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At least 613 records · Page 34

Star formation in X-ray cluster cooling flows

The basic equations for X-ray cluster cooling flows, including the effects of star formation, are derived. It is confirmed that cooling flows are thermally unstable, and it is found that the fastest-growing linear perturbations in the flow are radial, comoving, and isobaric. A local approximation for the star formation rate is developed which allows analytic solutions to be found for both isobaric and gravity-dominated cooling flows. These solutions are used to show how star formation reduces the gas density, thereby lowering the cooling rate, which raises the gas temperature relative to the non-star-forming case. The analytic solutions are also used to estimate the spatial distribution of newly formed stars. It is found that star formation must proceed at a nearly maximal rate if isothermal mass profiles are to result.

White, Raymond E., III↗

Perturbations of the Richardson number field by gravity waves

An analytic solution is presented for a stratified fluid of arbitrary constant Richardson number. By computer aided analysis the perturbation fields, including that of the Richardson number can be calculated. The results of the linear analytic model were compared with nonlinear simulations, leading to the following conclusions: (1) the perturbations in the Richardson number field, when small, are produced primarily by the perturbations of the shear; (2) perturbations of in the Richardson number field, even when small, are not symmetric, the increase being significantly larger than the decrease (the linear analytic solution and the nonlinear simulations both confirm this result); (3) as the perturbations grow, this asymmetry increases, but more so in the nonlinear simulations than in the linear analysis; (4) for large perturbations of the shear flow, the static stability, as represented by N2, is the dominating mechanism, becoming zero or negative, and producing convective overturning; and (5) the convectional measure of linearity in lee wave theory, NH/U, is no longer the critical parameter (it is suggested that (H/u sub 0) (du sub 0/dz) takes on this role in a shearing flow).

Wurtele, M. G.↗

Homogenized blocked arcs for multicriteria optimization of radiotherapy: Analytical and numerical solutions

Purpose: Homogenized blocked arcs are intuitively appealing as basis functions for multicriteria optimization of rotational radiotherapy. Such arcs avoid an organ-at-risk (OAR), spread dose out well over the rest-of-body (ROB), and deliver homogeneous doses to a planning target volume (PTV) using intensity modulated fluence profiles, obtainable either from closed-form solutions or iterative numerical calculations. Here, the analytic and iterative arcs are compared. Methods: Dose-distributions have been calculated for nondivergent beams, both including and excluding scatter, beam penumbra, and attenuation effects, which are left out of the derivation of the analytic arcs. The most straightforward analytic arc is created by truncating the well-known Brahme, Roos, and Lax (BRL) solution, cutting its uniform dose region down from an annulus to a smaller nonconcave region lying beyond the OAR. However, the truncation leaves behind high dose hot-spots immediately on either side of the OAR, generated by very high BRL fluence levels just beyond the OAR. These hot-spots can be eliminated using alternative analytical solutions ''C'' and ''L,'' which, respectively, deliver constant and linearly rising fluences in the gap region between the OAR and PTV (before truncation). Results: Measured in terms of PTV dose homogeneity, ROB dose-spread, and OAR avoidance, C solutions generate better arc dose-distributions than L when scatter, penumbra, and attenuation are left out of the dose modeling. Including these factors, L becomes the best analytical solution. However, the iterative approach generates better dose-distributions than any of the analytical solutions because it can account and compensate for penumbra and scatter effects. Using the analytical solutions as starting points for the iterative methodology, dose-distributions almost as good as those obtained using the conventional iterative approach can be calculated very rapidly. Conclusions: The iterative methodology is appropriate and useful for computing homogenized blocked arcs, as it produces better dose-distributions than the analytic approaches and their obvious extensions, and can more straightforwardly be used to generate homogenized arcs for concave OARs. However, the analytical solutions provide promising starting points for the iterative algorithm, leading to fast convergence.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

Development and Implementation of Radiation-Hydrodynamics Verification Test Problems

Analytic solutions to the radiation-hydrodynamic equations are useful for verifying any large-scale numerical simulation software that solves the same set of equations. The one-dimensional, spherically symmetric Coggeshall No.9 and No.11 analytic solutions, cell-averaged over a uniform-grid have been developed to analyze the corresponding solutions from the Los Alamos National Laboratory Eulerian Applications Project radiation-hydrodynamics code xRAGE. These Coggeshall solutions have been shown to be independent of heat conduction, providing a unique opportunity for comparison with xRAGE solutions with and without the heat conduction module. Solution convergence was analyzed based on radial step size. Since no shocks are involved in either problem and the solutions are smooth, second-order convergence was expected for both cases. The global L1 errors were used to estimate the convergence rates with and without the heat conduction module implemented.

97 MATHEMATICS AND COMPUTING↗

An analytical phytoplankton model and its application in the tidal freshwater James River

Here, this study constructs an analytical model to understand the phytoplankton dynamics in a one-dimensional system by integrating physical and biological processes. We present an analytical solution of phytoplankton distribution along the longitudinal direction of a tidal river. The solution has two components: water age and accumulative growth. The water age represents the overall effect of physical transport due to advection, while the accumulative growth represents the accumulation of phytoplankton biomass due to biological processes during the transport. In addition, an alternative solution with nonlinear reaction term is given to account for the feedback mechanism of nonlinear processes that affect phytoplankton growth rate. The nonlinear reaction term can be also used to approximate the effect of physical mixing on phytoplankton distribution. The analytical solutions can be further used to predict the temporal variation and spatial distribution of phytoplankton along the tidal river. They also serve as a mathematical tool to unite different field phytoplankton observations. We applied the model in the Tidal Freshwater Region (TFR) of James River under different dynamic conditions and the model results match well with the observations, which validates the theory on different time scales. Furthermore, an analysis is conducted for the Local Chlorophyll-a Maximum (LCM) in the TFR. It shows that the condition for LCM is biologically controlled, but its location is regulated by river flow and river geomorphology.

54 ENVIRONMENTAL SCIENCES↗

Evaluating geothermal and hydrogeologic controls on regional groundwater temperature distribution

Described is a one-dimensional (1-D) analytic solution is developed for heat transport through an aquifer system where the vertical temperature profile in the aquifer is nearly uniform. The general anisotropic form of the viscous heat generation term is developed for use in groundwater flow simulations. The 1-D solution is extended to more complex geometries by solving the equation for piece-wise linear or uniform properties and boundary conditions. A moderately complex example, the Eastern Snake River Plain (ESRP), is analyzed to demonstrate the use of the analytic solution for identifying important physical processes. For example, it is shown that viscous heating is variably important and that heat conduction to the land surface is a primary control on the distribution of aquifer and spring temperatures. Use of published values for all aquifer and thermal properties results in a reasonable match between simulated and measured groundwater temperatures over most of the 300 km length of the ESRP, except for geothermal heat flow into the base of the aquifer within 20 km of the Yellowstone hotspot. Previous basal heat flow measurements (110 mW/m 2 ) made beneath the ESRP aquifer were collected at distances of > 50 km from the Yellowstone Plateau, but a higher basal heat flow of 150 mW/m 2 is required to match groundwater temperatures near the Plateau. As a result, the ESRP example demonstrates how the new tool can be used during preliminary analysis of a groundwater system, allowing efficient identification of the important physical processes that must be represented during more-complex 2-D and 3-D simulations of combined groundwater and heat flow.

58 GEOSCIENCES↗

Test Problems for Programmed Burn [Slides]

This document presents updated FLAG results for the Kenamond programmed burn test problems. Problem definition, exact solutions for constant detonation velocity, and FLAG results using a Huygens solution are provided in Kenamond’s “Lund High Explosive Programmed Burn Model in the FLAG Code” LA-UR-09-08297. FLAG results using Line-of-Sight (LOS) preconditioning of the Huygens solution are presented in Kenamond’s “Line-of-Sight (LOS) High Explosive Programmed Burn Model in the FLAG Code” LA-UR-09-08296. FLAG has an old, serial, uniform-grid “mapped” 2D DSD solver and a new, inline, parallel, conformal mesh 2D/3D DSD solver. These solvers will be exercised with flat Dn-Kappa relations to facilitate comparison with analytic solutions. The remaining slides document FLAG results with 1) Huygens, 2) Huygens with LOS, 3) old DSD, and 4) new DSD. By design, these geometries are coarsely meshed. Plotting error with respect to the analytic solutions reveals sensitivities to element type, element aspect ratio, boundary conditions, detonator timing, and discontinuous detonation velocity. In all plots of this document, color is absolute error relative to the exact solution. In all plots of this document, contours are burntime isochrones of the numerical solution.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Uncertainty Analysis of Slug Calorimeters in the HyMETS Arc-Jet Facility

The objective of this work is to perform an uncertainty analysis of the deduced stagnation heat flux environment on a slug calorimeter for conditions that span the performance envelope of the Hypersonic Materials Environmental Test System arc-jet facility located at NASA Langley Research Center. Analytical solutions are developed for boundary-value problems on the slug element accounting for non-ideal effects, including spatial variation in the slug heat flux, multi-dimensional thermal conduction, and back-face losses, which departs from the state-of-the-art method derived from the American Society of Testing and Materials. Boundary-value problem definitions are informed by preliminary finite element thermal analysis of the slug calorimeter assembly (including both slug and housing) and just the slug element. The analytical solutions are presented in a general sense and in a truncated form from error analysis. Results are shown in optimizing and validating the analytical models against available slug back-face thermal data. The optimization results indicate that the appropriate epistemic uncertainty of the deduced stagnation heat flux on the slug calorimeter is at most±2.5% for both a high-and low-enthalpy test condition. In addition, a numerical approach is used to determine the aleatory (probabilistic) uncertainty component in the slug stagnation heat flux by applying a marching least-squares slope routine through the steady-state portion of the slug back-face thermal response. Results indicate a compromise between the number of samples and the filter frequency of slug back-face thermal data points when evaluating the standard deviation of the deduced stagnation heat flux statistics. When combining the mixed uncertainty, both aleatory and epistemic, the interval of uncertainty in the deduced stagnation heat flux is determined to be up to ±4%, which is at least a 60% reduction from the standard uncertainty used in the state-of-the-art method.

uncertainty↗

A class of exact solutions for population balances with arbitrary internal coordinates

We develop a novel transformation that maps the linear, nonhomogeneous, multidimensional population balance equation (PBE) into an advection equation that is readily solved using the method of characteristics. The PBEs targeted by this transformation exclude aggregation, breakage, and time dependent birth and death rates. In addition, internal coordinates are assumed to grow independently of each other. The ensuing general formulation is then used to recover closed‐form analytical solutions for problems with monosurface and bulk‐diffusion growth‐rates as well as Gaussian‐type nucleation. For completeness, we derive the multidimensional Green's functions for our approach. This is followed by a brief discussion on how the proposed framework may be used for code verification of moment methods such as the quadrature method of moments and the direct quadrature method of moments. Finally, a sample Mathematica code is provided to derive analytical solutions for the single‐internal‐coordinate case given user‐specified growth, birth, and death rates. © 2015 American Institute of Chemical Engineers AIChE J, 61: 1691–1698, 2015

Saad, Tony↗

The behavior of beams of relativistic non-thermal electrons under the influence of collisions and synchrotron losses

For many astrophysical situations, such as in solar flares or cosmic gamma-ray bursts, continuum gamma rays with energies up to hundreds of MeV were observed, and can be interpreted to be due to bremsstrahlung radiation by relativistic electrons. The region of acceleration for these particles is not necessarily the same as the region in which the radiation is produced, and the effects of the transport of the electrons must be included in the general problem. Hence it is necessary to solve the kinetic equation for relativistic electrons, including all the interactions and loss mechanisms relevant at such energies. The resulting kinetic equation for non-thermal electrons, including the effects of Coulomb collisions and losses due to synchrotron emission, was solved analytically in some simple limiting cases, and numerically for the general cases including constant and varying background plasma density and magnetic field. New approximate analytic solutions are presented for collision dominated cases, for small pitch angles and all energies, synchrotron dominated cases, both steady-state and time dependent, for all pitch angles and energies, and for cases when both synchrotron and collisional energy losses are important, but for relativistic electrons. These analytic solutions are compared to the full numerical results in the proper limits. These results will be useful for calculation of spectra and angular distribution of the radiation (x rays, gamma-rays, and microwaves) emitted via synchrotron or bremsstrahlung processes by the electrons. These properties and their relevance to observations will be observed in subsequent papers.

Mctiernan, James M.↗

The behavior of beams of relativistic nonthermal electrons under the influence of collisions and synchrotron losses

For many astrophysical situations, such as in solar flares or cosmic gamma-ray bursts, continuum gamma rays with energies up to hundreds of MeV were observed, and can be interpreted to be due to bremsstrahlung radiation by relativistic electrons. The region of acceleration for these particles is not necessarily the same as the region in which the radiation is produced, and the effects of the transport of the electrons must be included in the general problem. Hence it is necessary to solve the kinetic equation for relativistic electrons, including all the interactions and loss mechanisms relevant at such energies. The resulting kinetic equation for non-thermal electrons, including the effects of Coulomb collisions and losses due to synchrotron emission, was solved analytically in some simple limiting cases, and numerically for the general cases including constant and varying background plasma density and magnetic field. New approximate analytic solutions are presented for collision dominated cases, for small pitch angles and all energies, synchrotron dominated cases, both steady-state and time dependent, for all pitch angles and energies, and for cases when both synchrotron and collisional energy losses are important, but for relativistic electrons. These analytic solutions are compared to the full numerical results in the proper limits. These results will be useful for calculation of spectra and angular distribution of the radiation (X rays, gamma-rays, and microwaves) emitted via synchrotron or bremsstrahlung processes by the electrons. These properties and their relevance to observations will be observed in subsequent papers.

Mctiernan, James M.↗

A Bayesian Approach for Extracting Fluorescence Lifetimes from Sparse Data Sets and Its Significance for Imaging Experiments

The measurement of fluorescence lifetimes, especially in small sample volumes, presents the dual challenge of probing a small number of fluorophores and fitting the concomitant sparse data set to the appropriate excited-state decay function. A common method of analysis, such as the maximum likelihood (ML) technique, assumes a uniform probability distribution of the parameters describing the fluorescence decay function. An improvement is thus suggested by implementing a suitable nonuniform distribution, as is provided by a Bayesian framework, where the distribution of parameters is obtained from both their prior knowledge and the evidence-based likelihood of an event for a given set of parameters. We have also considered the Dirichlet prior distribution, whose mathematical form enables analytical solutions of the fitting parameters to be rapidly obtained. If Gaussian and exponential prior distributions are judiciously chosen, they reproduce the experimental target lifetime to within 20% with as few as 20 total photon counts for the data set, as does the Dirichlet prior distribution. But because of the analytical solutions afforded by the Dirichlet prior distribution, it is proposed to employ a Dirichlet prior to search parameter space rapidly to provide, if necessary, appropriate parameters for subsequent employment of a Gaussian or exponential prior distribution

97 MATHEMATICS AND COMPUTING↗

An Approximate Solution for the Circumsolar Flow Field of a Sun Moving Through the Local Interstellar Medium (LISM)

The solar system is in relative motion with respect to the ambient interstellar medium. The supersonic solar wind is expected to pass through the termination shock, thus the solar wind plasma eventually has to enter into an asymptotic outflow geometry appropriately adopted to this counterflow situation. Many attempts have been done to simulate the interaction between the solar wind and the LISM numerically. In this paper we generalize a Parker type analytical solution of the counterflow. The idea is to introduce a special kind of compressibility of the solar wind flow. With the assumption that only a transversal component of the density gradient normal to the flow lines exists we are able to calculate a full set of hydrodynamical quantities describing the circumsolar flow field of a Sun moving through the LISM. The equations governing the velocity and density fields lead to analytical solutions which can be taken as good approximations to the more general case of compressible plasma flows.

Ratkiewicz, Romana E.↗

High-order parabolic beam approximation for aero-optics

The parabolic beam equations are solved using high-order compact differences for the Laplacians and Runge-Kutta integration along the beam path. The solution method is verified by comparison to analytical solutions for apertured beams and both constant and complex index of refraction. An adaptive 4th-order Runge-Kutta using an embedded 2nd-order method is presented that has demonstrated itself to be very robust. For apertured beams, the results show that the method fails to capture near aperture effects due to a violation of the paraxial approximation in that region. Initial results indicate that the problem appears to be correctable by successive approximations. A preliminary assessment of the effect of turbulent scales is undertaken using high-order Lagrangian interpolation. The results show that while high fidelity methods are necessary to accurately capture the large scale flow structure, the method may not require the same level of fidelity in sampling the density for the index of refraction. The solution is used to calculate a phase difference that is directly compared with that commonly calculated via the optical path difference. Propagation through a supersonic boundary layer shows that for longer wavelengths, the traditional method to calculate the optical path is less accurate than for shorter wavelengths. While unlikely to supplant more traditional methods for most aero-optics applications, the current method can be used to give a quantitative assessment of the other methods as well as being amenable to the addition of more physics.

97 MATHEMATICS AND COMPUTING↗

Analytical and Numerical Solution for a Solidifying Liquid Alloy Slab

Numerical and analytical solutions are presented for the temperature and concentration distributions during the solidification of a binary liquid alloy slab. The slab is taken to be of a finite depth but infinite in the horizontal direction. The solidification process is started by withdrawing a fixed amount of heat from the lower surface of the slab. The upper surface of the slab is subjected to both radiation and convective conditions. The solution gives the concentration and temperature profiles and the interface position as a function of time. Due to the smallness of the mass diffusion coefficient in the solid, the numerical solution method breaks down whenever the ratio of the diffusivities in the solid and the liquid falls below a certain value. An analytical method is developed which gives accurate solution for any value of the diffusivity ratio.

Antar, B. N.↗

Kinetics of metastable atoms and non-Maxwellian electrons in two-temperature plasmas

Numerical and analytical solutions of the electron Boltzmann equation in two-temperature steady-state helium plasma are studied in a broad range of conditions T(a) = 5,000-20,000 K, T(e) = 10,000-20,000 K; N(a) = 10 to the 10th - 10 to the 18th per cu cm. The WKB analytical solution is found to be satisfactory in most situations. The deviation of the electron distribution from Maxwellian and a possibility of raising of the tail of the distribution in presence of sources of fast electrons is also discussed.

Kunc, J. A.↗

Quantitative Characterization of Impacts of Coupled Geomechanics and Flow on Safe and Permanent Geological Storage of CO 2 in Fractured Aquifers

This is the final scientific report for the award DE-FE0023305, entitled “Quantitative Characterization of Impacts of Coupled Geomechanics and Flow on Safe and Permanent Geological Storage of CO 2 in Fractured Aquifers.” The work has been divided into six tasks. In Task 2, we characterized rock properties, which are important when developing a quantitative approach for understanding and predicting geomechanical effects on large-scale CO 2 injection and long-term storage in the subsurface. Rock properties of interest for this characterization include porosity, permeability, elastic constants, strength, and heat capacity. We measured rock properties for three different rock types: concrete, sandstone and shale. These properties were obtained from the acoustic measurement, permeability and porosity measurement, Brazilian test, the uniaxial compressive test, and heat capacity measurement. Then, using the Brazilian technique to fracture samples, we measured their permeability under brine injection for varying effective stresses, followed by similar measurements of permeability under sc-CO 2 injection. Permeability to brine and sc-CO 2 decreased as the effective stress increased. The apparent permeability to sc-CO 2 was an order of magnitude less than that for brine, a consequence of CO 2 being the non-wetting fluid. In Task 3, we developed understandings and correlations for CO 2 injection pressure induced fracturing. We initially fractured four eight inch cubes of concrete with brine to establish a standard test procedure. Then, additional concrete samples were fractured using brine. Confining stresses were exerted on the samples and brine was injected at a constant rate into a borehole that was previously drilled into the concrete cube. The borehole pressure was measured and fracture initiation was identified as a peak in the pressure profile. That was followed by fracture propagation, which was identified as a plateau in the pressure profile, and finally the fracture reached the sample surface. The sample surfaces were photographed before and after fracturing, which enabled determination of where the fracture reached the sample surface. Afterwards, aqueous food dye solution was injected into the fractured concrete samples to color the fracture planes. The borehole was filled with food dye solution in advance, a gas pulse was transmitted to the borehole, and the borehole was then shut in to allow the food dye to be driven into existing fractures. Then, the concrete samples were broken down by high-pressure gas injection to reveal the geometry and morphology of the hydraulic fractures. In addition, acoustic measurements were conducted on multiple locations on each face of the concrete samples before and after injection for comparison. This comparison was used to confirm changes in the internal structure of the concrete sample, namely the formation of fractures. There were six concrete samples fractured with brine. Afterwards, concrete samples were fracture using sc-CO 2 . The main change in experimental procedure was the inclusion of temperature control, since in the field, we mostly encounter temperature and pressure conditions above the supercritical point of CO 2 . The concrete samples were pre-heated in an oven to elevate their temperature, and thermal tape was used to heat the CO 2 to a temperature above its supercritical temperature. Two concrete samples were fractured using liquid CO 2 , four concrete samples were fractured using gaseous CO 2 , and twenty two concrete samples were fractured using sc-CO 2 . For the concrete samples fractured using sc-CO 2 , three had pre-existing hydraulic fractures on the surface near the wellbore, and two were water saturated (as opposed to containing only air). Two concrete samples were composite samples, with a high permeability ball in the center, in order to represent CO 2 injection into a high permeability zone surrounded by a low permeability sealing formation. Later, we fractured five shale samples, obtained from the Niobrara shale outcrop, with sc-CO 2 . The injection rate was constant, confining stresses were applied, the shale samples were pre-heated in an oven to the desired temperature, and the sample was then fractured. An analysis of the failure type was done for many of the concrete samples, based on the predicted break down pressure for tensile and shear failure, and it was found that most samples fractured due to shear failure. Also, the presence of induced fractures has little to no effect on the fracture initiation pressure, but significantly changes the fracture morphology. The CO 2 fracturing in water saturated samples behaved much similar to brine induced fracturing. There was a significant difference of breakdown pressure for injection of CO 2 and brine, where the breakdown pressure for CO 2 induced fracturing is generally around the minimum horizontal stress and that for brine is much higher. Finally, the fracture orientation for most samples with large confining stress differential was perpendicular to the minimum horizontal stress. When the stress differential, especially the difference between the two directions with the smallest stresses was small, the fracture orientation was determined not only by the minimum stress direction, but also the magnitude of the confining stress, the tensile strength of the rock, and the breakdown pressure. In Task 4, we modified our coupled flow-geomechanical models to model fracture growth and propagation in storage formations and caprocks. These flow-geomechanical models are TOUGH2-CSM and TOUGH2-FLAC. The TOUGH2-CSM fluid and heat flow formulation is based on the TOUGH2 formulation of mass and energy conservation equations that govern fluid and heat flow in general multiphase, multicomponent, multi-porosity systems. The TOUGH2-CSM simplified geomechanical formulation is based on the linear theory of elasticity applied to multi-porosity non-isothermal (thermo-multi-poroelastic) media. We previously derived, from the fundamental relations of the linear theory of elasticity, an equation relating mean stress, pore pressures, temperatures, and body force (the Mean Stress equation) that we added to the TOUGH2-CSM fluid and heat flow equations. We extended the TOUGH2-CSM simplified geomechanical formulation to calculate the entire stress tensor by deriving equations for stress tensor components from derivatives of the Cartesian thermo-multi-poroelastic Navier equation components. In addition, we derived equations for stress tensor components in rz-coordinates. We solved these geomechanical equations using the integral finite difference method. This method utilizes momentum fluxes obtained by “factoring” a divergence operator out of each geomechanical equation. In Task 5, we did a literature survey to determine suitable correlations for fracture initiation, growth, and propagation. After surveying numerous papers, we modified TOUGH2-CSM to model stress dependent fracture initiation and growth using the Mohr-Coulomb criterion for shear failure in faults and caprock and the condition of negative minimum effective stress for tensile failure. Tensile failure results in a fracture whose width depends on the difference between fracture pressure and minimum stress and whose propagation is based on the difference between the stress intensity factor at the fracture tip and the rock toughness. The TOUGH-FLAC model is based on the FLAC3D ubiquitous joint model, extended for the modeling of permeability changes induced by changes in effective normal stress across fractures as well permeability changes caused by shearing along fractures. In Task 6, we validated our coupled flow-geomechanical models using analytical solutions and problems from the literature. We validated the TOUGH2-CSM modifications using analytical solutions to the displacement from a uniform load on a semi-infinite elastic medium and the two-dimensional Mandel-Cryer effect. Those were followed by a sample problem to test our stress tensor calculations, which was for injection into a single-phase reservoir with constant properties. Comparisons of our simulator to published results were done on for the depletion of a single-phase reservoir with stress dependent porosity, two examples from the In Salah gas project, an axisymmetric baserock–reservoir–caprock system with a normal faulting stress regime that had thermally induced shear stresses, simulations of CO 2 leakage through caprock in a two-dimensional reservoir in Cartesian coordinates, and predictions of caprock failure from CO 2 injection into an axisymmetric reservoir. Finally, we simulated CO 2 injection pressure-induced fracturing from one of our laboratory studies, Sample 39, and obtained a fracture profile. We also validated the TOUGH2-FLAC model for fluid driven fracture growth against solutions based on the KGD model and a case with an inclined fracture that was loaded from the boundaries to achieve wing-crack propagation. The strain softening tensile behavior and softening of modulus considering a simple damage approach was verified by a simulation representing hydraulic fracturing stress measurement around a vertical well. Finally, the model was validated against deep fracture zone opening and surface uplift at In Salah with inverses analysis used to improve the match of simulation with field data. In Task 7, we developed a scheme based on inverse modeling that can be used to identify caprock leakage. Injection of fluid into a reservoir results in a time varying pressure profile that depends on the properties of the rock and the fluid. The presence of significant caprock leakage would effect this pressure profile and we used inverse modeling, namely the Levenberg-Marquardt method, to determine the leakage location from the pressure profile. A two-dimensional and a three-dimensional example based on published data were presented. The published simulation was run and its results were considered as “field” data. In the two-dimensional example, a simulation of CO 2 injection into a two-dimensional aquifer-caprock system, the caprock had a vertical fault through which leakage could occur. The location of this leakage was obtained from our inverse modeling scheme. In the three-dimensional example, the simulation domain had four geological layers, one of which was the injection zone that contained a horizontal injection well. We simulated fractures as high permeability gridblocks, introduced a fracture that spanned the caprock depth, and ran a simulation to obtain “field” data. Then, using our inverse modeling scheme, we obtained the location of that fracture.

58 GEOSCIENCES↗

The “Fresnel Equations” for Diffuse radiation on Inclined photovoltaic Surfaces (FEDIS)

Here the well-known Fresnel equations solve for the reflection and transmission of light for precise incident angles. The transmission of diffuse radiation incident on a planar or domed surface is often needed for real-world applications. Due to the complexity of the Fresnel equations, the analytical solution of the integration has hitherto been unobtainable over the last centuries. Therefore, this problem was numerically solved by integrating the angular transmittances in space often leading to substantial computing burden and bias in the results. To efficiently estimate the solar energy resource for a glass-covered photovoltaic (PV) module, we derive an analytical solution of diffuse transmission based on the rigorous integration of an alternate form of the Fresnel equations. The approach leads to a simple yet accurate relative transmittance model that reconciles the solar energy sensed by pyranometers and PV panels. With limited and clearly stated approximations, the complex mathematical derivation resulted in an elegant solution. An experiment using 1-year of data at the National Renewable Energy Laboratory's (NREL's) Solar Radiation Research Laboratory (SRRL) shows that the new model dramatically decreases the disparity between the solar radiation measurements by a Kipp and Zonen CM Pyranometer 22 (CMP22) and an IMT reference cell on a 1-axis tracking system. The solution in this paper can be widely used in scientific and engineering research, development, and applications wherever the Fresnel equations are used.

14 SOLAR ENERGY↗