A fast algorithm for the estimation of the equivalent hydraulic conductivity of heterogeneous media

Abstract

Abstract. Fast upscaling of hydraulic conductivity is a recurrent problem in modeling flow through heterogeneous porous media. We propose a new renormalization technique. It is based on the iterative application of the Cardwell and Parsons [1945] bounds on elementary groups of cells. The combination of the bounds with a heuristic formula allows anisotropy to be taken into account. The new technique is tested and compared with other fast techniques. Among the tested techniques the two most reliable ones are the tensorial renormalization and the new simplified renormalization. The numerical efficiency of the simplified renormalization leads us to recommend it when a diagonal tensor of equivalent conductivity is sufficient. IntroductionModeling groundwater flow and transport requires a realistic description of the spatial distribution of hydraulic conductivity to capture flow paths and to make realistic forecasts of groundwater behavior. As discussed by de Marsily et al. [1998] and Koltermann and Gorelick [1996], statistical and genetic tools can be used to describe the underground distribution of hydraulic conductivity. Geological models generated by these tools often produce a very high spatial resolution which cannot be used directly in groundwater flow and transport models given currently available computer resources. It is necessary to adopt a much coarser description. This upscaling comprises the calculation of spatial averages of the hydraulic conductivity over blocks of the geological model that can be used directly in the groundwater model. This problem is different from the problem of finding a unique effective conductivity for the whole aquifer. Renard and de Marsily [1997] and Wen and G6mez-Herngndez [1996] present extensive reviews of upscaling theories, analytical results, and numerical techniques.In this paper, we develop a fast real-space renormalization algorithm to calculate block conductivities. The essence of renormalization is to apply composition rules to groups of cells at the local scale to produce "composite" cells and then to successively reapply the same rules to the resulting composite cells until the desired coarse-scale resolution is achieved. The basic assumption is that the upscaling rules are scale invariant. When the probability distribution of the local-scale conductivity is known, it is possible to follow the evolution of the prob-

Is high-resolution inverse characterization of heterogeneous river bed hydraulic conductivities needed and possible?

“…For scenario D also the initial K fields of the different ensembles were made uncertain and K was jointly updated together with h and L. The initial K fields were the same for all four ensembles and were generated by adding perturbation fields to the calibrated K field of the reference runs. These perturbation fields were generated by SGS with the code GCOSIM3D (Gómez-Hernández and Journel, 1993) on a very fine grid (1 m × 1 m × 1 cm) and then upscaled to the model grid through simplified renormalization (Renard et al, 2000). The geostatistical parameters for SGS (nugget: 0 log 10 (m s −1 ); sill: 0.584 log 10 (m 2 s −2 ); range in horizontal direction: 99 m; range in vertical direction: 3.2 m) were derived from real-world measurements of hydraulic conductivities for the Hardhof site (approx.…”

Efficiency of a two-step upscaling method for permeability evaluation at Darcy and pore scales

“…Renormalization consists in obtaining a macro-scale equivalent permeability by a series of fine-mesh permeabilities aggregations [18]. Improvements have been made to treat any anisotropy in the permeability fields [19]. The renormalization method reduces computational costs but it does not give a direct realization of the flow path, it does not handle high contrasts between neighboring permeabilities, and it is not directly applicable to real pore-space geometry.…”

A modeling study of heterogeneity and surface water-groundwater interactions in the Thomas Brook catchment, Annapolis Valley (Nova Scotia, Canada)

“…Sophisticated upscaling, homogenization, renormalization, and other techniques are useful in reconstructing geological complexity from available data, and numerous theoretical sensitivity studies on simplified or idealized systems have been conducted to investigate the importance of heterogeneity on flow and transport processes (e.g., Sykes et al, 1985;Kabala and Milly, 1990;Sánchez-Vila et al, 1995;Renard et al, 2000;Cushman et al, 2002). For the real catchment studied in this work, a more straightforward approach was pursued, starting with a very simple representation of the Thomas Brook catchment and then gradually increasing the level of geological complexity in the model, based on field measurements, maps, and other databases.…”