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Space transformations are proposed as a mathematically meaningful and practically comprehensive approach to simulate multidimensional random fields. Within this context the turning bands method of simulation is reconsidered and improved in both the space and frequency domains.
Elliptic partial differential equations can be solved using the Galerkin-finite element method to generate the approximating algebraic equations, and an electrical network to solve the resulting matrices. Some element configurations require the use of networks containing negative resistances which, while physically realizable, are more expensive and time-consuming to construct.
No abstract available.
Identification of the internal flow dynamics in catchments is difficult because of the lack of information in precipitation -stream discharge time series alone. Two experimental catchments, Hydrohill and Nandadish, near Nanjing in China, have been set up to monitor internal flows reaching the catchment stream at various depths, from the surface runoff to the bedrock. With analysis of the precipitation against these internal discharges, it is possible to quantify the time constants and volumes associated with various flowpaths in both catchments.
The U.S. Army Corps of Engineers' plan to control the natural chloride pollution in the Wichita River basin includes the construction of Truscott Brine Lake on a tributary of the North Wichita River. In connection with the proposed brine lake, the U.S. Geological Survey was requested to: (1) Define the existing ground-water conditions in the shallow fresh-water system of the project area; and (2) project the post-construction effects of the proposed lake on the fresh-water aquifer, especially in relation to hydraulic-head changes but also with respect to possible changes in the chemical quality of the ground water. The fresh-water aquifer in the project area is a shallow water-table system with relatively fresh water (generally a calcium sulfate type) that contains approximately 500 to 5,000 milligrams per liter of dissolved solids. The aquifer consists of Permian rocks with very small values of hydraulic conductivity and overlies a brine system (sodium chloride type) that is even less permeable. A thin transition zone separates the fresh-water aquifer and the brine system. Small quantities of infiltration from precipitation throughout the area's watershed constitute the recharge to the aquifer. Discharge from the aquifer consists of the small base flow along creeks; well discharge is negligible. Two-dimensional mathematical computer models were developed for aquifer simulation of: (1) Steady-state conditions in a fresh-water system and (2) transient conditions in a brine- fresh-water system where the density effects of the brine are considered. The main results 'of projecting the effects of the proposed Truscott Brine Lake on the fresh-water aquifer are: (1) Hydraulic head rises of 5 to 40 feet would be confined to areas near the proposed dam and along the lake shoreline, and (2) migration of salt water downstream from the dam generally would be limited to less than 1 mile and apparently would not reach equilibrium during the 100-year duration of the project. The modeling efforts did not include possible effects related to hydrodynamic dispersion in the brine- fresh-water system. Possible changes in the hydraulic conductivity of the aquifer, due to physical and chemical interactions in the brine and fresh-water environments, also were not considered.
Recursive algorithms for estimating the states of nonlinear physical systems are developed. This requires some key hypotheses regarding the structure of the underlying processes. Members of this class of random processes have several desirable properties for the nonlinear estimation of random signals. An assumption is made about the form of the estimator, which may then take account of a wide range of applications. Under the above assumption, the estimation algorithm is mathematically suboptimal but effective and computationally attractive. It may be compared favorably to Taylor series-type filters, nonlinear filters which approximate the probability density by Edgeworth or Gram-Charlier series, as well as to conventional statistical linearization-type estimators. To link theory with practice, some numerical results for a simulated system are presented, in which the responses from the proposed and the extended Kalman algorithms are compared.
Numerical simulations of sequences of earthquakes and aseismic slip (SEAS) have made great progress over past decades to address important questions in earthquake physics. However, significant challenges in SEAS modeling remain in resolving multiscale interactions between earthquake nucleation, dynamic rupture, and aseismic slip, and understanding physical factors controlling observables such as seismicity and ground deformation. The increasing complexity of SEAS modeling calls for extensive efforts to verify codes and advance these simulations with rigor, reproducibility, and broadened impact. In 2018, we initiated a community code-verification exercise for SEAS simulations, supported by the Southern California Earthquake Center. Here, we report the findings from our first two benchmark problems (BP1 and BP2), designed to verify different computational methods in solving a mathematically well-defined, basic faulting problem. We consider a 2D antiplane problem, with a 1D planar vertical strike-slip fault obeying rate-and-state friction, embedded in a 2D homogeneous, linear elastic halfspace. Sequences of quasi-dynamic earthquakes with periodic occurrences (BP1) or bimodal sizes (BP2) and their interactions with aseismic slip are simulated. The comparison of results from 11 groups using different numerical methods show excellent agreements in long-term and coseismic fault behavior. In BP1, we found that truncated domain boundaries influence interseismic stressing, earthquake recurrence, and coseismic rupture, and that model agreement is only achieved with sufficiently large domain sizes. In BP2, we found that complexity of fault behavior depends on how well physical length scales related to spontaneous nucleation and rupture propagation are resolved. Poor numerical resolution can result in artificial complexity, impacting simulation results that are of potential interest for characterizing seismic hazard such as earthquake size distributions, moment release, and recurrence times. These results inform the development of more advanced SEAS models, contributing to our further understanding of earthquake system dynamics.
This report presents MODFLOW-2005, which is a new version of the finite-difference ground-water model commonly called MODFLOW. Ground-water flow is simulated using a block-centered finite-difference approach. Layers can be simulated as confined or unconfined. Flow associated with external stresses, such as wells, areal recharge, evapotranspiration, drains, and rivers, also can be simulated. The report includes detailed explanations of physical and mathematical concepts on which the model is based, an explanation of how those concepts are incorporated in the modular structure of the computer program, instructions for using the model, and details of the computer code. The modular structure consists of a MAIN Program and a series of highly independent subroutines. The subroutines are grouped into 'packages.' Each package deals with a specific feature of the hydrologic system that is to be simulated, such as flow from rivers or flow into drains, or with a specific method of solving the set of simultaneous equations resulting from the finite-difference method. Several solution methods are incorporated, including the Preconditioned Conjugate-Gradient method. The division of the program into packages permits the user to examine specific hydrologic features of the model independently. This also facilitates development of additional capabilities because new packages can be added to the program without modifying the existing packages. The input and output systems of the computer program also are designed to permit maximum flexibility. The program is designed to allow other capabilities, such as transport and optimization, to be incorporated, but this report is limited to describing the ground-water flow capability. The program is written in Fortran 90 and will run without modification on most computers that have a Fortran 90 compiler.
In this study, developments in the theory of stochastic simulation are discussed. The unifying element is the notion of Radon projection in Euclidean spaces. This notion provides a natural way of reconstructing the real process from a corresponding process observable on a reduced dimensionality space, where analysis is theoretically easier and computationally tractable. Within this framework, the concept of space transformation is defined and several of its properties, which are of significant importance within the context of spatially correlated processes, are explored. The turning bands operator is shown to follow from this. This strengthens considerably the theoretical background of the geostatistical method of simulation, and some new results are obtained in both the space and frequency domains. The inverse problem is solved generally and the applicability of the method is extended to anisotropic as well as integrated processes. Some ill-posed problems of the inverse operator are discussed. Effects of the measurement error and impulses at origin are examined. Important features of the simulated process as described by geomechanical laws, the morphology of the deposit, etc., may be incorporated in the analysis. The simulation may become a model-dependent procedure and this, in turn, may provide numerical solutions to spatial-temporal geologic models. Because the spatial simu??lation may be technically reduced to unidimensional simulations, various techniques of generating one-dimensional realizations are reviewed. To link theory and practice, an example is computed in detail. ?? 1987 International Association for Mathematical Geology.
RAQSIM, a generalized flow model of a groundwater system using finite-element methods, is documented to explain how it works and to demonstrate that it gives valid results. Three support programs that are used to compute recharge and discharge data required as input to RAQSIM are described. RAQSIM was developed to solve transient, two-dimensional, regional groundwater flow problems with isotropic or anisotropic conductance. The model can also simulate radially-symmetric flow to a well and steady-state flow. The mathematical basis, program structure, data input and output procedures, organization of data sets, and program features and options of RAQSIM are discussed. An example , containing listings of data and results and illustrating RAQSIM 's capabilities, is discussed in detail. Two test problems also are discussed comparing RAQSIM 's results with analytical procedures. The first support program described, the PET Program, uses solar radiation and other climatic data in the Jensen-Haise method to compute potential evapotranspiration. The second support program, the Soil-Water Program, uses output from the PET Program, soil characteristics, and the ratio of potential to actual evapotranspiration for each crop to compute infiltration, storage, and removal of water from the soil zone. The third program, the Recharge-Discharge Program, uses output from the Soil-Water Program together with other data to compute recharge and discharge from the groundwater flow system. For each support program, a program listing and examples of the data and results for the Twin Platte-Middle Republican study are provided. In addition, a brief discussion on how each program operates and on procedures for running and modifying these programs are presented. (Author 's abstract)
Understanding karst aquifers, for purposes of their management and protection, poses unique challenges. Karst aquifers are characterized by groundwater flow through conduits (tertiary porosity), and (or) layers with interconnected pores (secondary porosity) and through intergranular porosity (primary or matrix porosity). Since the late 1960s, advances have been made in the development of numerical computer codes and the use of mathematical model applications towards the understanding of dual (primary [matrix] and secondary [fractures and conduits]) porosity groundwater flow processes, as well as characterization and management of karst aquifers. The Floridan aquifer system (FAS) in Florida and parts of Alabama, Georgia, and South Carolina is composed of a thick sequence of predominantly carbonate rocks. Karst features are present over much of its area, especially in Florida where more than 30 first-magnitude springs occur, numerous sinkholes and submerged conduits have been mapped, and numerous circular lakes within sinkhole depressions are present. Different types of mathematical models have been applied for simulation of the FAS. Most of these models are distributed parameter models based on the assumption that, like a sponge, water flows through connected pores within the aquifer system and can be simulated with the same mathematical methods applied to flow through sand and gravel aquifers; these models are usually referred to as porous-equivalent media models. The partial differential equation solved for groundwater flow is the potential flow equation of fluid mechanics, which is used when flow is dominated by potential energy and has been applied for many fluid problems in which kinetic energy terms are dropped from the differential equation solved. In many groundwater model codes (basic MODFLOW), it is assumed that the water has a constant temperature and density and that flow is laminar, such that kinetic energy has minimal impact on flow. Some models have been developed that incorporate the submerged conduits as a one-dimensional pipe network within the aquifer rather than as discrete, extremely transmissive features in a porous-equivalent medium; these submerged conduit models are usually referred to as hybrid models and may include the capability to simulate both laminar and turbulent flow in the one-dimensional pipe network. Comparisons of the application of a porous-equivalent media model with and without turbulence (MODFLOW-Conduit Flow Process mode 2 and basic MODFLOW, respectively) and a hybrid (MODFLOW-Conduit Flow Process mode 1) model to the Woodville Karst Plain near Tallahassee, Florida, indicated that for annual, monthly, or seasonal average hydrologic conditions, all methods met calibration criteria (matched observed groundwater levels and average flows). Thus, the increased effort required, such as the collection of data on conduit location, to develop a hybrid model and its increased computational burden, is not necessary for simulation of average hydrologic conditions (non-laminar flow effects on simulated head and spring discharge were minimal). However, simulation of a large storm event in the Woodville Karst Plain with daily stress periods indicated that turbulence is important for matching daily springflow hydrographs. Thus, if matching streamflow hydrographs over a storm event is required, the simulation of non-laminar flow and the location of conduits are required. The main challenge in application of the methods and approaches for developing hybrid models relates to the difficulty of mapping conduit networks or having high-quality datasets to calibrate these models. Additionally, hybrid models have long simulation times, which can preclude the use of parameter estimation for calibration. Simulation of contaminant transport that does not account for preferential flow through conduits or extremely permeable zones in any approach is ill-advised. Simulation results in other karst aquifers or other parts of the FAS may differ from the comparison demonstrated herein.
Successful use of predictive mathematical models requires verification of the accuracy of the models by applying them to existing situations where the prediction can be compared with reality. A Corps of Engineers' modification of a deep reservoir thermal stratification model developed by Water Resources Engineers, Inc., was applied to two existing Bureau of Reclamation reservoirs for verification. Diffusion coefficients used for the Corps' Detroit Reservoir were found to apply to Horsetooth Reservoir in Colorado, for which very food computer input data were available. The Detroit diffusion coefficients gave a reasonable simulation of Flaming Gorge Reservoir in Wyoming and Utah, which has very complex and variable physical characteristics and for which only average-quality computer input data were available.
A computer program for simulating one-dimensional, unsteady temperature and solute transport in a river has been developed and documented for general use. The solution approach to the convective-diffusion equation uses a moving reference frame (Lagrangian) which greatly simplifies the mathematics of the solution procedure and dramatically reduces errors caused by numerical dispersion. The solution procedure has the further advantages, relative to conventional Eulerian solution schemes, of being easy to understand in the physical sense, of being extremely stable numerically, and of providing an accounting system which is very useful for model calibration. The model documentation is presented as a series of four programs of increasing complexity. The conservative transport model can be used to route a single conservative substance, such as dye, through a reach of a river. The simplified temperature model is used to predict water temperature in rivers, either with or without thermal loading, when few meteorological data are available. Only equilibrium temperature and windspeed are required. It is suggested that air temperature can be used to approximate equilibrium temperature. The complete temperature model is highly accurate but requires rather complete meteorological data. Finally, the 10-parameter model can be used to route as many as 10 interacting constituents through a river reach. The mathematical description of the interaction between the constituents, which does not need to be linear, is generally up to the user to supply. An example problem is solved for a three-parameter system involving temperature, dissolved oxygen, and biochemical oxygen demand. For simplicity, all models are developed and presented assuming steady non-uniform flow. Generalization of the models to allow unsteady flow is extremely simple, involving the addition of no more than 18 cards to the program deck. The report is concluded by describing this generalization for any of the models. Before using the models with unsteady flow, a flow model must be used to calculate and store the necessary flow data at each cross section and time step. Such a flow model is available and documented.
A frequency dependent computer model based on the two-dimensional vertically averaged shallow-water equations is described for general purpose application in tidally dominated embayments. This model simulates the response of both tides and tidal currents to user-specified geometries and boundary conditions. The mathematical formulation and practical application of the model are discussed in detail. Salient features of the model include the ability to specify: (1) stage at the open boundaries as well as within the model grid, (2) velocities on open boundaries (river inflows and so forth), (3) spatially variable wind stress, and (4) spatially variable bottom friction. Using harmonically analyzed field data as boundary conditions, this model can be used to make real time predictions of tides and tidal currents. (USGS)
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.
Pore-fluid pressure plays a crucial role in debris flows because it counteracts normal stresses at grain contacts and thereby reduces intergranular friction. Pore-pressure feedback accompanying debris deformation is particularly important during the onset of debrisflow motion, when it can dramatically influence the balance of forces governing downslope acceleration. We consider further effects of this feedback by formulating a new, depth-averaged mathematical model that simulates coupled evolution of granular dilatancy, solid and fluid volume fractions, pore-fluid pressure, and flow depth and velocity during all stages of debris-flow motion. To illustrate implications of the model, we use a finite-volume method to compute one-dimensional motion of a debris flow descending a rigid, uniformly inclined slope, and we compare model predictions with data obtained in large-scale experiments at the USGS debris-flow flume. Predictions for the first 1 s of motion show that increasing pore pressures (due to debris contraction) cause liquefaction that enhances flow acceleration. As acceleration continues, however, debris dilation causes dissipation of pore pressures, and this dissipation helps stabilize debris-flow motion. Our numerical predictions of this process match experimental data reasonably well, but predictions might be improved by accounting for the effects of grain-size segregation.
Judy's Branch watershed, a small basin (8.64 square miles) in the St. Louis Metro East region in Illinois, was selected as a pilot site to determine suspended-sediment yields and stream-channel processes in the bluffs and American Bottoms (expansive low-lying valley floor in the region). Suspended-sediment and stream-chan-nel data collected and analyzed for Judy's Branch watershed are presented in this report to establish a baseline of data for water-resource managers to evaluate future stream rehabilitation and manage-ment alternatives. The sediment yield analysis determines the amount of sediment being delivered from the watershed and two subwatersheds: an urban tributary and an undeveloped headwater (pri-marily agricultural). The analysis of the subwater-sheds is used to compare the effects of urbanization on sediment yield to the river. The stream-channel contribution to sediment yield was determined by evaluation of the stream-channel processes operat-ing on the streambed and banks of Judy's Branch watershed. Bank stability was related to hydrologic events, bank stratigraphy, and channel geometry through model development and simulation. The average suspended-sediment yield from two upland subwatersheds (drainage areas of 0.23 and 0.40 sq.mi. was 1,163 tons per square mile per year (tons/sq.mi.-year) between July 2000 and June 2004. The suspended-sediment yield at the Route 157 station was 2,523 tons/sq.mi.-year, near the outlet of Judy's Branch watershed (drainage area = 8.33 sq.mi.). This is approximately 1,360 tons/sq.mi.-year greater than the average at the upland stations for the same time period. This result is unexpected in that, generally, the suspended-sediment yield decreases as the watershed area increases because of sediment stored in the channel and flood plain. The difference indicates a possible increase in yield from a source, such as bank retreat, and supports the concept that land-use changes increase stream-flows that may in turn result in higher rates of bank retreat. Utilizing both bank-rod data and resurveyed cross-section data, it was determined that approxi-mately half of the suspended- sediment yield at Route 157 during July 2000-June 2004 came from bank retreat. Given that bank retreat can be a substantial portion of the sediment yield, understanding bank stability processes is important. Bank stability can be assessed mathematically by computing the factor of safety, which is defined by the ratio of the shear strength (resisting force) along the failure surface and the shear stress (driving gravitational force). Once the factor of safety falls below one, the bank theoretically becomes unstable. Bank-stability conditions were related to hydrologic events, bank type, and channel geometry through model develop-ment and simulation. The most common type of bank in the watershed consists of cohesive alluvial soil deposits overlying a stiff glacial till. A stabil-ity chart for different bank types was developed using a bank-stability analysis. Banks steeper than 70 degrees and higher than from 10 to 11.5 feet (depending on bank type) become at risk for mass failure in the watershed under conditions that pro-mote saturation of the bank and a sudden drop in the river level.
Based on experimental evidence and using mathematical modeling, inter-phase mass transfer processes of CO 2 exsolving from and dissolving into water in heterogeneous porous media are investigated under two fundamentally different flow conditions: in a quasi one dimensional vertical column and in a two-dimensional tank with a lateral background water flow, both at laboratory scale. In both cases, the CO 2 dissolved in water under a given overpressure is injected for a certain period at the bottom of the tank, exsolves, and migrates upwards. A layer of fine sand is present in the tanks designed to mimic geological scenarios of accumulation and trapping of exsolved CO 2 in shallow aquifers. Then, clean water is injected and the accumulated CO 2 is dissolved back into the flowing water. The study aims to point out the differences in the mass transfer processes between the quasi-1D and 2D cases using a mathematical model of two-phase compositional flow in heterogeneous porous media calibrated to the experimental datasets, and expose strategies that should be explored in future research. Additionally, temperature variations observed during the 2D experiments allow for analysis of isothermal versus non-isothermal effects on the processes of multiphase CO 2 evolution. The mathematical model is discretized and solved using the mixed hybrid finite element method in 2D that allows for the simulation of both advection- and diffusion-dominated processes accurately.