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Stochastic analyses were performed to examine sequential first-order monomolecular reactions at the microscopic scale and both Fickian and non-Fickian plume reactive transport at the macroscopic scale. An analytical solution was derived for the chemical master equation (CME) for a closed system of irreversible first-order monomolecular reactions. Taking a Lagrangian reference frame of particles migrating from a source, analyses show that the relative concentration of each species in the deterministic analytical solution for 1-D steady state plug flow with first-order sequential degradation is mathematically equivalent to the mean of a multinomial distribution of plume particles moving at constant velocity with sequential transformations described by transition probabilities of a discrete state, continuous-time Markov chain. In order to examine the coupling of reaction and transport terms in subdiffusive-reactive transport equations, a closed-form multispecies analytical solution also was derived for steady state advection, dispersion, and sequential first-order reaction. Using a 1-D continuous-time random walk (CTRW) embedded in Markov chains, computationally efficient Monte Carlo simulations of particle movement were performed to more fully examine effects of subdiffusive-reactive transport with an application to steady state, sequentially degrading multispecies plumes at a site in Palm, Bay, FL. The simulation results indicated that non-Fickian steady state plumes can resemble Fickian plumes because linear reactions truncate the waiting time between particle jumps, which removes lower velocity particles from the broad spectrum of velocities in highly heterogeneous media. Results show that fitting of Fickian models to plume concentration data can lead to inaccurate estimates of rate constants because of the wide distribution of travel times in highly heterogeneous media.

[1] Over the decade 1996–2006, acoustic sources located off central California (1996–1999) and north of Kauai (1997–1999, 2002–2006) transmitted to receivers distributed throughout the northeast and north central Pacific. The acoustic travel times are inherently spatially integrating, which suppresses mesoscale variability and provides a precise measure of ray-averaged temperature. Daily average travel times at 4-day intervals provide excellent temporal resolution of the large-scale thermal field. The interannual, seasonal, and shorter-period variability is large, with substantial changes sometimes occurring in only a few weeks. Linear trends estimated over the decade are small compared to the interannual variability and inconsistent from path to path, with some acoustic paths warming slightly and others cooling slightly. The measured travel times are compared with travel times derived from four independent estimates of the North Pacific: (1) climatology, as represented by the World Ocean Atlas 2005 (WOA05); (2) objective analysis of the upper-ocean temperature field derived from satellite altimetry and in situ profiles; (3) an analysis provided by the Estimating the Circulation and Climate of the Ocean project, as implemented at the Jet Propulsion Laboratory (JPL-ECCO); and (4) simulation results from a high-resolution configuration of the Parallel Ocean Program (POP) model. The acoustic data show that WOA05 is a better estimate of the time mean hydrography than either the JPL-ECCO or the POP estimates, both of which proved incapable of reproducing the observed acoustic arrival patterns. The comparisons of time series provide a stringent test of the large-scale temperature variability in the models. The differences are sometimes substantial, indicating that acoustic thermometry data can provide significant additional constraints for numerical ocean models.

Broadband acoustic signals transmitted from a moored 250-Hz source to a 3-km-long vertical line array of hydrophones 1000 km distant in the north central Pacific Ocean were used to determine the amount of information available from tomographic techniques used in the vertical plane connecting a source-receiver pair. A range-independent, pure acoustic inverse to obtain the sound speed field using travel time data from the array is shown to be possible by iterating from climatological data without using any information from concurrent environmental measurements. Range-dependent inversions indicate resolution of components of oceanic variability with horizontal wavelengths shorter than 50 km, although the limited spatial resolution of concurrent direct measurements does not provide a strong cross-validation, since the typical cast spacing of 20–25 km gives a Nyquist wavelength of 40–50 km. The small travel time signals associated with high-wavenumber ocean variability place stringent but achievable requirements on travel time measurement precision. The forward problem for the high-wavenumber components of the model is found to be subject to relatively large linearization errors, however, unless the sound speed field at wavelengths greater than about 50 km is known from other measurements or from a two-dimensional tomographic array. The high-ocean-wavenumber resolution that is in principle available from tomographic measurements is therefore achievable only under restricted conditions.

A management model is developed that determines the optimum economic recoverability of a particular hot-water geothermal reservoir undergoing exploitation for electric power generation. The management model integrates a physical model of the reservoir that predicts the areas of pressure decline due to withdrawals, and pressure rise due to reinjection of spent fluid, with a model of a two-stage steam turbine power plant that determines the quantity of electricity generated for a rate of hot-water extraction. Capital costs, variable costs and annual fixed costs are obtained for the reservoir development, extraction and reinjection, the transmission system, and the power plant. Revenues are determined for electrical power production. Application of the management model to a simplified, yet realistic example reservoir demonstrates that the methodology developed in this report can be used for analyzing the management of an integrated geothermal reservoir power plant system. For the example reservoir, 12 potential sites are developed, five for extraction wells and seven for injection wells. The wells on these sites are used to develop up to 27 MW of electrical power over a 20-year time interval.

A three-dimensional finite element model is developed for the simulation of saltwater intrusion in single and multiple coastal aquifer systems with either a confined or phreatic top aquifer. The model formulation is based on two governing equations, one for fluid flow and the other for salt transport. Density coupling of these equations is accounted for and handled using a Picard sequential solution algorithm with special provisions to enhance convergence of the iterative solution. Flexibility in the formulation allows for either three-dimensional simulations or quasi three-dimensional simulations, where flow and transport in aquitards are treated using one-dimensional analytical and/or numerical approximations. Spatial discretization of three-dimensional regions is performed using a vertical slicing approach designed to accommodate complex geometry with irregular boundaries, layering, and/or lateral discontinuity. This approach is effectively combined with the use of simple linear elements such as rectangular and triangular prisms, and composite hexahedra and pentahedra made up of tetrahedra. For these elements, computation of element matrices can be performed efficiently using influence coefficient formulas that avoid numerical integration. New transport influence coefficient formulas are presented for rectangular and triangular prism elements. Matrix assembly is performed slice by slice, and the matrix solution is achieved using a slice successive relaxation scheme. This permits a fairly large number of nodal unknowns (of the order of five to ten thousand) to be handled conveniently on small or medium-size minicomputers. Flexibility of the formulation and matrix handling procedures also allows two-dimensional and axisymmetric problems to be solved efficiently using single slice representations. Four examples are presented to demonstrate the model verification and utility. These problems represent a fair range of physical conditions. Where possible, simulation results are compared with previously published solutions.

The available data base and hydrogeologic information for the Chem-Dyne hazardous waste site, located in southwestern Ohio, are typical of many Superfund sites. Data are localized because investigations conducted at these sites are for the purpose of defining contaminant plumes. Little or no data are available to characterize the regional flow systems which impart a controlling influence on the rate and direction of contaminant migration. To simulate groundwater flow and contaminant transport in this setting, a telescopic mesh refinement (TMR) approach is appropriate. This approach provides the means of accurately incorporating regional controlling factors into smaller model domains and also increased grid resolution in areas of critical importance. At the Chem-Dyne site a finite-difference model was applied at three scales: regional, local, and site. This application of TMR integrates the regional and local flow system characteristics to analyze the effectiveness of a proposed remedial action at the site scale. The TMR modeling approach permits effective flow and transport model construction and calibration to provide quantitative analysis of system response to a groundwater extraction-injection well clean-up system. Calibration of the flow models demonstrates the regional importance of induced river infiltration to groundwater pumping centers and the control of surface water features on the direction of plume migration. The site-scale transport model reveals that a significant portion of the volatile organic contaminants are not captured by the preliminary proposed system.

Two alternative mathematical models are presented that are suitable for numerical simulation of geothermal reservoirs. The general mathematical model describes the three-dimensional flow of single-component water (both one and two phase) and the transport of heat in porous media. It is composed of two nonlinear partial differential equations, posed in terms of fluid pressure and enthalpy, and appropriate boundary conditions. An alternative quasi-three-dimensional model is derived by partial integration (in the vertical dimension) of the three-dimensional equations. The reservoir is assumed to have good vertical communication so that vertical equilibrium (gravity segregation) between steam and water is achieved. The resulting equations, posed in terms of vertically averaged pressure and enthalpy, include effects of an inclined, variable-thickness reservoir and mass and energy leakage to confining beds.

The Baca geothermal reservoir and adjacent aquifers in the Jemez Mountains of New Mexico comprise an integrated hydrogeologic system. Analysis of the geothermal reservoir either under natural conditions or subject to proposed development should account for the mass (water) and energy (heat) balances of adjacent aquifers as well as the reservoir itself. A three-dimensional model based on finite difference approximations is applied to this integrated system. The model simulates heat transport associated with the flow of steam and water through an equivalent porous medium. The Baca geothermal reservoir is dominated by flow in fractures and distinct strata, but at the scale of application the equivalent porous media concept is appropriate. The geothermal reservoir and adjacent aquifers are simulated under both natural conditions and proposed production strategies. Simulation of natural conditions compares favorably with observed pressure, temperature, and thermal discharge data. The history matching simulations show that the results used for comparison are most sensitive to vertical permeability and the area of an assumed high-permeability zone connecting the reservoir to a deep hydrothermal source. Simulations using proposed production strategies and optimistic estimates of certain hydrologic parameters and reservoir extent indicate that a 50-MW power plant could be maintained for a period greater than 30 years. This production, however, will result in significant decreases in the total water discharge to the Jemez River.

A mathematical model is presented for numerical simulation of solute transport in naturally fractured porous media. The model is based on the dual-porosity conceptual approach and is capable of simulating interactions between porous rock matrix and fractures. Unlike earlier dual-porosity transport models, which rely on parallel fracture assumptions, the present model is a unified model that is also able to represent blocky fractured systems (i.e., systems with suborthogonal fractures) by using a spherical idealization of matrix blocks. Two governing, partial differential equations are written for solute transport in the fractures and diffusion in the porous matrix blocks. These equations are coupled by mass flux terms (leakages). An efficient numerical scheme is presented for approximating the governing transport equations. The scheme effectively combines a two-dimensional, upstream-weighted, finite-element approximation for transport in the fractures with a one-dimensional Galerkin approximation for diffusion within the individual matrix blocks. Coupling of the two approximations is performed implicitly. The systems of algebraic equations resulting from both finite-element approximations are solved by using sequential solution algorithms designed especially for this type of problem. Stability and accuracy of the numerical scheme are checked by applying the model to a number of test problems and comparing results with available analytical solutions. In all cases the numerical scheme is found to be highly stable and capable of producing reliable results with the use of relatively coarse spatial and temporal discretizations. Finally, the utility of the model is demonstrated by applying it to a hypothetical, yet realistic, problem.

A quasi three-dimensional, areal model based on finite-difference approximations is applied to the hydrothermal field at Wairakei, New Zealand. The model simulates heat transport associated with the flow of steam and water through porous media. An analysis is made of the production aquifer under steady-state and transient flow conditions, allowing vertical flow of heat and fluid through confining beds. Computed steady-state results correlate well with observed data and indicate that portions of the aquifer had a steam cap prior to exploitation. Computed transient results also match observed data and support the hypothesis that the production aquifer is recharged through underlying confining beds. The limiting factor on production is the amount of mass available, both from storage and leakage. Although simulation results indicate that the field can maintain production rates to the year 2000, such long range predictions are unreliable due to the lack of information on the leakage properties of the confining beds.

This paper presents a two-phase flow model based on a three-dimensional, finite-difference formulation. As three-dimensional simulations can require substantial computer effort, a numerical technique that takes advantage of vector and parallel processing computer architecture is developed. The model is posed in terms of water saturation and nonwetting fluid pressure. It uses three-phase capillary pressure and relative permeability relationships to permit simulation within or below the unsaturated zone. A modified formulation of slice successive overtaxation (an iterative matrix solution technique) is introduced. This technique is designed to use parallel processing capabilities of new computers. The model is applied to immiscible fluid flow at two chemical waste landfills near Niagara Falls, New York. At both sites, denser than water, nonaqueous liquids (NAPLs) are present in the groundwater regimes in relatively large quantities. The model applications address several technical concerns at the two sites, including the effectiveness of clay as a geologic barrier to NAPL migration owing to capillary pressure forces, the three-dimensional aspects of dense NAPL flow, and the sensitivity of NAPL recovery in pumping wells due to various hydrogeologic and fluid properties. The results of the applications show that (1) even under a downward hydraulic gradient, natural differences in capillary pressure relationships for different lithologies can prevent downward migration of NAPL, (2) without any lithologic-capillary barrier, an upward hydraulic gradient induced by a de watering system can prevent downward migration of NAPL, (3) NAPL recovery at wells is sensitive to relative permeability, a relationship that requires field calibration in many settings, and (4) the three-dimensional aspects of two-phase flow and hydrogeologic heterogeneity require explicit treatment in many settings.

A finite element model is presented for simulation of nuclide decay chain transport in a naturally fractured porous medium system. The model is capable of representing the physical system using a dual-porosity approach, a discrete fracture approach, or a combination of these two approaches. Advection and hydrodynamic dispersion in the fractures, as well as diffusion in the porous matrix and chain reactions of solute species, can be taken into account simultaneously. An efficient finite element solution technique has been developed to solve a coupled system of governing partial differential equations. Via this technique, spatial discretizations and solutions of systems of algebraic equations for nodal concentration values in fracture and porous matrix domains can be performed separately. The present finite element model has been verified against analytical solutions. Two test problems involving transport of a chain of three components in unfractured and fractured porous media are presented to demonstrate the accuracy and efficiency of the proposed finite element technique. Results indicate that the present numerical model is capable of producing good predictions of breakthrough curves using relatively coarse spatial and temporal discretizations. A major advantage of the present transport model over previous transport models is that the latter are based on a numerical approach that employs overall discretization and simultaneous solution of the entire set of algebraic equations for concentration values in the fracture and porous matrix block domains. Such an approach is not computationally efficient when compared with the present numerical approach, which employs not only separate discretizations of the fracture and porous matrix domains but also direct sequential solutions of much smaller subsets of algebraic equations. Another important practical aspect is that the present model enables more complex problems, involving transverse diffusion into porous matrix and two-dimensional transport in the fracture flow plane, to be handled without resorting to a fully three-dimensional grid. In practical applications, CPU time and cost involved in performing a fully three-dimensional analysis of multiple species transport may be prohibitive.

A single-phase simulation model was applied to the hot-water hydrothermal field at Wairakei, New Zealand. A two-dimensional areal analysis was made of the production aquifer under steady state and transient flow conditions, allowing vertical flow of heat and fluid through an overlying confining bed. Calculated temperature and pressure patterns correlate well with observed patterns until approximately 1963, when increasing quantities of steam in the production aquifer invalidated the assumption of single-phase flow. For further simulation of the Wairakei reservoir the numerical model will need to be extended to incorporate phase change and three-dimensional flow. Preliminary results, however, indicate that the response of hot-water hydrothermal systems to exploitation can be simulated by using a mathematical reservoir model based on a Galerkin-finite element approach.

An upstream-weighted residual finite element formulation is developed for the numerical simulation of solute transport in variably saturated porous media. Included in this formulation is an influence coefficient scheme for element matrix evaluation and a mass balance computation scheme. These schemes are not only simple to implement but also cost effective. Three examples are provided to verify the numerical techniques and demonstrate the utility of the computer model. These examples indicate that mass balance information serves a useful purpose as a screening device for examining the numerical solution. They also show certain behavior of the mass balance error as well as numerical difficulties associated with the manner in which the finite element formulation of the transport equation is performed. Finally, information on the CPU time and cost required to solve the three example problems is provided to show the cost effectiveness of the present transport model. Although the present analysis considers only two dimensions, the schemes described can be extended to three dimensions, where the time and cost savings are more significant.

Two numerical models are introduced for simulating three-dimensional, two-phase fluid flow and heat transport in geothermal reservoirs. The first model is based on a three-dimensional formulation of the governing equations for geothermal reservoirs. Since the resulting two partial differential equations, posed in terms of fluid pressure and enthalpy, are highly nonlinear and inhomogeneous, they require numerical solution. The three-dimensional numerical model uses finite difference approximations, with fully implicit Newton-Raphson treatment of nonlinear terms and a block (vertical slice) successive iterative technique for matrix solution. Newton-Raphson treatment of nonlinear terms permits the use of large time steps, while the robust iterative matrix method reduces computer execution time and storage for large three-dimensional problems. An alternative model is derived by partial integration (in the vertical dimension) of the three-dimensional equations. This second model explicitly assumes vertical equilibrium (gravity segregation) between steam and water and can be applied to reservoirs with good vertical communication. The resulting equations are posed in terms of depth-averaged pressure and enthalpy and are solved by a two-dimensional finite difference model that uses a stable sequential solution technique, direct matrix methods, and Newton-Raphson iteration on accumulation and source terms. The quasi-three-dimensional areal model should be used whenever possible, because it significantly reduces computer execution time and storage and it requires less data preparation. The areal model includes effects of an inclined, variable-thickness reservoir and mass and energy leakage to confining beds. The model works best for thin (

The effects of a finite-thickness skin (low-permeability zone surrounding the well bore face) on the response of slug tests is investigated by using a numerical model and a simple analytical solution. The results show that, for skins of finite thicknesses, estimates of hydraulic conductivity provided by slug tests can be more representative of the skin than of the surrounding formation. When a finite-thickness skin is present, the slug test response is shifted along the horizontal axis, making estimates of hydraulic conductivity unreliable. This result is different from that obtained by using an analytical solution (Ramey et al., 1975) for a skin of infinitesimal thickness.

The AGU Hydrology Section Executive Committee Meeting was called to order at approximately 4 P.M. on Monday, May 18, 1987, by Hydrology Section President Marshall Moss. In attendance were President-Elect George Pinder, Secretary Jim Mercer, Ron Cummings, Helen Joyce Peters, Peter Eagleson, Stephen Burges, Jim Wallis, Jurate Landwehr, Don Nielson, Ken Bencala, Pete Loucks, Jery Stedinger, Dennis Lettenmaier, Lenny Konikow, Ken Potter, John Wilson, Ivan Johnson, and Judy Holoviak.

The goal of the authors in writing Geothermal Reservoir Engineering was to bring together all the currently available information on geothermal reservoir engineering published before late 1981. The available information not only includes geothermal publications but also related publications in the areas of petroleum reservoir engineering and groundwater hydrology. Such a text would be a needed and worthwhile addition to many libraries, if the goal of the authors were achieved. Unfortunately, Geothermal Reservoir Engineering falls short of its potential contribution.

The current conventional theory for transport of solutes in groundwater was fully developed by the 1960s. Attention then focused on the development of analytical solutions to the governing equation. By the late 1960s, the increasing efficiency of digital computers, coupled with the restrictive assumptions required for analytical solutions, led to a major emphasis on developing deterministic, distributed-parameter, numerical simulation models. Several numerical models that solved the conventional solute-transport equation were developed in the early 1970s and initially applied to seawater-intrusion problems. Toward the middle and late 1970s, many applications involved hazardous and radioactive waste disposal problems. Almost all of these applications involved miscible transport. With the increased application of transport models to field problems, certain deficiencies in transport theory were identified. The deficiencies include the mathematical descriptions of hydrodynamic dispersion, reaction processes, and flow and transport in fractured rocks. Therefore, in the latter half of the 1970s and the beginning of the 1980s, considerable research was and continues to be directed toward improving our understanding of these processes in the saturated zone.