Frederick Delay

Time domain random walk method to simulate transport by advection‐dispersion and matrix diffusion in fracture networks

A method is proposed to calculate in one step the residence time of a particle by advection-dispersion and matrix diffusion in a bond of a fracture network. The calculation is very rapid and avoids the discretization of Eulerian methods or the multiple leaps of classical Lagrangian approaches. The method is accurate in most flow conditions prevailing in fracture networks. Therefore, the method will be useful to evaluate the conditions in which the different transport mechanisms are of influence at the scale of the entire network.

Interpretation of out-diffusion experiments on crystalline rocks using random walk modeling

Matrix diffusion in saturated rocks with very low permeability is one of the major mechanisms of solute transport. Laboratory out-diffusion experiments on rock samples may provide an estimate of the bulk diffusion coefficient. However, numerous results have shown that this average parameter does not really depict the complex mechanism of diffusion as a function of the internal heterogeneity of crystalline rocks. Two-dimensional images of the porosity distribution in a granite sample were obtained by impregnation with a radioactive resin and autoradiography. Some examples based on these images and synthetic images were used to perform numerical simulations of out-diffusion using two different random walk methods. The simulated shapes of the out-diffusion curves depend on the spatial distribution of the porosity and on the pore connectivity with the border of the sample. Such relations might explain the multiple nested slopes or the convex shapes often observed on real experimental curves.

Random Walk Methods for Modeling Hydrodynamic Transport in Porous and Fractured Media from Pore to Reservoir Scale

Random walk (RW) methods are recurring Monte Carlo methods used to model convective and diffusive transport in complex heterogeneous media. Many applications can be found, including fluid mechanic, hydrology and chemical reactors modeling. These methods are easy to implement, very versatile and flexible enough to become appealing for many applications because they generally overlook or deeply simplify the building of explicit complex meshes required by deterministic methods. RW provides a good physical understanding of the interactions between the space scales of heterogeneities and the transport phenomena under consideration. In addition, they can result in efficient upscaling methods, especially in the context of flow and transport in fractured media. In the present study, we review the applications of RW to several situations that cope with diverse spatial scales and different insights into upscaling problems. The advantages and downsides of RW are also discussed, thus providing a few avenues for further works and applications.

An Approach to Transport in Heterogeneous Porous Media Using the Truncated Temporal Moment Equations: Theory and Numerical Validation

In the last decade, the characterization of transport in porous media has benefited largely from numerical advances in applied mathematics and from the increasing power of computers. However, the resolution of a transport problem often remains cumbersome, mostly because of the time-dependence of the equations and the numerical stability constraints imposed by their discretization. To avoid these difficulties, another approach is proposed based on the calculation of the temporal moments of a curve of concentration versus time. The transformation into the Laplace domain of the transport equations makes it possible to develop partial derivative equations for the calculation of complete moments or truncated moments between two finite times, and for any point of a bounded domain. The temporal moment equations are stationary equations, independent of time, and with weaker constraints on their stability and diffusion errors compared to the classical advection–dispersion equation, even with simple discrete numerical schemes. Following the complete theoretical development of these equations, they are compared firstly with analytical solutions for simple cases of transport and secondly with a well-performing transport model for advective–dispersive transport in a heterogeneous medium with rate-limited mass transfer between the free water and an immobile phase. Temporal moment equations have a common parametrization with transport equations in terms of their parameters and their spatial distribution on a grid of discretization. Therefore, they can be used to replace the transport equations and thus accelerate the achievement of studies in which a large number of simulations must be carried out, such as the inverse problem conditioned with transport data or for forecasting pollution hazards.

Empirical Orthogonal Functions Analysis Applied to the Inverse Problem in Hydrogeology: Evaluation of Uncertainty and Simulation of New Solutions

To fulfil the need to generate more realistic solutions, stochastic inverse simulations in hydrogeology are now constrained on both piezometric head and hydraulic conductivity data. These inverse techniques, often based on geostatistics, allow modifications of an initial solution conditioned only on hydraulic conductivity data to arrive at a final solution that also matches observed heads. By repeating the process as many times as necessary with different initial solutions, one generates an ensemble of final solutions thereby addressing the uncertainty of the inverse problem. This requires a method able to handle the whole ensemble and to work on its relevant characteristics. From this standpoint, the analysis by Empirical Orthogonal Functions (EOF) appears promising. The method builds an orthogonal decomposition of the covariance matrix, calculated over the whole set of solutions, and the areas in space where the first functions have a greater influence corresponding to locations of maximum uncertainty in the solutions. These locations depend both on the hydraulic characteristics of the flow problem and on the spatial distribution of available data. The EOF analysis is used on a synthetic problem that mimics a possible behavior of the Culebra aquifer of the Waste Isolation Pilot Plant (WIPP, New Mexico). The method also allows new solutions to be generated at lower computational cost by a random composition of the functions obtained by the EOF analysis. These new solutions keep the main characteristics of the initial ensemble and because they can be conditioned, they return very good results when they are used to solve the direct problem.

A comparison of two Bayesian approaches for uncertainty quantification

Statistical calibration of model parameters conditioned on observations is performed in a Bayesian framework by evaluating the joint posterior probability density function (pdf) of the parameters. The posterior pdf is very often inferred by sampling the parameters with Markov Chain Monte Carlo (MCMC) algorithms. Recently, an alternative technique to calculate the so-called Maximal Conditional Posterior Distribution (MCPD) appeared. This technique infers the individual probability distribution of a given parameter under the condition that the other parameters of the model are optimal. Whereas the MCMC approach samples probable draws of the parameters, the MCPD samples the most probable draws when one of the parameters is set at various prescribed values. In this study, the results of a user-friendly MCMC sampler called DREAM(ZS) and those of the MCPD sampler are compared. The differences between the two approaches are highlighted before running a comparison inferring two analytical distributions with collinearity and multimodality. Then, the performances of both samplers are compared on an artificial multistep outflow experiment from which the soil hydraulic parameters are inferred. The results show that parameter and predictive uncertainties can be accurately assessed with both the MCMC and MCPD approaches.

A dual flowing continuum approach to model denitrification experiments in porous media colonized by biofilms.

We present a modeling exercise of solute transport and biodegradation in a coarse porous medium widely colonized by a biofilm phase. Tracer tests in large laboratory columns using both conservative (fluorescein) and biodegradable (nitrate) solutes are simulated by means of a dual flowing continuum approach. The latter clearly distinguishes concentrations in a flowing porous phase from concentrations conveyed in the biofilm. With this conceptual setting, it becomes possible to simulate the sharp front of concentrations at early times and the flat tail of low concentrations at late times observed on the experimental breakthrough curves. Thanks to the separation of flow in two phases at different velocities, dispersion coefficients in both flowing phases keep reasonable values with some physical meaning. This is not the case with simpler models based on a single continuum (eventually concealing dead-ends), for which inferred dispersivity may reach the unphysical value of twice the size of the columns. We also show that the behavior of the dual flowing continuum is mainly controlled by the relative fractions of flow passing in each phase and the rate of mass transfer between phases. These parameters also condition the efficiency of nitrate degradation, the degradation rate in a well-seeded medium being a weakly sensitive parameter. Even though the concept of dual flowing continuum appears promising for simulating transport in complex porous media, its inversion onto experimental data really benefits from attempts with simpler models providing a rough pre-evaluation of parameters such as porosity and mean fluid velocity in the system.

Assessment of Retention Processes for Transport in a Fractured System at Äspö (Sweden) Granitic Site: from Short‐Time Experiments to Long‐Time Predictive Models

We present results obtained within the Task 6 modeling project hosted by the Swedish Nuclear Fuel and Waste Management Company (SKB) Task Force on Numerical Modeling of Flow and Solute Transport in fractured granitic rock. Modeling the transport of radionuclides in natural fractured media is done to characterize and assess the performance of a potential deep geologic repository. The Task 6 modeling exercise objective provides a bridge between models based on detailed site investigation data and calibrated against tracer experiments (month scale), and models corresponding to a hypothetical repository postclosure time scale (hundred thousands of years). The latter postclosure models capture the most significant features and processes of radionuclide transport, and allow for a sensitivity analysis of uncertain parameters. Two features from the tracer experiments (TRUE-1 and True Block Scale-TBS) are studied. The first is a 10 m single fracture, the second is a 200 m semisynthetic fractured block. The study focuses on matrix zone heterogeneity, its influence on retention processes, and its level of identification from tracer tests. Results show that tracer tests poorly constrain the systems. In addition, we build a model suited to performance assessment for single fracture geometry and present a smeared-fracture approach to model transfers in a fractured block. The model is calibrated and validated on synthetic test cases and applied to the Task 6 fractured block. Results show that this approach is appropriate for performance-assessment time-scale modeling. Nevertheless, the approach requires further development to be applied to systems incorporating small-scale heterogeneities affecting flow and transport conditions.

Characterization of reciprocity gaps from interference tests in fractured media through a dual porosity model

We analyze drawdown reciprocity gaps emerging in interference tests performed in a confined fissured karstic formation. Modeling the system as a dual porosity continuum allows characterizing the dynamics of the relative contribution of the connected fractures and the rock matrix to the total flow rate extracted at the pumping wells. Observed lack of reciprocity of drawdowns can then be linked to the occurrence of processes that are not accounted for in the classical flow models based on a single-continuum representation of the system through flow equations grounded on Darcys law only. We show that interpreting the system as a dual porosity continuum can cause drawdown reciprocity gaps to emerge as a consequence of local effects associated with an identifiable contribution of the matrix to the total fluid extracted at the well location during pumping. These theoretical results are then employed to identify the contribution to the flow being supplied to the pumping well by the low conductivity matrix constituting the host rock formation, in contrast to that provided by the fractures. An application to data from two interference tests performed at the Hydrogeological Experimental Site (HES) in Poitiers, France, illustrates the approach. We show that, whenever the matrix is assumed to provide a contribution to the total flow rate extracted, nonreciprocity is expected, the latter being linked to the occurrence of a differential drawdown between fracture and matrix at the pumping well. This difference decreases with time in the example presented, displaying a power law late time behavior, with nonreciprocity effects persisting up to remarkably long times.

Global sensitivity analysis and Bayesian parameter inference for solute transport in porous media colonized by biofilms

The concept of dual flowing continuum is a promising approach for modeling solute transport in porous media that includes biofilm phases. The highly dispersed transit time distributions often generated by these media are taken into consideration by simply stipulating that advection-dispersion transport occurs through both the porous and the biofilm phases. Both phases are coupled but assigned with contrasting hydrodynamic properties. However, the dual flowing continuum suffers from intrinsic equifinality in the sense that the outlet solute concentration can be the result of several parameter sets of the two flowing phases. To assess the applicability of the dual flowing continuum, we investigate how the model behaves with respect to its parameters. For the purpose of this study, a Global Sensitivity Analysis (GSA) and a Statistical Calibration (SC) of model parameters are performed for two transport scenarios that differ by the strength of interaction between the flowing phases. The GSA is shown to be a valuable tool to understand how the complex system behaves. The results indicate that the rate of mass transfer between the two phases is a key parameter of the model behavior and influences the identifiability of the other parameters. For weak mass exchanges, the output concentration is mainly controlled by the velocity in the porous medium and by the porosity of both flowing phases. In the case of large mass exchanges, the kinetics of this exchange also controls the output concentration. The SC results show that transport with large mass exchange between the flowing phases is more likely affected by equifinality than transport with weak exchange. The SC also indicates that weakly sensitive parameters, such as the dispersion in each phase, can be accurately identified. Removing them from calibration procedures is not recommended because it might result in biased estimations of the highly sensitive parameters.