abstract. We employ an approach based on the ensemble Kalman filter coupled with stochastic moment equations (MEs-EnKF) of groundwater flow to explore the dependence of conductivity estimates on the type of available information about hydraulic heads in a three-dimensional randomly heterogeneous field where convergent flow driven by a pumping well takes place. To this end, we consider three types of observation devices corresponding to (i) multi-node monitoring wells equipped with packers (Type A) and (ii) partially (Type B) and (iii) fully (Type C) screened wells. We ground our analysis on a variety of synthetic test cases associated with various configurations of these observation wells. Moment equations are approximated at second order (in terms of the standard deviation of the natural logarithm, Y, of conductivity) and are solved by an efficient transient numerical scheme proposed in this study. The use of an inflation factor imposed to the observation error covariance matrix is also analyzed to assess the extent at which this can strengthen the ability of the MEs-EnKF to yield appropriate conductivity estimates in the presence of a simplified modeling strategy where flux exchanges between monitoring wells and aquifer are neglected. Our results show that (i) the configuration associated with Type A monitoring wells leads to conductivity estimates with the (overall) best quality, (ii) conductivity estimates anchored on information from Type B and C wells are of similar quality, (iii) inflation of the measurement-error covariance matrix can improve conductivity estimates when a simplified flow model is adopted, and (iv) when compared with the standard Monte Carlo-based EnKF method, the MEs-EnKF can efficiently and accurately estimate conductivity and head fields.

Grid convergence for numerical solutions of stochastic moment equations of groundwater flow

We provide qualitative and quantitative assessment of the results of a grid convergence study in terms of (a) the rate/order of convergence and (b) the grid convergence index, GCI, associated with the numerical solutions of moment equations (MEs) of steady-state groundwater flow. The latter are approximated at second order (in terms of the standard deviation of the natural logarithm, Y, of hydraulic conductivity). We consider (1) the analytical solutions of Riva et al. (Transp Porous Med 45(1):139–193, 2001) for steady-state radial flow in a randomly heterogeneous conductivity field, which we take as references; and (2) the numerical solutions of the MEs satisfied by the (ensemble) mean and (co)variance of hydraulic head and fluxes. Based on 45 numerical grids associated with differing degrees of discretization, we find a supra-linear rate of convergence for the mean and (co)variance of hydraulic head and for the variance of the transverse component of fluxes, the variance of radial fluxes being characterized by a sub-linear convergence rate. Our estimated values of GCI suggest that an accurate computation of mean and (co)variance of head and fluxes requires a space discretization comprising at least 8 grid elements per correlation length of Y, an even finer discretization being required for an accurate representation of the second-order component of mean heads.

EnKF coupled with groundwater flow moment equations applied to Lauswiesen aquifer, Germany

Summary We describe a field application of a new data assimilation method recently proposed by Panzeri et al. (2013, 2014). The method couples a modified Ensemble Kalman Filter (EnKF) algorithm with stochastic moment equations (MEs) governing space–time variations of (theoretical ensemble) mean and covariance values of groundwater flow state variables (hydraulic heads and fluxes). Whereas traditional EnKF entails Monte Carlo (MC) simulations and suffers from inbreeding, our approach obviates both. Synthetic case studies have shown the ME-based approach to be computationally efficient and accurate when compared to MC-based results. Here we use our ME-based method to assimilate drawdown data recorded during cross-hole pumping tests in the heterogeneous alluvial Lauswiesen aquifer near Tubingen, Germany. Our results include an estimate of log transmissivity distribution throughout the aquifer and corresponding measures of estimation error. We validate our calibrated model by using it to predict drawdowns recorded during another pumping test at the site and compare its performance with that of standard MC-based EnKF.

Comparison of Ensemble Kalman Filter groundwater-data assimilation methods based on stochastic moment equations and Monte Carlo simulation

Traditional Ensemble Kalman Filter (EnKF) data assimilation requires computationally intensive Monte Carlo (MC) sampling, which suffers from filter inbreeding unless the number of simulations is large. Recently we proposed an alternative EnKF groundwater-data assimilation method that obviates the need for sampling and is free of inbreeding issues. In our new approach, theoretical ensemble moments are approximated directly by solving a system of corresponding stochastic groundwater flow equations. Like MC-based EnKF, our moment equations (ME) approach allows Bayesian updating of system states and parameters in real-time as new data become available. Here we compare the performances and accuracies of the two approaches on two-dimensional transient groundwater flow toward a well pumping water in a synthetic, randomly heterogeneous confined aquifer subject to prescribed head and flux boundary conditions.

Data assimilation and parameter estimation via ensemble Kalman filter coupled with stochastic moment equations of transient groundwater flow

Key Points Embedding stochastic moment equations of flow in ensemble Kalman filter Avoiding need for Monte Carlo simulations in ensemble Kalman filter Avoiding need for batch inversion of Moment Equations

Exact sensitivity matrix and influence of the number of pilot points in the geostatistical inversion of moment equations of groundwater flow

We present novel equations for the exact sensitivity matrix of the (ensemble) mean hydraulic head under steady state groundwater flow conditions. These equations are embedded in a geostatistical inverse procedure to condition approximations of stochastic moment equations of flow on measured hydraulic conductivities and heads. Our formulation allows considerable improvement of the methodology proposed by Hernandez et al. (2003, 2006) and renders the inversion of moment equations feasible for a large number of unknown hydraulic parameters. The spatial distribution of the natural logarithm, Y, of conductivity is parameterized within the pilot points framework. Whereas prior values of Y at pilot points are obtained by a variant of kriging, posterior estimates at pilot points are obtained through a maximum likelihood fit of computed to measured heads. The maximum likelihood function also includes a regularization term. By means of a synthetic example and upon adopting formal model information criteria we explore the influence of (1) the number of pilot points and (2) the order of approximation of the governing mean flow equation on our ability to properly estimate the log conductivity and head fields and identify the relative weight of the regularization term and the parameters of the underlying Y variogram. We find that none of the adopted information criteria can identify the optimum number of pilot points and the plausibility weight and variogram parameters values can be determined by the Kashyap's Bayesian measure.

Inverse analysis of stochastic moment equations for transient flow in randomly heterogeneous media

We present a nonlinear stochastic inverse algorithm that allows conditioning estimates of transient hydraulic heads, fluxes and their associated uncertainty on information about hydraulic conductivity ( K) and hydraulic head ( h) data collected in a randomly heterogeneous confined aquifer. Our algorithm is based on Laplace-transformed recursive finite-element approximations of exact nonlocal first and second conditional stochastic moment equations of transient flow. It makes it possible to estimate jointly spatial variations in natural log-conductivity ( Y = ln K ) , the parameters of its underlying variogram, and the variance–covariance of these estimates. Log-conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete “pilot points”. Whereas prior values of Y at pilot point are obtained by generalized kriging, posterior estimates at pilot points are obtained through a maximum likelihood fit of computed and measured transient heads. These posterior estimates are then projected onto the computational grid by kriging. Optionally, the maximum likelihood function may include a regularization term reflecting prior information about Y. The relative weight assigned to this term is evaluated separately from other model parameters to avoid bias and instability. We illustrate and explore our algorithm by means of a synthetic example involving a pumping well. We find that whereas Y and h can be reproduced quite well with parameters estimated on the basis of zero-order mean flow equations, all model quality criteria identify the second-order results as being superior to zero-order results. Identifying the weight of the regularization term and variogram parameters can be done with much lesser ambiguity based on second- than on zero-order results. A second-order model is required to compute predictive error variances of hydraulic head (and flux) a posteriori. Conditioning the inversion jointly on conductivity and hydraulic head data results in lesser predictive uncertainty than conditioning on conductivity or head data alone.

Laplace-Transform Finite Element Solution of Nonlocal and Localized Stochastic Moment Equations of Transport

Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1238-1255) developed exact first and second nonlocal moment equations for advective-dispersive transport in finite, randomly heterogeneous geologic media. The velocity and concentration in these equations are generally nonstationary due to trends in heterogeneity, conditioning on site data and the influence of forcing terms. Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) solved the Laplace transformed versions of these equations recursively to second order in the standard deviation σY of (natural) log hydraulic conductivity, and iteratively to higher-order, by finite elements followed by numerical inversion of the Laplace transform. They did the same for a space-localized version of the mean transport equation. Here we recount briefly their theory and algorithms; compare the numerical performance of the Laplace-transform finite element scheme with that of a high-accuracy ULTIMATE-QUICKEST algorithm coupled with an alternating split operator approach; and review some computational results due to Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) to shed light on the accuracy and computational efficiency of their recursive and iterative solutions in comparison to conditional Monte Carlo simulations in two spatial dimensions. AMS subject classifications: 65Z05, 65Z05, 68U20, 76S05

Inverse stochastic moment analysis of steady state flow in randomly heterogeneous media

Nonlocal stochastic moment equations have been used successfully to analyze steady state and transient flow in randomly heterogeneous media conditional on measured values of medium properties. We present a nonlinear geostatistical inverse algorithm for steady state flow that makes it possible to further condition such analyses on measured values of state variables. Our approach accounts for all scales of spatial variability resolvable by the computational grid. It is based on recursive finite element approximations of exact nonlocal first and second conditional moment equations. Hydraulic conductivity is parameterized geostatistically on the basis of measured values at discrete locations and unknown values at discrete “pilot points.” Prior estimates of pilot point values are obtained (optionally) by generalized kriging. Posterior estimates at pilot points and (optionally) at measurement points are obtained by calibrating mean flow against measured values of head. The parameters are projected onto a computational grid via kriging. Maximum likelihood calibration allows estimating not only hydraulic but also (optionally) unknown variogram parameters with or without prior information about the former. The approach yields covariance matrices for parameter estimation as well as head and flux prediction errors, the latter being obtained a posteriori. We implement our inverse approach on highly and mildly heterogeneous media under superimposed mean uniform and convergent flows in a bounded two‐dimensional domain. Our examples illustrate how conductivity and head data act separately and jointly to reduce parameter estimation errors and to model predictive uncertainty. We also evaluate the functional form of the log conductivity variogram and its parameters using likelihood‐based model discrimination criteria.

Assessment of uncertainty associated with the estimation of well catchments by moment equations

Non-local stochastic moment equations are used successfully to analyze groundwater flow in randomly heterogeneous media. Here we present a moment equations-based approach to quantify the uncertainty associated with the estimation of well catchments. Our approach is based on the development of a complete second order formalism which allows obtaining the first statistical moments of the trajectories of conservative solute particles advected in a generally non-uniform groundwater flow. Approximate equations of moments of particles’ trajectories are then derived on the basis of a second order expansion in terms of the standard deviation of the aquifer log hydraulic conductivity. Analytical expressions are then obtained for the predictors of locations of mean stagnation points, together with their associated uncertainties. We implement our approach on heterogeneous media in bounded two-dimensional domains, with and without including the effect of conditioning on hydraulic conductivity information. The impact of domain size, boundary conditions, heterogeneity and non-stationarity of hydraulic conductivity on the prediction of a well catchment is explored. The results are compared against Monte Carlo simulations and semi-analytical solutions available in the literature. The methodology is applicable to both infinite and bounded domains and is free of distributional assumptions (and so applies to both Gaussian and non-Gaussian log hydraulic conductivity fields) and formally includes the effect of conditioning on available information.

Conditional Stochastic Moment Equations for Uncertainty Analysis of Flow in Heterogeneous Reservoirs

SummaryWe assume that permeability is a random space function defined by its mean and covariance. The stochastic nature of the permeability description leads to uncertainty in flow-related quantities such as pressure, saturation, and production rate. We extended our statistical moment equation (SME) approach to accommodate conditioning. The conditional statistical moment equations (CSME) framework is a direct approach for quantifying the uncertainty in flow performance caused by uncertainty in the reservoir description. It is quite different from Monte Carlo Simulation (MCS). In MCS, the performance uncertainty is obtained through a statistical post-processing of flow simulations, one for each of a large number of equiprobable realizations of the reservoir description. We developed a CSME computational tool for flow in heterogeneous domains, which we use here to analyze the behavior of the second statistical moment of pressure and velocity in the presence of permeability measurements. We study the effect of both the number and spatial arrangement of available measurements on the computed variances of pressure and velocity. This numerical CSME tool allows us to quantify the value of existing and future information, and that helps in the evaluation of existing projects and in steering future developments. We present several examples that demonstrate how to choose the best sampling locations to obtain maximum reduction in prediction uncertainty. We compare our results with high-resolution MCS.

Perturbation-based moment equation approach for flow in heterogeneous porous media: applicability range and analysis of high-order terms

We present detailed comparisons between high-resolution Monte Carlo simulation (MCS) and low-order numerical solutions of stochastic moment equations (SMEs) for the first and second statistical moments of pressure. The objective is to quantify the difference between the predictions obtained from MCS and SME. Natural formations with high permeability variability and large spatial correlation scales are of special interest for underground resources (e.g. oil and water). Consequently, we focus on such formations. We investigated fields with variance of log-permeability, σ Y 2, from 0.1 to 3.0 and correlation scales (normalized by domain length) of 0.05 to 0.5. In order to avoid issues related to statistical convergence and resolution level, we used 9000 highly resolved realizations of permeability for MCS. We derive exact discrete forms of the statistical moment equations. Formulations based on equations written explicitly in terms of permeability ( K-based) and log-transformed permeability ( Y-based) are considered. The discrete forms are applicable to systems of arbitrary variance and correlation scales. However, equations governing a particular statistical moment depend on higher moments. Thus, while the moment equations are exact, they are not closed. In particular, the discrete form of the second moment of pressure includes two triplet terms that involve log-permeability (or permeability) and pressure. We combined MCS computations with full discrete SME equations to quantify the importance of the various terms that make up the moment equations. We show that second-moment solutions obtained using a low-order Y-based SME formulation are significantly better than those from K-based formulations, especially when σ Y 2>1. As a result, Y-based formulations are preferred. The two triplet terms are complex functions of the variance level and correlation length. The importance (contribution) of these triplet terms increases dramatically as σ Y 2 increases above one. We also show that one of the triplet terms is much more important than the other. When comparing K-based MCS with Y-based SME, model differences must be taken into consideration. These differences (model errors) are due to embedded assumptions and differences in implementing the discrete forms of the equations.