Robert W. Ritzi

Behavior of Indicator Variograms and Transition Probabilities in Relation to the Variance in Lengths of Hydrofacies

When using indicator geostatistics to represent the distribution of hydrofacies or lithofacies, the range and curvature of the autotransition probabilities or the autovariogram are related to the variation in facies lengths. As the coefficients of variation for the length of each facies increase toward unity, the effective range increases while the indicator-correlation structure evolves from a periodic linear structure to a somewhat periodic spherical structure to an aperiodic exponential structure. Multimodal distributions of facies lengths can give rise to autotransition probabilities or autovariograms that appear to have nested structures. With the understanding of these relationships it is possible to choose a model form and model parameters for the autotransition probability or autovariogram based upon the facies proportions and the modality, mean, and variance in length of the facies. This is illustrated with data from a case study.

Improving Permeability Semivariograms with Transition Probability Models of Hierarchical Sedimentary Architecture Derived from Outcrop-Analog Studies

Received 21 July 2004; revised 22 February 2005; accepted 23 March 2005; published 30 July 2005. [1] As analogs for aquifers, outcrops of sedimentary deposits allow sedimentary units to be mapped, permeability to be measured with high resolution, and sedimentary architecture to be related to the univariate and spatial bivariate statistics of permeability. Sedimentary deposits typically can be organized into hierarchies of unit types and associated permeability modes. The types of units and the number of hierarchical levels defined on an outcrop might vary depending upon the focus of the study. Regardless of how the outcrop sediments are subdivided, a composite bivariate statistic like the permeability semivariogram is a linear summation of the autosemivariograms and cross semivariograms for the unit types defined, weighted by the proportions and transition probabilities associated with the unit types. The composite sample semivariogram will not be representative unless data locations adequately define these transition probabilities. Data reflecting the stratal architecture can often be much more numerous than permeability measurements. These lithologic data can be used to better define transition probabilities and thus improve the estimates of the composite permeability semivariogram. In doing so, bias created from the incomplete exposure of units can be reduced by a Bayesian approach for estimating unit proportions and mean lengths. We illustrate this ^

Comparing statistical models of physical heterogeneity in buried‐valley aquifers

The hypothesis that physical heterogeneity has similarities in separate aquifers created by similar depositional environments is tested by comparing statistical characteristics of facies assemblages. The comparisons are made for a number of data-rich sites in two buried-valley aquifers in the North American midcontinent: the White River aquifer in Indiana and the Miami Valley aquifer in Ohio. These were proglacial valleys that directed drainage away from Quaternary ice margins and were filled with glaciofluvial sediments: predominantly sand and gravel (s) lithofacies, with interbedded mud and diamicton (m) lithofacies. At scales encompassing assemblages of both lithofacies m and s, permeability is strongly bimodal. We find that it is useful to compare statistics that characterize the proportions, geometry, and spatial distribution of each facies. The results give rise to a general model for heterogeneity in valley-fill sediments along the proglacial sluiceway in both aquifers. The proportion of facies m is ∼15%. The mean thickness of facies m is 3.5 m and of the order of 10 m for facies s. The coefficient of variation in thickness for either facies is of the order of 1, with thickness ranging over orders of magnitude. Correspondingly, the vertical autotransition probabilities are exponential, and they are relatively symmetric with effective range of the order of 10 m. The lateral facies lengths are indicated to vary over orders of magnitude and to be multimodally distributed, with mean lengths of the order of 102 m, effective range in correlation structure of the order of 103 m, and lateral anisotropy ratio <2. There is some variation in how the facies m are vertically embedded within the facies s. The White River aquifer and areas in the Miami aquifer have facies proportions relatively stationary with elevation. In other areas of the Miami, there are near-horizontal zones having relatively higher or lower proportions. However, such variations on the general model give rise to similar statistics for mass transport within the context of a relevant remediation problem and thus would lead to a similar conclusion or decision. Thus one general model is applicable to both aquifers in this context. In a broader sense, we have illustrated a method by which other examples developed from data-rich sites can be compared.

Hydrofacies distribution and correlation in the Miami Valley Aquifer System

This study combines geostatistical analyses with geologic interpretations to further the quantitative understanding of physical heterogeneity within glaciofluvial aquifers. The stochastic simulation of aquifers requires quantitative measures of heterogeneity, including both cumulative distribution and spatial correlation functions. The heterogeneity in glaciofluvial aquifers is typified by low-permeability facies (e.g., till or lacustrine clay) juxtaposed with high-permeability facies (e.g., sand and gravel outwash). The Miami Valley aquifer system was examined at multiple sites for the spatial distribution and correlation of these two hydrofacies. Binary indicator geostatistics were used to quantitatively determine, at each site, the relative volume of each hydrofacies, their spatial distribution, the major principal direction of their spatial correlation, the minor principal direction, and the correlation range in these directions. The percent by volume of the system that is aquitard material decreases down the valley, from 31% to 12%. Each site has an elevation zone with more aquitard material relative to other elevations at that site. The percent aquitard material in these zones decreases down the valley from 45% to 22%. The maximum principal direction of spatial correlation in the aquitard zones generally is NE-SW, subparallel to the trend of the bedrock valley, with a range of the order of 0.75 km and minimum/maximum anisotropy ratio of 0.4. The locations exceeding 0.5 probability of aquitard occurrence generally occur on the valley margins. Thus, among the sites investigated, there is a trend down the valley in the ratio of aquitard volume to aquifer volume, and the spatial correlation and distribution of aquitard material are similar within the aquitard zones. Furthermore, these findings are consistent with aspects of the subglacial and proglacial depositional environments responsible for the facies assemblage and thus are likely to be applicable to other parts of the aquifer system and other aquifer systems where similar geologic processes are inferred to have existed.

Influence of small‐scale fluvial architecture on CO2 trapping processes in deep brine reservoirs

A number of important candidate CO2 reservoirs exhibit sedimentary architecture reflecting fluvial deposition. Recent studies have led to new conceptual and quantitative models for sedimentary architecture in fluvial deposits over a range of scales that are relevant to CO2 injection and storage. We used a geocellular modeling approach to represent this multiscaled and hierarchical sedimentary architecture. With this model, we investigated the dynamics of CO2 plumes, during and after injection, in such reservoirs. The physical mechanism of CO2 trapping by capillary trapping incorporates a number of related processes, i.e., residual trapping, trapping due to hysteresis of the relative permeability, and trapping due to hysteresis of the capillary pressure. Additionally, CO2 may be trapped due to differences in capillary entry pressure for different textural sedimentary facies (e.g., coarser-grained versus finer-grained cross sets). The amount of CO2 trapped by these processes depends upon a complex system of nonlinear and hysteretic characteristic relationships including how relative permeability and capillary pressure vary with brine and CO2 saturation. The results strongly suggest that representing small-scale features (decimeter to meter), including their organization within a hierarchy of larger-scale features, and representing their differences in characteristic relationships can all be critical to understanding trapping processes in some important candidate CO2 reservoirs.

Identifying geochemical processes by inverse modeling of multicomponent reactive transport in the Aquia aquifer

Modeling reactive geochemical transport in the subsurface is a powerful tool for understanding and interpreting geochemical processes in aquifer systems. Different conceptual models can include different combinations of geochemical processes. A limitation of current inverse models is that they are based only on one conceptual model, which may lead to statistical bias and underestimation of uncertainty. We present a stepwise inverse modeling methodology that can include any number of conceptual models and thus consider alternate combinations of processes, and it can provide a quantitative basis for selecting the best among them. We applied the inverse methodology to modeling the geochemical evolution in the Aquia aquifer (Maryland, USA) over 105 yr. The inverse model accounts for aqueous complexation, acid-base and redox reactions, cation exchange, proton surface complexation, and mineral dissolution and precipitation; identifi es relevant geochemical processes; and estimates key reactive transport parameters from available hydrogeochemical data. Inverse modeling provides optimum estimates of transmissivities, leakage rates, dispersivities, cation exchange capacity (CEC), cation selectivities, and initial and boundary concentrations of selected chemical components. Inverse modeling with multiple conceptual models helps to identify the most likely physical and chemical processes in the paleohydrology and paleogeochemistry of the Aquia aquifer. Identifi cation criteria derived from information theory are used to select the best among ten candidate conceptual models. In the fi nal model, both proton surface complexation and methane oxidation are identifi ed as relevant geochemical processes.

The Estimation of Fluid Flow Properties from the Response of Water Levels in Wells to the Combined Atmospheric and Earth Tide Forces

The water level in an open well tapping a confined formation is influenced by natural forces including the solid Earth tide (SET) and atmospheric pressure variation (APV). The spectral method is used to analytically derive a model for well response to both random and periodic components of the combined SET and APV forcings (CSA). An inverse theory and algorithm are developed in order to provide improved results when using the model to estimate the hydraulic parameters associated with a given formation. An examination of the response surface of the estimation criterion reveals a uniqueness problem in estimating storativity (S). Since there is little correlation between the transmissivity (T) and S estimators, a good estimate for T is still possible independent of having accurate knowledge of S. An estimate of T is possible only if the data contain sufficient information so that the analysis occurs within an identifiability window. The CSA estimation methodology is compared to individual SET and APV schemes. The CSA scheme gives the greatest probability that sufficient information is contained in a data record so that T is indeed identifiable. The results of applications to synthetic data indicate that the CSA scheme gives a T estimate with the most precision and also that it requires collecting fewer observations. These results are discussed in light of practical considerations when designing data collection procedures.

The Influence of Streambed Heterogeneity on Hyporheic Flow in Gravelly Rivers

Deposits of open-framework gravel occurring in gravelly streambeds can exert a significant influence on hyporheic flow. The influence was quantified using a numerical model of the hyporheic zone. The model included open-framework gravel stratasets represented with commonly observed characteristics including a volume fraction of about one-third of the streambed sediment, a hydraulic conductivity two orders of magnitude greater than other strata present, and a spatial connectivity forming preferential-flow pathways. The influence of open-framework gravel stratasets on hyporheic flow was much greater than the influence of the channel morphology including meanders, point bars, dunes, and ripples. Seventy percent of the total hyporheic exchange occurred across 30% of the channel boundary at locations of open-framework gravel stratasets. The maximum local interfacial flux rates occurred at these locations, and were orders of magnitude greater than those at other locations. The local flux rates varied by six orders of magnitude over the channel boundary. The composite flow rate through the model with open-framework gravel stratsets was an order of magnitude greater than that through an equivalent but homogeneous model.

Relating reactive solute transport to hierarchical and multiscale sedimentary architecture in a Lagrangian-based transport model: 1. Time-dependent effective retardation factor

This series of papers addresses the transport of reactive solutes in groundwater. In part 1, the time-dependent effective retardation factor, R∼eff(t), of reactive solutes undergoing equilibrium sorption is linked to hierarchical stratal architecture using a Lagrangian-based transport model. The model is based on hierarchical expressions of the spatial covariance of the log distribution coefficient, Ξ=ln⁡(Kd), and the spatial cross covariance between Ξ and the log permeability, Y=ln⁡(k). The spatial correlation structure in these covariance expressions is the probability of transitioning across strata types of different scales, and they are parameterized by independent and quantifiable physical attributes of sedimentary architecture including univariate statistics for Y, Ξ, and the proportions and lengths of facies. Nothing is assumed about Y- Ξ point correlation; it is allowed to differ by facies type. The duration of the time-dependent change in R∼eff(t) is a function of the effective ranges of the cross-transition probability structures (i.e., the ranges of indicator correlation structures) for each scale of stratal architecture. The plume velocity and the effective retardation stabilize at a large-time limit after the plume centroid has traveled a distance that encompasses the effective ranges of these cross-transition probability structures. The well-documented perchloroethene (PCE) tracer test at the Borden research site is used to illustrate the model. The model gives a viable explanation for the observed PCE plume deceleration, and thus the observed R∼eff(t) can be explained by the process of linear equilibrium sorption and the heterogeneity in k and Kd. In part 2 [Soltanian et al., 2015a], reactive plume dispersion, as quantified by the particle displacement variance is linked to stratal architecture using a Lagrangian-based transport model.

Understanding the impact of open-framework conglomerates on water–oil displacements: the Victor interval of the Ivishak Reservoir, Prudhoe Bay Field, Alaska

The Victor Unit of the Ivishak Formation in the Prudhoe Bay Oilfield is characterized by high net-to-gross fluvial sandstones and conglomerates. The highest permeability is found within sets of cross-strata of open-framework conglomerate (OFC). These cross-strata are preserved within unit-bar deposits and assemblages of unit-bar deposits within compound (braid)-bar deposits, and may form thief zones limiting enhanced oil recovery. We incorporate recent research that has quantified important attributes of preserved sedimentary architecture into high-resolution models. Waterflooding experiments using these models demonstrate the control that such architecture has on oil production rate, water breakthrough time, and spatial and temporal distribution of residual oil saturation. We found that when the pressure gradient is orientated perpendicular to the palaeoflow direction, the total oil production and the water breakthrough time are larger, and the remaining oil saturation is smaller, than when it is orientated parallel to palaeoflow. The pressure difference between production and injection wells does not affect sweep efficiency, although the spatial distribution of oil remaining in the reservoir critically depends on this value. Oil sweep efficiency decreases slightly with increase in the proportion of OFC cross-strata. Whether or not clusters of connected OFC span the domain does not visibly affect sweep efficiency.