Steven M. Gorelick

Multiple‐Rate Mass Transfer for Modeling Diffusion and Surface Reactions in Media with Pore‐Scale Heterogeneity

Mass transfer between immobile and mobile zones is a consequence of simultaneous processes. We develop a “multirate” model that allows modeling of small-scale variation in rates and types of mass transfer by using a series of first-order equations to represent each of the mass transfer processes. The multirate model is incorporated into the advective-dispersive equation. First, we compare the multirate model to the standard first-order and diffusion models of mass transfer. The spherical, cylindrical, and layered diffusion models are all shown to be specific cases of the multirate model. Mixtures of diffusion from different geometries and first-order rate-limited mass transfer can be combined and represented exactly with the multirate model. Second, we develop solutions to the multirate equations under conditions of no flow, fast flow, and radial flow to a pumping well. Third, using the multirate model, it is possible to accurately predict rates of mass transfer in a bulk sample of the Borden sand containing a mixture of different grain sizes and diffusion rates. Fourth, we investigate the effects on aquifer remediation of having a heterogeneous mixture of types and rates of mass transfer. Under some circumstances, even in a relatively homogeneous aquifer such as at Borden, the mass transfer process is best modeled by a mixture of diffusion rates.

Heterogeneity in Sedimentary Deposits: A Review of Structure‐Imitating, Process‐Imitating, and Descriptive Approaches

Numerical models that solve governing equations for subsurface fluid flow and transport are commonly applied to analyze quantitatively the effects of heterogeneity. These models require maps of spatially variable hydraulic properties. Because complete three-dimensional information about hydraulic properties is never obtainable, numerous methods have been developed to interpolate between data values and use geologic, hydrogeologic, and geophysical information to create images of aquifer properties. Image creation approaches fall into three general categories: structure-imitating, process-imitating, and descriptive. Structure-imitating methods rely on one or more of the following to constrain the geometry of spatial patterns in geologic media: correlated random fields, probabilistic rules, and deterministic constraints developed from facies relations. Structure-imitating methods include spatial statistical algorithms and geologically based sedimentation pattern-matching approaches. Process-imitating models include aquifer model calibration methods and geologic process models. Aquifer model calibration methods use governing equations for subsurface fluid flow and transport to relate hydraulic properties to heads and solute information through history and steady state data matching. Geologic process models combine fundamental laws of conservation of mass and momentum with sediment transport equations to simulate spatial patterns in grain size distributions. At the sedimentary basin scale, multiprocess models include thermomechanical mechanisms of basin subsidence. Descriptive methods couple geologic observations with facies relations to divide an aquifer into zones of characteristic hydraulic properties. All approaches are capable of reproducing heterogeneity over a range of scales and considering some types of geologic information. Some approaches are strictly spatial while some are linked to the time evolution of sedimentation. Some approaches can be conditioned on measurements. Recent advances aimed at infusing geologic information into images of the subsurface include extracting more information from sedimentological facies models, incorporating qualitative geologic information into random field generators and simulating depositional processes. Classes of research missing from the literature include multiprocess models that incorporate diagenesis and three-dimensional surface water flow, hybrid methods that combine features of existing approaches, and approaches that can make use of all available geologic, geophysical, and hydrologic data.

Earthquake triggering and large-scale geologic storage of carbon dioxide

Despite its enormous cost, large-scale carbon capture and storage (CCS) is considered a viable strategy for significantly reducing CO2 emissions associated with coal-based electrical power generation and other industrial sources of CO2 [Intergovernmental Panel on Climate Change (2005) IPCC Special Report on Carbon Dioxide Capture and Storage. Prepared by Working Group III of the Intergovernmental Panel on Climate Change, eds Metz B, et al. (Cambridge Univ Press, Cambridge, UK); Szulczewski ML, et al. (2012) Proc Natl Acad Sci USA 109:5185–5189]. We argue here that there is a high probability that earthquakes will be triggered by injection of large volumes of CO2 into the brittle rocks commonly found in continental interiors. Because even small- to moderate-sized earthquakes threaten the seal integrity of CO2 repositories, in this context, large-scale CCS is a risky, and likely unsuccessful, strategy for significantly reducing greenhouse gas emissions.

Rate‐limited mass transfer or macrodispersion: Which dominates plume evolution at the macrodispersion experiment (MADE) site?

We present a model of solute transport that explains the large-scale behavior of the solute tracer-test plumes at the Macrodispersion Experiment (MADE) site as the result of advection and rate-limited mass transfer between mobile and small-scale immobile domains. This model does not consider the process of dispersion and yet provides an alternative explanation of the evolution of the observed concentration profiles. Compared to the macrodispersion model, the mass transfer model better represents the change in mobile dissolved mass with time, the peak of the concentration profile, and the profile asymmetry. Specifically, unlike the macrodispersion model, the mass transfer model explains the facts that the observed mass of the plume was greater than the injected mass in early snap shots of the plume and less than the injected mass at late times. We suggest that the injected mass advects through the mobile domain and diffuses into and out of the immobile domain. The immobile domain consists of a combination of low-permeability zones on the scale of centimeters to decimeters (the Darcy-scale immobile domain), and intragranular porosity, dead-end pores, and surface sorption (the pore-scale immobile domain). We suggest that the mobile domain was sampled preferentially when water was extracted. Therefore, at early times, relatively clean water in the immobile domain was not sampled and incorrectly assumed to contain high solute concentrations. Similarly, the mass at late times was underestimated because solute trapped in the pore-scale immobile domain was not extracted during sampling and therefore ignored. The combination of advection and slow mass transfer is consistent with the fact that the peak of the plume migrated only ∼5 m by the termination of the experiment, as well as the different behavior of bromide and tritium tracers.

Temporal Moment-Generating Equations: Modeling Transport and Mass Transfer in Heterogeneous Aquifers

We present an efficient method for determining temporal moments of concentration for a solute subject to first-order and diffusive mass transfer in steady velocity fields. The differential equations for the moments of all orders have the same form as the steady state nonreactive transport equation. Thus temporal moments can be calculated by a solute transport code that was written to simulate nonreactive steady state transport, even though the actual transport system is reactive and transient. Higher-order moments are found recursively from lower-order moments. For many cases a small number of moments sufficiently describe the movement of a solute plume. The first four moments describe the accumulated mass, mean, spread, and skewness of the concentration histories at all locations. Actual concentration histories at any location can be approximated from the moments by applying the principle of maximum entropy, a constraint consistent with the physical process of dispersion. The forms of the moment-generating equations for different mass transfer models provide insight into reactive transport through heterogeneous aquifers. For the mass transfer models we considered, the zeroth moment in a heterogeneous aquifer is independent of the mass transfer coefficients. Thus, if the velocity field is known, the mass transported past any point, or out any boundary, can be calculated without knowledge of the spatial pattern of mass transfer coefficients and, in fact, without knowledge of whether mass transfer is occurring. Also, for both first-order and diffusive mass transfer models, the mean arrival time depends on the distribution coefficient but is independent of the values of the rate coefficients, regardless of the spatial variability of groundwater velocity and mass transfer coefficients.

Fractional packing model for hydraulic conductivity derived from sediment mixtures

Petrophysical relations are derived to predict porosity and hydraulic conductivity from grain size distributions considering particle packing in sediment mixtures. First, we develop a fractional packing model for porosity that considers the fraction of intrapore fines that occur as the fines content increases. Then, a fractional packing Kozeny-Carman relation for hydraulic conductivity is developed by examining which particle sizes dominate the pore structure, and which averaging procedure best represents the mean grain diameter in any given sediment mixture. The relations developed here perform well for a wide range of sediment mixtures regardless of confining pressure. Graphs are presented that show hydraulic conductivity versus weight fraction of fines for mixtures of coarse- and fine-grained sediment commonly observed in the field, such as clayey gravel and silty sand. These graphs show that the wide range of hydraulic conductivity values reported for sediment mixtures can display a 5 order of magnitude variation over a few percent fines. Finally, a field scale application using grain size distributions from a quantitative depositional model shows that these petrophysical relations successfully predict more than 90% of hydraulic conductivity values to within 1 order of magnitude over 7 orders of magnitude of spatial variability.

Time-lapse imaging of saline-tracer transport in fractured rock using difference-attenuation radar tomography

[1] Accurate characterization of fractured-rock aquifer heterogeneity remains one of the most challenging and important problems in groundwater hydrology. We demonstrate a promising strategy to identify preferential flow paths in fractured rock using a combination of geophysical monitoring and conventional hydrogeologic tests. Cross-well differenceattenuation ground-penetrating radar was used to monitor saline-tracer migration in an experiment at the U.S. Geological Survey Fractured Rock Hydrology Research Site in Grafton County, New Hampshire. Radar data sets were collected every 10 min in three adjoining planes for 5 hours during each of 12 tracer tests. An innovative inversion method accounts for data acquisition times and temporal changes in attenuation during data collection. The inverse algorithm minimizes a combination of two functions. The first is the sum of weighted squared data residuals. Second is a measure of solution complexity based on an a priori space-time covariance function, subject to constraints that limit radarattenuation changes to regions of the tomograms traversed by high difference-attenuation ray paths. The time series of tomograms indicate relative tracer concentrations and tracer arrival times in the image planes; from these we infer the presence and location of a preferential flow path within a previously identified zone of transmissive fractures. These results provide new insights into solute channeling and the nature of aquifer heterogeneity at the site. INDEX TERMS: 0910 Exploration Geophysics: Data processing; 0915 Exploration Geophysics: Downhole methods; 1829 Hydrology: Groundwater hydrology; 1832 Hydrology: Groundwater transport; 1894 Hydrology: Instruments and techniques; KEYWORDS: radar tomography, fractured rock, ground-penetrating radar, geophysics, hydrogeophysics

Estimating lithologic and transport properties in three dimensions using seismic and tracer data: The Kesterson aquifer

The identification of aquifer heterogeneities, particularly flow paths and barriers, has become a critical research topic in hydrology. Cross-well seismic tomography may provide the needed resolution when used in conjunction with hydraulic head and tracer concentration measurements. We demonstrate a field application and sensitivity analysis of the split inversion method (SIM), which combines seismic, hydraulic, and tracer data to estimate the three-dimensional zonation of aquifer properties along with the hydraulic properties for these zones. For the Kesterson aquifer in the San Joaquin Valley, California, we first invert seismic travel times measured between six well pairs to obtain seismic slowness (1/seismic velocity) cross sections, or tomograms. We then use conditional simulation to provide three-dimensional seismic slowness realizations. Next, the SIM is used to split several realizations into three lithologic zones and assign hydraulic properties to the zones to best match six tracer concentration histories.

Mapping Hydraulic Conductivity: Sequential Conditioning with Measurements of Solute Arrival Time, Hydraulic Head, and Local Conductivity

We present a method for estimating the spatial pattern of aquifer hydraulic conductivity from three types of measurements: solute arrival time quantiles, hydraulic heads, and direct local measurements. Results indicate that arrival times and heads provide different, but complementary, information about the large features of the conductivity field which serve as flow paths and barriers. We compare the information provided by head and arrival time measurements by plotting the correlation between measurement and conductivity over the entire domain. We also compare the value of head and arrival time data by estimating the conductivity with only one type of data, and by incorporating the different data types into the estimation procedure in different sequences. Using quantiles of arrival time, rather than concentrations, has three advantages: (1) Arrival times are independent of the amount of dilute solute introduced into the aquifer. (2) Under some conditions, by measuring the median arrival time, the accuracy of the conductivity field estimate is not degraded by poor knowledge of the local dispersivity. (3) The estimation procedure is greatly simplified by relying on a single quantile to represent the critical information contained in a breakthrough curve that is constructed from many concentration measurements. For the case examined here, incorporating arrival time quantiles that describe the tails of the breakthrough curves did not significantly improve our estimate of the conductivity field. An accurate map of the hydraulic conductivity of an aquifer is vital for predicting groundwater flow and solute transport through the aquifer. Typically, the spatial pattern of conduc- tivity is qualitatively inferred from a few local values of con- ductivity, head, and concentration measured at wells which are sparsely distributed, leaving large volumes of the aquifer un- sampled. When monitoring wells fail to penetrate important flow channels or barriers, these features must be deduced from measurements of head and concentration at nearby wells. A number of studies (Kitanidis and Vomvoris, 1983; Hoek- sema and Kitanidis, 1984; Dagan, 1985; Sun and Yeh, 1992) have shown how measurements of hydraulic head can be used in conjunction with measurements of local transmissivity to estimate the spatial pattern of aquifer transmissivity. These studies solve the inverse problem with a procedure that com- bines solution of the flow equation with linear estimation. Some recent studies (Graham and McLaughlin, 1989a, b; Ru- bin, 1991a, b) describe how measurements of solute concen- trations can be used to predict or estimate concentrations at other times and locations. Although spatially variable conduc- tivity is the underlying cause of uncertainty in these studies, the procedures are not directed toward estimating the conductivity field. Wagner (1992) used both head and concentration mea- surements to estimate the value of conductivity within pre- scribed zones. Woodbury and Smith (1988) combined steady state temperature and head measurements to estimate spa-

The nature and causes of the global water crisis: Syndromes from a meta-analysis of coupled human-water studies

Freshwater scarcity has been cited as the major crisis of the 21st century, but it is surprisingly hard to describe the nature of the global water crisis. We conducted a meta- analysis of 22 coupled human– water system case studies, using qualitative comparison analysis (QCA) to identify water resource system outcomes and the factors that drive them. The cases exhibited different outcomes for human wellbeing that could be grouped into a six “syndromes ”: groundwater depletion, ecological destruction, drought-driven conflicts, unmet subsistence needs, resource capture by elite, and water reallocation to nature. For syndromes that were not successful adaptations, three characteristics gave cause for concern: (1) unsustainability —a decline in the water stock or ecosystem function that could result in a long-term steep decline in future human wellbeing; (2) vulnerability —high variability in water resource availability combined with inadequate coping capacity, leading to temporary drops in human wellbeing; (3) chronic scarcity —persistent inadequate access and hence low conditions of human wellbeing. All syndromes could be explained by a limited set of causal factors that fell into four categories: demand changes, supply changes, governance systems, and infrastructure/technology. By considering basins as members of syndrome classes and tracing common causal pathways of water crises, water resource analysts and planners might develop improved water policies aimed at reducing vulnerability, inequity, and unsustainability of freshwater systems.