Function Erfc Research Articles

Mechanical formation damage management in oil wells with finite extent hydraulic fractures and source effects using an asymptotic-perturbation method

The petroleum industry has used hydraulic fracturing technology to solve various problems and provide an understanding of many complex issues. This work presents a new analytical model to evaluate the effective permeability response during the oil’s production through a vertical well fully penetrating a pressure-sensitive reservoir with a finite extent hydraulic fracture. The nonlinear hydraulic diffusivity equation (NHDE) is solved analytically through an integro-differential model coupled with Green’s function (GF) related to an oil source plan representing a finite extent hydraulic fracture. A new effective hydraulic diffusivity deviator factor is also derived to represent the permeability loss as a function of the pore pressure throughout the well-reservoir life cycle. For this oil flow modeling, the general solution is expressed by the sum of the linear solution (constant permeability) plus a corrective term given by the combination of the exponential function Ei(xD,yD,tD) and the complementary error function erfc(xD,yD,tD). The proposed model is implemented in Matlab software to evaluate the effect of the effective deviator factor in the permeability response over the oil’s production period. The model calibration is performed through a numerical oil flow simulator, widely used in the petroleum industry, and the results were accurate.

Чи деградують термоелектричні генераторні модулі внаслідок дифузії нікелю

The author builds a strict diffusion theory of electrical contact resistance by studying the process of non-stationary diffusion of metal into the thermoelectric material, as a result of which a transition contact layer is formed. The distribution of metal atoms in the transition layer and the law of its growth are strictly determined from the condition of the balance of the substance quantity. It is shown that the relative distribution of nickel in the transition layer is described by a function erfc(6,771x), where erfc(…) is the so-called complementary error integral, and x is the normalized depth measured from the metal-thermoelectric material separation boundary. At the same time, the transition contact layer grows with time t according to the law x0 = 13,552(Dt)1/2, where D is the diffusion coefficient of the metal in the thermoelectric material, and its thickness is related to the thickness of the spent metal layer by the ratio x0 = 12h, where h the thickness of the spent nickel layer. Based on the obtained regularities, the temperature dependence of the electrical contact resistance of the transition layer and its time dependence (degradation law) were strictly calculated for the first time. It is shown that for the thickness of the spent nickel layer in the range of 5–20 μm and in the temperature range of 300–500 K, the electrical contact resistance varies in the range from 3,8•10–6 to 3,4•10–5 Ω•cm2 and increases over time according to the square root law. It is also shown that there is no reason to fear a significant decrease in the thermoelectric figure of merit of the generator material as a result of nickel concentration leveling through the thickness of the thermoelectric leg after its surface layer is depleted.

Analytical model to effective permeability loss monitoring in hydraulically fractured oil wells in pressure-sensitive reservoirs

This work develops a new analytical model to evaluate the effective permeability response during the oil’s production through a fractured vertical oil well fully penetrating a pressure-sensitive reservoir. The nonlinear hydraulic diffusivity equation (NHDE) is solved analytically through an integro-differential model coupled to Green’s function (GF) related to a semi-infinite oil source plane that represents a hydraulic fracture. The NHDE is perturbed using a first-order asymptotic series expansion method, and the solution is derived in terms of a new effective permeability pseudo-pressure function, considering the matrix and fracture permeability roles. A new hydraulic diffusivity deviation factor is also derived to represent the effective permeability loss as a function of the pore pressure throughout the well-reservoir life cycle. For this oil flow modeling, the general solution is expressed by the sum of the linear solution (constant permeability) plus a corrective term given by the combination of the exponential function and the complementary error function erfc(yD,tD). The model calibration is performed through a porous media numerical oil flow simulator, named IMEX®, widely used in the petroleum industry and scientific works. The results presented high accuracy when compared to the numerical simulator, thus showing that the developed model allows representing the permeability pressure-sensitive response.

Pertechnetate/Perrhenate Surface Complexation on Bamboo Engineered Biochar.

The work deals with the evaluation of biochar samples prepared from Phyllostachys Viridiglaucescens bamboo. This evaluation consists of the characterization of prepared materials’ structural properties, batch and dynamic sorption experiments, and potentiometric titrations. The batch technique was focused on obtaining basic sorption data of 99mTcO4− on biochar samples including influence of pH, contact time, and Freundlich isotherm. ReO4−, which has very similar chemical properties to 99mTcO4−, was used as a carrier in the experiments. Theoretical modeling of titration curves of biochar samples was based on the application of surface complexation models, namely, so called Chemical Equilibrium Model (CEM) and Ion Exchange Model (IExM). In this case it is assumed that there are two types of surface groups, namely, the so-called layer and edge sites. The dynamic experimental data of sorption curves were fitted by a model based on complementary error function erfc(x).

间歇性生态输水塔里木河下游断面地下水位变化模拟

PDF HTML阅读 XML下载 导出引用 引用提醒 间歇性生态输水塔里木河下游断面地下水位变化模拟 DOI: 10.5846/stxb201710111813 作者: 作者单位: 中国科学院新疆生态与地理研究所,新疆水利水电科学研究院水资源研究所,新疆塔里木河流域干流管理局,中国科学院新疆生态与地理研究所,中国科学院新疆生态与地理研究所 作者简介: 通讯作者: 中图分类号: 基金项目: 国家自然科学基金项目（U1403182，U1603342）；新疆优秀青年科技人才培养项目（qn2015yx033） Simulation of sectional groundwater level variation in the lower reaches of Tarim River under intermittent ecological water conveyance Author: Affiliation: Xinjiang Institute of Ecology and Geography，CAS,Xinjiang Institute of Water Resources and Hydropower Research,Management Bureau of the main stream of Xinjiang Tarim River Basin,Xinjiang Institute of Ecology and Geography，CAS,Xinjiang Institute of Ecology and Geography，CAS Fund Project: 摘要 | 图/表 | 访问统计 | 参考文献 | 相似文献 | 引证文献 | 资源附件 | 文章评论 摘要:地下水作为干旱区生态系统的重要组成部分，对植物生长具有重要作用。以塔里木河下游英苏断面2011-2015年逐月地下水埋深变化为研究内容，基于非稳定流理论，以河道附近水位边界条件作为初始求解条件，以余误差函数erfc（x）为求解函数，综合考虑地下水变化时间滞后效应及潜水蒸发作用，旨在构建间歇性输水河道地下水埋深变化模型。结果表明：（1）生态输水期间，在短期内，临近河道地下水埋深波动较大，而远离河道地下水埋深波动并未受到较大的影响，生态输水作用影响河岸附近地下水埋深变化存在一定时段的滞后期，且随着距离河道的增加，滞后期增长。（2）以河道为基点的地下水非稳定流运动及潜水蒸发作用是影响沿岸地下水变化的诸多环境因子中最敏感的因子。（3）模拟结果表明基于非稳定流理论对塔里木河下游地下水变化模拟效果较好，达到了地下水模拟精度的要求，对塔里木河下游断面尺度上地下水恢复状况研究及后期生态恢复评价具有重要的理论意义与参考价值。 Abstract:As an important component of arid ecosystems, groundwater plays an important role in plant growth. We used the unsteady flow theory as the method, residual error function erfc(x) as the solution function, and time lag effect of groundwater depth change and phreatic evaporation as the basic condition to construct a monthly average groundwater depth change model from 2011 to 2015 of the lower reaches of the Tarim River Yingsu-section based on intermittent water conveyance. The results indicated that:First, during the period of ecological water conveyance, groundwater depth near the river channel fluctuates greatly in a short time, whereas fluctuation in groundwater depth was not greatly affected far from the river. This shows the influence of ecological water transport on groundwater depth near the river bank, which has a certain lag period, and the lag time increases gradually as distance from the river increases. Among all monitoring wells, the C4 monitoring well showed the shortest lag period, whereas the C7 monitoring well showed the longest lag period. Second, the unsteady flow of groundwater and phreatic evaporation were the most sensitive factors influencing the change in groundwater depth. The action of unsteady flow showed the greatest influence on groundwater depth during monthly variation and annual variation. The effect of phreatic water evaporation on the change in groundwater depth was small in the short term; however, the total influence of groundwater depth over many years cannot be ignored. Third, in core monitoring of the lower reaches of Tarim River, Populus-Tamarix chinensis-meadow mixed zone, meadow, meadow-desert mixed zone, and semi-desert were observed within 1050 m of the Yingsu-section, which require different groundwater depth. The simulation of groundwater depth changes in the lower reaches of Tarim River are based on the unsteady flow theory, with fitting accuracies (R2) at 300, 500, 750, and 1050 m of 0.701, 0.654, 0.701, and 0.625, respectively. These values meet the requirements for groundwater simulation accuracy and have important theoretical significance and reference values for studying relationships between groundwater and vegetation restoration in the lower reaches of Tarim River. This research will provide a reference for the simulation of groundwater level in arid area, which can be used to simulate the change of groundwater level at any location within the influence range of the river. Further exploration will improve the efficiency of ecological evaluation at a certain extent. 参考文献 相似文献 引证文献

Chloride ingress into marine exposed concrete: A comparison of empirical- and physically- based models

In establishing the reliability of performance-related design methods for concrete – which are relevant for resistance against chloride-induced corrosion - long-term experience of local materials and practices and detailed knowledge of the ambient and local micro-climate are critical. Furthermore, in the development of analytical models for performance-based design, calibration against test data representative of actual conditions in practice is required. To this end, the current study presents results from full-scale, concrete pier-stems under long-term exposure to a marine environment with work focussing on XS2 (below mid-tide level) in which the concrete is regarded as fully saturated and XS3 (tidal, splash and spray) in which the concrete is in an unsaturated condition. These exposures represent zones where concrete structures are most susceptible to ionic ingress and deterioration. Chloride profiles and chloride transport behaviour are studied using both an empirical model (erfc function) and a physical model (ClinConc). The time dependency of surface chloride concentration (Cs) and apparent diffusivity (Da) were established for the empirical model whereas, in the ClinConc model (originally based on saturated concrete), two new environmental factors were introduced for the XS3 environmental exposure zone. Although the XS3 is considered as one environmental exposure zone according to BS EN 206-1:2013, the work has highlighted that even within this zone, significant changes in chloride ingress are evident. This study aims to update the parameters of both models for predicting the long term transport behaviour of concrete subjected to environmental exposure classes XS2 and XS3.

Comparison of algorithms for Doppler broadening pointwise tabulated cross sections

In this paper, three algorithms for Doppler broadening pointwise tabulated cross sections are compared on the basis of speed and accuracy. The use of a Gauss–Legendre quadrature for numerical integration on subintervals of the convolution integral for which the cross section is smooth was found to reduce run time by 48% compared to the exact method used in NJOY and other cross section processing codes. However, a comparable decrease in run time was also achieved simply by using a more efficient implementation of the erfc function. The use of a Gauss–Hermite quadrature for performing the convolution integral resulted in an order of magnitude decrease in run time for broadening. The accuracy of the Gauss–Hermite quadrature was also studied, and it was found that quadrature orders greater than 15 reach an asymptotic level of error. Lastly, an on-the-fly implementation of Doppler broadening in OpenMC using a Gauss–Hermite quadrature resulted in at least a factor of 15 increase in run time relative to a simulation without Doppler broadening for the BEAVRS benchmark problem, underscoring the need for advanced methods for on-the-fly temperature treatment for Monte Carlo codes.

On spectral quadrature for linear-scaling Density Functional Theory

We provide a unified description of the Fermi operator expansion and recursion methods within the technique of spectral quadrature. Through rigorous error estimates, we prove that this approach is linear-scaling, stable and exponentially convergent. We use this analysis to determine the influence of smearing, band-gap, position of Fermi energy, and spectral width of the Hamiltonian on the convergence rates obtained in practical calculations. Additionally, we establish that super-geometric convergence can be achieved when the erfc function is used for smearing. We validate the spectral quadrature method and the accuracy of our analysis by means of selected examples.

Invertible Exponential-Type Approximations for the Gaussian Probability Integral Q(x) with Applications

In this article, we present very tight exponential-type approximations for the Gaussian probability integral Q(.) and/or the complementary error function erfc(.) which are subsequently used to devise the conditional symbol error probability (CEP) formulas for several classes of digital modulation schemes that are both invertible and in a desirable exponential form. These invertible CEP formulas are of interest in the optimization of discrete-rate adaptive modulation designs and for computation of the symbol/bit error outage performance metric. Our framework also facilitates the derivation of simple and tight closed-form approximation formulas for the average symbol error rate performance metric for a wide range of digital modulation schemes over generalized fading channels via the moment generating function approach. The accuracies of our new closed-form approximations have been validated with that of the exact expressions in an integral form.

A Note on New Exact Solutions for Some Unsteady Flows of Brinkman- Type Fluids over a Plane Wall

Flows of a Brinkman fluid due to a plane boundary moving in its plane are studied using Laplace transform. The solutions that have been obtained for the velocity are presented in simple forms in terms of the complementary error function erfc(.). They satisfy all imposed initial and boundary conditions and can easily be reduced to the similar solutions for Newtonian fluids

A Note on the Error Function

A new exact representation of the error function of real arguments justifies an accurate and simple analytical approximation. Two of the most widely used functions in physical sciences are the error function erf(x) and its related complimentary error function erfc(x). These functions occur extensively in problems relating to diffusion, heat conduction, and probability. When the argument x is real, rational approximations for these functions provide a high accuracy. In addition, the Faddeeva function, which is a variant of the error function for the complex argument z = x + iy, is used extensively in nuclear physics and spectroscopy.

Numerical simulation of the PGNA signal from chlorine diffusion gradients in concrete

A nondestructive test method for detecting chlorides in concrete has been developed based on prompt gamma neutron activation (PGNA). Its performance has been modeled using a hybrid MCNP/optical ray tracing approach. Since the chlorides often come from de-icing salts applied to the concrete surface, the Cl concentration has a non-linear depth profile which is typically modeled by the erfc function. The signals from this distribution have been simulated for several significant Cl capture peaks to estimate the erfc function parameters.

New representations and bounds for the generalized marcum Q-function via a geometric approach, and an application

The generalized Marcum Q-function of order m, Qm(a, b), is interpreted geometrically as the probability of a 2m-dimensional, real, Gaussian random vector Z2m, whose mean vector has a Frobenius norm of a, lying outside of a hyperball BO,b 2m of 2m dimensions, with radius b, and centered at the origin O. Based on this new geometric view, some new representations and closed-form bounds are derived for Qm(a, b). For the case that m is an odd multiple of 0.5, a new closed-form representation is derived, which involves only simple exponential and ERFC functions. For the case that m is an integer, a pair of new, finite-integral representations for Qm(a, b) is derived. Some generic exponential bounds and ERFC bounds are also derived by computing the probability of Z2m lying outside of various bounding geometrical shapes whose surfaces tightly enclose, or are tightly enclosed by the surface of BO,b 2m. These bounding shapes consist of an arbitrarily large number of parts. As their closeness of fit with BO,b 2m improves, our generic bounds approach the exact value of Qm(a, b). The function Qm(a, b) is proved to be an increasing function of its order when 2m is a positive integer. Thus, Qm+0.5(a, b) and Qm-0.5(a, b) can be used as tight upper and lower bounds, respectively, on Qm(a,b). Their average is a good approximation to Qm(a, b). An application of our new representations and bounds is also given.

Computing and bounding the first-order Marcum Q-function: a geometric approach

A geometric interpretation of the first-order Marcum Q-function, Q(a,b), is introduced as the probability that a complex, Gaussian random variable with real mean a, takes on values outside of a disk C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O,b</sub> of radius b centered at the origin O. This interpretation engenders a fruitful approach for deriving new representations and tight, upper and lower bounds on Q(a,b). The new representations obtained involve finite-range integrals with pure exponential integrands. They are shown to be simpler and more robust than their counterparts in the literature. The new bounds obtained include the generic exponential bounds which involve an arbitrarily large number of exponential functions, and the simple erfc bounds which involve just a few erfc functions, together with exponential functions in some cases. The new generic exponential bounds approach the exact value of Q(a,b) as the number of exponential terms involved increases. These generic exponential bounds evaluated with only two terms and the new simple erfc bounds are much tighter than the existing exponential bounds in most cases, especially when the arguments a and b are large. Thus, in many applications requiring further analytical manipulations of Q(a,b), these new bounds can lead to some closed-form results which are better than the results available so far.

Gain and power optimization of the wireless optical system with multilevel modulation

When used in an outdoor environment to expedite networking access, the performance of wireless optical communication systems is affected by transmitter sway. In the design of such systems, much attention has been paid to developing power-efficient schemes. However, the bandwidth efficiency is also an important issue. One of the most natural approaches to promote bandwidth efficiency is to use multilevel modulation. This leads to multilevel pulse amplitude modulation in the context of intensity modulation and direct detection. We develop a model based on the four-level pulse amplitude modulation. We show that the model can be formulated as an optimization problem in terms of the transmitter power, bit error probability, transmitter gain, and receiver gain. The technical challenges raised by modeling and solving the problem include the analytical and numerical treatments for the improper integrals of the Gaussian functions coupled with the erfc function. The results demonstrate that, at the optimal points, the power penalty paid to the doubled bandwidth efficiency is around 3 dB.

Evaluating the Normal Distribution

This article provides a little table-free C function that evaluates the normal distribution with absolute error less than 8 x 10 -16 . A small extension provides relative error near the limit available in double precision: 14 to 16 digits, the limits determined mainly by the computer's ability to evaluate exp(-t) for large t. Results are compared with those provided by calls to erf or erfc functions, the best of which compare favorably, others do not, and all appear to be much more complicated than need be to get either absolute accuracy less than 10 -15 or relative accuracy to the exp()-limited 14 to 16 digits. Also provided: A short history of the error function erf and its intended use, as well as, in the browse files attachment, various erf or erfc versions used for comparison.

Investigation of boron diffusion in 6H-SiC

p -type doping of 6H-SiC was implemented by diffusion of boron at temperatures higher than 1900 °C. The doping profiles were clearly divided into steep (zone I) and long-tail (zone II) regions. The boron diffusions in both regions are well fitted by erfc functions but with different diffusion coefficients, which are an activation energy (EA) of 6.1 eV and a prefactor (D0) of 3.2 cm2/s for zone I and 4.6 eV and 0.1 cm2/s for zone II, respectively. Further, it has been confirmed that the boron acceptors in zone I are primarily located at shallow energy levels (∼300 meV) and the ones in zone II are located at deep energy levels (∼700 meV).

Soda-lime glass with gradient of refraction index (GRIN)

In a soda-lime glass provided by Companhia Vidraria Santa Marina, subsidiary of French Saint-Gobain, we induced a gradient of refraction of index (GRIN) by exchange of Na+ in glass by Li+ at 550ºC for 12, 24, 36 and 48 hours and varying bath temperature at 525ºC, 550ºC, 575ºC and 600ºC for fixed time of 48 hours. deltan = 0.0107 ± 0.0005 and GRIN depth of about 2.4mm were obtained for the ion exchange at 550ºC for 48 hours. The GRIN profile was fitted with er fc(x) function and then obtained the diffusion coefficient D = (1.6 ± 0.3) · 10-6 mm²/s.

Numerical studies of anomalous fast diffusion in metallic alloys and semiconductors

The so-called anomalous fast diffusion in metallic alloys and semi-conductors is often analysed within an interstitial–substitutional model. The equations used for modelling these mechanisms are reaction–diffusion type whose analytical solutions are available only under drastic simplifications. The dissociative variety of this model is simulated using both finite difference (FD) and Monte Carlo (MC) methods. In MC simulation, diffusion of different species (interstitial impurities B i, substitutional B s, and vacancies V) and reaction jumps occur according to the suitable probabilities which are jump frequencies-dependent. Whereas in FD method, an implicit scheme is used to solve the system of non-linear partial differential equations. In both cases, the finite source conditions have been considered. A good agreement between results obtained by the two methods is found. On the other hand, the double-stages of simulated profiles are found to have similar shapes to those obtained experimentally in Nb(Co) and in GaAs(Zn). The first stage is well analysed by a Gaussian function, whereas the second one is well represented by an erfc type function. Furthermore, a detailed study of the two stages leads to a qualitative agreement with Stolwijk's analysis in two limited cases where the diffusion is vacancy- or foreign interstitial-controlled. However, the effective diffusion coefficients present a quantitative departure from those obtained by Stolwijk's expressions.

Order and chaos in Hofstadter'sQ(n) sequence

A number of observations are made on Hofstadter's integer sequence defined by Q(n) = Q(n − Q(n − 1)) + Q(n − Q(n − 2)), for n > 2, and Q(1) = Q(2) = 1. On short scales, the sequence looks chaotic. It turns out, however, that the Q(n) can be grouped into a sequence of generations. The k-th generation has 2k members that have “parents” mostly in generation k − 1 and a few from generation k − 2. In this sense, the sequence becomes Fibonacci type on a logarithmic scale. The variance of S(n) = Q(n) − n/2, averaged over generations, is ≅2αk, with exponent α = 0.88(1). The probability distribution p*(x) of x = R(n) = S(n)/nα, n ≫ 1, is well defined and strongly non-Gaussian, with tails well described by the error function erfc. The probability distribution of xm = R(n) − R(n − m) is given by pm(xm) = λm p*(xm/λm), with λm → √2 for large m. © 1999 John Wiley & Sons, Inc.