Paolo Salandin

Simulation of dispersion in heterogeneous porous formations: Statistics, first‐order theories, convergence of computations

This paper discusses the results of numerical analysis of dispersion of passive solutes in two-dimensional heterogeneous porous formations. Statistics of flow and transport variables, the accuracy and the role of approximations implicit in existing first-order theories, and the convergence of computational results are investigated. The results suggest that quite different rates of convergence with Monte Carlo runs hold for different spatial moments and that over 1000 realizations are required to stabilize second moments even for relatively mild heterogeneity (sigma(Y)2 < 1.6). This has implications for the extent of the spatial domain for single-realization numerical studies of the same type. A comparison of the variance of plumes with the results of linear theories (0.05 < sigma(Y)2 < 1.6) shows an unexpectedly broad validity field for the theoretical solution obtained from a suitable linearization of flow and transport. Reformulation of the same problem linearizing in tum the flow or the transport equations shows opposite deviations from the linear theory. The interesting consequence is that the errors induced by linearizations in the flow or the transport equations have different signs, and their effects on the moments of dispersing plumes are compensating, thereby yielding consistent formulations. Unexpected features of the statistics of probability distributions of longitudinal and transverse velocities and travel times are also computed and discussed.

SOLUTE TRANSPORT IN HIGHLY HETEROGENEOUS AQUIFERS

The paper deals with the transport of nonreactive solute in heterogeneous formations with prescribed statistical properties of the hydraulic log conductivity Y=ln K. Available solutions obtained from perturbation methods are limited to first- and second-order solutions, valid only for small values of the log conductivity variance σ2Y. Published numerical investigations give comparative results with finite values of σ2Y, but some discrepancies among the results generate doubts about the capability of numerical methods to capture high-order effects for media with large variance values. When these large σ2Y values are encountered in natural formations, the related nonlinear effects could be significant in the velocity statistics and in the overall dispersion process. The nonlinearity consequences are here investigated in two-dimensional isotropic porous media by the Monte Carlo technique coupled with a finite element analysis. The analysis includes σ2Y values from 0.05 to 4 enhancing the relevance of the nonlinear effects in the dispersion tensor solution. To dissipate the doubts related to the numerical approach, the accuracy of the solutions was defined by checking the influence of the factors which can affect the solution and by giving an estimation of the related errors. The numerical results confirm the validity of the first-order and second-order analyses when σ2Y → 0; the second-order solution captures the nonlinear effects in a small range of log conductivity variance close to zero. The nonlinear terms neglected in the first-order formulation for higher σ2Y values give (1) late travel time longitudinal dispersion values greater than the linear solution, (2) notably non-Gaussian distribution of the Lagrangian velocity and particle displacements, and (3) travel times to approach the asymptotic Fickian regime longer than those obtained using the linear solution.

A Note on Transport in Stratified Formations by Flow Tilted With Respect to the Bedding

The tendency to ergodicity of transport through heterogeneous stratified formations by a flow tilted with respect to the bedding is examined in this note. The idealized model of evenly stratified formations resembles recharge areas in naturally layered sedimentary geological structures over short distances, and transport features of more complex heterogeneous structures. Two cases are considered herein: the ergodic limit and the nonergodic regime. In the former case the theory predicts a constant asymptotic value of longitudinal dispersivity controlled by the log transmissivity integral scale. In the latter case, asymptotic results of an analytic nature are derived for the limit case of large travel times. Monte Carlo simulations are performed to study the plume evolutions for a wide range of heterogeneities and of initial size of the solute body transverse to the bedding. Results are compared with the analytical solution. It is concluded that, in the realistic case of finite initial transverse size of the plumes, ergodicity is not obeyed. Ergodic conditions, in our experiments, were not achieved even for a solute body whose dimension was 400 times the log transmissivity correlation scale. In such cases, theoretical and numerical evidence suggests that in nonergodic regimes the longitudinal dispersion coefficient tends asymptotically to zero for any initial size of the solute body.

Coupled and uncoupled hydrogeophysical inversions using ensemble Kalman filter assimilation of ERT-monitored tracer test data

Recent advances in geophysical methods have been increasingly exploited as inverse modeling tools in groundwater hydrology. In particular, several attempts to constrain the hydrogeophysical inverse problem to reduce inversion errors have been made using time-lapse geophysical measurements through both coupled and uncoupled (also known as sequential) inversion approaches. Despite the appeal and popularity of coupled inversion approaches, their superiority over uncoupled methods has not been proved conclusively; the goal of this work is to provide an objective comparison between the two approaches within a specific inversion modeling framework based on the ensemble Kalman filter (EnKF). Using EnKF and a model of Lagrangian transport, we compare the performance of a fully coupled and uncoupled inversion method for the reconstruction of heterogeneous saturated hydraulic conductivity fields through the assimilation of ERT-monitored tracer test data. The two inversion approaches are tested in a number of different scenarios, including isotropic and anisotropic synthetic aquifers, where we change the geostatistical parameters used to generate the prior ensemble of hydraulic conductivity fields. Our results show that the coupled approach outperforms the uncoupled when the prior statistics are close to the ones used to generate the true field. Otherwise, the coupled approach is heavily affected by “filter inbreeding” (an undesired effect of variance underestimation typical of EnKF), while the uncoupled approach is more robust, being able to correct biased prior information, thanks to its capability of capturing the solute travel times even in presence of inversion artifacts such as the violation of mass balance. Furthermore, the coupled approach is more computationally intensive than the uncoupled, due to the much larger number of forward runs required by the electrical model. Overall, we conclude that the relative merit of the coupled versus the uncoupled approach cannot be assumed a priori and should be assessed case by case.

Dispersion tensor evaluation in heterogeneous media for finite Peclet values.

In natural formations the transport process at the local scale is characterized by the spatial heterogeneity of hydraulic conductivity and by the pore-scale dispersion. Usually, theoretical investigations consider only the effect due to the spatial variation of hydraulic conductivity, because the first prevails over the latter by some order of magnitude. Nevertheless, the pore-scale dispersion has a noteworthy impact on the concentration variance evaluation and on the ergodicity condition, so that its influence on the overall dispersion processes may be relevant. The note illustrates a theoretical procedure that allows one to define the global dispersion tensor at the local scale by taking into account the pore-scale effects as evaluated in laboratory columns. To reach this goal, the pore-scale and the local-scale velocity fluctuations are coupled via a rigorous analytical procedure in the three-dimensional (3-D) domain, and it is demonstrated that the porous media heterogeneity modifies the pore-scale dispersion effects as measured in laboratory columns. The impact of the hydraulic conductivity heterogeneity on the pore-scale dispersion at the local scale is due to (1) the path line sinuosity and (2) the module of the local velocity variation. A quantitative evaluation of these effects is made according to a first-order analysis, i.e., assuming that the spatial fluctuations of the hydraulic conductivity are small, and, in the 2-D case, a comparison with nonlinear results is made also. The results demonstrate that the only path line sinuosity coupled with the pore-scale anisotropy enhances the transversal mixing and reduces the longitudinal one, but the global heterogeneity effect, considering also the velocity module fluctuations, gives a generalized increase of the local dispersion tensor components.

High density distributed strain sensing of landslide in large scale physical model

This paper describes the application of a commercial distributed optical fiber sensing system to a large scale physical model of landslide. An optical fiber cable, deployed inside the landslide body, is interrogated by means of optical frequency domain reflectometry with very high spatial density. A shallow landslide is triggered in the physical model by artificial rainfall and the evolution of the strain is measured up to the slope failure. Precursory signs of failure are detected well before the collapse, providing insights to the failure dynamic.

COPING WITH UNCERTAINTY IN THE RELIABILITY EVALUATION OF WATER DISTRIBUTION SYSTEMS

The application of the reliability analysis techniques to the water distribution systems aims to properly achieve in probabilistic terms the fulfilment of the nodal demand in a conceptual and physical context subject to different causes of uncertainties. The availability of system components, like pumps or pipes subjected to failure, the uncertainty in the spatial - temporal behaviour of nodal demand, the pipe roughness, the reservoir level as well as the availability of supply resources contribute to the definition of the system reliability. Nevertheless the complexity of a reliability model increases with the number of different uncertainties, and usually, for each specific case, only few parameters are assumed as random while the remaining are fixed as deterministic. This fact always implies more or less limitations in the system description also when deterministic assumptions are based on reasonable hypotheses. To overcome this shortcoming, a mixed analytical - numerical approach, able to take into account both the mechanical failure of system components and the random spatial - temporal distribution of nodal demands, is here developed. The goal is achieved by joining a Monte Carlo numerical approach based on the head-driven simulation with a First Order Second Moment (FOSM) closed form solution (Xu and Goulter, 1998). The Monte Carlo approach lets to analyze the series of partial shutdowns related to the renewal process of pipes and the time evolution of the demand, while the spatial variability of the latter and the uncertainty in the pipe roughness are described by the FOSM approach. An illustrative example developed in the well-known case of the Anytown network, demonstrates that the uncertainty related to the spatial nodal demand and pipe roughness may have a relevant impact on the reliability evaluation of a distribution network whose pipes are subject to the mechanical failure.

Losses identification in water distribution networks through EnKF and ES

This study proposes a method for the identification of the spatial distribution of water losses in water distribution networks through the use of pressure head measurements. The proper identification of areas most prone to water losses reduces the costs associated with acoustic surveys both in terms of number of pipes to be examined and working time. To get the best estimate of the water losses spatial distribution, data assimilation techniques based on the Kalman Filter approach (Ensemble Kalman Filter and Ensemble Smoother) are coupled with the hydraulic network model (EPANET). The coupled model performances are investigated on the Anytown benchmark system with both a known and an unknown consumption pattern. A method to identify the most effective network monitoring locations is also proposed. Despite the fact that the method is tested on a single synthetic network, the results suggest that the tool is promising for water losses identification.

Electrical Resistivity Tomography Time-lapse Monitoring of Three-dimensional Synthetic Tracer Test Experiments

Time-lapse electrical resistivity tomography (ERT) represents a powerful tool for subsurface solute transport characterization since a full picture of the spatio-temporal evolution of the process can be obtained. However, the quantitative interpretation of tracer tests is difficult because of the uncertainty related to the geo-electrical inversion and the constitutive models linking geophysical and hydrological quantities. Here a new approach based on the Lagrangian formulation of transport and the ensemble Kalman filter (EnKF) data assimilation technique is applied to assess the spatial distribution of hydraulic conductivity K by incorporating time-lapse cross-hole ERT data. Under the assumption that the solute spreads as a passive tracer, for high Peclet numbers the spatial moments of the evolving plume are dominated by the spatial distribution of the hydraulic conductivity. The assimilation of the electrical conductivity 4D images allows updating of the hydrological state as well as the spatial distribution of K. Thus, delineation of the tracer plume and estimation of the local aquifer heterogeneity can be achieved at the same time by means of this interpretation of time-lapse electrical images from tracer tests.

Stochastic Analysis Of Biodegradable Solutes In Natural Aquifers

The procedure and some results about transport of biodegradable solutes in heterogeneous formations are discussed. Following a Lagrangian framework the biodegradation process along the single streamline is numerically solved by the method of the characteristics, where the heterogeneity is taken in account by the particle displacement statistics concerning the passive solute. This technique gives relevant advantages, since the space-time evolution of reactive solutes can be deduced on the knowledge of the passive solute displacement statistics with a limited computational effort. Moreover the numerical procedure allows us to define the validity of the biodegradable solute transport solutions in heterogeneous porous media based on the linear analysis.