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We performed a set of numerical simulations to characterize the interplay of fracture network topology, upscaling, and mesh refinement on flow and transport properties in fractured porous media. We generated a set of generic three-dimensional discrete fracture networks at various densities, where the radii of the fractures were sampled from a truncated power-law distribution, and whose parameters were loosely based on field site characterizations. We also considered five network densities, which were defined using a dimensionless version of density based on percolation theory. Once the networks were generated, we upscaled them into a single continuum model using the upscaled discrete fracture matrix model presented by Sweeney et al. (2019). We considered steady, isothermal pressure-driven flow through each domain and then simulated conservative, decaying, and adsorbing tracers using a pulse injection into the domain. For each simulation, we calculated the effective permeability and solute breakthrough curves as quantities of interest to compare between network realizations. We found that selecting a mesh resolution such that the global topology of the upscaled mesh matches the fracture network is essential. If the upscaled mesh has a connected pathway of fracture (higher permeability) cells but the fracture network does not, then the estimates for effective permeability and solute breakthrough will be incorrect. False connections cannot be eliminated entirely, but they can be managed by choosing appropriate mesh resolution and refinement for a given network. Adopting octree meshing to obtain sufficient levels of refinement leads to fewer computational cells (up to a 90% reduction in overall cell count) when compared to using a uniform resolution grid and can result in a more accurate continuum representation of the true fracture network. [ABSTRACT FROM AUTHOR]

The inherent uncertainty of subsurface fracture characteristics requires an ensemble-based approach where multiple network realizations are generated to represent a single physical system. However, the computational cost of these simulations is often prohibitive for carrying out an adequate number of simulations to obtain stable statistics for many quantities of interest, including the first passage time distribution (FPTD) of particles passing through the system. We characterize how variability induced by stochastic representations of subsurface fracture networks propagates into the FPTD. We simulate flow and transport on a large ensemble of three-dimensional fracture networks, observe the quantiles of the first passage time distributions, and characterize the network structure using coarse graph-based representations. Through analysis of the first passage times and graphs, we identify key geostructural features which explain variation in the FPTD. These features integrate hydrological fracture properties (permeability) with topological attributes. Using these features, we fit both parametric and nonparametric regression models to predict FPTD with uncertainty, compare the relative performance of each model in terms of error and coverage, and discuss possible model extensions. Models are validated using a held-out set of networks independently generated under different parameter settings. These nonparametric regression models can flexibly account for nonlinear relationships while allowing prediction intervals to adjust accurately depending on input feature values. [ABSTRACT FROM AUTHOR]

While several studies have linked network and in-fracture scale properties to conservative transport behavior in subsurface fractured media, studies on reactive transport cases remain relatively underdeveloped. In this study, we explore the behavior of an irreversible kinetic reaction during the interaction of two solute plumes, one consisting of species A and the other species B. When the plumes converge, these species react kinetically to form a new species C via A + B → k C . This reactive system is studied using a three-dimensional discrete fracture network (DFN) model coupled with reactive Lagrangian particle tracking. We find that the interplay of network topology and chemical properties of the reactive solutes controls reactive transport processes. The network topology drives species A and B together, and the chemical properties dictate whether and how quickly a reaction occurs. Results demonstrate that reactions are most likely to occur in high-velocity fractures that make up the network backbone. The interplay between species' chemical properties and transport is characterized by a non-dimensional Damköhler (Da) number. We show that the spatial distribution of reactions is sensitive to Da, which subsequently influences late-time tailing behavior in outlet breakthrough time distributions. The results of this study provide initial insights into how an irreversible reaction occurs during transport in a fracture network, using a methodology that can be applied to study reactive transport in a wide range of fractured media environments and contexts. Article Highlights: Discrete Fracture Network models are used to study reactive transport behavior. We consider the irreversible chemical reaction A + B → C Reactions primarily occur in the network backbone and reaction locations are sensitive to chemical properties. [ABSTRACT FROM AUTHOR]

Fluid flow in fractures has the potential to drastically change the economic, environmental, and safety risks associated with a subsurface operation. This work focuses on the analysis of experimental shear fracturing and subsequent shear displacement of a shale specimen that contains multiple pre-existing natural fractures. A triaxial direct shear apparatus with concurrent µCT imaging allowed continuous monitoring of sample permeability and changes in fracture geometry during fracture creation, displacement, and changes in stress. Steady-state, low Reynolds number fluid flow simulations were used to determine the changes of each individual fracture's hydraulic aperture due to displacement and effective stress. The development of anisotropy in the hydraulic aperture of a generated shear fracture was determined for shear displacements of 0.38, 0.74, and 2.04 mm. At high confining stress (30 MPa), shear displacement significantly reduced the hydraulic aperture of the shear fracture to below flow detectable limits and X-ray imaging resolution (23 × 23 × 23 µm3). The shear generated fracture was found to be more sensitive to changes in effective stress compared to the pre-existing natural fractures. Existing models derived from fluid flow equations accounting for stochastic fracture roughness were able to partially fit the simulated flow properties of the pre-existing fractures, but no model was found to fit the flow properties of the newly formed shear fracture. These results are the first report of simultaneous changes in the geometry and flow properties of surrounding, parallel natural fractures as a shear fracture is generated with varying stress and shear displacement conditions. Article Highlights: Flow anisotropy in a shear generated fracture varies based on shear displacement. Fracture flow channelization increases with shear displacement. Flow in a newly generated shear fracture has a larger response to changes in stress than pre-existing natural fractures. [ABSTRACT FROM AUTHOR]

We present a novel workflow for forecasting production in unconventional reservoirs using reduced-order models and machine-learning. Our physics-informed machine-learning workflow addresses the challenges to real-time reservoir management in unconventionals, namely the lack of data (i.e., the time-frame for which the wells have been producing), and the significant computational expense of high-fidelity modeling. We do this by applying the machine-learning paradigm of transfer learning, where we combine fast, but less accurate reduced-order models with slow, but accurate high-fidelity models. We use the Patzek model (Proc Natl Acad Sci 11:19731–19736, https://doi.org/10.1073/pnas.1313380110, 2013) as the reduced-order model to generate synthetic production data and supplement this data with synthetic production data obtained from high-fidelity discrete fracture network simulations of the site of interest. Our results demonstrate that training with low-fidelity models is not sufficient for accurate forecasting, but transfer learning is able to augment the knowledge and perform well once trained with the small set of results from the high-fidelity model. Such a physics-informed machine-learning (PIML) workflow, grounded in physics, is a viable candidate for real-time history matching and production forecasting in a fractured shale gas reservoir. [ABSTRACT FROM AUTHOR]

We present a topology-based method for mesh-partitioning in three-dimensional discrete fracture network (DFN) simulations that takes advantage of the intrinsic multi-level nature of a DFN. DFN models are used to simulate flow and transport through low-permeability fractured media in the subsurface by explicitly representing fractures as discrete entities. The governing equations for flow and transport are numerically integrated on computational meshes generated on the interconnected fracture networks. Modern high-fidelity DFN simulations require high-performance computing on multiple processors where performance and scalability depends partially on obtaining a high-quality partition of the mesh to balance work-loads and minimize communication across all processors. The discrete structure of a DFN naturally lends itself to various graph representations, which can be thought of as coarse-scale representations of the computational mesh. Using this concept, we develop two applications of the multilevel graph partitioning algorithm to partition the mesh of a DFN. In the first, we project a partition of the graph based on the DFN topology onto the mesh of the DFN and in the second, this DFN-based projection is used as the initial condition for further partitioning refinement of the mesh. We compare the performance of these methods with standard multi-level graph partitioning using graph-based metrics (cut, imbalance, partitioning time), computational-based metrics (FLOPS, iterations, solver time), and total run time. The DFN-based and the mesh-based partitioning methods are comparable in terms of the graph-based metrics, but the time required to obtain the partition is several orders of magnitude faster using the DFN-based partitions. The computation-based metrics show comparable performance between both methods so, in combination, the DFN-based partitions are several orders of magnitude faster than the mesh-based partition. Moreover, the method which uses the DFN-partition solution as the initial condition of the mesh partition provided cut and imbalance values that were close to the mesh-based partition but in a fraction of the time. In turn, this hybrid method outperformed both of the other methods in terms of the total run time. [ABSTRACT FROM AUTHOR]

The phenomenon of flow-channeling, or the existence of preferential pathways for flow in fracture networks, is well known. Identification of the channels ("backbone") allows for system reduction and computational efficiency in simulation of flow and transport through fracture networks. However, the purpose of machine learning techniques for backbone identification in fractured media is two-pronged system reduction for computational efficiency in simulation of flow and transport as well as physical insight into the phenomenon of flow channeling. The most critical aspect of this problem is the need to have a truly "physics-informed" technique that respects the constraint of connectivity. We present a method that views a network as a union of connected paths with each path comprising a sequence of fractures. Thus, the fundamental unit of selection becomes a sequence of fractures, classified based on attributes that characterize the sequence. In summary, this method represents a parametrization of the sample space that ensures every selected sample sub-network (which is the union of all selected sequences of fractures) always respects the constraint of connectivity, demonstrating that it is a truly physics-informed method. The algorithm has a user-defined parameter which allows control of the backbone size when using the random forest or logistic regression classifier. Even when the backbones are less than 30% in size (leading to computational savings), the backbones still capture the behavior of the breakthrough curve of the full network. Moreover, there is no need to check for path connectedness in the backbones unlike previous methods since the backbones are guaranteed to be connected. [ABSTRACT FROM AUTHOR]

We develop a computational geometry-based upscaling approach to accurately capture the dynamic processes occurring within and between subsurface fracture networks and the surrounding porous matrix named the Upscaled Discrete Fracture Matrix model (UDFM). Fracture attributes (orientations and apertures) are upscaled and combined with matrix attributes (permeability and porosity) into properties of an unstructured spatially variable Delaunay tetrahedral continuum mesh. The resolution of the mesh depends on proximity to the fractures to preserve the geometric and topological integrity of the underlying fracture network as well as increasing the accuracy of gradients in the dynamics between the fractures and matrix. The UDFM Delaunay mesh and its dual Voronoi mesh can be used in existing multiphysics simulators for flow, solute/heat mass transport, and geomechanics, thereby eliminating the need for the additional development of numerical methods to couple processes in fracture/matrix systems. The model is verified by comparing flow, transport, and coupled heat-mass flow simulations on the provided meshes against analytical and numerical benchmarks. We also provide an additional example to demonstrate the applicability of the UDFM to complex heteregenous fractured media. Overall, the UDFM is accurate in all of the cases considered here and presents an attractive alternative to other modeling strategies that require novel numerical methods or multi-dimensional meshes. [ABSTRACT FROM AUTHOR]

We present a set of multiphase flow simulations where supercritical CO 2 (scCO 2 ) displaces water at hydrostatic conditions within three-dimensional discrete fracture networks that represent paths for potential leakage through caprock above CO 2 storage reservoirs. The simulations are performed to characterize and compare the relative impact of hydraulic and structural heterogeneity in fractured media on the initial movement of scCO 2 through these caprock formations. In one scenario, intrinsic fracture permeabilities are varied stochastically within a fixed network structure. In another scenario, we generate multiple independent, identically distributed network realizations with varying fracture network densities to explore a wide range of geometric and topological configurations. Analysis of the simulations indicates that network structure, specifically connectivity and the presence of hanging fractures, plays a larger role in controlling the displacement of water by scCO 2 than variations in local hydraulic properties. We identify active surface area of the network as a single-phase feature that could provide a lower bound on the percentage of the network surface area reached by scCO 2 . [ABSTRACT FROM AUTHOR]

Discrete fracture networks (DFN) are often used to model flow and transport in fractured porous media. The accurate resolution of flow and transport behavior on a large DFN involving thousands of fractures is computationally expensive. This makes uncertainty quantification studies of quantities of interest such as travel time through the network computationally intractable, since hundreds to thousands of runs of the DFN model are required to get bounds on the uncertainty of the predictions. Prior works on the subject demonstrated that the complexity of a DFN could be reduced by considering a sub-network of it (often termed a "backbone" sub-network), one whose flow and transport properties were then shown to be similar to that of the full network. The technique is tantamount to partitioning the complete set of fractures of a network into two disjoint sets, one of which is the backbone sub-network while the other its complement. It is in this context that we present a system-reduction technique for DFNs using supervised machine learning via a Random Forest Classifier that selects a backbone sub-network from the full set of fractures. The in-sample errors (in terms of precision and recall scores) of the trained classifier are found to be very accurate indicators of the out-of-sample errors, thus exhibiting that the classifier generalizes well to test data. Moreover, this system-reduction technique yields sub-networks as small as 12% of the full DFN that still recover transport characteristics of the full network such as the peak dosage and tailing behavior for late times. Most importantly, the sub-networks remain connected, and their size can be controlled by a single dimensionless parameter. Furthermore, measures of KL-divergence and KS-statistic for the breakthrough curves of the sub-networks with respect to the full network show physically realistic trends in that the measures decrease monotonically as the size of the sub-networks increase. The computational efficiency gained by this technique depends on the size of the sub-network, but large reductions in computational time can be expected for small sub-networks, yielding as much as 90% computational savings for sub-networks that are as small as 10-12% of the full network. [ABSTRACT FROM AUTHOR]

We propose a multi-fidelity system reduction technique that uses weighted graphs paired with three-dimensional discrete fracture network (DFN) modelling for efficient simulation of subsurface flow and transport in fractured media. DFN models are used to simulate flow and transport in subsurface fractured rock with low-permeability. One method to alleviate the heavy computational overhead associated with these simulations is to reduce the size of the DFN using a graph representation of it to identify the primary flow sub-network and only simulate flow and transport thereon. The first of these methods used unweighted graphs constructed solely on DFN topology and could be used for accurate predictions of first-passage times. However, these techniques perform poorly when predicting later stages of the mass breakthrough. We utilize a weighted-graph representation of the DFN where edge weights are based on hydrological parameters in the DFN that allows us to exploit the kinematic quantities derivable a posteriori from the flow solution obtained on the graph representation of the DFN to perform system reduction and predict the later stages of the breakthrough curve with high fidelity. We also propose and demonstrate the use of an adaptive pruning algorithm with error control that produces a pruned DFN sub-network whose predicted mass breakthrough agrees with the original DFN within a user-specified tolerance. The method allows for the level of accuracy to be a user-controlled parameter. [ABSTRACT FROM AUTHOR]

Structural and topological information play a key role in modeling flow and transport through fractured rock in the subsurface. Discrete fracture network (DFN) computational suites such as DFNWORKS (Hyman et al. Comput. Geosci. 84, 10-19 2015) are designed to simulate flow and transport in such porous media. Flow and transport calculations reveal that a small backbone of fractures exists, where most flow and transport occurs. Restricting the flowing fracture network to this backbone provides a significant reduction in the network’s effective size. However, the particle-tracking simulations needed to determine this reduction are computationally intensive. Such methods may be impractical for large systems or for robust uncertainty quantification of fracture networks, where thousands of forward simulations are needed to bound system behavior. In this paper, we develop an alternative network reduction approach to characterizing transport in DFNs, by combining graph theoretical and machine learning methods. We consider a graph representation where nodes signify fractures and edges denote their intersections. Using random forest and support vector machines, we rapidly identify a subnetwork that captures the flow patterns of the full DFN, based primarily on node centrality features in the graph. Our supervised learning techniques train on particle-tracking backbone paths found by DFNWORKS, but run in negligible time compared to those simulations. We find that our predictions can reduce the network to approximately 20% of its original size, while still generating breakthrough curves consistent with those of the original network. [ABSTRACT FROM AUTHOR]

The publication of this article unfortunately contained a mistake. The assignment to the affiliations of author Satish Karra was not correct [ABSTRACT FROM AUTHOR]

We analyze the statistical scaling of structural attributes of virtual porous microstructures that are stochastically generated by thresholding Gaussian random fields. Characterization of the extent at which randomly generated pore spaces can be considered as representative of a particular rock sample depends on the metrics employed to compare the virtual sample against its physical counterpart. Typically, comparisons against features and/patterns of geometric observables, e.g., porosity and specific surface area, flow-related macroscopic parameters, e.g., permeability, or autocorrelation functions are used to assess the representativeness of a virtual sample, and thereby the quality of the generation method. Here, we rely on manifestations of statistical scaling of geometric observables which were recently observed in real millimeter scale rock samples [] as additional relevant metrics by which to characterize a virtual sample. We explore the statistical scaling of two geometric observables, namely porosity ( ϕ) and specific surface area ( SSA), of porous microstructures generated using the method of Smolarkiewicz and Winter [] and Hyman and Winter []. Our results suggest that the method can produce virtual pore space samples displaying the symptoms of statistical scaling observed in real rock samples. Order q sample structure functions (statistical moments of absolute increments) of ϕ and SSA scale as a power of the separation distance (lag) over a range of lags, and extended self-similarity (linear relationship between log structure functions of successive orders) appears to be an intrinsic property of the generated media. The width of the range of lags where power-law scaling is observed and the Hurst coefficient associated with the variables we consider can be controlled by the generation parameters of the method. [ABSTRACT FROM AUTHOR]

The aim of this study was to apply and evaluate a recently developed prioritization model which uses the synoptic approach to geographically prioritize watersheds in which Best Management Practices (BMPs) can be implemented to reduce water quality problems resulting from erosion and sedimentation. The model uses a benefit-cost framework to rank candidate watersheds within an ecoregion or river basin so that BMP implementation within the highest ranked watersheds will result in the most water quality improvement per conservation dollar invested. The model was developed to prioritize BMP implementation efforts in ecoregions containing watersheds associated with the USDA-NRCS Conservation Effects Assessment Project (CEAP). We applied the model to HUC-8 watersheds within the southeastern Coastal Plain ecoregion (USA) because not only is it an important agricultural area but also because it contains a well-studied medium-sized CEAP watershed which is thought to be representative of the ecoregion. The results showed that the three HUC-8 watersheds with the highest rankings (most water quality improvement expected per conservation dollar invested) were located in the southern Alabama, northern Florida, and eastern Virginia. Within these watersheds, measures of community attitudes toward conservation practices were highly ranked, and these indicators seemed to push the watersheds to the top of the rankings above other similar watersheds. The results, visualized as maps, can be used to screen and reduce the number of watersheds that need further assessment by managers and decision-makers within the study area. We anticipate that this model will allow agencies like USDA-NRCS to geographically prioritize BMP implementation efforts. [ABSTRACT FROM AUTHOR]

Understanding the best way to allocate limited resources is a constant challenge for water quality improvement efforts. The synoptic approach is a tool for geographic prioritization of these efforts. It uses a benefit-cost framework to calculate indices for functional criteria in subunits (watersheds, counties) of a region and then rank the subunits. The synoptic approach was specifically designed to incorporate best professional judgment in cases where information and resources are limited. To date, the synoptic approach has been applied primarily to local or regional wetland restoration prioritization projects. The goal of this work was to develop a synoptic model for prioritizing watersheds within which suites of agricultural best management practices (BMPs) can be implemented to reduce sediment load at the watershed outlets. The model ranks candidate watersheds within an ecoregion or river basin so that BMP implementation within the highest ranked watersheds will result in the most sediment load reduction per conservation dollar invested. The model can be applied anywhere and at many scales provided that the selected suite of BMPs is appropriate for the evaluation area's biophysical and climatic conditions. The model was specifically developed as a tool for prioritizing BMP implementation efforts in ecoregions containing watersheds associated with the USDA-NRCS conservation effects assessment project (CEAP). This paper presents the testing of the model in the little river experimental watershed (LREW) which is located near Tifton, Georgia, USA and is the CEAP watershed representing the southeastern coastal plain. The application of the model to the LREW demonstrated that the model represents the physical drivers of erosion and sediment loading well. The application also showed that the model is quite responsive to social and economic drivers and is, therefore, best applied at a scale large enough to ensure differences in social and economic drivers across the candidate watersheds. The prioritization model will be used for planning purposes. Its results are visualized as maps which enable resource managers to identify watersheds within which BMP implementation would result in the most water quality improvement per conservation dollar invested. [ABSTRACT FROM AUTHOR]

Presents a synoptic assessment intended to maximize the benefits to wetland species biodiversity gained through Clean Water Act regulatory efforts within the sub-basins in Missouri, Iowa, Nebraska and Kansas. Consideration of the risk of wetland species extirpation; Incorporation of landscape and anthropogenic factors; Suggestion that different management strategies should allow managers to cast their local decisions in the context of regional scale maintenance of wetland species biodiversity.

There are two issues in indicator development that have not been adequately addressed: (1) how to select an optimal combination of potentially redundant indicators that together best represent an endpoint, given cost constraints; (2) how to identify and evaluate indicators when the endpoint is unmeasured. This paper presents an approach to identifying and evaluating combinations of indicators when the mathematical relationships between the indicators and an endpoint may not be quantified, a limitation common to many ecological assessments. The approach uses the framework of Structural Equation Modeling (SEM), which combines path analysis with measurement models, to formalize available information about potential indicators and to evaluate their potential adequacy for representing an endpoint. Unlike traditional applications of SEM which require data on all variables, our approach – judgement-based SEM (JSEM) – can utilize expert judgement regarding the strengths and shapes of indicator-endpoint relationships. JSEM is applied in two stages. First, a conceptual model that relates variables in a network of direct and indirect linkages is developed, and is used to identify indicators relevant to an endpoint. Second, an index of indicator strength – i.e., the strength of the relationship between the endpoint and a set of indicators – is calculated from estimates of correlation between the modeled variables, and is used to compare alternative sets of indicators. The second stage is most appropriate for large, long-term assessments. Although JSEM is not a statistical technique, basing JSEM on SEM provides a structure for validating the conceptual model and for refining the index of indicator strength as data become available. Our main objective is to contribute to a rigorous and consistent selection of indicators even when knowledge about the ability of indicators to represent an endpoint is limited to expert judgement. [ABSTRACT FROM AUTHOR]

ABSTRACT / Past methods of prioritizing sites for protection and restoration have focused on lists of criteria or algorithms. These methods lack a common underlying framework, such that the process is explicit and repeatable, assumptions are highlighted, and commonalities and differences among prioritizations can be readily assessed. Our objective in this paper is to provide such a framework for cases where the goal of setting priorities is to maximize the ecological benefit gained from limited resources. We provide simple and general models that can be used to prioritize sites based on the projected ecological benefit per unit restoration or protection effort and to estimate the total projected benefit of restoring or protecting a set of sites. These models, which are based on an expression of the functional relationship between an end point and effort, hold up under a variety of situations and provide a common language for prioritization. We then discuss procedures for estimating model terms—calculations from regression curves when data are available, and use of judgement indicators when data are relatively limited. Finally, we present two case studies that apply the models and examine selected past prioritizations in the context of our framework. [ABSTRACT FROM AUTHOR]

Fractured systems are ubiquitous in natural and engineered applications as diverse as hydraulic fracturing, underground nuclear test detection, corrosive damage in materials and brittle failure of metals and ceramics. Microstructural information (fracture size, orientation, etc.) plays a key role in governing the dominant physics for these systems but can only be known statistically. Current models either ignore or idealize microscale information at these larger scales because we lack a framework that efficiently utilizes it in its entirety to predict macroscale behavior in brittle materials. We propose a method that integrates computational physics, machine learning and graph theory to make a paradigm shift from computationally intensive high-fidelity models to coarse-scale graphs without loss of critical structural information. We exploit the underlying discrete structure of fracture networks in systems considering flow through fractures and fracture propagation. We demonstrate that compact graph representations require significantly fewer degrees of freedom (dof) to capture micro-fracture information and further accelerate these models with Machine Learning. Our method has been shown to improve accuracy of predictions with up to four orders of magnitude speedup. [ABSTRACT FROM AUTHOR]

The placement of wetland restoration projects in a landscape to optimize the functional performance of wetlands on a regional scale is often overlooked. To address this problem, the U.S. Environmental Protection AgencyOs Landscape Function Project developed the synoptic approach to assign restoration priority to landscape subunits according to selected functional criteria. The approach provides a flexible, ecologically-based framework for allocating limited restoration-resources and preserving valued wetland functions on a landscape scale. We conducted a synoptic assessment for the Prairie Pothole Region of the north-central U.S. to demonstrate application of the method for our assessment criterion P the marginal decrease in total downstream flood volume per restoration dollar. A criterion is often not directly measurable but can be represented by an index composed of measurements onrelated variables. In a synoptic assessment, these measured variables, referred to as indicators, are limited to variables for which dataare existing, accessible, and uniformly available for the entire region. We developed a conceptual model to guide the development of an index of the assessment criterion. We then ranked landscape subunits based on index values and mapped the ranks to show relative priority for restoration among landscape subunits. We conducted a series of analyses to justify selection of indicators and some of our assumptions.The approach offers multiple options for processing and displaying information for use by wetland managers. [ABSTRACT FROM AUTHOR]

Indicators are used to draw conclusions about ecological endpoints when these endpoints cannot be measured directly. In many cases, inferences about an endpoint are only possible because assumptions have been made about the relationship between indicator and endpoint; we refer to such indicators as judgement indicators. The validity of inferences made using a judgement indicator can be gauged by examining the known or assumed form of the general relationship between indicator and endpoint. The rules for this kind of inference are a consequence of scale invariance, which originates from measurement theory. For simple indicators comprised of a single indicator measurement, the inferences allowed - equivalence, rank, equality of intervals, and equality of ratios - depend on whether the data are nominal, ordinal, interval, or ratio scaled. For composite indicators containing two or more simple indicators, inferences are also affected by the mathematical form of combination; e.g., whether the terms are summed or multiplied.Standardizing simple or composite indicators can allow inferences about the relative importance of observations, based on the natural range of occurrence. Scale invariance is a particularly important consideration in landscape assessments, since these often make use of judgement indicators. [ABSTRACT FROM AUTHOR]