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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Optimization using pathwise algorithmic derivatives of electromagnetic shower simulations

Among the well-known methods to approximate derivatives of expectancies computed by Monte-Carlo simulations, averages of pathwise derivatives are often the easiest one to apply. Computing them via algorithmic differentiation typically does not require major manual analysis and rewriting of the code, even for very complex programs like simulations of particle-detector interactions in high-energy physics. However, the pathwise derivative estimator can be biased if there are discontinuities in the program, which may diminish its value for applications. This work integrates algorithmic differentiation into the electromagnetic shower simulation code HepEmShow based on G4HepEm, allowing us to study how well pathwise derivatives approximate derivatives of energy depositions in a sampling calorimeter with respect to parameters of the beam and geometry. We found that when multiple scattering is disabled in the simulation, means of pathwise derivatives converge quickly to their expected values, and these are close to the actual derivatives of the energy deposition. Additionally, we demonstrate the applicability of this novel gradient estimator for stochastic gradient-based optimization in a model example.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

A universal implementation of radiative effects in neutrino event generators

Due to the similarities between electron-nucleus (eA) and neutrino-nucleus scattering (νA), eA data can contribute key information to improve cross-section modeling in eA and hence in νA event generators. However, to compare data and generated events, either the data must be radiatively corrected or radiative effects need to be included in the event generators. We implemented a universal radiative corrections program that can be used with all reaction mechanisms and any eA event generator. Our program includes real photon radiation by the incident and scattered electrons, and virtual photon exchange and photon vacuum polarization diagrams. It uses the “extended peaking” approximation for electron radiation and neglects charged hadron radiation. This method, validated with GENIE, can also be extended to simulate νA radiative effects. This work facilitates data-event-generator comparisons used to improve νA event generators for the next-generation of neutrino experiments. Program Title: emMCRadCorr CPC Library link to program files:https://doi.org/10.17632/hmsxg82vnf.1 Developer's repository link:https://github.com/e4nu/emMCRadCorr Licensing provisions: AGPLv3 Programming language:C++ Nature of problem: Radiative effects can significantly modify the event kinematics and the resulting cross-sections. Such effects must be accounted for when comparing event generators to eA data. Existing radiative correction codes are tailored to specific processes and topologies, and are limited to a restricted phase space defined by the spectrometer acceptance. Therefore, a more general approach is required to apply radiative corrections to semi-inclusive and exclusive eA measurements. Solution method: Our program incorporates real photon radiation from both the incident and scattered electrons, as well as virtual photon exchange and photon vacuum polarization effects. It employs the “extended peaking” approximation for electron radiation while neglecting contributions from charged hadron radiation. The code is fully decoupled from event generator codes and can be used for all event generators in the market.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

A provably stable numerical method for the anisotropic diffusion equation in confined magnetic fields

We present a novel numerical method for solving the anisotropic diffusion equation in magnetic fields confined to a periodic box which is accurate and provably stable. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing both boundary conditions and parallel diffusion operator obtained by field line tracing. We prove that the fully-discrete approximation is unconditionally stable. Discrete energy estimates are shown to match the continuous energy estimate given the correct choice of penalty parameters. A nonlinear penalty parameter is shown to provide an effective method for tuning the parallel diffusion penalty and significantly minimises rounding errors. Several numerical experiments, using manufactured solutions, the “NIMROD benchmark” problem and a single island problem, are presented to verify numerical accuracy, convergence, and asymptotic preserving properties of the method. Finally, we present a magnetic field with chaotic regions and islands and show the contours of the anisotropic diffusion equation reproduce key features in the field.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

Enhancing photoionization rate calculations in low-temperature plasmas using spectral methods

Photoionization plays a central role in the development of streamer discharges and other non-equilibrium plasma phenomena. It creates seed electrons, which are essential for positive streamer propagation, allowing the ionization front to move forward. Because of this, accurate modeling of photoionization is very important for predicting streamer behavior and plasma evolution. The photoionization process in air (N 2 – O 2 mixture) is often described by the Zheleznyak model (1982). This model is usually solved through Helmholtz-type equations that approximate the Zheleznyak photoionization model (Zheleznyak et al. 1982) as Partial Differential Equations (PDEs). Conventional numerical methods, such as the Finite Difference Method (FDM) or Finite Volume Method (FVM), are widely used to solve these equations. Although they are prevalent, the computational cost of these methods due to their need for matrix operations and iterative solver is demanding. To address this challenge, this work develops a spectral solver based on the Fast Fourier Transform (FFT) combined with Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST) to calculate the photoionization rate efficiently in an axisymmetric cylindrical domain. This method naturally satisfies the boundary conditions used in the model and converts the PDE into algebraic ones in spectral space. Thus, avoids the need for iterative matrix solvers. When compared with FDM results, it is demonstrated that the new solver not only maintains accuracy, but also reduces the computational cost, showing a performance increase of approximately 100 compared to FDM over a wide range of problem sizes. The method is parallelized using Message Passing Interface (MPI) and has been integrated into a fluid plasma model for streamer simulation. Here, this FFT-based approach provides a fast and reliable alternative for calculating photoionization in fluid models, helping large-scale plasma simulations run faster and efficiently, and allows higher-resolution simulation without extra computational cost.

Axisymmetric system↗
DOE OSTI2025

Assessing electrification readiness in U.S. single-family homes based on a nationwide survey of electrical panel capacities

Electrification of residential buildings is a key strategy for increasing the use of renewable energy sources. Central to this transition is understanding the capacity of existing electrical infrastructure—specifically electrical panels—to safely and effectively manage increased electricity demands from electrification technologies. However, comprehensive nationwide data on electrical panel capacities in U.S. single-family homes is currently lacking. To address this gap, we conducted a nationwide survey of single-family homes, collecting detailed data on electrical panel capacities, breaker slot availability, major electric and gas appliances, electrical panel models, and home characteristics such as construction year and floor area. Photographic documentation was used to verify electrical panel data and appliance information. Results show that approximately 60% of surveyed homes have electrical panels rated at ≥200 amperes (A), indicating that a significant portion of the existing housing stock can accommodate additional electric loads. However, 31% of homes possess panels rated at ≤100 A, potentially restricting their ability to adopt new electric appliances without significant upgrades. Panel capacities positively correlate with both home size and construction year, with newer and larger homes generally better suited for electrification. Homes with higher-capacity panels tend to have fewer gas appliances, reflecting a gradual shift toward electric technologies. Conversely, homes with lower-capacity panels frequently rely on multiple gas appliances, highlighting substantial electrification challenges. Additionally, approximately 3% of surveyed homes had potentially hazardous electrical panel models, emphasizing important safety considerations in the residential electrification process. Our findings underscore the need for targeted policies, financial incentives, and infrastructure investments designed specifically to address infrastructural and safety barriers, particularly in older and smaller homes, to support equitable and efficient electrification across the U.S. residential sector.

Gul, Sadia↗
DOE OSTI2025

The short-term comprehensive impact of the phase-out of global coal combustion on air pollution and climate change

With the continuous intensification of global warming, the reduction and ultimate phase-out of coal combustion is an inevitable trend in the future global energy transformation. This study comprehensively analyzed the impact of phasing out coal combustion on global emissions and concentrations of air pollutants, radiative fluxes, meteorology and climate using Community Earth System Model 2 (CESM2). The results indicate that after the global phase-out of coal combustion, there is a marked decrease in the concentrations of sulfur dioxide (SO 2 ), nitrogen oxides (NO x ) and fine particulate matter (PM 2.5 ), with some regions experiencing a reduction of exceeding 50%. There is no significant change in global ozone (O 3 ) concentration. There are decreasing AOD and positive radiative fluxes globally in the short term, though the cloud contributes minor negative radiative fluxes. The global air temperature may increase by approximately (0.02 ± 0.15) °C on average with regional and seasonal variations, and the precipitation may potentially increase by approximately (2.7 ± 40.6) mm yr −1 globally and over 20% in equatorial regions in the short term. But combined with the decreasing trend of cloud water content in the Northern Hemisphere, it indicates a potential increase in the extremity of precipitation events. In conclusion, this study provides references for global control of air pollution, mitigation strategies of climate change, and transformation of energy structures under the objective of “carbon neutrality”, such as focusing on the negative climate impacts of exacerbating regional warming and increasing extreme precipitation resulting from the rapid reduction of aerosols in the short term.

01 COAL, LIGNITE, AND PEAT↗
DOE OSTI2025

Adaptive Interface-PINNs (AdaI-PINNs) for inverse problems: Determining material properties for heterogeneous systems

Here, we determine spatially varying discontinuous material properties using a domain-decomposition based physics-informed neural networks (PINNs) framework named the Adaptive Interface-PINNs or AdaI-PINNs (Roy et al., 2024). We propose the use of distinct neural networks for the field variables and material properties within each material, utilizing adaptive activation functions. While the neural networks across different materials share the same weights and biases, their activation functions are uniquely tailored using a hyperparameter that influences the slope of the activation function. The proposed framework is tested on several one-dimensional and two-dimensional benchmark examples, and its performance is compared with conventional PINNs and existing domain-decomposition PINNs frameworks, namely, the Multi-domain physics-informed neural network (M-PINN), and the eXtended physics-informed neural networks (XPINNs). The results demonstrate that the proposed approach can determine randomly distributed discontinuous material properties with an L 2 error of $\mathscr{O}$ (10 -3 ) for the material property and the root-mean-square error of $\mathscr{O}$ (10 -3 ) for the primary variable while the other approaches yield errors that are approximately two orders of magnitude larger (that is, $\mathscr{O}$ (10 -1 )). Moreover, the spatial distribution of material properties obtained using the proposed framework is in close agreement with the true distribution, whereas the other approaches fare much worse. Additionally, the proposed approach is approximately 40% faster than its competitors, indicating its potential as a robust alternative for solving inverse problems in heterogeneous materials.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Approach to startup inventory for viable commercial fusion power plant

With the increasing efforts to commercialize fusion power, private and government organizations are investing heavily in the development of technology to support a viable fusion power plant. Deuterium-Tritium (DT) fueled reactors are more prevalent than other proposed designs, requiring tritium processing and handling technology for safe operations and self-sufficiency. Further, each fusion power plant will need a specific-to-design startup inventory of tritium to begin operations. This startup inventory is required prior to breeding and is the minimum tritium inventory required to fill each processing component in the fuel cycle, to offset radioactive decay losses, and to avoid a zero-fuel situation for continuous operation. We present an approach to calculate the startup tritium inventory for a 500 MW th reactor, with considerations for reserve inventory for maintenance and commissioning. A baseline startup inventory was calculated to be approximately 327 gs. This value was obtained using modest assumptions about the technology and operating parameters of a fusion power plant. The required operating reserve inventory or the inventory necessary to keep a fusion power plant operational using only direct internal recycling for 24 h for the same plant design is approximately 642 gs. The approach and findings of this paper will enable fusion energy stakeholders to better utilize the existing scarce global tritium supply.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Coupled feldspar dissolution and secondary mineral precipitation in batch systems: 6. Labradorite dissolution, calcite growth, and clay precipitation at 60 °C and pH 8.2–8.4

Here, we conducted experiments on concurrent labradorite dissolution, calcite precipitation, and clay precipitation in batch reactor systems and tracked reaction processes using multiple isotope tracers. Labradorite was chosen for its role as a major and reactive component in basalt; the experiments thus directly impact our understanding of CO 2 storage in basalt aquifers and enhanced rock weathering. We doped initial solutions with 29 Si, 43 Ca, and Ca 13 CO 3 (s). Experiments were conducted at 60 °C and pH ~ 8.3 for up to 840 h, with isotope ratios in the experimental aqueous solutions measured using MC-ICP-MS. Unidirectional rates of labradorite dissolution near equilibrium were approximately two orders of magnitude slower than far-from-equilibrium rates reported in the literature. Calcite growth occurred near equilibrium and the rates were limited by the labradorite dissolution rates. In the steady state phase, the interplay of these three heterogeneous reactions—labradorite dissolution, calcite growth, and clay precipitation—results in a coupled system that approaches a near-equilibrium state. The system does not reach true equilibrium because labradorite continues to dissolve, albeit at a much slower rate near equilibrium. The overall reaction can be approximated as, Na 0.4 Ca 0.6 Al 1.6 Si 2.4 O 8 + 0.6HCO 3 - + 1·.7H 2 O + 0.4H + → 0.4Na + + 0.6CaCO 3(s) + 0.5Al 2 Si 2 O 5 (OH) 4(s) + 0.6Al(OH) 4 - + 1.4SiO 2 o (aq). The experimental results show that using short-term far-from-equilibrium rate constants would lead to an overestimation of feldspar weathering rates at the Earth’s surface (e.g., basalt weathering and enhanced rock weathering) and CO 2 mineralization in basalt aquifers.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Basin-scale analysis of Mokelumne River Formation for multi-well CO 2 injection

Large-scale carbon sequestration will likely require multiple projects injecting CO 2 into the same subsurface formation, raising concerns about safe operation and efficient use of storage capacity. This study evaluates the long-term response of the Mokelumne River Formation in California’s Sacramento Basin to multi-megaton CO 2 injection using three geologic models of the formation and the open-source simulator GEOS. The analysis focuses on three aspects of reservoir performance: (1) average pressure increase and dissolved CO 2 mass after 30 years for varying well counts and injection rates, (2) pressure interference in a multi-well configuration, and (3) dynamic storage capacity with identification of overpressure-prone regions. The results show that average formation pressure increases linearly with injected mass, while CO 2 dissolution exhibits mixed scaling: approximately linear with the number of wells but sublinear with injection rate, indicating that distributing injection across more wells enhances dissolution more effectively than increasing per-well rates. Pressure-interference effects are significant, with lower-permeability conditions delaying their onset but amplifying their magnitude at later times. Dynamic capacity, defined by the first occurrence of pressure exceeding the local overburden-based limit anywhere in the formation, varies across geologic models and assumed overburden pressure gradients. A lower fidelity geologic model predicts nearly twice the storage capacity of the two higher fidelity models, which consistently estimate approximately 1 Gt under the upper-bound overburden pressure gradient considered for the Sacramento Basin. In all model scenarios, overpressure develops away from injection wells, particularly in higher-elevation regions, highlighting the importance of basin-scale modelling for identifying risks beyond the immediate well vicinity.

Basin-scale↗
DOE OSTI2024

Superconductivity in WBe 2

WBe 2 , which occurs in space group 194, with hexagonal symmetry P6 3 /mmc, is prepared by arc-melting at temperatures above 2200 C, where Be vapor loss is significant. This study is motivated by recent work on MoB 2 and WB 2 , both superconductors (T c =32 and 17 K respectively) under high (∼70 GPa) pressure. In order to avoid the known Be-rich superconducting phases (WBe 13 and WBe 22 ) in the complex phase diagram, both known to be superconducting at 4.1 K, the sample was prepared with a slight (∼5 %) excess of W. The resultant sample, prepared using high purity (99.999 %) Be, is essentially single phase WBe 2 , with some spread in its superconducting properties due to the known homogeneity range. (WBe 2 forms in space group 194 between approximately W 1.02 Be 1.98 and W 0.88 Be 2.12 .) Characterization was carried out with x-ray diffraction, electrical resistivity, ρ, in zero and applied magnetic fields, and specific heat. The resistivity in zero and applied fields and specific heat data indicate that our sample of WBe 2 is a bulk superconductor at ambient pressure with a T c onset in ρ at 1.05 K, and T c (ρ→0) at ∼0.86 K. Furthermore, there is no signature of superconductivity in ρ at 4.1 K, indicating successful avoidance of WBe 13 and WBe 22 . The ρ data in field indicate an upper critical field of approximately 400 gauss.

Electrical transport↗
DOE OSTI2024

RandONets: Shallow networks with random projections for learning linear and nonlinear operators

Deep neural networks have been extensively used for the solution of both the forward and the inverse problem for dynamical systems. However, their implementation necessitates optimizing a high-dimensional space of parameters and hyperparameters. This fact, along with the requirement of substantial computational resources, pose a barrier to achieving high numerical accuracy, but also interpretability. Here, to address the above challenges, we present Random Projection-based Operator Networks (RandONets): shallow networks with random projections and tailor-made numerical analysis methods that learn accurately and fast linear and nonlinear operators. Building on previous works, we prove that RandOnets are universal approximators of linear and nonlinear operators. Due to their simplicity, RandONets provide a one-step transformation of the input space, facilitating interpretability. For the evaluation of their performance, we focus on operators of PDEs. We show, that RandONets outperform by several orders of magnitude, both in terms of numerical approximation accuracy and computational cost, the “vanilla” DeepONets. Hence, we believe that our method will trigger further developments in the field of scientific machine learning, for the development of new ‘’light”schemes that will provide high accuracy while reducing dramatically the computational cost. A MATLAB toolbox for RandONets, including demos, is available on GitHub at https://github.com/GianlucaFabiani/RandONets.

Interpretable machine learning↗
DOE OSTI2025

Multi-plane moment-of-fluid interface reconstruction in 3D

Moment-of-fluid (MOF) methods for interface reconstruction approximate the region occupied by material in each mesh element only through reference to its geometric moments. Here, we present a 3D MOF method that represents the material (POM) in each cell as the convex intersection of the cell and multiple half-spaces, each selected to minimize the least-squares error between computed moments of the approximated material and provided reference moments. This optimization problem is highly non-linear and non-convex, making the numerical result very sensitive to the initial guess. To create an effective initial guess in each cell, we construct an ellipsoid from 0th–2nd order reference moments such that its shape corresponds with that of the POM. Within this ellipsoid we inscribe a polyhedron, and initialize the minimization problem with the half-spaces defined by each of its faces. The inscribed polyhedron has minimally 4 faces, and using up to 3rd order moments permits optimization over up to 20 unknown values. We therefore define MOF methods that utilize 4, 5, or 6 half-spaces, correspondingly initialized with the faces of a single inscribed tetrahedron, triangular prism, or hexahedron. Stability of the non-linear optimization is further improved with a prepossessing step that normalizes the reference moments according to the axes of the reference ellipsoid. Using this approach, the non-linear least-squares solver reliably converges to a near-global minimum from a single initial guess. We demonstrate accuracy and robustness using single-cell and multi-cell examples over a wide spectrum of geometry. In particular, we demonstrate our ability to exactly reproduce several important and complex features defined by up to four half-spaces, such as corners, filaments, filament tips, and embedded material in the cell.

3D interface reconstruction↗
DOE OSTI2025

Solving high-dimensional partial integral differential equations: The finite expression method

Partial integro-differential equations (PIDEs) have broad applications in the sciences, from electro-magnetism to options pricing. Here, in this paper, we introduce a new finite expression method (FEX) to solve PIDEs. This approach builds upon the original FEX and its inherent advantages with new advances: 1) A novel method of parameter grouping is proposed to reduce the number of coefficients in high-dimensional function approximation; 2) A Taylor series approximation method is implemented to significantly improve the computational efficiency and accuracy of the evaluation of the integral terms of PIDEs. The new FEX based method, denoted FEX-PG to indicate the addition of the parameter grouping (PG) step to the algorithm, provides both high accuracy and interpretable numerical solutions, with the outcome being an explicit equation that facilitates intuitive understanding of the underlying solution structures. These features are often absent in traditional methods, such as finite element methods (FEM) and finite difference methods, as well as in deep learning-based approaches. To benchmark our method against recent advances, we apply the new FEX-PG to solve benchmark PIDEs in the literature. In high-dimensional settings, FEX-PG exhibits strong and robust performance, achieving relative errors on the order of single precision machine epsilon, significantly outperforming existing approaches based on neural networks.

Combinatorial optimization↗
DOE OSTI2025

A method for bounding high-order finite element functions: Applications to mesh validity and bounds-preserving limiters

We introduce a novel method for bounding high-order multi-dimensional polynomials in finite element approximations. The method involves precomputing optimal piecewise-linear bounding boxes for polynomial basis functions, which can then be used to locally bound any combination of these basis functions. This approach can be applied to any element/basis type at any approximation order, can provide local (i.e., subcell) extremum bounds to a desired level of accuracy, and can be evaluated efficiently on-the-fly in simulations. Furthermore, we show that this approach generally yields more accurate bounds in comparison to traditional methods based on convex hull properties (e.g., Bernstein polynomials). Furthermore, the efficacy of this technique is shown in applications such as mesh validity checks and optimization for high-order curved meshes, where positivity of the element Jacobian determinant can be ensured throughout the entire element, and continuously bounds-preserving limiters for hyperbolic systems, which can enforce maximum principle bounds across the entire solution polynomial.

Bounding box↗
DOE OSTI2026

Non -degenerate marginal-likelihood calibration with application to quantum characterization

Here, we propose a marginal likelihood strategy within the Kennedy-O’Hagan (KOH) Bayesian framework, where a Gaussian process (GP) models the discrepancy between a physical system and its simulator. Our approach introduces a novel marginalized likelihood by integrating out the degenerate eigenspace of the covariance matrix, rather than approximating the original likelihood. Unlike approximation methods that compromise accuracy for computational efficiency, our method defines an exact likelihood—distinct from the original but preserving all relevant information. This formulation achieves computational efficiency and stability, even for large datasets where the covariance matrix nears degeneracy. Applied to the characterization of a superconducting quantum device at Lawrence Livermore National Laboratory, the approach enhances the predictive accuracy of the Lindblad master equations for modeling Ramsey measurement data by effectively quantifying uncertainties consistent with the quantum data.

general physics↗
DOE OSTI2024

Valence instability and collapse of ferromagnetism in EuB 6 at high pressures

Despite the simplicity of their cubic crystal lattice, rare-earth hexaborides display complex physical properties including a (long debated) onset of metallization via magnetic polaron formation at T c1 approximate to 15 K preceding ferromagnetic ordering at T c2 approximate to 12 K. Here, in this work, we used applied pressure to tune the interplay between electronic structure and magnetism in EuB6. We probed the magnetism, valence, and structure of EuB 6 under quasi-hydrostatic pressures up to 30 GPa using X-ray techniques. Our findings show evidence for collapse of ferromagnetism above 20 GPa following a monotonic increase of mean Eu valence. While X-ray diffraction measurements in the paramagnetic state at room temperature show that the lattice retains cubic symmetry, a measurable quadrupole interaction seen by time-domain synchrotron Mössbauer spectroscopy suggests a lowering of symmetry associated with magnetic ordering, becoming more prominent across the magnetic transition. The interplay between conduction band electron count and magnetism observed under applied pressure in EuB 6 opens possibilities for fine-tuning metallization and magnetic properties of similar Eu-based semi-metal systems.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2025

Osmotic control of the spacing of parallel shear cracks in shale growing subcritically in geologic past

The geological genesis of natural cracks in sedimentary rocks such as shale is a problem that needs to be understood to improve the technology of hydraulic fracturing as well as deep sequestration of harmful fluids. Why are the vertical natural cracks roughly parallel and equidistant, and why is the spacing roughly 10 cm rather than 1 cm or 100 cm? Fracture mechanics of critical cracks cannot answer this question. Neither can the material heterogeneity. The growth of critical parallel cracks is impossible because the relative crack face displacements would immediately localize into one crack, leading to an earthquake. The cracks must have formed, on the tectonic time scale, by a slow growth of subcritical shear cracks governed by the Charles-Evans law. The idea advanced here is that what controls the crack spacing is the balance between the reduction, due to shear dilatancy, of the concentration of ions such as Na + and Cl - in each fracture process zone (PFZ), which decelerates the cracks, and the restoration of ion concentration by diffusion of ions from the space between the cracks into the FPZ. This diffusion of water is driven mainly by the osmotic pressure gradient, which offsets the deceleration and depends strongly on the crack spacing. A simple analytical solution of the steady state is rendered possible by approximating the ion concentration profiles between adjacent cracks by parabolic arcs. Applying this theory to Woodford shale yields the approximate crack spacing of 10 cm, which is realistic. Furthermore, the stability of unlimited parallel mode II frictional crack growth is proven by examining the second variation of the free energy. Water concentration drop in the FPZ due to shear dilatancy and its restoration by water diffusion from the inter-crack space have similar effect, although probably much weaker.

42 ENGINEERING↗