Outer Boundary Conditions Research Articles

A robust deep structured prediction model for petroleum reservoir characterization using pressure transient test data

A robust deep learning model consisting of long short-term memory and fully connected neural networks has been proposed to automatically interpret homogeneous petroleum reservoirs having infinite, no flow, and constant pressure outer boundary conditions. The pressure change data recorded during the well test operation along with its derivative is input into the model to perform the classification for identifying the reservoir model and, further, regression to estimate output parameter. Gaussian noise was added to analytical models while generating the synthetic training data. The hyperparameters were regulated to perform model optimization, resulting in a batch size of 64, Adam optimization algorithm, learning rate of 0.01, and 80:10:10 data split ratio as the best choices of hyperparameters. The performance accuracy also increased with an increase in the number of samples during training. Suitable classification and regression metrics have been used to evaluate the performance of the models. The paper also demonstrates the prediction performance of the optimized model using simulated and actual oil well pressure drawdown test cases. The proposed model achieved minimum and maximum relative errors of 0.0019 and 0.0308, respectively, in estimating output for the simulated test cases and relative error of 0.0319 for the real test case.

The Constructive Solution of the Fractal Composite Reservoir with Stress-Sensitivity Formation

The concentric two-zone composite reservoir model is a boundary value problems (BVPs) of modified Bessel equations. In this paper, we propose a constructive method to solve the BVPs for the system of modified Bessel equations with Robin mixed outer boundary condition and apply it to solve a two-zone fractal composite reservoir seepage model with stress-sensitivity formation. By using Pedrosa variable substitution, regular perturbation technique, Laplace transform, and Stehfest numerical inversion technique, the unified expression for the solutions of the reservoir model with three outer boundary (infinite, impermeable, and constant pressure) conditions is constructed. Type curves of bottom-hole pressure and pressure derivative are drawn, and sensitivity analysis of reservoir parameters are carried out. In comparison with the traditional approach, the solutions of this model are simple and regular, with continued fraction form, the constructive method is efficient and easy to operate. The application of this method avoids the complicated and trivial derivative operation and the use of Cramer’s rule to solve the system of linear equations. It can help to better understand the relationship between the solutions of the reservoir model and the inner and outer boundary conditions. The constructive method can be applied not only to solve the fractal composite reservoir model but also to solve more general reservoir model, BVPs of fluid diffusion, heat conduction, and so on.

Analysis of flow characteristics of the dual media shale gas reservoir with the elastic outer boundary

In order to more accurately describe the seepage characteristics of shale gas reservoirs, in this paper, an elastic outer boundary condition is introduced, and a new dual-media shale gas seepage model is established to describe the seepage characteristics of shale gas reservoirs more accurately, while considering the adsorption and desorption process. Combining Laplace transformation and Similar Structure Method, the solution of the percolation model is obtained in Laplace space. Furthermore, the solutions are inverted into real space with the Stehfest numerical inversion method. Type curves of dimensionless pressure and dimensionless pressure derivative can be used to evaluate the reservoir characteristics. The results show that the conventional three kinds of outer boundary conditions (infinite, constant pressure and closed) are actually three special cases of elastic outer boundary. The introduction of elastic outer boundary conditions not only expands the scope of data interpretation, but also closer to the actual situation of the outer boundary of the reservoir. The theory of similar structure greatly simplifies the complex process of solving the model and will have a far-reaching impact on the development of well test analysis software in the future.

New boundary conditions for the approximate flux-limited diffusion radiative transfer in circumstellar environments

Context. In order to constrain the models describing circumstellar environments, it is necessary to solve the radiative transfer equation in the presence of absorption and scattering, coupled with the equation for radiative equilibrium. However, solving this problem requires much CPU time, which makes the use of automatic minimisation procedures to characterise these environments challenging. Aims. In this context, the use of approximate methods is of primary interest. One promising candidate method is the flux-limited diffusion (FLD), which recasts the radiative transfer problem into a non-linear diffusion equation. One important aspect for the accuracy of the method lies in the implementation of appropriate boundary conditions (BCs). We present new BCs for the FLD approximation in circumstellar environments that we apply here to spherically symmetric envelopes. Methods. At the inner boundary, the entering flux (coming from the star and from the envelope itself) may be written in the FLD formalism and provides us with an adequate BC. At the free outer boundary, we used the FLD formalism to constrain the ratio of the mean radiation intensity over the emerging flux. In both cases we derived non-linear mixed BCs relating the surface values of the mean specific intensity and its gradient. We implemented these conditions and compared the results with previous benchmarks and the results of a Monte Carlo radiative transfer code. A comparison with results derived from BCs that were previously proposed in other contexts is presented as well. Results. For all the tested cases, the average relative difference with the benchmark results is below 2% for the temperature profile and below 6% for the corresponding spectral energy distribution or the emerging flux. We point out that the FLD method together with the new outer BC also allows us to derive an approximation for the emerging flux. This feature avoids additional formal solutions for the radiative transfer equation in a set of rays (ray-tracing computations). Conclusions. The FLD approximation together with the proposed new BCs performs well and captures the main physical properties of the radiative equilibrium in spherical circumstellar envelopes.

SPECFEM2D-DG, an open-source software modelling mechanical waves in coupled solid–fluid systems: the linearized Navier–Stokes approach

SUMMARYWe introduce SPECFEM2D-DG, an open-source, time-domain, hybrid Galerkin software modelling the propagation of seismic and acoustic waves in coupled solid–fluid systems. For the solid part, the visco-elastic system from the routinely used SPECFEM2D software is used to simulate linear seismic waves subject to attenuation. For the fluid part, SPECFEM2D-DG includes two extensions to the acoustic part of SPECFEM2D, both relying on the Navier–Stokes equations to model high-frequency acoustics, infrasound and gravity waves in complex atmospheres. The first fluid extension, SPECFEM2D-DG-FNS, was introduced in 2017 by Brissaud, Martin, Garcia, and Komatitsch; it features a nonlinear Full Navier–Stokes (FNS) approach discretized with a discontinuous Galerkin numerical scheme. In this contribution, we focus only on introducing a second fluid extension, SPECFEM2D-DG-LNS, based on the same numerical method but rather relying on the Linear Navier–Stokes (LNS) equations. The three main modules of SPECFEM2D-DG all use the spectral element method (SEM). For both fluid extensions (FNS and LNS), two-way mechanical coupling conditions preserve the Riemann problem solution at the fluid–solid interface. Absorbing outer boundary conditions (ABCs) derived from the perfectly matched layers’ approach is proposed for the LNS extension. The SEM approach supports complex topographies and unstructured meshes. The LNS equations allow the use of range-dependent atmospheric models, known to be crucial for the propagation of infrasound at regional scales. The LNS extension is verified using the method of manufactured solutions, and convergence is numerically characterized. The mechanical coupling conditions at the fluid–solid interface (between the LNS and elastodynamics systems of equations) are verified against theoretical reflection-transmission coefficients. The ABCs in the LNS extension are tested and prove to yield satisfactory energy dissipation. In an example case study, we model infrasonic waves caused by quakes occurring under various topographies; we characterize the acoustic scattering conditions as well as the apparent acoustic radiation pattern. Finally, we discuss the example case and conclude by describing the capabilities of this software. SPECFEM2D-DG is open-source and is freely available online on GitHub.

Variations in Finite-difference Potential Fields

Abstract The potential field (PF) solution of the solar corona is a vital modeling tool for a wide range of applications, including minimum energy estimates, coronal magnetic field modeling, and empirical solar wind solutions. Given its popularity, it is important to understand how choices made in computing a PF may influence key properties of the solution. Here we study PF solutions for the global coronal magnetic field on 2012 June 13, computed with our high-performance finite-difference code POT3D. Solutions are analyzed for their global properties and locally around NOAA AR 11504, using the net open flux, open-field boundaries, total magnetic energy, and magnetic structure as metrics. We explore how PF solutions depend on (1) the data source, type, and processing of the inner boundary conditions; (2) the choice of the outer boundary condition height and type; and (3) the numerical resolution and spatial scale of information at the lower boundary. We discuss the various qualitative and quantitative differences that naturally arise by using different maps as input, and we illustrate how coronal morphology and open flux depend most strongly on the outer boundary condition. We also show how large-scale morphologies and the open magnetic flux are remarkably insensitive to model resolution, while the surface mapping and embedded magnetic complexity vary considerably. This establishes important context for past, current, and future applications of the PF for coronal and solar wind modeling.

Goodness-of-fit test for isochrone fitting in the Gaia era

Context. The increasing precision in observational data made available by recent surveys means that the reliability of stellar models can be tested. For this purpose, a firm theoretical basis is crucial for evaluating the agreement of the data and theoretical predictions. Aims This paper presents a rigorous derivation of a goodness-of-fit statistics for colour-magnitude diagrams (CMD). We discuss the reliability of the underlying assumptions and their validity in real-world testing. Methods. We derived the distribution of the sum of squared Mahalanobis distances of stellar data and theoretical isochrone for a generic set of data and models. We applied this to the case of synthetic CMDs that were constructed to mimic real data of open clusters in the Gaia sample. Then, we analysed the capability of distinguishing among different sets of input physics and parameters that were used to compute the stellar models. To do this, we generated synthetic clusters from isochrones computed with these perturbed quantities, and we evaluated the goodness-of-fit with respect to the reference unperturbed isochrone. Results. We show that when r magnitudes are available for each of the N observational objects and p hyperparameters are estimated in the fit, the error distribution follows a χ2 distribution with (r − 1)N − p degrees of freedom. We show that the linearisation of the isochrone between support points introduces negligible deviation from this result. We investigated the possibility of detecting the effects on stellar models that are induced when the following physical quantities were varied: convective core overshooting efficiency, 14N(p, γ)15O reaction rate, microscopic diffusion velocities, outer boundary conditions, and colour transformation (bolometric corrections). We conducted the analysis at three different ages, 150 Myr, 1 Gyr, and 7 Gyr, and accounted for errors in photometry from 0.003 mag to 0.03 mag. The results suggest that it is possible to detect the effect induced by only some of the perturbed quantities. The effects induced by a change in the diffusion velocities or in the 14N(p, γ)15O reaction rate are too small to be detected even when the smallest photometric uncertainty is adopted. A variation in the convective core overshooting efficiency was detectable only for photometric errors of 0.003 mag and only for the 1 Gyr case. The effects induced by the outer boundary conditions and the bolometric corrections are the largest; the change in outer boundary conditions is detected for photometric errors below about 0.01 mag, while the variation in bolometric corrections is detectable in the whole photometric error range. As a last exercise, we addressed the validity of the goodness-of-fit statistics for real-world open cluster CMDs, contaminated by field stars or unresolved binaries. We assessed the performance of a data-driven cleaning of observations, aiming to select only single stars in the main sequence from Gaia photometry. This showed that this selection is possible only for a very precise photometry with errors of few millimagnitudes.

A study on the proper location of observation wells for various boundary conditions for geothermal reservoirs

This study deals with the determination of the region within a geothermal reservoir that best provides the volumetric average reservoir pressure. This is performed in order to able to determine the proper location for an observation well through which the volumetric average reservoir pressure could be recorded. In this study a numerical work is carried out for determining the proper location for an observation well to be placed for various configurations of reservoir outer boundary conditions and well configurations. Even though either a single well or two wells are used, the results may be generalized to cases where there are multiple wells in the reservoir by using the production/injection well locations as the region where production/injection occurs. The locations of the wells and boundary conditions have been chosen as to cover many of the possible configurations that could take place in practice. The boundary conditions used in this study are of two types; a sealed outer boundary and a constant pressure outer boundary for mimicking the effects of a recharge source. Wells are considered to be either producer or injector wells where a constant rate production or injection scheme is applied. The effects of faults are also considered in this study where the faults are modeled as a very high permeability medium. The results of this study provide various maps for various boundary and well configurations from which the engineer may be able to determine qualitatively where to place the observation well. This study is original in the sense that to the authors knowledge no such study exists in the literature. This study will be filling the gap.

A Semi-Analytical Analysis of Gas Slippage Effect in Pressure Transient Behavior of Non-Condensate Gas Reservoirs with the Different Boundary Condition

The conventional models utilized to study flow behavior in a Non-Condensate gas reservoir did not consider the effects of gas slippage and different outer boundary conditions. For gas wells with a constant production flow rate in a bounded oil or gas reservoir, commonly, the outer boundary conditions are infinite boundary conditions or zero flux. In this study, the dimensionless pseudo pressure and dimensionless derivative of pseudo pressure in the presence of 4 different outer boundary conditions (infinite reservoir, constant pressure, exponential, and power-law) besides effects of gas slippage, wellbore storage, and skin factor are analyzed. To do this, the dimensionless pseudo pressure partial differential equation of radial flow was derived from the combination of continuity equation with Darcy’s law, the equation of state, compressibility equation, and dimensionless parameters. Then the derived partial differential equation is solved analytically in the Laplace domain. The obtained results of this work have important significance to understand the effects of different conditions on the transient pressure behavior of Non-Condensate gas reservoirs.

Simultaneous estimation of fluid temperature and convective heat transfer coefficient by sequential function specification method

In pressurized water reactor (PWR) with multiple loops, abnormal working conditions occur when coolant pumps do not work in some loops. In these closed-loop pipelines, thermal stratification is formed under the effect of natural convection cooling and this poses a threat to the safe operation of the reactor. In this paper, the sequential function specification method (SFSM) is implemented to simultaneously predict the spatially and temporally varying internal fluid temperature and convective heat transfer coefficient of two-dimensional pipe under thermal stratification. In the direct problem, the finite volume method (FVM) is adopted to solve the governing equations subjected to initial and boundary conditions. In the inverse problem, the least-square method is employed to obtain the heat flux of the inner wall, and then Newton's law of cooling and the natural-convection correlation are adopted to obtain the fluid temperature and convective heat transfer coefficient. Numerical experiments of natural convection cooling are carried out under different outer wall boundary conditions to demonstrate the effectiveness of the proposed inversion method. The results of temperature estimation are consistent with the results of natural cooling numerical experiment and the relative error is below 4%, whereas the convective heat transfer coefficient is slightly worse due to the difference in the extraction positions. The effects of the number of involved future time steps and the measurement noise on the accuracy of the inverse analysis are also investigated in detail. Our results show that the fluid temperature and convective heat transfer coefficient of natural cooling may be effectively estimated using the proposed inverse method.

Clumpy Wind Accretion in Cygnus X-1

Abstract Cygnus X-1 is one of the brightest X-ray sources observed and shows the X-ray intensity variations on timescales from milliseconds to months in both the soft and hard X-rays. The accretion onto the black hole is believed to be wind fed due to focused stellar wind from the binary companion HDE-226868. We aim to understand the physical mechanism responsible for the short timescale X-ray variability (<100 s) of the source in its hard/low state. We compute the 2D relativistic hydrodynamic simulation of the low angular momentum accretion flow with a time-dependent outer boundary condition that reflects the focused, clumpy wind from the supergiant in this X-ray binary system. We follow the dynamical evolution of our model for about 100 s and present the results showing an oscillatory shock, being a potential explanation of variability observed in hard X-rays. The simulated model with shock solutions is in good agreement with the observed power density spectra of the source.

Lattice Boltzmann method for simulation of diffusion magnetic resonance imaging physics in multiphase tissue models.

We report an implementation of the lattice Boltzmann method (LBM) to integrate the Bloch-Torrey equation, which describes the evolution of the transverse magnetization vector and the fate of the signal of diffusion magnetic resonance imaging (dMRI). Motivated by the need to interpret dMRI experiments in biological tissues, and to offset the small time-step limitation of classical LBM, a hybrid LBM scheme is introduced and implemented to solve the Bloch-Torrey equation. A membrane boundary condition is presented which is able to accurately represent the effects of thin curvilinear membranes typically found in biological tissues. As implemented, the hybrid LBM scheme accommodates piece-wise uniform transport, dMRI parameters, periodic and mirroring outer boundary conditions, and finite membrane permeabilities on non-boundary-conforming inner boundaries. By comparing with analytical solutions of limiting cases, we demonstrate that the hybrid LBM scheme is more accurate than the classical LBM scheme. The proposed explicit LBM scheme maintains second-order spatial accuracy, stability, and first-order temporal accuracy for a wide range of parameters. The parallel implementation of the hybrid LBM code in a multi-CPU computer system, as well as on GPUs, is straightforward and efficient. Along with offering certain advantages over finite element or Monte Carlo schemes, the proposed hybrid LBM constitutes a flexible scheme that can by easily adapted to model more complex interfacial conditions and physics in heterogeneous multiphase tissue models and to accommodate sophisticated dMRI sequences.

Investigation of the orbit-spinning behaviors of the outer ring in a full ceramic ball bearing-steel pedestal system in wide temperature ranges

Owing to the difference in thermal deformation of the ceramic outer ring and steel pedestal, the outer ring experiences orbital and spinning motions in the fit clearance in wide temperature ranges, which has significant impacts on the bearing performance. In this paper, a dynamic model considering the temperature-related clearance is proposed, and the interactions between bearing and pedestal are calculated with a loose outer ring boundary condition. The orbit and spinning speeds are obtained, and the parameters concerning the speed ratio are used to evaluate the dynamic behaviors of the outer ring. Parametric analyses of temperature, rotation speed, and external load are conducted, and the results are verified through an experiment. The results show that changes in orbit-spinning behaviors are produced by variations in contact forces and area, and increased axial preload improves the stability of the bearing system. The results provide insights on bearing dynamics, and are useful for the design of ceramic bearing systems.

A novel well test model for fractured vuggy carbonate reservoirs with the vertical bead-on-a-string structure

The bead-on-a-string structure has been widely observed in many fractured vuggy carbonate reservoirs, especially in Shunbei oilfield, northwestern China. To date, there are still no appropriate well test methods for analyzing these reservoirs with the vertical bead-on-a-string structure. This paper proposes a new theoretical well test model, which considers the gravity effect and the natural fractures connecting the vuggs, to analyze this kind of reservoirs. First, we establish the mathematical model of fractured vuggy reservoirs with the vertical bead-on-a-string structure. With the different outer boundary conditions, Laplace transform and inversion are employed to solve the well test model. Then, the type curves for the well test model are plotted and discussed. It is found that the gravity effect can be equivalent to a type of constant pressure boundary condition. Take the three-region-two-cave as an example, type curves can be divided into nine distinct flow regimes, including wellbore storage, skin, linear flow I and II, transition flow I and II, vug storage I and II, and boundary flow. The transition flow and vug storage period are named as the vug storage effect. Subsequently, the effects of relevant parameters are studied. It is concluded that the depth of the drop during vug storage effect increases with increasing dimensionless vug storage constant CvD and dimensionless vug height zD2-zD1; the greater dimensionless gravity effect coefficient GD leads to the smaller equivalent boundary distance of the constant pressure boundary condition; the smaller the mobility ratio M12, the earlier the appearance of the second linear flow regime; the larger the storage ratio ω12, the more distinct the vug storage effect. Finally, two field case studies are carried out by using the proposed method. The locations and volumes of the caves are estimated, which shows that the proposed theoretical well test model has a great potential of obtaining the formation and cave parameters accurately. However, there is not available data that enable an accurate description of the locations and volumes of the caves by now, the proposed model still needs further field verifications.

A two-step semi-analytical solution method for the pressure response during cold water injection for a vertical well in a warm bounded reservoir

Development of a mathematical solution for injection/falloff problem of an oil-water system for transient data analysis is complicated due to the presence of two mobile phases that create a moving discontinuity. The solutions that have been developed in the past are based on a series of assumptions that make the models limited. The previous models have neglected one or more of the key reservoir and well parameters such as the variation of saturation in the invaded zone, temperature effects, wellbore storage, skin and outer boundary conditions. In this work, a generalized two-step semi-analytical solution method is presented to precisely simulate the pressure transient behaviour for injection and falloff tests following cold water injection for a fully-penetrating vertical well in a warm and bounded oil reservoir in the presence of skin and wellbore storage. The formulated model uses two simplistic steps to solve the complex mathematical problem and improve computational efficiency. The usage of the Laplace-transform finite-difference method to solve the system of equations eliminates the need for temporal discretization, thereby resolving issues related to convergence and time iterations by allowing an unlimited timestep size without the loss of stability and accuracy. The accuracy and consistency of the proposed solution method are assessed by comparing the results with that of ECLIPSE 100 and an excellent agreement is observed. The results suggest that the injection period pressure behaviour is vastly influenced by the injected water propagation under cold conditions, whereas, the falloff period pressure behaviour provides information of the water bank under cold conditions and the uninvaded zone under hot conditions. The comparison of the non-isothermal and isothermal pressure behaviour shows that the isothermal model can be improved using modified fluid viscosities to match the non-isothermal pressure behaviour that not only could minimize the mathematical complications, but also the computation time.

Transient Flow Theory of Multiple-Fractured Horizontal Wells with Complex Mechanisms in Shale Gas Reservoirs

Shale reservoirs have the characterizations of low porosity, low permeability, and hydrocarbon organic matter self-generation and self-storage, resulting in its complex flow mechanisms. Compared with fractured vertical wells, multiple-fractured horizontal wells are widely used due to their advantages of effectively increasing the well control range and further expanding the drainage area. To further study the multiscale flow mechanisms of shale gas, a flow model was established that considered viscous flow in microfractures and inorganic pores, the diffusion of Knudsen in nanoscale porosity, the coexistence of slippage, adsorption-desorption effects under infinity, and closed outer boundary conditions; based on the continuous point source solution, a multiple-fractured horizontal well flow model was established and solved by MATLAB programming. Then, the effects of various factors were investigated. The results show that the Knudsen diffusion and slippage coefficients mainly affect the apparent permeability of the matrix pores. The more the Knudsen diffusion and slippage coefficients are, the earlier the turbulent flow occurs and the higher the gas production is.

A Numerical Study of the Effects of Corotating Interaction Regions on Cosmic-Ray Transport

Abstract The intensity of galactic cosmic rays (GCRs) is modulated by solar activity on various timescales. In this study, we performed comprehensive numerical modeling of the solar rotational recurrent variation in GCRs caused by a corotation interaction region (CIR). A recently developed magnetohydrodynamic numerical model is adapted to simulate the background solar wind plasma with a CIR structure present in the inner heliosphere. As for the outer heliospheric plasma background, from 27 to 80 au, the Parker interplanetary magnetic field model is utilized. The output of these plasma and magnetic field models is incorporated into a comprehensive Parker-type transport model for GCRs. The local interstellar spectrum for galactic protons is transported to 80 au, specifying the outer boundary condition. The obtained solutions of this hybrid model, for studying the CIR effect, are as follows: (1) the onset of the decrease in the GCR intensity inside the CIR coincides with the increase of the solar wind speed with the intensity depression accompanied by a magnetic field and plasma density enhancement. Additionally, the CIR effect weakens with increasing heliocentric radial distance. (2) This decrease in GCR intensity also appears at different heliolatitudes and varies with changing latitude; the amplitude of the GCR depression exhibits a maximum in the low-latitude region. (3) The CIR affects GCR transport at different energy levels as well. Careful analysis has revealed a specific energy dependence of the amplitude of the recurrent GCR variation in the range of 30–2000 MeV.

Revisiting the Sun’s Strong Differential Rotation along Radial Lines

Abstract Current state-of-the-art models of the solar convection zone consist of solutions to the Navier–Stokes equations in rotating, 3D spherical shells. Such models are highly sensitive to the choice of boundary conditions. Here we present two suites of simulations differing only in their outer thermal boundary condition, which is either one of fixed entropy (FE) or fixed flux (FF; corresponding to a fixed gradient in the entropy). We find that the resulting differential rotation is markedly different between the two sets. The FF simulations have strong differential rotation contrast and isocontours tilted along radial lines (in good agreement with the Sun’s interior rotation revealed by helioseismology), whereas the FE simulations have weaker contrast and contours tilted in the opposite sense. We examine in detail the force balances in our models and find that the poleward transport of heat by Busse columns drives a thermal wind responsible for the different rotation profiles. We conclude that the Sun’s strong differential rotation along radial lines may result from the solar emissivity being invariant with latitude (which is similar to the FF condition in our models) and the poleward transport of heat by Busse columns. In future work on convection in the solar context, we strongly advise modelers to use an FF outer boundary condition.

Bayesian Inference of Quasi‐Linear Radial Diffusion Parameters using Van Allen Probes

AbstractThe Van Allen radiation belts in the magnetosphere have been extensively studied using models based on radial diffusion theory, which is derived from a quasi‐linear approach with prescribed inner and outer boundary conditions. The 1D diffusion model requires the knowledge of a diffusion coefficient and an electron loss timescale, which is typically parameterized in terms of various quantities such as the spatial (L) coordinate or a geomagnetic index (e.g., Kp). These terms are typically empirically derived, not directly measurable, and hence are not known precisely, due to the inherent nonlinearity of the process and the variable boundary conditions. In this work, we demonstrate a probabilistic approach by inferring the values of the diffusion and loss term parameters, along with their uncertainty, in a Bayesian framework, where identification is obtained using the Van Allen Probe measurements. Our results show that the probabilistic approach statistically improves the performance of the model, compared to the empirical parameterization employed in the literature.

The elastic boundary value problem of extended modified Bessel equation and its application in fractal homogeneous reservoir

By introducing the elastic boundary value condition, the elastic boundary value problem of extended modified Bessel equation is proposed, we can use the following method to solve it. First, two linear independent solutions of extended modified Bessel equation are obtained. Second, the generating function of solution is constructed. Third, the kernel function of solution is constructed using the elastic right value condition. Finally, the solution is obtained by assembling coefficients with the left boundary value condition. As for its application, a fractal homogeneous reservoir seepage model under the elastic outer boundary condition is established, and solution of the model is obtained. Influences of reservoir parameters on characteristic curves corresponding to dimensionless bottom hole pressure and its derivative are analyzed, which provide a new theoretical basis for exploring the flow law of oil. It can be found that seepage model under the elastic outer boundary condition regards three idealized outer boundary conditions (infinite, constant pressure and closed) and homogeneous reservoir seepage model without considering fractal as special cases, so it can reflect real situation of reservoir better and it is helpful to the development of related well test analysis software.