Michael A. Celia

Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts: 2. Analysis of spatial moments for a nonreactive tracer

A large-scale natural gradient tracer test was conducted to examine the transport of reactive and nonreactive tracers in a sand and gravel aquifer on Cape Cod, Massachusetts. As part of this test the transport of bromide, a nonreactive tracer, was monitored for about 280 m and quantified using spatial moments. The calculated mass of bromide for each sampling date varied between 85% and 105% of the injected mass using an estimated porosity of 0.39, and the center of mass moved at a nearly constant horizontal velocity of 0.42 m per day. A nonlinear change in the bromide longitudinal variance was observed during the first 26 m of travel distance, but afterward the variance followed a linear trend, indicating the longitudinal dispersivity had reached a constant value of 0.96 m. The transverse dispersivities were much smaller; transverse horizontal dispersivity was 1.8 cm, and transverse vertical dispersivity was about 1.5 mm.

An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation

Abstract Many numerical methods use characteristic analysis to accommodate the advective component of transport. Such characteristic methods include Eulerian-Lagrangian methods (ELM), modified method of characteristics (MMOC), and operator splitting methods. A generalization of characteristic methods can be developed using an approach that we refer to as an Eulerian-Lagrangian localized adjoint method (ELLAM). This approach is a space-time extension of the optimal test function (OTF) method. The method provides a consistent formulation by defining test functions as specific solutions of the localized homogeneous adjoint equation. All relevant boundary terms arise naturally in the ELLAM formulation, and a systematic and complete treatment of boundary condition implementation results. This turns out to have significant implications for the calculation of boundary fluxes. An analysis of global mass conservation leads to the final ELLAM approximation, which is shown to possess the conservative property. Numerical calculations demonstrate the behaviour of the method with emphasis on treatment of boundary conditions. Discussion of the method includes ideas on extensions to higher spatial dimensions, reactive transport, and variable coefficient equations.

Large‐scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts: 3. Hydraulic conductivity variability and calculated macrodispersivities

Hydraulic conductivity (K) variability in a sand and gravel aquifer on Cape Cod, Massachusetts, was measured and subsequently used in stochastic transport theories to estimate macrodispersivities. Nearly 1500 K measurements were obtained by borehole flowmeter tests and permeameter analyses of cores. The geometric mean for the flowmeter tests (0.11 cm/s) is similar to that estimated from other field tests. The mean for the permeameter tests (0.035 cm/s) is significantly lower, possibly because of compaction of the cores. The variance for the flowmeter (0.24) is also greater than that for the permeameter (0.14). Geostatistical analyses applying negative exponential models with and without nuggets reveal similar spatial correlation structures for the two data sets. Estimated correlation scales range from 2.9 to 8 m in the horizontal and from 0.18 to 0.38 m in the vertical. Estimates of asymptotic longitudinal dispersivity (b.35–0.78 m) are similar in magnitude to that observed in the natural gradient tracer test (0.96 m) previously conducted at this site.

A functional relationship between capillary pressure, saturation and interfacial area as revealed by a pore-scale network model.

The constitutive relationships required for the parameterization of multiphase flow and transport problems are of critical importance to hydrologic modeling. Recently, a hypothesis has been developed that predicts a functional relationship between capillary pressure, saturation, and interfacial area. A network model was developed to test this hypothesis. Microscale physical processes were simulated and volume averaging was used to derive the macroscopic measures of saturation andfluid-fluid interfacial area per volume of porous media. Results indicate that a smooth, though complex, functional relationship exists at the continuum scale. These results have direct relevance to constitutive theory and the modeling of nonaqueous phase liquid dissolution processes.

Dynamic Effect in the Capillary Pressure–Saturation Relationship and its Impacts on Unsaturated Flow

Capillary pressure plays a central role in the description of water flow in unsaturated soils. While capillarity is ubiquitous in unsaturated analyses, the theoretical basis and practical implications of capillarity in soils remain poorly understood. In most traditional treatments of capillary pressure, it is defined as the difference between pressures of phases, in this case air and water, and is assumed to be a function of saturation. Recent theories have indicated that capillary pressure should be given a more general thermodynamic definition, and its functional dependence should be generalized to include dynamic effects. Experimental evidence has slowly accumulated in the past decades to support a more general description of capillary pressure that includes dynamic effects. A review of these experiments shows that the coefficient arising in the theoretical analysis can be estimated from the reported data. The calculated values range from 10 4 to 10 7 kg (m s) −1 . In addition, recently developed pore-scale models that simulate interface dynamics within a network of pores can also be used to estimate the appropriate dynamic coefficients. Analyses of experiments reported in the literature, and of simulations based on pore-scale models, indicate a range of dynamic coefficients that spans about three orders of magnitude. To examine whether these coefficients have any practical effects on larger-scale problems, continuum-scale simulators may be constructed in which the dynamic effects are included. These simulators may then be run to determine the range of coefficients for which discernable effects occur. Results from such simulations indicate that measured values of dynamic coefficients are within one order of magnitude of those values that produce significant effects in field simulations. This indicates that dynamic effects may be important for some field situations, and numerical simulators for unsaturated flow should generally include the additional term(s) associated with dynamic capillary pressure.

Similarity solutions for fluid injection into confined aquifers

Fluid injection into the deep subsurface, such as injection of carbon dioxide (CO 2 ) into deep saline aquifers, often involves two-fluid flow in confined geological formations. Similarity solutions may be derived for these problems by assuming that a sharp interface separates the two fluids, by imposing a suitable no-flow condition along both the top and bottom boundaries, and by including an explicit solution for the pressure distribution in both fluids. When the injected fluid is less dense and less viscous than the resident fluid, as is the case for CO 2 injection into a resident brine, gravity override produces a fluid flow system that is captured well by the similarity solutions. The similarity solutions may be extended to include slight miscibility between the two fluids, as well as compressibility in both of the fluid phases. The solutions provide the location of the interface between the two fluids, as well as drying fronts that develop within the injected fluid. Applications to cases of supercritical CO 2 injection into deep saline aquifers demonstrate the utility of the solutions, and comparisons to solutions from full numerical simulations show the ability to predict the system behaviour.

Recent advances in pore scale models for multiphase flow in porous media

In the last decade, multi-phase flow in porous media has become a prominent topic in hydrologic research. This has been motivated by the widespread occurrence of subsurface contamination problems involving sparingly soluble liquids, often referred to as non-aqueous phase liquids (NAPLs). NAPL contamination problems require analysis of the simultaneous movement of at least two fluid phases, NAPL and water. This is somewhat different than the traditional multiphase flow problem in hydrology, namely water movement in unsaturated soils. In the traditional treatment of unsaturated systems, the movement of air, the non-aqueous phase, is not of interest, and its movement is ignored. Both NAPL-water and air-water systems are similar in that capillary forces, acting at the pore scale, usually dictate the pore-scale distribution of each fluid phase. Pore-scale fluid distributions then dictate the continuum-scale properties such as relative saturation and relative permeability.

Practical Modeling Approaches for Geological Storage of Carbon Dioxide

The relentless increase of anthropogenic carbon dioxide emissions and the associated concerns about climate change have motivated new ideas about carbon-constrained energy production. One technological approach to control carbon dioxide emissions is carbon capture and storage, or CCS. The underlying idea of CCS is to capture the carbon before it emitted to the atmosphere and store it somewhere other than the atmosphere. Currently, the most attractive option for large-scale storage is in deep geological formations, including deep saline aquifers. Many physical and chemical processes can affect the fate of the injected CO2, with the overall mathematical description of the complete system becoming very complex. Our approach to the problem has been to reduce complexity as much as possible, so that we can focus on the few truly important questions about the injected CO2, most of which involve leakage out of the injection formation. Toward this end, we have established a set of simplifying assumptions that allow us to derive simplified models, which can be solved numerically or, for the most simplified cases, analytically. These simplified models allow calculation of solutions to large-scale injection and leakage problems in ways that traditional multicomponent multiphase simulators cannot. Such simplified models provide important tools for system analysis, screening calculations, and overall risk-assessment calculations. We believe this is a practical and important approach to model geological storage of carbon dioxide. It also serves as an example of how complex systems can be simplified while retaining the essential physics of the problem.

Modeling support of functional relationships between capillary pressure, saturation, interfacial area and common lines

Computational pore-scale network models describe two-phase porous media flow systems by resolving individual interfaces at the pore scale, and tracking these interfaces through the pore network. Coupled with volume averaging techniques, these models can reproduce relationships between measured variables like capillary pressure, saturation, and relative permeability. In addition, these models allow nontraditional porous media variables to be quantified, such as interfacial areas and common line lengths. They also allow explorations of possible relationships between these variables, as well as testing of new theoretical conjectures. Herein we compute relationships between capillary pressure, saturation, interfacial areas, and common line lengths using a pore-scale network model. We then consider a conjecture that definition of an extended constitutive relationship between capillary pressure, saturation, and interfacial area eliminates hysteresis between drainage and imbibition; such hysteresis is commonly seen in the traditional relationship between capillary pressure and saturation. For the sample pore network under consideration, we find that hysteresis can essentially be eliminated using a specific choice of displacement rules; these rules are within the range of experimental observations for interface displacements and therefore are considered to be physically plausible. We find that macroscopic measures of common line lengths behave similarly to fluid‐fluid interfacial areas, although the functional dependencies on capillary pressure and saturation diAer to some extent. ” 2001 Elsevier Science Ltd. All rights reserved.

A mass conservative numerical solution for two-phase flow in porous media with application to unsaturated flow

A numerical algorithm for simulation of two-phase flow in porous media is presented. The algorithm is based on a modified Picard linearization of the governing equations of flow, coupled with a lumped finite element approximation in space and dynamic time step control. Numerical results indicate that the algorithm produces solutions that are essentially mass conservative and oscillation free, even in the presence of steep infiltrating fronts. When the algorithm is applied to the case of air and water flow in unsaturated soils, numerical results confirm the conditions under which Richardss equation is valid. Numerical results also demonstrate the potential importance of air phase advection when considering contaminant transport in unsaturated soils. Comparison to several other numerical algorithms shows that the modified Picard approach offers robust, mass conservative solutions to the general equations that describe two-phase flow in porous media.