H14B-01 INVITED 16:00h
Characterization of retention processes in crystalline rock
Tracer transport in randomly fractured porous media results from a combination of random advection and a variety of exchange (retention) processes. A generalized dual porosity model is presented for this coupling, which explicitly accounts for transitional and exchange disorder, and a formal solution is provided; the solution is applicable to both sparsley and densely fractured rocks. Illustration examples are focused on sparsley fractured rocks that are highly heterogeneous and are of prime interest in environmental applications. We study the relative effect of advection and retention heterogeneity on tracer first passage times through a rock volume, comparing our generalized with the classical dual porosity model. The classical model generally underestimates retention significantly. A simple measure and criteria is established for assessing the relative effect of advection and retention heterogeneity. It is found that for tracers with a wide range of retention and decay properties, transport is controlled by retention heterogeneity, with advection heterogeneity being of secondary importance. This finding can have important practical implications for predicting large-scale, long-term tracer transport in fractured geological media.
H14B-02 INVITED 16:15h
Robust Coarse Scale Modeling of Flow and Transport in Subsurface Formations
Subsurface flows are affected by permeability heterogeneity over a range of length scales. It is difficult to fully resolve all of the scales that impact flow and transport through such systems, so models for subgrid effects are often required. In this work, new methods for the coarse scale modeling of flow (i.e., total flow rate for specified wellbore pressures) and transport (movement of injected fluid) in highly heterogeneous deterministic systems are described and applied. The technique for the improved upscaling of flow, referred to as adaptive local-global upscaling, efficiently provides coarse scale transmissibilities (transmissibility is the numerical analog of permeability) that are adapted to a specific flow scenario. Global coarse scale simulations are used to determine the boundary conditions for the local calculation of upscaled properties and a thresholding procedure is introduced for efficiency and to minimize the appearance of spurious transmissibility values. A specialized near-well upscaling is included naturally in the procedure and transmissibilities can be efficiently recomputed when new wells are introduced. The method is applied to both 2D and 3D systems and its improved accuracy over existing procedures is demonstrated. The adaptive local-global transmissibility upscaling is then combined with models for transport. Two approaches are considered for the transport problem. First, a multiscale (dual-grid) technique is presented, in which the fine scale velocity, reconstructed from the coarse scale pressure solution, is used for the solution of advective contaminant transport on the fine scale. This approach provides excellent accuracy but it requires that two grids be used for the simulation. A new generalized convection-diffusion model for transport (so named because of the form of the subgrid terms), in which both flow and transport (single-phase or two-phase) are solved on the coarse grid, is described next. The method is shown to provide significantly improved results compared to those attainable with standard procedures. It is also shown to be well-suited to difficult problems in which well rates or locations change in time.
H14B-03 16:30h
Scaling up a Conceptual Model of Matrix Diffusion from Laboratory to Field
For the movement of solutes in fractured rocks, it remains challenging to scale up laboratory experiment results and conceptual models to field scale studies. Heterogeneities in the field are often multiscale and more complex than those captured in undisturbed soil and rock columns. The objective of this research is to scale up a conceptual model derived from laboratory soil column experiments to a field tracer release study. The study site is located at Oak Ridge National Laboratory in eastern Tennessee, USA. For years, undisturbed soil columns obtained near the field site have been used in the laboratory to elucidate the effect of matrix diffusion on the movement of radioactive chemicals through fractured rocks and macroporous soils. Conceptual models thus derived have not been rigorously tested in the field until a recent field tracer release has been completed and the experiment results are analyzed. In particular, it has not been clearly demonstrated that matrix diffusion, a diffusive mass transfer process, is the dominating mechanism of moving solutes between fractures and the rock matrix under natural flow conditions. Previous studies have identified that, under transient flow conditions, advective mass transfer as a result of inter-pore-structure hydraulic gradients may also contribute to the movement of solutes between the fracture and rock matrix. Our results suggest that, at the field scale, matrix diffusion indeed dominates the movement of solutes into the rock matrix within the area of developed fracture networks that constitute a preferred flow path. However, it is likely that movement of solutes into bedrocks surrounding the preferred flow path is mainly a result of advection and dispersion through the microfractures in the bedrocks.
H14B-04 16:45h
Correct Characterization of Passive Tracer Dispersion in Porous Columns: Experiments vs. Theory
Breakthrough curves (BTC) of a passive tracer in macroscopically ``homogeneous'' granular materials (well-sorted, unconsolidated sands or glass beads) were measured in a series of column experiments. % In parallel, classical experiments on dispersion of a passive tracer in fully and partially saturated porous columns were re-examined. % All of these BTCs exhibit anomalous (non-Fickian) features: early and late arrival times are observed to differ systematically from theoretical predictions based on solution of the advective-dispersion equation (ADE) for uniform porous media. % We propose that even in these small-scale, ``homogeneous'' porous medium columns, subtle and residual pore-scale disorder effects can account for these observations. % In a Continuous Time Random Walk (CTRW) framework, we determined an ensemble-averaged distribution of particle transfer rates (based on a Master Equation for the local flux-averaged concentration) which accounts for these effects. % Solutions of the resulting CTRW transport equations yield BTCs that are in excellent agreement with the entire series of observations. % The CTRW formulation also specifies the dependence of the effective macroscopic parameters on measurable quantities. % The theoretical predictions are in excellent agreement with the observations. % It is critical to understand that as a consequence of our results, the ADE should not be taken as the starting point of any upscaling technique. % Our analyses demonstrate that existing measurements and interpretations of tracer dispersion experiments in laboratory experiments should be carefully re-considered in the framework of these recent advances in conceptual understanding and quantification. % These results have also important implications for modeling the transport of contaminants in large-scale, highly-heterogeneous, hydrogeological systems.
H14B-05 17:00h
Effective Conductivity of 2D Isotropic Formations: Performance of the Effective Medium Approximation
The effective conductivity of isotropic formations is investigated using a combination of the effective medium approximation and accurate numerical simulations. The 2D isotropic heterogeneous medium contains circular inclusions of uniform radii and of two different conductivities, K1 and K2. The inclusions are submerged into a homogeneous matrix. The matrix conductivity was set to an effective conductivity computed using the effective medium approximation. The matrix represents inclusions of the same conductivities, but of much smaller radii. The conglomerate of inclusions is placed in a circular domain and subject to uniform flow from infinity. The average velocity inside the conglomerate differs from that at infinity. The difference in the velocities is used to infer the actual effective conductivity. For our binary medium, the latter depends on three parameters: the conductivity ratio K1/K2, the total area fraction n of the inclusions and the relative area fraction n1. For n1=0.5 (equal numbers of inclusions of conductivity K1 and K2), the classical Matheron-Dykhne exact result was recovered; the actual effective conductivity equals the geometric mean of inclusions' conductivities. For an asymmetrical distribution (n1 different from 0.5) the effective medium approximation is shown to be quite accurate even for relatively large values of the total area fraction or the log-conductivity variance; the actual effective conductivity and that computed using the effective medium approximation are very close. The cases of a wide conductivity distribution and different radii are briefly discussed.
H14B-06 17:15h
An Analytical Form for the Effective Permeability Tensor
Upscaling soil hydraulic properties is one of the most challenging problems in Mathematics and Geophysics. Its application runs from Petroleum Engineering, Hydrology and, Soil Science among many others. The idea is to rewrite a partial differential equation describing a physical process in a homogenized or effective form, which takes into account the spatial heterogeneity. We derive an appropriate analytical form, by using the method of homogenization and two-scale asymptotic expansion based on our recently proposed, analytical solution for a sub-problem. Up to now, finding the effective tensor by this method, even though very accurate mathematically, it was very computational demanding. This analytical form provides a huge step towards understanding and modeling multiscale processes. We present numerical simulations for different types of permeability fields including random media and demonstrate that separable and layered media are particular cases.
H14B-07 17:30h
Binary 3-D Markov Chain Random Fields: Finite-size Scaling Analysis of Percolation Properties
Percolation phenomena in random media have been extensively studied in a wide variety of fields in physics, chemistry, engineering, bio-, earth-, and environmental sciences. Most work has focused on uncorrelated random fields. The critical behavior in media with short-range correlations is thought to be identical to that in uncorrelated systems. However, the percolation threshold, pc, which is 0.3116 in uncorrelated media, has been observed to vary with the correlation scale and also with the random field type. Here, we present percolation properties and finite-size scaling effects in three-dimensional binary cubic lattices represented by correlated Markov-chain random fields and compare them to those in sequential Gaussian and sequential indicator random fields. We find that the computed percolation threshold in correlated random fields is significantly lower than in the uncorrelated lattice and decreases with increasing correlation scale. The rate of decrease rapidly flattens out for correlation lengths larger than 2-3 grid-blocks. At correlation scales of 5-6 grid blocks, pc is found to be 0.126 for the Markov chain random fields and slightly higher for sequential Gaussian and indicator random fields. The universal scaling constants for mean cluster size, backbone fraction, and connectivity are found to be consistent with results on uncorrelated lattices. For numerical studies, it is critical to understand finite-size effects on the percolation and associated phase connectivity properties of lattices. We present detailed statistical results on the percolation properties in finite sized lattice and their dependence on correlation scale. We show that appropriate grid resolution and choice of simulation boundaries is critical to properly simulate correlated natural geologic systems, which may display significant finite-size effects.