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USGS2003

A numerical simulation of magma motion, crustal deformation, and seismic radiation associated with volcanic eruptions

The finite difference method is used to calculate the magma dynamics, seismic radiation, and crustal deformation associated with a volcanic eruption. The model geometry consists of a cylindrical reservoir and narrow cylindrical conduit embedded in a homogeneous crust. We consider two models of eruption. In the first model, a lid caps the vent and the magma is overpressurized prior to the eruption. The eruption is triggered by the instantaneous removal of the lid, at which point the exit pressure becomes equal to the atmospheric pressure. In the second model, a plug at the reservoir outlet allows pressurization of only the magmatic fluid in the reservoir before the eruption. Magma transfer between the reservoir and conduit is triggered by the instantaneous removal of the plug, and the eruption occurs when the pressure at the conduit orifice exceeds the material strength of the lid capping the vent. In both models, magma dynamics are expressed by the equations of mass and momentum conservation in a compressible fluid, in which fluid expansion associated with depressurization is accounted for by a constitutive law relating pressure and density. Crustal motions are calculated from the equations of elastodynamics. The fluid and solid are dynamically coupled by applying the continuity of wall velocities and normal stresses across the conduit and reservoir boundaries. Free slip is allowed at the fluid-solid boundary. Both models predict the gradual depletion of the magma reservoir, which causes crustal deformation observed as a long-duration dilatational signal. Superimposed on this very-long-period (VLP) signal generated by mass transport are long-period (LP) oscillations of the magma reservoir and conduit excited by the acoustic resonance of the reservoir-conduit system during the eruption. The volume of the reservoir, vent size, and magma properties control the duration of VLP waves and dominant periods of LP oscillations. The second model predicts that when the magmatic fluid reaches the vent, a high-pressure pulse occurs at this location in accordance with the basic theory of compressible fluid dynamics. This abrupt pressure increase just beneath the vent is consistent with observed seismograms in which pulse-like Rayleigh waves excited by a shallow source are dominant. The strength of the lid plays an important role in the character of the seismograms and in defining the type of eruption observed.

Geophysical Journal International↗

USGS1980

The frontal method in hydrodynamics simulations

The frontal solution method has proven to be an effective means of solving the matrix equations resulting from the application of the finite element method to a variety of problems. In this study, several versions of the frontal method were compared in efficiency for several hydrodynamics problems. Three basic modifications were shown to be of value: 1. Elimination of equations with boundary conditions beforehand, 2. Modification of the pivoting procedures to allow dynamic management of the equation size, and 3. Storage of the eliminated equations in a vector. These modifications are sufficiently general to be applied to other classes of problems.

Computers and Fluids↗

USGS1988

Documentation of a steady-state saltwater-intrusion model for three-dimensional ground-water flow, and user's guide

A finite-difference model that simulates three-dimensional flow of groundwater was modified to simulate steady flow of freshwater in a multiple-aquifer system containing freshwater and static saltwater. The two fluids are assumed to be immiscible, with constant but different densities, and are separated by a sharp interface. The interface position computed by the model for a test problem was in good agreement with the analytic solution for this problem. The model was developed to simulate seawater intrusion in coastal aquifers, but it could be used to simulate flow in any aquifer system that is bounded by saltwater. This report describes modifications made to the existing numerical model and the method of locating an interface , and contains a user 's guide for the model. (USGS)

Open-File Report↗

USGS2009

Shallow seismic structure of Kunlun fault zone in northern Tibetan Plateau, China: Implications for the 2001 M s8.1 Kunlun earthquake

The shallow seismic velocity structure of the Kunlun fault zone (KLFZ) was jointly deduced from seismic refraction profiling and the records of trapped waves that were excited by five explosions. The data were collected after the 2001 Kunlun M s8.1 earthquake in the northern Tibetan Plateau. Seismic phases for the in-line record sections (26 records up to a distance of 15 km) along the fault zone were analysed, and 1-D P - and S -wave velocity models of shallow crust within the fault zone were determined by using the seismic refraction method. Sixteen seismic stations were deployed along the off-line profile perpendicular to the fault zone. Fault-zone trapped waves appear clearly on the record sections, which were simulated with a 3-D finite difference algorithm. Quantitative analysis of the correlation coefficients of the synthetic and observed trapped waveforms indicates that the Kunlun fault-zone width is 300 m, and S -wave quality factor Q within the fault zone is 15. Significantly, S -wave velocities within the fault zone are reduced by 30–45 per cent from surrounding rocks to a depth of at least 1–2 km, while P -wave velocities are reduced by 7–20 per cent. A fault-zone with such P - and S -low velocities is an indication of high fluid pressure because Vs is affected more than Vp . The low-velocity and low- Q zone in the KLFZ model is the effect of multiple ruptures along the fault trace of the 2001 M s8.1 Kunlun earthquake.

Tibetan Plateau↗

USGS2003

Finite element model predictions of static deformation from dislocation sources in a subduction zone: Sensitivities to homogeneous, isotropic, Poisson-solid, and half-space assumptions

Dislocation models can simulate static deformation caused by slip along a fault. These models usually take the form of a dislocation embedded in a homogeneous, isotropic, Poisson-solid half-space (HIPSHS). However, the widely accepted HIPSHS assumptions poorly approximate subduction zone systems of converging oceanic and continental crust. This study uses three-dimensional finite element models (FEMs) that allow for any combination (including none) of the HIPSHS assumptions to compute synthetic Green's functions for displacement. Using the 1995 Mw = 8.0 Jalisco-Colima, Mexico, subduction zone earthquake and associated measurements from a nearby GPS array as an example, FEM-generated synthetic Green's functions are combined with standard linear inverse methods to estimate dislocation distributions along the subduction interface. Loading a forward HIPSHS model with dislocation distributions, estimated from FEMs that sequentially relax the HIPSHS assumptions, yields the sensitivity of predicted displacements to each of the HIPSHS assumptions. For the subduction zone models tested and the specific field situation considered, sensitivities to the individual Poisson-solid, isotropy, and homogeneity assumptions can be substantially greater than GPS. measurement uncertainties. Forward modeling quantifies stress coupling between the Mw = 8.0 earthquake and a nearby Mw = 6.3 earthquake that occurred 63 days later. Coulomb stress changes predicted from static HIPSHS models cannot account for the 63-day lag time between events. Alternatively, an FEM that includes a poroelastic oceanic crust, which allows for postseismic pore fluid pressure recovery, can account for the lag time. The pore fluid pressure recovery rate puts an upper limit of 10-17 m2 on the bulk permeability of the oceanic crust. Copyright 2003 by the American Geophysical Union.

Journal of Geophysical Research B: Solid Earth↗

USGS2001

Documentation and verification of VST2D; a model for simulating transient, Variably Saturated, coupled water-heat-solute Transport in heterogeneous, anisotropic 2-Dimensional, ground-water systems with variable fluid density

This report describes a model for simulating transient, Variably Saturated, coupled water-heatsolute Transport in heterogeneous, anisotropic, 2-Dimensional, ground-water systems with variable fluid density (VST2D). VST2D was developed to help understand the effects of natural and anthropogenic factors on quantity and quality of variably saturated ground-water systems. The model solves simultaneously for one or more dependent variables (pressure, temperature, and concentration) at nodes in a horizontal or vertical mesh using a quasi-linearized general minimum residual method. This approach enhances computational speed beyond the speed of a sequential approach. Heterogeneous and anisotropic conditions are implemented locally using individual element property descriptions. This implementation allows local principal directions to differ among elements and from the global solution domain coordinates. Boundary conditions can include time-varying pressure head (or moisture content), heat, and/or concentration; fluxes distributed along domain boundaries and/or at internal node points; and/or convective moisture, heat, and solute fluxes along the domain boundaries; and/or unit hydraulic gradient along domain boundaries. Other model features include temperature and concentration dependent density (liquid and vapor) and viscosity, sorption and/or decay of a solute, and capability to determine moisture content beyond residual to zero. These features are described in the documentation together with development of the governing equations, application of the finite-element formulation (using the Galerkin approach), solution procedure, mass and energy balance considerations, input requirements, and output options. The VST2D model was verified, and results included solutions for problems of water transport under isohaline and isothermal conditions, heat transport under isobaric and isohaline conditions, solute transport under isobaric and isothermal conditions, and coupled water-heat-solute transport. The first three problems considered in model verification were compared to either analytical or numerical solutions, whereas the coupled problem was compared to measured laboratory results for which no known analytic solutions or numerical models are available. The test results indicate the model is accurate and applicable for a wide range of conditions, including when water (liquid and vapor), heat (sensible and latent), and solute are coupled in ground-water systems. The cumulative residual errors for the coupled problem tested was less than 10–8 cubic centimeter per cubic centimeter, 10-5 moles per kilogram, and 102 calories per cubic meter for liquid water content, solute concentration and heat content, respectively. This model should be useful to hydrologists, engineers, and researchers interested in studying coupled processes associated with variably saturated transport in ground-water systems.

Water-Resources Investigations Report↗

USGS1983

Analysis of spurious oscillation modes for the shallow water and Navier-Stokes equations

The origin and nature of spurious oscillation modes that appear in mixed finite element methods are examined. In particular, the shallow water equations are considered and a modal analysis for the one-dimensional problem is developed. From the resulting dispersion relations we find that the spurious modes in elevation are associated with zero frequency and large wave number (wavelengths of the order of the nodal spacing) and consequently are zero-velocity modes. The spurious modal behavior is the result of the finite spatial discretization. By means of an artificial compressibility and limiting argument we are able to resolve the similar problem for the Navier-Stokes equations. The relationship of this simpler analysis to alternative consistency arguments is explained. This modal approach provides an explanation of the phenomenon in question and permits us to deduce the cause of the very complex behavior of spurious modes observed in numerical experiments with the shallow water equations and Navier-Stokes equations. Furthermore, this analysis is not limited to finite element formulations, but is also applicable to finite difference formulations.

Computers and Fluids↗

USGS2010

Seismic source mechanism of degassing bursts at Kilauea volcano, Hawaii: Results from waveform inversion in the 10–50 s band

The current (March 2008 to February 2009) summit eruptive activity at Kilauea Volcano is characterized by explosive degassing bursts accompanied by very long period (VLP) seismic signals. We model the source mechanisms of VLP signals in the 10–50 s band using data recorded for 15 bursts with a 10‐station broadband network deployed in the summit caldera. To determine the source centroid location and source mechanism, we minimize the residual error between data and synthetics calculated by the finite difference method for a point source embedded in a homogeneous medium that takes topography into account. The VLP signals associated with the bursts originate in a source region ∼1 km below the eastern perimeter of Halemaumau pit crater. The observed waveforms are well explained by the combination of a volumetric component and a vertical single force component. For the volumetric component, several source geometries are obtained which equally explain the observed waveforms. These geometries include (1) a pipe dipping 64° to the northeast; (2) two intersecting cracks including an east striking crack (dike) dipping 80° to the north, intersecting a north striking crack (another dike) dipping 65° to the east; (3) a pipe dipping 58° to the northeast, intersecting a crack dipping 48° to the west–southwest; and (4) a pipe dipping 57° to the northeast, intersecting a pipe dipping 58° to the west–southwest. Using the dual‐crack model as reference, the largest volume change obtained among the 15 bursts is ∼24,400 m 3 , and the maximum amplitude (peak to peak) of the force is ∼20 GN. Each burst is marked by a similar sequence of deflation and inflation, trailed by decaying oscillations of the volumetric source. The vertical force is initially upward, synchronous with source deflation, then downward, synchronous with source reinflation, followed by oscillations with polarity opposite to the volumetric oscillations. This combination of force and volume change is attributed to pressure and momentum changes induced during a fluid dynamic source mechanism involving the ascent, expansion, and burst of a large slug of gas within the upper ∼150 m of the magma conduit. As the slug expands upon approach to the surface and more liquid becomes wall supported by viscous shear forces, the pressure below the slug decreases, inducing conduit deflation and an upward force on the Earth. The final rapid slug expansion and burst stimulate VLP and LP oscillations of the conduit system, which slowly decay due to viscous dissipation and elastic radiation. Consideration of the fluid dynamic arguments leads us to prefer the dual‐crack VLP source model as it is the only candidate model capable of producing plausible values of length scales and pressure changes. The magnitudes of the vertical forces observed in the 15 bursts appear consistent with slug masses of 10 4 to 10 6 kg.

Hawaii↗

USGS2000

An unstructured grid, three-dimensional model based on the shallow water equations

A semi-implicit finite difference model based on the three-dimensional shallow water equations is modified to use unstructured grids. There are obvious advantages in using unstructured grids in problems with a complicated geometry. In this development, the concept of unstructured orthogonal grids is introduced and applied to this model. The governing differential equations are discretized by means of a semi-implicit algorithm that is robust, stable and very efficient. The resulting model is relatively simple, conserves mass, can fit complicated boundaries and yet is sufficiently flexible to permit local mesh refinements in areas of interest. Moreover, the simulation of the flooding and drying is included in a natural and straightforward manner. These features are illustrated by a test case for studies of convergence rates and by examples of flooding on a river plain and flow in a shallow estuary. Copyright ?? 2000 John Wiley & Sons, Ltd.

International Journal for Numerical Methods in Flu↗

USGS2011

Adaptive finite volume methods with well-balanced Riemann solvers for modeling floods in rugged terrain: Application to the Malpasset dam-break flood (France, 1959)

The simulation of advancing flood waves over rugged topography, by solving the shallow-water equations with well-balanced high-resolution finite volume methods and block-structured dynamic adaptive mesh refinement (AMR), is described and validated in this paper. The efficiency of block-structured AMR makes large-scale problems tractable, and allows the use of accurate and stable methods developed for solving general hyperbolic problems on quadrilateral grids. Features indicative of flooding in rugged terrain, such as advancing wet-dry fronts and non-stationary steady states due to balanced source terms from variable topography, present unique challenges and require modifications such as special Riemann solvers. A well-balanced Riemann solver for inundation and general (non-stationary) flow over topography is tested in this context. The difficulties of modeling floods in rugged terrain, and the rationale for and efficacy of using AMR and well-balanced methods, are presented. The algorithms are validated by simulating the Malpasset dam-break flood (France, 1959), which has served as a benchmark problem previously. Historical field data, laboratory model data and other numerical simulation results (computed on static fitted meshes) are shown for comparison. The methods are implemented in GEOCLAW, a subset of the open-source CLAWPACK software. All the software is freely available at. Published in 2010 by John Wiley & Sons, Ltd.

International Journal for Numerical Methods in Flu↗

USGS2005

Semi-implicit finite difference methods for three-dimensional shallow water flow

A semi-implicit finite difference method for the numerical solution of three-dimensional shallow water flows is presented and discussed. The governing equations are the primitive three-dimensional turbulent mean flow equations where the pressure distribution in the vertical has been assumed to be hydrostatic. In the method of solution a minimal degree of implicitness has been adopted in such a fashion that the resulting algorithm is stable and gives a maximal computational efficiency at a minimal computational cost. At each time step the numerical method requires the solution of one large linear system which can be formally decomposed into a set of small three-diagonal systems coupled with one five-diagonal system. All these linear systems are symmetric and positive definite. Thus the existence and uniquencess of the numerical solution are assured. When only one vertical layer is specified, this method reduces as a special case to a semi-implicit scheme for solving the corresponding two-dimensional shallow water equations. The resulting two- and three-dimensional algorithm has been shown to be fast, accurate and mass-conservative and can also be applied to simulate flooding and drying of tidal mud-flats in conjunction with three-dimensional flows. Furthermore, the resulting algorithm is fully vectorizable for an efficient implementation on modern vector computers.

International Journal for Numerical Methods in Flu↗

USGS2004

PHAST--a program for simulating ground-water flow, solute transport, and multicomponent geochemical reactions

The computer program PHAST simulates multi-component, reactive solute transport in three-dimensional saturated ground-water flow systems. PHAST is a versatile ground-water flow and solute-transport simulator with capabilities to model a wide range of equilibrium and kinetic geochemical reactions. The flow and transport calculations are based on a modified version of HST3D that is restricted to constant fluid density and constant temperature. The geochemical reactions are simulated with the geochemical model PHREEQC, which is embedded in PHAST. PHAST is applicable to the study of natural and contaminated ground-water systems at a variety of scales ranging from laboratory experiments to local and regional field scales. PHAST can be used in studies of migration of nutrients, inorganic and organic contaminants, and radionuclides; in projects such as aquifer storage and recovery or engineered remediation; and in investigations of the natural rock-water interactions in aquifers. PHAST is not appropriate for unsaturated-zone flow, multiphase flow, density-dependent flow, or waters with high ionic strengths. A variety of boundary conditions are available in PHAST to simulate flow and transport, including specified-head, flux, and leaky conditions, as well as the special cases of rivers and wells. Chemical reactions in PHAST include (1) homogeneous equilibria using an ion-association thermodynamic model; (2) heterogeneous equilibria between the aqueous solution and minerals, gases, surface complexation sites, ion exchange sites, and solid solutions; and (3) kinetic reactions with rates that are a function of solution composition. The aqueous model (elements, chemical reactions, and equilibrium constants), minerals, gases, exchangers, surfaces, and rate expressions may be defined or modified by the user. A number of options are available to save results of simulations to output files. The data may be saved in three formats: a format suitable for viewing with a text editor; a format suitable for exporting to spreadsheets and post-processing programs; or in Hierarchical Data Format (HDF), which is a compressed binary format. Data in the HDF file can be visualized on Windows computers with the program Model Viewer and extracted with the utility program PHASTHDF; both programs are distributed with PHAST. Operator splitting of the flow, transport, and geochemical equations is used to separate the three processes into three sequential calculations. No iterations between transport and reaction calculations are implemented. A three-dimensional Cartesian coordinate system and finite-difference techniques are used for the spatial and temporal discretization of the flow and transport equations. The non-linear chemical equilibrium equations are solved by a Newton-Raphson method, and the kinetic reaction equations are solved by a Runge-Kutta or an implicit method for integrating ordinary differential equations. The PHAST simulator may require large amounts of memory and long Central Processing Unit (CPU) times. To reduce the long CPU times, a parallel version of PHAST has been developed that runs on a multiprocessor computer or on a collection of computers that are networked. The parallel version requires Message Passing Interface, which is currently (2004) freely available. The parallel version is effective in reducing simulation times. This report documents the use of the PHAST simulator, including running the simulator, preparing the input files, selecting the output files, and visualizing the results. It also presents four examples that verify the numerical method and demonstrate the capabilities of the simulator. PHAST requires three input files. Only the flow and transport file is described in detail in this report. The other two files, the chemistry data file and the database file, are identical to PHREEQC files and the detailed description of these files is found in the PHREEQC documentation.

Techniques and Methods↗