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.

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. Introduction Qualitative relations between the porous media hydraulic properties of porosity and permeability and the petrographicproperties of grain diameter, shape, sorting, and packing are well established [Chilingar, 1964; Wolf and Chilingarian, 1976; Beard and Weft, 1973; Blatt et al., 1980]. Laboratory and field experiments have shown that grain diameter and sorting are the key petrographic factors affecting porosity and permeability in unconsolidated siliciclastic sediments [Chilingarian and Wolf, 1976; Blatt et al., 1980; Wolf and Chilingarian, 1976; Beard and Weft, 1973]. Less well established are quantitative relations between key petrographic factors and hydraulic properties. It is known that estimation of subsurface material propert, ies made with quantitative petrophysical relations is often subject to much uncertainty [Taylor et al., 1987]. Two basic approaches have been applied to transform grain size distributions to hydraulic properties: (1) average hydraulic properties can be based on average grain diameters using tabulated values or nomographs; and (2) hydraulic properties can be calculated from grain size distributions using quantitative relations developed either empirically or from the theory of flow through porous media. The grain size approach requires tables of hydraulic conductivity and porosity versus average grain size [Freeze and Cherry, 1979; Marsily, 1986; Domenico and Schwartz, 1990]. Such tables do not consider sorting, porosity variations, and intrapore finegrained materials and cements. To include sorting, nomographs have been developed that plot porosity and permeability versus grain size and sorting for coarse to very fine sands [Masch and Denny, 19...

Large-scale process simulation was used to reconstruct the geologic evolution during the past 600,000 years of an alluvial fan in northern California. In order to reproduce the sedimentary record, the simulation accounted for the dynamics of river flooding, sedimentation, subsidence, land movement that resulted from faulting, and sea level changes. Paleoclimatic trends induced fluctuations in stream flows and dominated the development of the sedimentary deposits. The process simulation approach serves as a quantitative means to explore the genesis of sedimentary architecture and its link to past climatic conditions and fault motion.

We have developed groundwater flow models to explore the possible relationship between wastewater injection and the 12 November 2014 Mw 4.8 Milan, Kansas earthquake. We calculate pore pressure increases in the uppermost crust using a suite of models in which hydraulic properties of the Arbuckle Formation and the Milan earthquake fault zone, the Milan earthquake hypocenter depth, and fault zone geometry are varied. Given pre‐earthquake injection volumes and reasonable hydrogeologic properties, significantly increasing pore pressure at the Milan hypocenter requires that most flow occur through a conductive channel (i.e., the lower Arbuckle and the fault zone) rather than a conductive 3‐D volume. For a range of reasonable lower Arbuckle and fault zone hydraulic parameters, the modeled pore pressure increase at the Milan hypocenter exceeds a minimum triggering threshold of 0.01 MPa at the time of the earthquake. Critical factors include injection into the base of the Arbuckle Formation and proximity of the injection point to a narrow fault damage zone or conductive fracture in the pre‐Cambrian basement with a hydraulic diffusivity of about 3–30 m2/s. The maximum pore pressure increase we obtain at the Milan hypocenter before the earthquake is 0.06 MPa. This suggests that the Milan earthquake occurred on a fault segment that was critically stressed prior to significant wastewater injection in the area. Given continued wastewater injection into the upper Arbuckle in the Milan region, assessment of the middle Arbuckle as a hydraulic barrier remains an important research priority.

A distributed parameter ground‐water management model was developed using finite‐difference approximations of the governing partial differential equation for ground‐water flow in a confined aquifer. The resulting system of simultaneous equations was embedded in an optimization model, which used hydraulic heads and pumpages as unknown state variables and decision variables, respectively. The model was applied to several hypothetical examples of varying size and complexity and to one real‐world example, the Las Vegas Valley aquifer. The grid spacing, time increment, bounds placed on pumpage, and the number of constraints were found to affect not only the execution time, but also whether the model would execute at all. In order to accommodate long‐term management goals, a lumped transient model should be used which incorporates all time periods the model encompasses. Some objective functions, such as the maximization of the head, can be executed in a stepwise manner in which the final heads from the current period are used as the initial heads for the next period. The embedding technique proved useful for small management problems, but had numerical difficulties with the large real‐world problems of considerable heterogeneity. Unless the embedding technique can become computationally efficient and stable, it should be bypassed in favor of the response matrix approach.