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Comparison of two probability distributions used to model sizes of undiscovered oil and gas accumulations: Does the tail wag the assessment?

Undiscovered oil and gas assessments are commonly reported as aggregate estimates of hydrocarbon volumes. Potential commercial value and discovery costs are, however, determined by accumulation size, so engineers, economists, decision makers, and sometimes policy analysts are most interested in projected discovery sizes. The lognormal and Pareto distributions have been used to model exploration target sizes. This note contrasts the outcomes of applying these alternative distributions to the play level assessments of the U.S. Geological Survey's 1995 National Oil and Gas Assessment. Using the same numbers of undiscovered accumulations and the same minimum, medium, and maximum size estimates, substitution of the shifted truncated lognormal distribution for the shifted truncated Pareto distribution reduced assessed undiscovered oil by 16% and gas by 15%. Nearly all of the volume differences resulted because the lognormal had fewer larger fields relative to the Pareto. The lognormal also resulted in a smaller number of small fields relative to the Pareto. For the Permian Basin case study presented here, reserve addition costs were 20% higher with the lognormal size assumption. ?? 2002 International Association for Mathematical Geology.

Mathematical Geology

A compositional multiphase model for groundwater contamination by petroleum products: 1. Theoretical considerations

A mathematical model is developed to describe the fate of hydrocarbon constituents of petroleum products introduced to soils as an immiscible liquid from sources such as leaking underground storage tanks and ruptured pipelines. The problem is one of multiphase transport (oil (immiscible), air, and water phases) of a reactive contaminant with constituents such as benzene, toluene, and xylene found in refined petroleum products like gasoline. In the unsaturated zone, transport of each constituent can occur as a solute in the water phase, vapor in the air phase, and as an unaltered constituent in the oil phase. Additionally, the model allows for adsorption. Molecular transformations, microbially mediated or abiotic, are incorporated as sink terms in the conservation of mass equations. An equilibrium approximation, applicable to any immiscible organic contaminant is applied to partition constituent mass between the air, oil, water, and adsorbed phases for points in the region where the oil phase exists. Outside the oil plume the equilibrium approximation takes on a simpler form to partition constituent mass between the air, water, and adsorbed phases only. Microbial degradation of petroleum products is first discussed in a general model, then the conservation of mass equation for oxygen is incorporated into the analysis which takes advantage of the key role played by oxygen in the metabolism of hydrocarbon utilizing microbes in soil environments. Approximations to two subproblems, oil plume establishment in the unsaturated zone, and solute and vapor transport subsequent to immiscible plume establishment are then developed from the general model.

Water Resources Research

A three-dimensional method-of-characteristics solute-transport model (MOC3D)

This report presents a model, MOC3D, that simulates three-dimensional solute transport in flowing ground water. The model computes changes in concentration of a single dissolved chemical constituent over time that are caused by advective transport, hydrodynamic dispersion (including both mechanical dispersion and diffusion), mixing (or dilution) from fluid sources, and mathematically simple chemical reactions (including linear sorption, which is represented by a retardation factor, and decay). The transport model is integrated with MODFLOW, a three-dimensional ground-water flow model that uses implicit finite-difference methods to solve the transient flow equation. MOC3D uses the method of characteristics to solve the transport equation on the basis of the hydraulic gradients computed with MODFLOW for a given time step. This implementation of the method of characteristics uses particle tracking to represent advective transport and explicit finite-difference methods to calculate the effects of other processes. However, the explicit procedure has several stability criteria that may limit the size of time increments for solving the transport equation; these are automatically determined by the program. For improved efficiency, the user can apply MOC3D to a subgrid of the primary MODFLOW grid that is used to solve the flow equation. However, the transport subgrid must have uniform grid spacing along rows and columns. The report includes a description of the theoretical basis of the model, a detailed description of input requirements and output options, and the results of model testing and evaluation. The model was evaluated for several problems for which exact analytical solutions are available and by benchmarking against other numerical codes for selected complex problems for which no exact solutions are available. These test results indicate that the model is very accurate for a wide range of conditions and yields minimal numerical dispersion for advection-dominated problems. Mass-balance errors are generally less than 10 percent, and tend to decrease and stabilize with time.

Water-Resources Investigations Report

Parallel Fortran-MPI software for numerical inversion of the Laplace transform and its application to oscillatory water levels in groundwater environments

A parallel Fortran-MPI (Message Passing Interface) software for numerical inversion of the Laplace transform based on a Fourier series method is developed to meet the need of solving intensive computational problems involving oscillatory water level's response to hydraulic tests in a groundwater environment. The software is a parallel version of ACM (The Association for Computing Machinery) Transactions on Mathematical Software (TOMS) Algorithm 796. Running 38 test examples indicated that implementation of MPI techniques with distributed memory architecture speedups the processing and improves the efficiency. Applications to oscillatory water levels in a well during aquifer tests are presented to illustrate how this package can be applied to solve complicated environmental problems involved in differential and integral equations. The package is free and is easy to use for people with little or no previous experience in using MPI but who wish to get off to a quick start in parallel computing. ?? 2004 Elsevier Ltd. All rights reserved.

Environmental Modelling and Software

Evaluation of six methods for estimating magnitude and frequency of peak discharges on urban streams in New York

Six methods of estimating peak discharges of urban streams were compared and evaluated for applicability to urban streams in New York. Discharge and frequency values developed from a series of synthesized annual flood records were compared with values obtained from the six methods. The synthesized flood records were computed from rainfall-runoff models of 11 urban basins in three counties across the State. Four of these basins had a sufficient period of record to enable rainfall-runoff modeling of two different 5-year periods so that increases in peak flow due to increased urbanization could also be used for comparison of the six methods. A graphical analysis and three types of mathematical analyses were made to evaluate the closeness of fit and bias of the methods. All methods showed a tendency to overestimate synthetic urban flood-magnitude values, but the two adjust rural flood-frequency estimates on a nationwide basis showed smallest standard errors of estimate and bias. The standard errors for these two methods ranged from 44 to 57 percent over the six recurrence intervals (2, 5, 10, 25, 50, and 100 year), and the bias ranged from +28 to +53 percent. The bias , however, is probably due to errors inherent in using synthetic records and in applying the New York rural flood-frequency equations to urban basins with small drainage areas. (USGS)

Water-Resources Investigations Report

Optimization techniques for integrating spatial data

Two optimization techniques ta predict a spatial variable from any number of related spatial variables are presented. The applicability of the two different methods for petroleum-resource assessment is tested in a mature oil province of the Midcontinent (USA). The information on petroleum productivity, usually not directly accessible, is related indirectly to geological, geophysical, petrographical, and other observable data. This paper presents two approaches based on construction of a multivariate spatial model from the available data to determine a relationship for prediction. In the first approach, the variables are combined into a spatial model by an algebraic map-comparison/integration technique. Optimal weights for the map comparison function are determined by the Nelder-Mead downhill simplex algorithm in multidimensions. Geologic knowledge is necessary to provide a first guess of weights to start the automatization, because the solution is not unique. In the second approach, active set optimization for linear prediction of the target under positivity constraints is applied. Here, the procedure seems to select one variable from each data type (structure, isopachous, and petrophysical) eliminating data redundancy. Automating the determination of optimum combinations of different variables by applying optimization techniques is a valuable extension of the algebraic map-comparison/integration approach to analyzing spatial data. Because of the capability of handling multivariate data sets and partial retention of geographical information, the approaches can be useful in mineral-resource exploration. ?? 1995 International Association for Mathematical Geology.

Mathematical Geology

Building on crossvalidation for increasing the quality of geostatistical modeling

The random function is a mathematical model commonly used in the assessment of uncertainty associated with a spatially correlated attribute that has been partially sampled. There are multiple algorithms for modeling such random functions, all sharing the requirement of specifying various parameters that have critical influence on the results. The importance of finding ways to compare the methods and setting parameters to obtain results that better model uncertainty has increased as these algorithms have grown in number and complexity. Crossvalidation has been used in spatial statistics, mostly in kriging, for the analysis of mean square errors. An appeal of this approach is its ability to work with the same empirical sample available for running the algorithms. This paper goes beyond checking estimates by formulating a function sensitive to conditional bias. Under ideal conditions, such function turns into a straight line, which can be used as a reference for preparing measures of performance. Applied to kriging, deviations from the ideal line provide sensitivity to the semivariogram lacking in crossvalidation of kriging errors and are more sensitive to conditional bias than analyses of errors. In terms of stochastic simulation, in addition to finding better parameters, the deviations allow comparison of the realizations resulting from the applications of different methods. Examples show improvements of about 30% in the deviations and approximately 10% in the square root of mean square errors between reasonable starting modelling and the solutions according to the new criteria.

Stochastic Environmental Research and Risk Assessm

Estimation of rod scale errors in geodetic leveling

Comparisons among repeated geodetic levelings have often been used for detecting and estimating residual rod scale errors in leveled heights. Individual rod-pair scale errors are estimated by a two-step procedure using a model based on either differences in heights, differences in section height differences, or differences in section tilts. It is shown that the estimated rod-pair scale errors derived from each model are identical only when the data are correctly weighted, and the mathematical correlations are accounted for in the model based on heights. Analyses based on simple regressions of changes in height versus height can easily lead to incorrect conclusions. We also show that the statistically estimated scale errors are not a simple function of height, height difference, or tilt. The models are valid only when terrain slope is constant over adjacent pairs of setups (i.e., smoothly varying terrain). In order to discriminate between rod scale errors and vertical displacements due to crustal motion, the individual rod-pairs should be used in more than one leveling, preferably in areas of contrasting tectonic activity. From an analysis of 37 separately calibrated rod-pairs used in 55 levelings in southern California, we found eight statistically significant coefficients that could be reasonably attributed to rod scale errors, only one of which was larger than the expected random error in the applied calibration-based scale correction. However, significant differences with other independent checks indicate that caution should be exercised before accepting these results as evidence of scale error. Further refinements of the technique are clearly needed if the results are to be routinely applied in practice.

Journal of Geophysical Research B: Solid Earth

Effective groundwater model calibration: With analysis of data, sensitivities, predictions, and uncertainty

Methods and guidelines for developing and using mathematical models Turn to Effective Groundwater Model Calibration for a set of methods and guidelines that can help produce more accurate and transparent mathematical models. The models can represent groundwater flow and transport and other natural and engineered systems. Use this book and its extensive exercises to learn methods to fully exploit the data on hand, maximize the model's potential, and troubleshoot any problems that arise. Use the methods to perform: Sensitivity analysis to evaluate the information content of data Data assessment to identify (a) existing measurements that dominate model development and predictions and (b) potential measurements likely to improve the reliability of predictions Calibration to develop models that are consistent with the data in an optimal manner Uncertainty evaluation to quantify and communicate errors in simulated results that are often used to make important societal decisions Most of the methods are based on linear and nonlinear regression theory. Fourteen guidelines show the reader how to use the methods advantageously in practical situations. Exercises focus on a groundwater flow system and management problem, enabling readers to apply all the methods presented in the text. The exercises can be completed using the material provided in the book, or as hands-on computer exercises using instructions and files available on the text's accompanying Web site. Throughout the book, the authors stress the need for valid statistical concepts and easily understood presentation methods required to achieve well-tested, transparent models. Most of the examples and all of the exercises focus on simulating groundwater systems; other examples come from surface-water hydrology and geophysics. The methods and guidelines in the text are broadly applicable and can be used by students, researchers, and engineers to simulate many kinds systems.

Book

Section 1. Simulation of surface-water integrated flow and transport in two-dimensions: SWIFT2D user's manual

A numerical model for simulation of surface-water integrated flow and transport in two (horizontal-space) dimensions is documented. The model solves vertically integrated forms of the equations of mass and momentum conservation and solute transport equations for heat, salt, and constituent fluxes. An equation of state for salt balance directly couples solution of the hydrodynamic and transport equations to account for the horizontal density gradient effects of salt concentrations on flow. The model can be used to simulate the hydrodynamics, transport, and water quality of well-mixed bodies of water, such as estuaries, coastal seas, harbors, lakes, rivers, and inland waterways. The finite-difference model can be applied to geographical areas bounded by any combination of closed land or open water boundaries. The simulation program accounts for sources of internal discharges (such as tributary rivers or hydraulic outfalls), tidal flats, islands, dams, and movable flow barriers or sluices. Water-quality computations can treat reactive and (or) conservative constituents simultaneously. Input requirements include bathymetric and topographic data defining land-surface elevations, time-varying water level or flow conditions at open boundaries, and hydraulic coefficients. Optional input includes the geometry of hydraulic barriers and constituent concentrations at open boundaries. Time-dependent water level, flow, and constituent-concentration data are required for model calibration and verification. Model output consists of printed reports and digital files of numerical results in forms suitable for postprocessing by graphical software programs and (or) scientific visualization packages. The model is compatible with most mainframe, workstation, mini- and micro-computer operating systems and FORTRAN compilers. This report defines the mathematical formulation and computational features of the model, explains the solution technique and related model constraints, describes the model framework, documents the type and format of inputs required, and identifies the type and format of output available.

Techniques and Methods

Enhancing the predictability of ecology in a changing world: A call for an organism-based approach

Ecology is usually very good in making descriptive explanations of what is observed, but is often unable to make predictions of the response of ecosystems to change. This has implications in a human-dominated world where a suite of anthropogenic stresses are threatening the resilience and functioning of ecosystems that sustain mankind through a range of critical regulating and supporting services. In ecosystems, cause-and-effect relationships are difficult to elucidate because of complex networks of negative and positive feedbacks. Therefore, being able to effectively predict when and where ecosystems could pass into different (and potentially unstable) new states is vitally important under rapid global change. Here, we argue that such better predictions may be reached if we focus on organisms instead of species, because organisms are the principal biotic agents in ecosystems that react directly on changes in their environment. Several studies show that changes in ecosystems may be accurately described as the result of changes in organisms and their interactions. Organism-based theories are available that are simple and derived from first principles, but allow many predictions. Of these we discuss Trait-based Ecology, Agent Based Models, and Maximum Entropy Theory of Ecology and show that together they form a logical sequence of approaches that allow organism-based studies of ecological communities. Combining and extending them makes it possible to predict the spatiotemporal distribution of groups of organisms in terms of how metabolic energy is distributed over areas, time, and resources. We expect that this “Organism-based Ecology” (OE) ultimately will improve our ability to predict ecosystem dynamics.

Frontiers in Applied Mathematics and Statistics

Modeling sandhill crane population dynamics

The impact of sport hunting on the Central Flyway population of sandhill cranes (Grus canadensis) has been a subject of controversy for several years. A recent study (Buller 1979) presented new and important information on sandhill crane population dynamics. The present report is intended to incorporate that and other information into a mathematical model for the purpose of assessing the long-range impact of hunting on the population of sandhill cranes.The model is a simple deterministic system that embodies density-dependent rates of survival and recruitment. The model employs four kinds of data: (1) spring population size of sandhill cranes, estimated from aerial surveys to be between 250,000 and 400,000 birds; (2) age composition in fall, estimated for 1974-76 to be 11.3% young; (3) annual harvest of cranes, estimated from a variety of sources to be about 5 to 7% of the spring population; and (4) age composition of harvested cranes, which was difficult to estimate but suggests that immatures were 2 to 4 times as vulnerable to hunting as adults.Because the true nature of sandhill crane population dynamics remains so poorly understood, it was necessary to try numerous (768 in all) combinations of survival and recruitment functions, and focus on the relatively few (37) that yielded population sizes and age structures comparable to those extant in the real population. Hunting was then applied to those simulated populations. In all combinations, hunting resulted in a lower asymptotic crane population, the decline ranging from 5 to 54%. The median decline was 22%, which suggests that a hunted sandhill crane population might be about three-fourths as large as it would be if left unhunted. Results apply to the aggregate of the three subspecies in the Central Flyway; individual subspecies or populations could be affected to a greater or lesser degree.

Special Scientific Report - Wildlife

State of stress in the lithosphere: Inferences from the flow laws of olivine

The experimental flow data for rocks and minerals are reviewed and found to fit a law of the form {Mathematical expression} where {Mathematical expression} This law reduces to the familiar power-law stress dependency at low stress and to an exponential stress dependency at high stress. Using the material flow law parameters for olivine, stress profiles with depth and strain rate are computed for a representative range of temperature distributions in the lithosphere. The results show that the upper 15 to 25 km of the oceanic lithosphere must behave elastically or fail by fracture and that the remainder deforms by exponential law flow at intermediate depths and by power-law flow in the rest. A model computation of the gravitational sliding of a lithospheric plate using olivine rheology exhibits a very sharp decoupling zone which is a consequence of the combined effects of increasing stress and temperature on the flow law, which is a very sensitive function of both. ?? 1977 Birkha??user Verlag.

Pure and Applied Geophysics PAGEOPH

August median streamflow on ungaged streams in eastern Aroostook County, Maine

Methods for estimating August median streamflow were developed for ungaged, unregulated streams in the eastern part of Aroostook County, Maine, with drainage areas from 0.38 to 43 square miles and mean basin elevations from 437 to 1,024 feet. Few long-term, continuous-record streamflow-gaging stations with small drainage areas were available from which to develop the equations; therefore, 24 partial-record gaging stations were established in this investigation. A mathematical technique for estimating a standard low-flow statistic, August median streamflow, at partial-record stations was applied by relating base-flow measurements at these stations to concurrent daily flows at nearby long-term, continuous-record streamflow- gaging stations (index stations). Generalized least-squares regression analysis (GLS) was used to relate estimates of August median streamflow at gaging stations to basin characteristics at these same stations to develop equations that can be applied to estimate August median streamflow on ungaged streams. GLS accounts for varying periods of record at the gaging stations and the cross correlation of concurrent streamflows among gaging stations. Twenty-three partial-record stations and one continuous-record station were used for the final regression equations. The basin characteristics of drainage area and mean basin elevation are used in the calculated regression equation for ungaged streams to estimate August median flow. The equation has an average standard error of prediction from -38 to 62 percent. A one-variable equation uses only drainage area to estimate August median streamflow when less accuracy is acceptable. This equation has an average standard error of prediction from -40 to 67 percent. Model error is larger than sampling error for both equations, indicating that additional basin characteristics could be important to improved estimates of low-flow statistics. Weighted estimates of August median streamflow, which can be used when making estimates at partial-record or continuous-record gaging stations, range from 0.03 to 11.7 cubic feet per second or from 0.1 to 0.4 cubic feet per second per square mile. Estimates of August median streamflow on ungaged streams in the eastern part of Aroostook County, within the range of acceptable explanatory variables, range from 0.03 to 30 cubic feet per second or 0.1 to 0.7 cubic feet per second per square mile. Estimates of August median streamflow per square mile of drainage area generally increase as mean elevation and drainage area increase.

Maine

Bounded fractional diffusion in geological media: Definition and Lagrangian approximation

Spatiotemporal Fractional-Derivative Models (FDMs) have been increasingly used to simulate non-Fickian diffusion, but methods have not been available to define boundary conditions for FDMs in bounded domains. This study defines boundary conditions and then develops a Lagrangian solver to approximate bounded, one-dimensional fractional diffusion. Both the zero-value and non-zero-value Dirichlet, Neumann, and mixed Robin boundary conditions are defined, where the sign of Riemann-Liouville fractional derivative (capturing non-zero-value spatial-nonlocal boundary conditions with directional super-diffusion) remains consistent with the sign of the fractional-diffusive flux term in the FDMs. New Lagrangian schemes are then proposed to track solute particles moving in bounded domains, where the solutions are checked against analytical or Eularian solutions available for simplified FDMs. Numerical experiments show that the particle-tracking algorithm for non-Fickian diffusion differs from Fickian diffusion in relocating the particle position around the reflective boundary, likely due to the non-local and non-symmetric fractional diffusion. For a non-zero-value Neumann or Robin boundary, a source cell with a reflective face can be applied to define the release rate of random-walking particles at the specified flux boundary. Mathematical definitions of physically meaningful nonlocal boundaries combined with bounded Lagrangian solvers in this study may provide the only viable techniques at present to quantify the impact of boundaries on anomalous diffusion, expanding the applicability of FDMs from infinite do mains to those with any size and boundary conditions.

Water Resources Research

Optimal hydrograph separation using a recursive digital filter constrained by chemical mass balance, with application to selected Chesapeake Bay watersheds

Quantitative estimates of base flow are necessary to address questions concerning the vulnerability and response of the Nation’s water supply to natural and human-induced change in environmental conditions. An objective of the U.S. Geological Survey National Water-Quality Assessment Project is to determine how hydrologic systems are affected by watershed characteristics, including land use, land cover, water use, climate, and natural characteristics (geology, soil type, and topography). An important component of any hydrologic system is base flow, generally described as the part of streamflow that is sustained between precipitation events, fed to stream channels by delayed (usually subsurface) pathways, and more specifically as the volumetric discharge of water, estimated at a measurement site or gage at the watershed scale, which represents groundwater that discharges directly or indirectly to stream reaches and is then routed to the measurement point. Hydrograph separation using a recursive digital filter was applied to 225 sites in the Chesapeake Bay watershed. The recursive digital filter was chosen for the following reasons: it is based in part on the assumption that groundwater acts as a linear reservoir, and so has a physical basis; it has only two adjustable parameters (alpha, obtained directly from recession analysis, and beta, the maximum value of the base-flow index that can be modeled by the filter), which can be determined objectively and with the same physical basis of groundwater reservoir linearity, or that can be optimized by applying a chemical-mass-balance constraint. Base-flow estimates from the recursive digital filter were compared with those from five other hydrograph-separation methods with respect to two metrics: the long-term average fraction of streamflow that is base flow, or base-flow index, and the fraction of days where streamflow is entirely base flow. There was generally good correlation between the methods, with some biased slightly high and some biased slightly low compared to the recursive digital filter. There were notable differences between the days at base flow estimated by the different methods, with the recursive digital filter having a smaller range of values. This was attributed to how the different methods determine cessation of quickflow (the part of streamflow which is not base flow). For 109 Chesapeake Bay watershed sites with available specific conductance data, the parameters of the filter were optimized using a chemical-mass-balance constraint and two different models for the time-dependence of base-flow specific conductance. Sixty-seven models were deemed acceptable and the results compared well with non-optimized results. There are a number of limitations to the optimal hydrograph-separation approach resulting from the assumptions implicit in the conceptual model, the mathematical model, and the approach taken to impose chemical mass balance (including tracer choice). These limitations may be evidenced by poor model results; conversely, poor model fit may provide an indication that two-component separation does not adequately describe the hydrologic system’s runoff response. The results of this study may be used to address a number of questions regarding the role of groundwater in understanding past changes in stream-water quality and forecasting possible future changes, such as the timing and magnitude of land-use and management practice effects on stream and groundwater quality. Ongoing and future modeling efforts may benefit from the estimates of base flow as calibration targets or as a means to filter chemical data to model base-flow loads and trends. Ultimately, base-flow estimation might provide the basis for future work aimed at improving the ability to quantify groundwater discharge, not only at the scale of a gaged watershed, but at the scale of individual reaches as well.

Scientific Investigations Report

August median streamflow on ungaged streams in eastern coastal Maine

Methods for estimating August median streamflow were developed for ungaged, unregulated streams in eastern coastal Maine. The methods apply to streams with drainage areas ranging in size from 0.04 to 73.2 square miles and fraction of basin underlain by a sand and gravel aquifer ranging from 0 to 71 percent. The equations were developed with data from three long-term (greater than or equal to 10 years of record) continuous-record streamflow-gaging stations, 23 partial-record streamflow- gaging stations, and 5 short-term (less than 10 years of record) continuous-record streamflow-gaging stations. A mathematical technique for estimating a standard low-flow statistic, August median streamflow, at partial-record streamflow-gaging stations and short-term continuous-record streamflow-gaging stations was applied by relating base-flow measurements at these stations to concurrent daily streamflows at nearby long-term continuous-record streamflow-gaging stations (index stations). Generalized least-squares regression analysis (GLS) was used to relate estimates of August median streamflow at streamflow-gaging stations to basin characteristics at these same stations to develop equations that can be applied to estimate August median streamflow on ungaged streams. GLS accounts for different periods of record at the gaging stations and the cross correlation of concurrent streamflows among gaging stations. Thirty-one stations were used for the final regression equations. Two basin characteristics-drainage area and fraction of basin underlain by a sand and gravel aquifer-are used in the calculated regression equation to estimate August median streamflow for ungaged streams. The equation has an average standard error of prediction from -27 to 38 percent. A one-variable equation uses only drainage area to estimate August median streamflow when less accuracy is acceptable. This equation has an average standard error of prediction from -30 to 43 percent. Model error is larger than sampling error for both equations, indicating that additional or improved estimates of basin characteristics could be important to improved estimates of low-flow statistics. Weighted estimates of August median streamflow at partial- record or continuous-record gaging stations range from 0.003 to 31.0 cubic feet per second or from 0.1 to 0.6 cubic feet per second per square mile. Estimates of August median streamflow on ungaged streams in eastern coastal Maine, within the range of acceptable explanatory variables, range from 0.003 to 45 cubic feet per second or 0.1 to 0.6 cubic feet per second per square mile. Estimates of August median streamflow per square mile of drainage area generally increase as drainage area and fraction of basin underlain by a sand and gravel aquifer increase.

Maine

Lognormal field size distributions as a consequence of economic truncation

The assumption of lognormal (parent) field size distributions has for a long time been applied to resource appraisal and evaluation of exploration strategy by the petroleum industry. However, frequency distributions estimated with observed data and used to justify this hypotheses are conditional. Examination of various observed field size distributions across basins and over time shows that such distributions should be regarded as the end result of an economic filtering process. Commercial discoveries depend on oil and gas prices and field development costs. Some new fields are eliminated due to location, depths, or water depths. This filtering process is called economic truncation. Economic truncation may occur when predictions of a discovery process are passed through an economic appraisal model. We demonstrate that (1) economic resource appraisals, (2) forecasts of levels of petroleum industry activity, and (3) expected benefits of developing and implementing cost reducing technology are sensitive to assumptions made about the nature of that portion of (parent) field size distribution subject to economic truncation. ?? 1985 Plenum Publishing Corporation.

Journal of the International Association for Mathe