Supervised Learning in Physical Networks: From Machine Learning to Learning Machines
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36 MATERIALS SCIENCE↗
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The Present and Future of QCD: QCD Town Meeting White Paper – An Input to the 2023 NSAC Long Range Plan
It is currently understood that there are four fundamental forces in nature: gravitational, electromagnetic, weak and strong forces. The strong force governs the interactions between quarks and gluons, elementary particles whose interactions give rise to the vast majority of visible mass in the universe. The mathematical description of the strong force is provided by the non-Abelian gauge theory Quantum Chromodynamics (QCD). While QCD is an exquisite theory, constructing the nucleons and nuclei from quarks, and furthermore explaining the behavior of quarks and gluons at all energies, remain to be complex and challenging problems. Such challenges, along with the desire to understand all visible matter at the most fundamental level, position the study of QCD as a central thrust of research in nuclear science. Experimental insight into the strong force can be gained using large particle accelerator facilities, which are necessary to probe the very short distance scales over which quarks and gluons interact. The Long Range Plans (LRPs) exercise of 1989 and 1996 led directly to the construction of two world-class facilities: the Continuous Electron Beam Accelerator Facility (CEBAF) at Jefferson Lab (JLab) that is focused on studying how the structure of hadrons emerges from QCD (cold QCD research), and the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Lab (BNL) that aims at the discovery and study of a new state of matter, the quark-gluon plasma (QGP), at extremely high temperatures (hot QCD research). These past investments have produced major advances. Nucleons and nuclei are being studied with increasing precision with a unified description of the partonic structure utilizing multi-dimensional imaging. Significant progress has been made, paving the way towards a complete picture of how quarks and gluons give rise to the mass, spin, and momentum of the nucleon. In hot QCD, the QGP is created in the collisions of nuclei at RHIC and the Large Hadron Collider (LHC) and is observed to behave like a fluid with very low specific shear viscosity; the current goals are to understand how the fluid behavior emerges from QCD and to characterize the temperature (and chemical potential) dependence of the properties of the QGP. As this White Paper is written, current experimental programs at CEBAF, RHIC and the LHC continue to provide exciting near term opportunities to capitalize on the investments in experimental equipment and accelerator operations. Most importantly, the QCD community looks forward to the construction of the Electron Ion Collider (EIC) as a major new facility to push forward QCD research in the next decades, with significant focus on exploring the properties of gluons, the mediators of the strong force.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Near-SUSY to non-SUSY crossover
Gauge theories can be solved exactly slightly away from the supersymmetric (SUSY) limit softly broken by anomaly mediation when the size of SUSY breaking is much smaller than the dynamical scale (𝑚 ≪ Λ). We show empirical evidence that the near-SUSY limit is continuously connected to the non-SUSY limit (𝑚 ≫ Λ) in SU(𝑁 𝑐 ) gauge theories with 𝑁 𝑓 quarks in the fundamental representation. The evidence includes the behavior of quark bilinear condensate and gluon condensates, light hadron spectra, and consistency with the large 𝑁 𝑐 limit including the Witten-Veneziano relations between the topological susceptibility and 𝑚 𝜂′ and 𝑚 𝜋 . In addition, we present new predictions when 𝑁 𝑓 /𝑁 𝑐 ≳ 𝑂(1).
Effective field theory↗
Specific heats for quantum BTZ black holes in extended thermodynamics
It was shown recently that extended black hole thermodynamics, where the cosmological constant is a dynamical variable, giving rise to a pressure p and its conjugate volume V , can be given a natural setting in the context of braneworld models. We study the specific heat capacities C p ( T ) and C V ( T ) of the quantum version of the Bañados, Teitelboim, and Zanelli (BTZ) black hole that lives in the induced gravity theory on the brane. There are multiple branches of solutions, and we explore and characterize key features of the possible behavior. We identify and study a critical point in the space of solutions where both specific heats diverge. In the regime of weak backreaction where we are close to an ordinary theory of gravity, the black hole is “subentropic,” but as backreaction is increased we note that there are parts of parameter space that has regions where it is “superentropic.” While a study of the sign of the specific heats does not always show a corresponding instability (conjectured in the literature), the presence of strong backreaction makes interpretation unclear. Published by the American Physical Society 2024
Johnson, Clifford V. (ORCID:0000000189645830)↗
Lifelike behavior of chemically oscillating mobile capsules
Inspired by the self-organization of unicellular species into multicellular organisms, we use theory and simulation to design a system of mobile, active microcapsules that produce chemicals according to a catalytic reaction network (CRN). In solution, the catalytic reactions generate a force that drives the flow of the surrounding fluid. The convective flow, in turn, transports the capsules to new chemical surroundings with each successive translation. Consequently, the chemical signal produced by a cluster of capsules is critically dependent on the proximity and spatial configuration of the neighboring capsules. If all of the capsules and chemical products involved in the CRN lie within sufficient proximity, then the system develops oscillatory behavior that drives the dynamic self-assembly of capsules into larger clusters that are capable of collective action. Finally, these simulations indicate potential chemo-mechanic mechanisms that enable single cellular units, such as ameba, to organize into multicellular life forms.
59 BASIC BIOLOGICAL SCIENCES↗
The pervasive impact of critical fluctuations in liquid–liquid extraction organic phases
Liquid-liquid extraction is an essential chemical separation technique where polar solutes are extracted from an aqueous phase into a nonpolar organic solvent by amphiphilic extractant molecules. A fundamental limitation to the efficiency of this important technology is third phase formation, wherein the organic phase splits upon sufficient loading of polar solutes. The nanoscale drivers of phase splitting are challenging to understand in the complex hierarchically structured organic phases. In this study, we demonstrate that the organic phase structure and phase behavior are fundamentally connected in a way than can be understood with critical phenomena theory. For a series of binary mixtures of trialkyl phosphate extractants with linear alkane diluents, we combine small angle x-ray scattering and molecular dynamics simulations to demonstrate how the organic phase mesostructure over a wide range of compositions is dominated by critical concentration fluctuations associated with the critical point of the third phase formation phase transition. These findings reconcile many longstanding inconsistencies in the literature where small angle scattering features, also consistent with such critical fluctuations, were interpreted as reverse micellar-like particles. Altogether, this study shows how the organic phase mesostructure and phase behavior are intrinsically linked, deepening our understanding of both and providing a new framework for using molecular structure and thermodynamic variables to control mesostructure and phase behavior in liquid-liquid extraction.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Hall coefficient and resistivity in the doped bilayer Hubbard model
Finding and understanding non-Fermi-liquid transport behaviors are at the core of condensed matter physics. Most of the existing studies in this field were devoted to the monolayer Hubbard model, which is the minimal model that captures the essential features of high-temperature superconductivity. Here, we discover another type of non-Fermi-liquid behavior emergent in the hole-doped bilayer Hubbard model, using dynamical mean-field theory with a full consideration of the short-range interlayer electron correlation. We find that at low temperatures, the Hall coefficient has a strong nonmonotonic dependence on temperature, leading to a double or quadruple reversal of its sign depending on the doping level. At the same time, the resistivity exhibits two plateaus rather than linearity in its temperature dependence. Finally, we show that these intriguing transport behaviors stem from the formation of coherent interlayer singlets, which scatter off gapped collective modes arising from short-range interlayer antiferromagnetic fluctuations.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Symmetric Kondo Lattice States in Doped Strained Twisted Bilayer Graphene
We use the topological heavy fermion (THF) model and its Kondo lattice (KL) formulation to study the possibility of a symmetric Kondo (SK) state in twisted bilayer graphene. Via a large-N approximation, we find a SK state in the KL model at fillings v = 0,±1,±2 where a KL model can be constructed. In the SK state, all symmetries are preserved and the local moments are Kondo screened by the conduction electrons. At the mean-field level of the THF model at v = 0,±1,±2,±3 we also find a similar symmetric state that is adiabatically connected to the symmetric Kondo state. We study the stability of the symmetric state by comparing its energy with the ordered (symmetry-breaking) states found in [H. Hu et al., Phys. Rev. Lett. 131, 026502 (2023)., Z.-D. Song and B. A. Bernevig, Phys. Rev. Lett. 129, 047601 (2022).] and find the ordered states to have lower energy at v = 0,±1,±2. However, moving away from integer fillings by doping the light bands, our mean-field calculations find the energy difference between the ordered state and the symmetric state to be reduced, which suggests the loss of ordering and a tendency toward Kondo screening. In order to include many-body effects beyond the mean-field approximation, we also performed dynamical mean-field theory calculations on the THF model in the nonordered phase. The spin susceptibility follows a Curie behavior at v = 0,±1,±2 down to ~2 K where the onset of screening of the local moment becomes visible. This hints to very low Kondo temperatures at these fillings, in agreement with the outcome of our mean-field calculations. At noninteger filling v = ±0.5,±0.8,±1.2 dynamical mean-field theory shows deviations from a 1/T susceptibility at much higher temperatures, suggesting a more effective screening of local moments with doping. Lastly, we study the effect of a C 3z -rotational-symmetry-breaking strain via mean-field approaches and find that a symmetric phase (that only breaks C 3z symmetry) can be stabilized at sufficiently large strain at v = 0,±1,±2. Our results suggest that a symmetric Kondo phase is strongly suppressed at integer fillings, but could be stabilized either at noninteger fillings or by applying strain.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Domain Wall Reactions in Multiple-Order Parameter Ferroelectrics
The motion of domain walls is crucial for ferroelectric switching. Conventionally, the switching dynamics is believed to be determined by the motion of one or a few low-energy domain wall types of dominant population. Here, we challenge this conventional idea in multiple-order-parameter ferroelectrics. Using hafnia as example, we show that the multiple-order-parameter nature not only provides various mobile domain walls and defectlike immobile domain walls, but also enables the domain wall reactions. In analogy with chemical reactions where substances react to form new substances, domain walls could also react to form other domain walls during switching. We identify several elementary domain wall reaction types including synthesis, decomposition, and exchange reactions. Domain walls are continually changed by these reactions during switching so that the switching behavior reflects the statistical average of many domain wall types with distinct mobility and stability. These reactions also lead to phenomenon like remanent nuclei and defect-site nucleation that facilitate switching and lower coercive field. Finally, the concept of domain wall reaction is not limited to hafnia but can be generalized to any multiple-order-parameter ferroelectric. As a result, this Letter conceptualizes domain wall reaction, expands the theory of ferroelectric switching, and suggests a practical way for defect engineering to control switching behavior.
36 MATERIALS SCIENCE↗
Real-fluid behavior in rapid compression machines: Does it matter?
Rapid compression machines (RCMs) have been extensively used to quantify fuel autoignition chemistry and validate chemical kinetic models at high-pressure conditions. Historically, the analyses of experimental and modeling RCM autoignition data have been conducted based on the adiabatic core hypothesis with ideal gas assumption, where real-fluid behavior has been completely overlooked, though this might be significant at common RCM test conditions. Here, this work presents a first-of-its-kind study that addresses two significant but overlooked questions for autoignition studies within RCMs in the fundamental combustion community: (i) experiment-wise, can unaccounted-for real-fluid behavior in RCMs affect the interpretation and analysis of RCM experimental data? and (ii) simulation-wise, can unaccounted-for real-fluid behavior in RCMs affect RCM autoignition modeling and the validation of chemical kinetic models? To this end, theories for real-fluid isentropic change are newly proposed and derived based on high-order Virial EoS, and are further incorporated into an effective-volume real-fluid autoignition modeling framework newly developed for RCMs. With detailed analyses, the strong real-fluid behavior in representative RCM tests is confirmed, which can greatly influence the interpretation of RCM autoignition experiments, particularly the determination of end-of-compression temperature and evolution of the adiabatic core in the reaction chamber. Furthermore, real-fluid RCM modeling results reveal that considerable error can be introduced into simulating RCM autoignition experiments when following the community-wide accepted effective-volume approach by assuming ideal-gas behavior, which can be as high as 64% in the simulated ignition delay time at compressed pressure of 125 bar and lead to contradictory validation results of chemical kinetic models. Therefore, we recommend the community to adopt frameworks with real-fluid behavior fully accounted for (e.g., the one developed in this study) to analyze and simulate past and future RCM experiments, so as to avoid misinterpretation of RCM autoignition experiments and eliminate the potential errors that can be introduced into the simulation results with the existing RCM modeling frameworks.
High-order Virial equation of state↗
Investigating the electrical transport properties and electronic structure of Zr 2 CuSb 3
The checkerboard lattice has been proposed to host topological flat bands as a result of destructive interference among its various electronic hopping terms. However, it has proven challenging to realize experimentally due to the difficulty of isolating this structure from any significant out-of-plane bonding while maintaining structural integrity. Here, in this study, single crystals of Zr 2 CuSb 3 , a potential candidate for the checkerboard lattice, were synthesized using the solution (self-flux) method, and their structure was confirmed via x-ray diffraction. Electrical-transport measurements indicate metallic behavior with electron-dominated carriers. Angle-resolved photoemission spectroscopy reveals multiple electron pockets and significant k z broadening due to its large c axis and low dispersion features in k z . Density-functional theory (DFT) calculations further disentangle the contributions from each high-symmetry plane, providing a comprehensive characterization of electronic behavior. The DFT calculations were then used to determine the orbital contributions of the bands and detect the out-of-plane bonding which prevented the flat bands from forming.
Downey, Eoghan [Univ. of Michigan, Ann Arbor, MI (↗
The Status and Ambitions of the US Heavy Element Program
The aim of this whitepaper is to highlight the current capabilities and priorities of the US Heavy Element community and to provide the framework for a coordinated advancement of nuclear science from these studies. This is an organized effort to reflect on what has been achieved in the field given the recommendations and initiatives of the 2015 NSAC Long Range Plan, and on what can be realized in the next decade given current and possibly expanded investments. Current investments have positioned the US community to be among the world leaders in studies of the nuclear and chemical properties of the heaviest elements. These include studies of reaction mechanisms, moving us ever closer to the “island of stability”, in spectroscopy, allowing us to better understand nuclear structure at these extreme proton numbers, in chemical behavior, looking to determine how these elements should be placed on the Periodic Table, in performing the first measurements where isotopes are directly identified by their mass numbers, and in laying the foundation for a potential US-led new element discovery experiment. At present, the highest priority of the US Heavy Element community is to capitalize on the current investments by supporting the operations of US facilities at optimal values. These facilities include the Argonne Tandem Linac Accelerator System at Argonne National Laboratory and other Department of Energy facilities such as the 88-Inch Cyclotron at Lawrence Berkeley National Laboratory, which has a dedicated superheavy element program, as well as university laboratories, including Texas A&M University. The High Flux Isotope Reactor at Oak Ridge National Laboratory is crucial to providing the radioactive isotopes required for heavy element science targets. This facility should be supported to provide the actinide materials that are essential for US-based science. Production of stable, rare isotopes for beam material, including 48 Ca, 50 Ti, 54 Cr and 58 Fe, at the Stable Isotope Production and Research Center is critical to continued research in heavy element science and should be a priority. The continued development of targets for heavy element science and retaining US-based expertise is critical for the heavy element community. This is an area that is currently under pressure. For example, the target laboratory at Argonne National Laboratory serves a broad community and is currently under threat due to loss of critical personnel. The skills needed to make targets for nuclear science and develop new targetry methods need to be supported long term at Argonne, Oak Ridge and Lawrence Livermore National Laboratories as well as maintaining the programs at Oregon State University and San José State University as vital pipelines for training students. Advances in theory are the foundation to understand how nuclei behave and to predict those behaviors in new circumstances. Progress in these studies will necessitate continued and new investment and access to high-performance computing. The future health of the heavy element field is dependent on the continuous support of talented early-career professionals at all levels. It is critical that opportunities continue to be created for the next generation to become established in heavy element research so that we can ensure the field is attracting and retaining the best minds for continued success. It is also clear that to ensure diversity of ideas, perspectives and techniques, we need to recruit diverse personnel that are trained at the best facilities. The heavy element community is in support of continued investment to programs with initiatives in diversity, equity, and inclusion. Looking to the next decade of research, support needs to maintain and grow US leadership in heavy element science. Specifically, new investments in state-of-the-art instrumentation will be essential to scientific development of the field and in expanding scientific knowledge. Advances in the next generation of electron cyclotron resonance ion sources, multi-reflection time-of-flight devices, laser spectroscopy, trapping methods and next generation alpha and gamma spectroscopy systems should be prioritized.
38 RADIATION CHEMISTRY, RADIOCHEMISTRY, AND NUCLEA↗
A Theoretical Study of NH 2 Radical Reactions with Propane and Its Kinetic Implications in NH 3 -Propane Blends’ Oxidation
The reaction of NH 2 radicals with C 3 H 8 is crucial for understanding the combustion behavior of NH 3 /C 3 H 8 blends. In this study, we investigated the temperature dependence of the rate coefficients for the hydrogen abstraction reactions of C 3 H 8 by NH 2 radicals using high-level theoretical approaches. The potential energy surface was constructed at the CCSD(T)/cc-pV(T, Q)//M06-2X/aug-cc-pVTZ level of theory, and the rate coefficients were computed using conventional transition state theory, incorporating the corrections for quantum tunneling and hindered internal rotors (HIR). The computed rate coefficients showed a strong curvature in the Arrhenius behavior, capturing the experimental literature data well at low temperatures. However, at T > 1500 K, the theory severely overpredicted the experimental data. The available theoretical studies did not align with the experiment at high temperatures, and the possible reasons for this discrepancy are discussed. At 300 K, the reaction of NH 2 with C 3 H 8 predominantly occurs at the secondary C-H site, which accounts for approximately 95% of the total reaction flux. However, the hydrogen abstraction reaction at the primary C-H site becomes the dominant reaction above 1700 K. A composite kinetic model was built, which incorporated the computed rate coefficients for NH 2 + C 3 H 8 reactions. The importance of NH 2 + C 3 H 8 reactions in predicting the combustion behavior of NH 3 /C 3 H 8 blends was demonstrated by kinetic modeling.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Photoemission and dynamical mean field theory study of electronic correlations in a t 2 g 5 metal SrRhO 3 thin film
Perovskite rhodates are characterized by intermediate strengths of both electronic correlation as well as spin-orbit coupling (SOC) and usually behave as moderately correlated metals. A recent publication on epitaxial SrRhO 3 thin films reported a bad-metallic behavior and suggested the occurrence of antiferromagnetism below 100 K. In this work, we have further studied this SrRhO 3 thin film by hard x-ray photoemission spectroscopy and found a very small density of states (DOS) at the Fermi level $(E_F)$, which is consistent with the previously reported bad-metallic behavior. However, the absence of DOS persists up to room temperature, which contradicts the explanation of antiferromagnetic transition at ~100 K. We also employed electronic structure calculations within the framework of density functional theory and dynamical mean-field theory. In contrast to the photoemission results, our calculations indicate metallic behaviors of both bulk SrRhO 3 and the SrRhO 3 thin film, and a stronger correlation effect was observed in the thin film than that in the bulk. The calculated uniform magnetic susceptibility is substantially larger in the thin film than that in the bulk. We also investigated the role of SOC and found only a moderate modulation of the band structure. Hence SOC is not expected to significantly affect the electronic correlation in SrRhO 3 . Extrinsic effects of finite-thickness effects beyond our calculations and localization effects may play important roles to induce the negligible spectral weight at $(E_F)$.
36 MATERIALS SCIENCE↗
Individual Data Sparsity in Smart Thermostat Big Data: Impacts on Modeling Thermostat Use Behavior Dynamics
This study explores the impacts of the sparsity of individual thermostat interaction data on modeling thermostat use behavior dynamics using a dataset of over 100,000 smart thermostats. In developing a data-driven model of Thermal Frustration Theory (TFT), we investigate the challenges and trade-offs in clustering occupant data to enhance predictive accuracy. Our findings reveal that a single, aggregated model fails to capture the diversity of occupant behaviors, resulting in extremely poor prediction performance. Conversely, excessive clustering exacerbates data sparsity, undermining model reliability. By identifying an optimal clustering strategy, we achieve a balance that significantly improves the prediction of manual setpoint changes during demand response (DR) events, enhancing energy management and occupant comfort
Fannon, David↗
Accurate and efficient parameterization of an atomic cluster expansion (ACE) potential for ammonia under extreme conditions
We present a machine learning interatomic potential for ammonia designed to capture its complex multiphase behavior, including both molecular and superionic phases. The potential is based on the atomic cluster expansion (ACE) formulation and has been parameterized to facilitate high-fidelity molecular dynamics simulations of ammonia under extreme conditions, for pressures up to 100 GPa and for temperatures above 500 K and up to 6000 K. A diverse range of configurations was generated through high-quality ab initio molecular dynamics simulations, covering insulating and superionic ice phases, liquid ammonia, molecular nitrogen (N 2 ) and hydrogen (H 2 ), and metastable compounds that form upon dissociation, including $NH^{+}_{4}$, $H^{+}_{3}$, N 2 H 4 , and N 3 H. We demonstrate that the ammonia ACE potential accurately reproduces experimental and density functional theory predicted isotherms and Hugoniots. Crucially, the potential is able to capture the intricate phase behavior of ammonia, including the transition from insulating molecular fluid to the superionic phase. This work provides a robust interatomic potential that can be used for large-scale, accurate simulations of ammonia under extreme thermodynamic conditions, offering a powerful tool for investigating its behavior in various phases and applications.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Activated relaxation in supercooled monodisperse atomic and polymeric WCA fluids: Simulation and ECNLE theory
Here, we combine simulation and Elastically Collective Nonlinear Langevin Equation (ECNLE) theory to study the activated relaxation in monodisperse atomic and polymeric Weeks–Chandler–Andersen (WCA) liquids over a wide range of temperatures and densities in the supercooled regime under isochoric conditions. By employing novel crystal-avoiding simulations, metastable equilibrium dynamics is probed in the absence of complications associated with size polydispersity. Based on a highly accurate structural input from integral equation theory, ECNLE theory is found to describe well the simulated density and temperature dependences of the alpha relaxation time of atomic fluids using a single system-specific parameter, a c , that reflects the nonuniversal relative importance of local cage and collective elastic barriers. For polymer fluids, the explicit dynamical effect of local chain connectivity is modeled at the fundamental dynamic free energy trajectory level based on a different parameter, N c , that quantifies the degree of intramolecular correlation of bonded segment activated barrier hopping. For the flexible chain model studied, a physically intuitive value of N c ≈ 2 results in good agreement between simulation and theory. A direct comparison between atomic and polymeric systems reveals that chain connectivity can speed up activated segmental relaxation due to weakening of equilibrium packing correlations but can slow down relaxation due to local bonding constraints. The empirical thermodynamic scaling idea for the alpha time is found to work well at high densities or temperatures but fails when both density and temperature are low. The rich and subtle behaviors revealed from simulation for atomic and polymeric WCA fluids are all well captured by ECNLE theory.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Verification of Coupled Hydraulic Fracturing Simulators Using Laboratory-Scale Experiments
In this work, we aim to verify the predictions of the numerical simulators, which are used for designing field-scale hydraulic stimulation experiments. Although a strong theoretical understanding of this process has been gained over the past few decades, numerical predictions of fracture propagation in low-permeability rocks still remains a challenge. Against this background, we performed controlled laboratory-scale hydraulic fracturing experiments in granite samples, which not only provides high-quality experimental data but also a well-characterized experimental set-up. Using the experimental pressure responses and the final fracture sizes as benchmark, we compared the numerical predictions of two coupled hydraulic fracturing simulators—CSMP and GEOS. Both the simulators reproduced the experimental pressure behavior by implementing the physics of Linear Elastic Fracture Mechanics (LEFM) and lubrication theory within a reasonable degree of accuracy. The simulation results indicate that even in the very low-porosity (1–2 %) and low-permeability (10 -18 m 2 - 10 -19 m 2 ) crystalline rocks, which are usually the target of EGS, fluid-loss into the matrix and unsaturated flow impacts the formation breakdown pressure and the post-breakdown pressure trends. Therefore, underestimation of such parameters in numerical modeling can lead to significant underestimation of breakdown pressure. The simulation results also indicate the importance of implementing wellbore solvers for considering the effect of system compressibility and pressure drop due to friction in the injection line. The varying injection rate as a result of decompression at the instant of fracture initiation affects the fracture size, while the entry friction at the connection between the well and the initial notch may cause an increase in the measured breakdown pressure.
15 GEOTHERMAL ENERGY↗