A locally conservative Eulerian–Lagrangian numerical method and its application to nonlinear transport in porous media

Eulerian-Lagrangian and Modified Method of Characteristics (MMOC) procedures provide computationally efficient techniques for approximating the solutions of transport-dominated diffusive systems. The original MMOC fails to preserve certain integral identities satisfied by the solution of the differential system; the recently introduced variant, called the MMOCAA, preserves the global form of the identity associated with conservation of mass in petroleum reservoir simulations, but it does not preserve a localized form of this identity. Here, we introduce an Eulerian-Lagrangian method related to these MMOC procedures that guarantees conservation of mass locally for the problem of two-phase, immiscible, incompressible flow in porous media. The computational efficiencies of the older procedures are maintained. Both the original MMOC and the MMOCAA procedures for this problem are derived from a nondivergence form of the saturation equation; the new method is based on the divergence form of the equation. A reasonably extensive set of computational experiments are presented to validate the new method and to show that it produces a more detailed picture of the local behavior in waterflooding a fractally heterogeneous medium. A brief discussion of the application of the new method to miscible flow in porous media is included. This revised version was published online in July 2006 with corrections to the Cover Date.     

Using a shifted Grünwald-Letnikov scheme for the Caputo derivative to study anomalous solute transport in porous medium

We propose an extension of the shifted Grünwald-Letnikov method to solve fractional partial differential equations in the Caputo sense with arbitrary fractional order derivative α and with an advective term. The method uses the relation between Caputo and Riemann-Liouville definitions, the shifted Grünwald-Letnikov, and the traditional backward and forward finite difference method. The stability of the method is investigated for the implicit and explicit scheme with homogeneous boundary conditions, and a stability criterion is found for the advective-dispersive equation. An application of the method is used to solve contaminant diffusion and advective-dispersive problems. The numerical solution for the fractional diffusion and fractional advection-dispersion is compared with their respective analytical solutions for different time and space grid refinements. The diffusion simulation exhibited a good fit between the analytical and numerical solutions, with the explicit scheme going from stable to unstable as the time and space refinement changes. The fractional advection-dispersion application produced small deviations from the analytical solution. These deviations, however, are analogous to the numerical dispersions encountered in conventional finite difference solutions of the advection-dispersion equation. The new method is also compared with the traditional L2 method. Notably, an example that involves asymmetrical fractional conditions, a fractional diffusivity that depends on time, and a source term show how the methods compare. Overall, this study assesses the quality and easiness of use of the numerical method.     

Compositional dual mesh method for single phase flow in heterogeneous porous media—application to CO<Subscript>2</Subscript> storage

Numerical simulation of fluid flow coupled with chemical reactions has been an active field in the hydrogeology community and many formulations have been programmed into different software. In recent years, this subject has attracted increasing interest in the reservoir simulation community, partly for the application of chemical methods for hydrocarbon extraction but also for research on the geological sequestration of CO₂. In this paper, an extension to the concept of dual mesh for reactive transport modeling is presented. This approach involves two meshes, a low-resolution mesh to resolve the pressure equation and a high-resolution mesh to transport the species and to calculate the geochemical equilibrium. The main objective is to preserve the fine scale heterogeneities to reach a more accurate field behavior simulation than conventional approach which consist in performing simulations on a coarser mesh. The method is applied to a simulation of CO₂ storage in the SPE10 model that keep a high resolution of the heterogeneities.     

Development of a Darcy-Brinkman model to simulate water flow and tracer transport in a heterogeneous karstic aquifer (Val d’Orléans, France)

Darcy’s law is the equation of reference widely used to model aquifer flows. However, its use to model karstic aquifers functioning with large pores is problematic. The physics occurring within the karstic conduits requires the use of a more representative macroscopic equation. A hydrodynamic model is presented which is adapted to the karstic aquifer of the Val d’Orléans (France) using two flow equations: (1) Darcy’s law, used to describe water flow within the massive limestone, and (2) the Brinkman equation, used to model water flow within the conduits. The flow equations coupled with the transport equation allow the prediction of the karst transfer properties. The model was tested by using six dye tracer tests and compared to a model that uses Darcy’s law to describe the flow in karstic conduits. The simulations show that the conduit permeability ranges from 5?×?10^?6 to 5.5?×?10^?5?m² and the limestone permeability ranges from 8?×?10^?11 to 6?×?10^?10?m². The dispersivity coefficient ranges from 23 to 53 m in the conduits and from 1 to 5 m in the limestone. The results of the simulations carried out using Darcy’s law in the conduits show that the dispersion towards the fractures is underestimated.     

The motion of deformable ellipsoids in power-law viscous materials: Formulation and numerical implementation of a micromechanical approach applicable to flow partitioning and heterogeneous deformation in Earth’s lithosphere

Earth’s lithosphere is heterogeneous in rheology on a wide range of observation scales. When subjected to a tectonic deformation, the incurred flow field can vary significantly from one rheologically distinct element to another and the flow field in an individual element is generally different from the bulk averaged flow field. Kinematic and mechanical models for high-strain zones provide the relations between prescribed tectonic boundary conditions and the resulting bulk flow field. They do not determine how structures and fabrics observed on local and small scales form. To bridge the scale gap between the bulk flow field and minor structures, Eshelby’s formalism extended for general power-law viscous materials is shown to be a powerful means. This paper first gives a complete presentation of Eshelby’s formalism, from the classic elastic inclusion problem, to Newtonian viscous materials, and to the most general case of a power-law viscous inhomogeneity embedded in a general power-law viscous medium. The formulation is then implemented numerically. The implications and potential applications of the approach are discussed. It is concluded that the general Eshelby formalism together with the self-consistent method is a powerful and physically sound means to tackle large plastic deformation of Earth’s lithosphere.     

A compound-specific n-alkane δC and δD approach for assessing source and delivery processes of terrestrial organic matter within a forested watershed in northern Japan

We measured molecular distributions and compound-specific hydrogen (δD) and stable carbon isotopic ratios (δ¹³C) of mid- and long-chain n-alkanes in forest soils, wetland peats and lake sediments within the Dorokawa watershed, Hokkaido, Japan, to better understand sources and processes associate with delivery of terrestrial organic matter into the lake sediments. δ¹³C values of odd carbon numbered C₂₃-C₃₃n-alkanes ranged from −37.2‰ to −31.5‰, while δD values of these alkanes showed a large degree of variability that ranged from −244‰ to −180‰. Molecular distributions in combination with stable carbon isotopic compositions indicate a large contribution of C3 trees as the main source of n-alkanes in forested soils whereas n-alkanes in wetland soil are exclusively derived from marsh grass and/or moss. We found that the n-alkane δD values are much higher in forest soils than wetland peat. The higher δD values in forest samples could be explained by the enrichment of deuterium in leaf and soil waters due to increased evapotranspiration in the forest or differences in physiology of source plants between wetland and forest. A δ¹³C vs. δD diagram of n-alkanes among forest, wetland and lake samples showed that C₂₅-C₃₁n-alkanes deposited in lake sediments are mainly derived from tree leaves due to the preferential transport of the forest soil organic matter over the wetland or an increased contribution of atmospheric input of tree leaf wax in the offshore sites. This study demonstrates that compound-specific δD analysis provides a useful approach for better understanding source and transport of terrestrial biomarkers in a C3 plant-dominated catchment.     

Diverse mode of phosphatic rocks in the environs of Proterozoic Bijawar and Sonrai basins and its relevance to uranium mineralisation — A case study from central India

The Bundelkhand massif comprising a variety of Archean-Paleoproterozoic granitoids along with low grade and high-grade metamorphites and located in the centre of the Indian Plate, underwent extension during Paleoproterozoic period, resulting in the formation of homotaxial intracratonic Bijawar and Sonrai basins in the south and Gwalior basin in the northern margin. The Bijawar and Sonrai basins are typified by their characteristic sediments and basic volcanic rocks. A feature common to both the basins, is the overwhelming occurrence of phosphatic rocks across stratigraphy and lithotype in the Bijawar basin and its confinement to the basal part of the sedimentary column in Sonrai basin. Most of these rocks are primarily of marine origin, and later subjected to periods of repeated phosphatic redistribution. Multiple episodes of such phosphatisation culminates in the proliferation and enrichment of phosphate in the upper Bijawar rocks of Bijawar basin (phosphatic breccia of Hirapur-Mardeora) and lower Bijawar rocks of Sonrai basin (phosphatic breccia of Lalitpur). Apart from these established phosphatic rocks in both the basins, quartz reefs occurring in the basement as well as the lower Bijawar Malhera Chert Breccia Formation in Bijawar basin at places are endowed with anomalously high phosphate content. The phosphatic component in all the lithotypes is in the form of apatite varying in form from microcrystalline to well formed coarser crystal aggregate comprising cement, veins and botroidal encrustations. Irrespective of its spatial, temporal and paragenetic position, it invariably registers weak to moderate radioactivity, due to the presence of uranium within it, as is evident from microprobe data. Although intra-grain and inter-grain distribution of uranium is found to be random and erratic, in general, it is observed that uranium tends to be enriched in the later generation phosphates, due to secondary process of dissolution and reprecipitation. The present paper, with fresh inputs from petrological, geochemical, minerochemical and isotope data pertaining to apatite from all these diverse units, not only explores the already established association of uranium and phosphate in these basins but also provides new insight to the phosphatic quartz reef within the basement and the phosphatised arenaceous sediments of the lower Bijawar Formation.     

Using impacts of deep-level mining to research karst hydrology—a Darcy-based approach to predict the future of dried-up dolomitic springs in the Far West Rand goldfield (South Africa). Part 1: a conceptual model of recharge and inter-compartmental flow

Some of the world’s deepest goldmines are located in the Far West Rand (FWR) goldfield operating below of up to 1.2-km-thick dolomites hosting some of the largest karst aquifers in South Africa. Associated impacts include the dewatering of the overlying karst aquifers as well as linking previously disconnected compartments by mining through aquicludes (dykes). The focus of the study is on predicting groundwater balances in re-watered aquifers after mining ceases as this will determine whether or not associated karst springs that dried-up due to dewatering will ever flow again. Critically revisiting, Swart et al. (Environ Geol 44:751–770, 2003a) who predict that all springs will flow again, this study uses significantly larger data sets and modified assumptions to increase the robustness of findings as the question is crucial for post-closure development. As a first of two papers, this part develops a conceptual model on the mega-compartment concept that predicts a flat water table across all linked compartments that would leave the springs dry. The model identifies the ratio between inflowing surface water (recharge) and underground water losses to downstream compartments via mined-through dykes (‘inter-compartmental groundwater flow’, IGF) as a key factor governing the elevation of the post-mining water table, creating the base for part 2, where the IGF and the post-mining water tables are determined using unique large data sets that have not been evaluated before.     

Using impacts of deep-level mining to research karst hydrology—a Darcy-based approach to predict the future of dried-up dolomitic springs in the Far West Rand goldfield (South Africa). Part 2: predicting inter-compartmental flow and final groundwater tables

Some of the world’s deepest goldmines operate below dolomitic karst aquifers in the Far West Rand (FWR) goldfield, South Africa. Associated impacts include the continuous dewatering of karst aquifers for over six decades and irreversible changes of the hydrogeological setting. Affecting an area of approximately 400 km² by drawing down the water table up to 700 m, these impacts, and the large amounts of data generated in the process, are used as unique research opportunities to better understand the complex karst hydrology. The focus of this study is on predicting final water table elevations in rewatered aquifers after mining ceases taking the fact that mines hydraulically linked previously disconnected aquifers into account. While part 1 of this series develops the conceptual model, this second part utilises large sets of pertinent data to calculate actual flow rates for predicting the fate of dried up springs after mine closure. Following a Darcy-based approach first applied by Swart et al. (Environ Geol 44:751–770, 2003a) it is not only predicted that the springs will flow again but also shown that linear relationships exist between flow rates through a combined system of karst-fractured aquifers overlying the mine void and the associated hydraulic head driving them. This suggests that—at this scale—porous media-based equations can be meaningfully used to predict flow in non-porous media.