General Green Function Research Articles

Impact of well placement on heat production from complex fracture networks in enhanced geothermal systems: A semi-analytical model

Understanding of the effective fracture path between a producer and an injector in complex fracture networks plays a key role in successful geothermal heat extraction from an enhanced geothermal system (EGS). Therefore, there is a need to investigate the impact of well placement on EGS heat extraction performance. An efficient thermal model is developed to simulate fluid and heat flow in fractured reservoirs with complex fracture networks. A general Green's function for fractures oriented at any angle is used, which reduces problem dimensions by one. The effect of fracture types on heat production performance is investigated and the impact of well placement on the efficiency of extracting geothermal energy is demonstrated. Results indicate that the rate of energy extraction depends on the number of backbone fractures connecting the injector and producer, emphasizing the need for well placement that creates more backbone fractures to allow for fluid circulation. These findings provide valuable guidance on well placement strategies to optimize geothermal heat extraction efficiency.

Effect of fracture geometry, topology and connectivity on energy recovery from enhanced geothermal systems

A combination of hydraulic fracturing and horizontal wells is now being used to tap geothermal energy from naturally fractured reservoirs. Fully grid-based numerical models are currently used to simulate heat recovery from enhanced geothermal systems (EGS). Such models require a fine unstructured mesh and are computationally expensive. In this paper we present a computationally efficient model that allows us to accurately simulate fracture propagation, fluid flow, and heat transfer in networks of natural fractures that may be created in naturally fractured geothermal reservoirs.The integrated simulator is developed by combining the displacement discontinuity method (DDM) for fracture propagation in naturally fractured reservoirs with a general Green's function solution for fluid and heat flow from the matrix to the fracture. This eliminates the need to discretize the matrix domain resulting in a very computationally efficient solution. A discrete fracture network (DFN) approach is used to represent the pre-existing natural fractures.The model is first validated against an analytical solution for fluid flow and heat transfer in a rock matrix with a single fracture. The computation time with and without discretizing the rock matrix shows a 100-fold reduction in computation cost with very little loss in accuracy. Parametric studies are conducted to investigate the effect of the distribution of natural fracture density, length, and orientation. The results show that the efficiency of tapping geothermal energy is impacted by geometrical and topological complexities of the fracture network and in particular the connectivity of backbone fractures. It is, therefore, important to optimize (not maximize) the connectivity and complexity of the backbone fracture network. The computationally efficient model presented here provides a practical tool for optimizing operational parameters for efficient geothermal production.

Tunable magnetic field effects on the near-field radiative heat transfer in planar three-body systems

Recently, the application of an external magnetic field to actively control the near-field radiative heat transfer (NFRHT) has emerged as an appealing and promising technique. Most prior studies have shown that the externally static magnetic field can only reduce the near-field flux transferred between two planar magneto-optical (MO) structures, yet so far the thermomagnetic effect in many-body systems consisting of such structures has not been revealed. Here, we investigate how the presence of an external static magnetic field modifies the nanoscale heat transfer in a MO many-body configuration comprising three noncontact slabs made of $n$-doped InSb. To this end, we first generalize a general Green's function approach for the calculation of the radiative energy transfer in many-body planar geometries composed of materials with complex optical anisotropy. Based on this approach, we show that the presence of a third MO body allows for either the reduction or even the enhancement of the NFRHT between MO slabs by applying external magnetic fields, which depends on the interplay between the zero-field surface waves and propagating hyperbolic modes induced by fields. Our study not only deepens the understanding of the active control of the nanoscale heat transfer via the applied magnetic field, but also paves the way for the magnetic regulation of energy fluxes in complex macroscopic many-body systems.

Wavelet Based Solutions to the Poisson and the Helmholtz Equations on the n-Dimensional Unit Sphere

We present a method of solving partial differential equations on the n-dimensional unit sphere using methods based on the continuous wavelet transform derived from approximate identities. We give an explicit analytical solution to the Poisson equation and to the Helmholtz equations. For the first one and for some special values of the parameter in the latter one, we derive a closed formula for the generalized Green function.

A computationally efficient model for fracture propagation and fluid flow in naturally fractured reservoirs

There is a need to develop computationally efficient models for oil and gas production from naturally fractured reservoirs. In this paper, we present an efficient, integrated fracturing-production simulator by combining a boundary element method (for fracture propagation) and a general Green's function solution (for fluid flow) that eliminates the need to discretize the matrix domain. First, the model is validated against analytical solutions and then compared with a fully numerical model. A comparison of results and computation time shows that our simulator significantly reduces the computation cost without any significant loss in accuracy. The simulator is then applied to investigate the effect of cluster spacing and pumping schedule on production from a hydraulically fractured horizontal well in a naturally fractured reservoir. The results show an optimal cluster spacing that can maximize the contact area between fractures and reservoirs while maintaining the highest production rate. Based on the chosen optimal cluster spacing, we observe that changes in the pumping schedule can have an impact on the production rate due to changes in the created fracture network.

General Opial Type Inequality and New Green Functions

In this paper we provide many new results involving Opial type inequalities. We consider two functions—one is convex and the other is concave—and prove a new general inequality on a measure space (Ω,Σ,μ). We give an new result involving four new Green functions. Our results include Grüss and Ostrowski type inequalities related to the generalized Opial type inequality. The obtained inequalities are of Opial type because the integrals contain the function and its integral representation. They are not a direct generalization of the Opial inequality.

Assessing thermodynamical properties of Al1−xGaxSb alloys and optical modes for Al1−xGaxSb/GaAs epifilms and (AlSb)m/(GaSb)n superlattices

A generalized Green's function (GF) theory is adopted in the framework of a realistic rigid-ion-model (RIM) to assess the composition, x-dependent lattice dynamics, and thermodynamical characteristics of ideal random Al1−xGaxSb alloys. For simulating phonons, the alloy parameters are achieved by interpolating the values of the RIM force constants between AlSb and GaSb without requiring any additional interactions. The outcomes of phonon dispersions ωj(q→), Debye temperature ΘD(T), and specific heat Cv(T) compare favorably well with the existing experimental and theoretical data. An established methodology of multilayer optics is also employed for modeling the far-infrared reflectance and transmission spectra of ultrathin GaSb/GaAs, AlSb/GaAs, Al1−xGaxSb/GaAs epilayers, and (AlSb)m/(GaSb)n/GaAs superlattices at near normal (θi = 0) incidence and oblique (θi ≠ 0) incidence. An accurate appraisal of the x-dependent longitudinal-optical [ωLO(Γ)] and transverse-optical [ωTO(Γ)] phonon splitting by Berreman's effect, along with the calculated GF results of localized vibrational mode (GaSb:Al) and gap mode (AlSb:Ga), is carefully integrated into the modified-random-iso-displacement model to validate the two-phonon mode behavior in Al1−xGaxSb ternary alloys.

Analytical approach to coupled bending-torsional vibrations of cracked Timoshenko beam

The explicit expressions of steady-state responses of a coupled bending-torsional Timoshenko beam with cracks and damping effect subjected to external harmonic loadings are presented in this paper. The Green's functions method is employed to obtain the analytical solutions. The beam is split to several segments due to the existence of cracks. General Green's functions of each segment with unknown boundary constants are given by using Laplace transform method. A mixture line-spring is introduced to obtain the compatibility conditions of the cracked cross-section between the segments. Based on the transfer matrix method, those constants in the Green's functions for each segment are determined with those matching conditions, as well as boundary relationships and end boundary conditions. Then, the solutions of the whole cracked beam with one or multiple cracks can be expressed in terms of the piecewise functions. The solutions of the multi-cracked beam cover those of the beam with only one crack and the uncracked beam. The theory developed can be used for the beam with classical boundary conditions, but for illustration, results for a cantilever cracked beam with T shape cross-section are illustrated, emphasizing the effect of cracks on the solutions. Comparisons between results in the published literature and those obtained from the developed method show a good agreement, which confirms the validity and accuracy of the proposed approach. The influences of crack depth, location and number on the natural frequencies are discussed. The changes of the steady-state responses of beam are investigated due to the existence of crack. Moreover, the symmetric property of the Green's functions and damping effect on the amplitude of steady-state responses of the cracked beam are studied particularly.

Analytical and numerical modeling of rock blasting operations using a two-dimensional elasto-dynamic Green's function

Rock media may be exposed to dynamic forces and time-dependent stresses. Dynamic phenomena in rock such as blasting and earthquakes need to propagate waves and dynamic stresses through the body. Therefore, stresses, strains, and displacements in rocks should be defined as time-dependent due to the dynamic phenomena. In this study, a general Green's function solution of an elastic wave propagation due to rock blasting has been analytically derived. The Navier's equations of motion have been assumed as governing equations and a general two-dimensional elastodynamic Green's function has been obtained in terms of displacement. The strain and stress fields related to the displacement Green's function have also been obtained using the theory of elasticity. Then, a generic two-dimensional problem containing an explosion in a blast hole has been solved using the resulting analytical solution. Moreover, a similar numerical modeling has been conducted using Finite Difference Method (FDM). Eventually, results from the analytical scheme have been compared with the finite difference numerical modeling and the valuable results from a previous experimental and numerical study.

Green's function for higher-order boundary value problems involving a Nabla Caputo fractional operator

ABSTRACTWe consider the discrete, fractional operator involving the nabla Caputo fractional difference, which can be thought of as an analogue to the self-adjoint differential operator. We show that solutions to difference equations involving this operator have expected properties, such as the form of solutions to homogeneous and nonhomogeneous equations. We also give a variation of constants formula via a Cauchy function in order to solve initial value problems involving . We also consider boundary value problems of any fractional order involving . We solve these BVPs by giving a definition of a Green's function along with a corresponding Green's Theorem. Finally, we consider a (2,1) conjugate BVP as a special case of the more general Green's function definition.

Green’s function for a sharpened and metal-coated dielectric probe

In apertureless scanning-probe optical microscopy and in the case of more traditional scanned optical probes coated with a metal that is thin near the probe tip (in lieu of an aperture), samples are probed via interaction between the probe and surface. In the nanometer-scale region between the tip and the sample, the field can be approximated by quasi-electrostatic analytics. Hence, the coated probe can be modeled as in the present case as a hyperboloid of revolution without the need for hyperboloidal wave functions in the near zone. The solutions to Laplace's equation and in general Green's function with the application of the boundary conditions, therefore, yield an appropriate approximation and allow a completely analytical solution for the resonance effects upon the probe tip to be obtained. The large field enhancements due to the sharpness of the tip and to surface plasmon fields may thus be analytically examined.

General Green's function formalism for layered systems: Wave function approach

The single-particle Green's function (GF) of mesoscopic structures plays a central role in mesoscopic quantum transport. The recursive GF technique is a standard tool to compute this quantity numerically, but it lacks physical transparency and is limited to relatively small systems. Here we present a numerically efficient and physically transparent GF formalism for a general layered structure. In contrast to the recursive GF that directly calculates the GF through the Dyson equations, our approach converts the calculation of the GF to the generation and subsequent propagation of a scattering wave function emanating from a local excitation. This viewpoint not only allows us to reproduce existing results in a concise and physically intuitive manner, but also provides analytical expressions of the GF in terms of a generalized scattering matrix. This identifies the contributions from each individual scattering channel to the GF and hence allows this information to be extracted quantitatively from dual-probe STM experiments. The simplicity and physical transparency of the formalism further allows us to treat the multiple reflection analytically and derive an analytical rule to construct the GF of a general layered system. This could significantly reduce the computational time and enable quantum transport calculations for large samples. We apply this formalism to perform both analytical analysis and numerical simulation for the two-dimensional conductance map of a realistic graphene p-n junction. The results demonstrate the possibility of observing the spatially-resolved interference pattern caused by negative refraction and further reveal a few interesting features, such as the distance-independent conductance and its quadratic dependence on the carrier concentration, as opposed to the linear dependence in uniform graphene.

Spin-orbit torques in Co/Pt(111) and Mn/W(001) magnetic bilayers from first principles

An applied electric current through a space-inversion asymmetric magnet induces spin-orbit torques (SOTs) on the magnetic moments, which holds much promise for future memory devices. We discuss general Green's function expressions suitable to compute the linear-response SOT in disordered ferromagnets. The SOT can be decomposed into an even and an odd component with respect to magnetization reversal, where in the limit of vanishing disorder the even SOT is given by the constant Berry curvature of the occupied states, while the odd part exhibits a divergence with respect to disorder strength. Within this formalism, we perform first-principles density-functional theory calculations of the SOT in Co/Pt(111) and Mn/W(001) magnetic bilayers. We find the even and odd torque components to be of comparable magnitude. Moreover, the odd torque depends strongly on an additional capping layer, while the even torque is less sensitive. We show that the even torque is nearly entirely mediated by spin currents in contrast to the odd torque, which can contain an important contribution not due to spin transfer. Our results are in agreement with experiments, showing that our linear-response theory is well-suited for the description of SOTs in complex ferromagnets.

From Andreev bound states to Majorana fermions in topological wires on superconducting substrates: A story of mutation

We study the proximity effect in a topological nanowire tunnel coupled to an s-wave superconducting substrate. We use a general Green's function approach that allows us to study the evolution of the Andreev bound states in the wire into Majorana fermions. We show that the strength of the tunnel coupling induces a topological transition in which the Majorana fermionic states can be destroyed when the coupling is very strong. Moreover, we provide a phenomenologial study of the effects of disorder in the superconductor on the formation of Majorana fermions. We note a non-trivial effect of a quasiparticle broadening term which can take the wire from a topological into a non-topological phase in certain ranges of parameters. Our results have also direct consequences for a nanowire coupled to an inhomogenous superconductor.

Thermal coherence properties of topological insulator slabs in time-reversal symmetry breaking fields

We first develop a general Green's-function method for the slab geometry, which can handle quite complex conversions between the different polarized radiation modes. Then by using the method, we study the near field thermal coherence properties of a slab of ${Z}_{2}$ topological insulator (TI) with a finite thickness. Under a strong enough external magnetic field, the gapless helical Dirac fermions on the TI surfaces can acquire an energy gap larger than the photon energy of the radiation field at the Dirac point. This gapped surface states can couple strongly with the waveguide modes in the bulk of the TI slab, which thus induces an electromagnetic resonance. Exactly due to the resonance, the coherence properties of the thermal radiation field can be modified dramatically. For ${Z}_{2}$ topological insulators, the parameter regime with the surface Hall conductivity half integer quantized is unique. We demonstrated that within the parameter regime, for a TI slab of a proper thickness, the thermal radiation energy density can be enhanced very much, and the coherence length of the thermal radiation field can be as long as twice that of SiC.

Influence of internal absorption and interference on the optical efficiency of thin-film GaN-InGaN light-emitting diodes

We present a first-principle method for quantitative modeling of optical energy flow and dissipation in thin-film (TF) light-emitting diodes (LEDs) based on highly general Green's function method. Unlike conventional models, the presented model simultaneously accounts for interference, near-field effects, and internal absorption in determining the radiance generated by a LED. We show that these effects have a profound influence on the efficiency of LEDs and strongly affect the light extraction efficiency (LEE) and the internal quantum efficiency. According to our results, the LEE of an InGaN-GaN TF-LED with untextured surfaces and typical active region (AR) thickness on the order of 10 nm is 67% while a LED with a thin AR exhibits only a LEE of 29%. Based on the numerical results, we discuss the factors that affect the overall efficiency and design considerations to optimize the structure of thin-film LEDs.

Nonequilibrium Transport through a Spinful Quantum Dot with Superconducting Leads

We study the nonlinear cotunneling current through a spinful quantum dot contacted by two superconducting leads. Applying a general nonequilibrium Green function formalism to an effective Kondo model, we study the rich variation in the IV characteristics with varying asymmetry in the tunnel coupling to source and drain electrodes. The current is found to be carried, respectively, by multiple Andreev reflections in the symmetric limit, and by spin-induced Yu-Shiba-Rusinov bound states in the strongly asymmetric limit. The interplay between these two mechanisms leads to qualitatively different IV characteristics in the crossover regime of intermediate symmetry, consistent with recent experimental observations of negative differential conductance and repositioned conductance peaks in subgap cotunneling spectroscopy.

Method Study on Generation-Dissipation of Seismic Pore Water Pressure in Saturated Sands

With considering the effect of the initial pore water pressure and the variety of seismic pore water pressure generation, a two-dimensional inhomogeneous compound diffusion equation for describing generation and dissipation of seismic pore water pressure are given by coupling the Terzaghi’s consolidation theory and the increasing mode of seismic pore water pressure in saturated sands. The analytical expression of the diffusion equation is obtained by using the method of the generalized Green function and the finite Fourier transformation.

Generalized Green Functions and Current Correlations in the TASEP

We study correlation functions of the totally asymmetric simple exclusion process (TASEP) in discrete time with backward sequential update. We prove a determinantal formula for the generalized Green function which describes transitions between positions of particles at different individual time moments. In particular, the generalized Green function defines a probability measure at staircase lines on the space-time plane. The marginals of this measure are the TASEP correlation functions in the space-time region not covered by the standard Green function approach. As an example, we calculate the current correlation function that is the joint probability distribution of times taken by selected particles to travel given distance. An asymptotic analysis shows that current fluctuations converge to the ${Airy}_2$ process.

Non-interferometric, non-iterative phase retrieval by Green’s functions

In this paper a non-interferometric, non-iterative method for phase retrieval by Green's functions is presented. The theory is based on the parabolic wave equation that describes propagation of light in the Fresnel approximation in homogeneous media. Green's first identity will be used to derive an algorithm for phase retrieval considering different boundary conditions. Finally it will be shown that a commonly used solution of the transport-of-intensity equation can be obtained as a special case of the more general Green's function formulation derived here.