Moment Equation Approach Research Articles

Quantification of uncertainty for fluid flow in heterogeneous petroleum reservoirs

Detailed description of the heterogeneity of oil/gas reservoirs is needed to make performance predictions of oil/gas recovery. However, only limited measurements at a few locations are usually available. This combination of significant spatial heterogeneity with incomplete information about it leads to uncertainty about the values of reservoir properties and thus, to uncertainty in estimates of production potential. The theory of stochastic processes provides a natural method for evaluating these uncertainties. In this study, we present a stochastic analysis of transient, single phase flow in heterogeneous reservoirs. We derive general equations governing the statistical moments of flow quantities by perturbation expansions. These moments can be used to construct confidence intervals for the flow quantities (e.g., pressure and flow rate). The moment equations are deterministic and can be solved numerically with existing solvers. The proposed moment equation approach has certain advantages over the commonly used Monte Carlo approach.

Moment-Equation Approach to Single Phase Fluid Flow in Heterogeneous Reservoirs

Summary In this paper, we study single phase, steady-state flow in bounded, heterogeneous reservoirs. We derive general equations governing the statistical moments of flow quantities by perturbation expansions. These moments may be used to construct confidence intervals for the flow quantities. Due to their mathematical complexity, we solve the moment differential equations (MDEs) by the numerical technique of finite differences. The numerical MDE approach renders the flexibility in handling complex flow configurations, different boundary conditions, various covariance functions of the independent variables, and moderately irregular geometry, all of which are important factors to consider for real-world applications. The other method with these flexibilities is Monte Carlo simulation (MCS) which has been widely used in the industry. These two approaches are complementary, and each has its own advantages and disadvantages. The numerical MDE approach is compared with published results of MCS and analytical MDE approaches and is demonstrated with two examples involving injection/production wells.

Nonstationary stochastic analysis of transient unsaturated flow in randomly heterogeneous media

In this study, a first‐order, nonstationary stochastic model of transient flow is developed which is applicable to the entire domain of a bounded vadose zone in the presence of sink/source. We derive general equations governing the statistical moments of the flow quantities by perturbation expansions. Owing to the mathematical complexity of the equations, in general we need to solve them numerically. The numerical moment equation approach, however, has the flexibility in handling different boundary conditions, flow configurations, input covariance structures, and soil constitutive relationships. The moment equations presented in this study are simpler and easier to solve than those in the literature for transient unsaturated flow. We solve these moment equations by the method of finite differences and demonstrate the developed model through some one‐ and two‐ dimensional examples under various transient conditions.

Non-Gaussian Equivalent Linearization for Nonstationary Random Vibration of a 2-DOF Hysteretic System.

In the previous paper an approximate analytical method was developed for calculating the nonstationary responses of a single DOF system with bilinear hysteresis subjected to amplitude modulated nonwhite random excitation. The method consists of the non-Gaussian equivalent linearization technique and the use of the moment equations approach. In the present paper the method is extended and applied to random vibration analysis of a two DOF hysteretic system. The mean square responses and the mean dissipated hysteretic energy are obtained and compared with the corresponding digital simulation results.

Estimation of Non-Gaussian Response Distribution of a Nonlinear System Subjected to Random Excitation. Application to Nonwhite Excitation with Nonrational Spectrum.

The non-Gaussian response distribution of a single degree-of-freedom system with nonsymmetric nonlinearities subjected to nonwhite random excitation with nonrational spectra is investigated. The nonstationary and stationary probability densities of stochastic responses are calculated by employing the method which consists of the non-Gaussian equivalent linearization technique and the use of the moment equations approach. The non-Gaussian probability density function for the response distribution is expressed in terms of the potential of the system and employed for determining the equivalent linear coefficients. The results are compared with the corresponding digital simulation results.

Non Gaussian closure techniques for the analysis of R‐FBI isolation system

AbstractThe Resilient‐Friction Base Isolator (R‐FBI) stochastic response under severe ground motion modelled as a stationary and non‐stationary zero mean stochastic white noise processes is performed. The moment equation approach is applied and the non‐normal response is obtained by means of a non‐Gaussian closure technique, based on the Gram‐Charlier asymptotic expansion of the response probability density function. Results are compared with the equivalent non linearization technique and with results obtained by means of Monte Carlo simulation.

Electron transport processes close to magnetic axis in tokamaks

Electron transport fluxes valid in the region close to magnetic axis in tokamaks are derived from the moment equation approach. It is found that bootstrap current does not vanish on the axis, electrical conductivity is reduced from its classical value, and particle and heat fluxes are enhanced over those in the conventional neoclassical theory. The reason for the deviation from the conventional theory is because the fraction of the trapped particles is finite in the region close to the magnetic axis. The existence of the bootstrap current on the magnetic axis may reduce or eliminate the need of the seed current.

Drift-orbit-driven flux and plasma viscosity in asymmetric toroidal plasmas

It is shown, based on the kinetic definition of the drift-orbit-driven flux, that the drift-orbit-driven flux is related to either the toroidal viscosity, or the poloidal viscosity, or a combination of both. This result is the same as that obtained from the less direct moment equation approach [Phys. Fluids 26, 3315 (1983)].

Moment equation description of interstellar hydrogen

The flow of interstellar hydrogen in the heliosphere can be studied using the moment equation approach. The Boltzmann equation is integrated over the velocity space to obtain the moment equations, the moment equations are then solved directly for the flow conditions. We present a closed system of moment equations. This approach can include anisotropic pressure when the distribution function is distorted into skewed ellipsoid.

Non-Gaussian Equivalent Linearization for Estimation of Stochastic Response Distribution of Nonlinear Systems.

A method is developed for estimating the response distribution of a nonlinear system subjected to nonwhite random excitation. The method consists of the non-Gaussian equivalent linearization technique and the use of the moment equations approach. The non-Gaussian probability density for the response distribution is expressed in terms of the potential of the system and employed for determining the equivalent linear coefficients. Numerical examples are given for a single degree-of freedom system with nonsymmetric nonlinearities. The rms and mean responses and the probability density of stochastic responses are obtained and compared with the corresponding digital simulation results.

On the definition of Pfirsch–Schlüter and bootstrap currents in toroidal systems

In the plasma physics literature there appear two different definitions of Pfirsch–Schlüter current. One of them is predominantly used in equilibrium calculations and satisfies the condition IT=0. The other definition appears commonly in transport calculations and requires that the surface average of the dot product of the Pfirsch–Schlüter current density with the magnetic field vanish, i.e., 〈JPS⋅B〉=0. The difference between the definitions is a surface function. Within the framework of the moment equation approach, the total parallel current is completely determined through a surface average of Ohm’s law; thus different definitions of Pfirsch–Schlüter current imply different expressions for the bootstrap current. Understanding the different implications of these two definitions is of particular importance when designing toroidal devices with minimized Pfirsch–Schlüter current or studying tokamaks with optimized bootstrap current. In this paper the definitions of Pfirsch–Schlüter and bootstrap current, as well as the expressions for the corresponding Pfirsch–Schlüter diffusion flux, are analyzed and discussed for the case of axisymmetric and nonaxisymmetric plasmas. Although in cases like a current-free stellarator or a large-aspect-ratio tokamak both definitions are equivalent, they are in general different, and in order to avoid misunderstandings it is therefore important to use only one. The most appropriate definition is IT=0. In this paper the equations for determining the bootstrap current within the framework of the fluid equations are also analyzed.

Stochastic response analysis of nonlinear systems under Gaussian inputs

In this paper the extension of Itô's rule for the case of vector real valued functions of the response of nonlinear systems excited by zero-mean Gaussian white noise processes is presented. A suitable particularization of the vector function, in order to obtain the statistical moments of every order to the response, is treated, obtaining the differential equations of the response moments in an elegant and compact form. Polynomial expansion and closure schemes are framed in the context outlined here in order to obtain an effective procedure from a computational point of view. An application to a trigonometric nonlinear system, solved in the literature by the stochastic averaging method, is treated here by the moment equation approach using the polynomial expansion of the nonlinear terms in order to evidence the validity of this approach.

Monte Carlo study of the escape of a minor species.

The understanding of some important space-physics problems (e.g., the Jeans escape of light atoms from a planetary atmosphere and the ion escape in the terrestrial polar wind) is related to the problem of the escape of a minor species through a nonuniform background. This latter problem was studied for different interparticle collision models (Maxwell molecule and hard sphere), for different minor-to major-species mass ratios, and for different values of the escape velocity (${\mathit{v}}_{\mathit{c}}$). The gravitational force was simulated by a critical escape velocity (${\mathit{v}}_{\mathit{c}}$), and the Lorentz forces were ignored. First, a simple variable change was used to transform the problem into a simpler one where the background medium is uniform. This simplified problem, which is similar to the ``Milne'' problem, was solved by a Monte Carlo simulation. In the collisionless region and for the case of zero escape velocity (${\mathit{v}}_{\mathit{c}}$=0), the velocity distribution function of the minor species (${\mathit{f}}_{\mathit{t}}$) showed large deviations from a Maxwellian, with large temperature anisotropy (parallel temperature less than the perpendicular temperature) and asymmetry in the parallel direction (upward tail), for both of the collision models. In the collision-dominated region the normalized density gradient was found to be independent of the mass ratio for the Maxwell molecule collision. For the hard-sphere cross section, it was reduced by a mass-ratio-dependent factor. This is due to the heat-flow contribution to the momentum balance, which vanishes only for the former collision model. The Monte Carlo results were compared with the moment-equations approach and the direct solution of the Boltzmann equation. This comparison confirms that the Monte Carlo method is a viable alternative and complementary to other techniques, especially in recent years, due to the large increase in the available computation power.

Calculation of Hamada coordinates for a large-aspect-ratio tokamak

The application of the well-known moment equation approach to neoclassical transport requires the use of Hamada coordinates in three-dimensional toroidal plasmas. The lack of analytical expressions and the difficulty associated with the numerical calculation of these coordinates has strongly limited the use of this approach. In this paper analytical calculations of Hamada coordinates for a large-aspect-ratio tokamak are presented. Knowledge of these coordinates for this relatively simple two-dimensional case will allow, for the first time, a general analytical application of the moment equation approach, and also a check of this approach against well-known results. In particular, it is shown here that the approach correctly reproduces the Pfirsch–Schlüter diffusion rate.

Numerical analysis of nonequilibrium electron transport in AlGaAs/InGaAs/GaAs pseudomorphic MODFETs

Nonequilibrium electron transport in InGaAs pseudomorphic MODFETs has been analyzed with the moment equations approach. In the model, the momentum and energy balance equations for the two-dimensional electrons in the InGaAs channel are solved with relaxation times generated from a Monte Carlo simulation. The two-dimensional electron wave functions and the quantized state energies in the InGaAs quantum well are calculated exactly from the Schrodinger equation along the direction perpendicular to the quantum well. Also included is a two-dimensional Poisson equation solver. In the calculation, all of the equations are solved iteratively until a self-consistent solution is achieved. The simulation results for a realistic device structure with a 0.5- mu m recessed gate show a significant overshoot velocity of 4.5*10/sup 7/ cm/s at a drain bias of 1.0 V. Electron temperature reaches a peak value of around 2500 K under the gate. In energy transport, the diffusive component of the energy flux is found to be dominant in the high-field region. >

Calculation of neoclassical fluxes in the plateau regime for nonaxisymmetric toroidal plasmas

An analytic calculation of the neoclassical fluxes in the plateau regime in the absence of momentum and energy sources is carried out with a general model field, B=B0[1+∑iεi cos(liθ−miζ)], following the moment-equation approach. It is shown that the radial particle and heat fluxes resulting from each harmonic (εi) are additive and have the same sign. Consequently, the neoclassical fluxes in a stellarator in the plateau regime are larger than those in the corresponding ‘‘equivalent’’ tokamak (same aspect ratio and rotational transform). Numerical calculations performed with a drift kinetic equation solver agree to within 10% with the analytic results for the examples considered.

Random vibrations of a beam structure with elastoplastic hysteresis.

An experiment is conducted by using a lead cantilever beam with a mass in order to examine the random vibrations of an hysteretic system. Power spectra and root mean square values of the stationary responses to two types of random excitation, i.e., white noise, and nonwhite noise with a dominant frequency, are measured. A theoretical analysis is carried out by assuming a bilinear hysteresis model and using the moment equation approach. A fairly good agreement is found between the theoretical and experimental results. It is shown that the effect on the response of energy dissipation due to hysteresis is observed even in the case of small amplitude excitation.

A pressure-gradient-driven tokamak ‘‘resistive magnetohydrodynamic’’ instability in the banana-plateau collisionality regime

The moment equation approach to neoclassical processes is used to derive the linearized electrostatic perturbed flows, currents, and resistive MHD-like equations for a tokamak plasma. The new features of the resultant ‘‘neoclassical magnetohydrodynamics,’’ which requires a multiple length scale analysis for the parallel eigenfunction, but is valid in the experimentally relevant banana-plateau regime of collisionality, are: (1) a global Ohm’s law that includes a fluctuating bootstrap current resulting from the ‘‘parallel’’ electron viscous damping (at rate μe) of the poloidal flow due to the perturbed radial pressure gradient; (2) reduction of the curvature effects to their flux surface average because Pfirsch–Schlüter currents cancel out the lowest-order geodesic curvature effects: (3) an increased polarization drift contribution with B−2, replaced by B−2Θ where BΘ is the poloidal magnetic field component. An electrostatic eigenmode equation is determined from ∇⋅J̃=0. For the unstable fluid-like eigenmodes, the new viscous damping effects dominate (by ε−3/2) over the curvature effects, but the growth rates still scale roughly like resistive-g or resistive-ballooning modes, γμτA∼n2/3S1/3Nβ2/3T ×( μe/νe)1/3. Diamagnetic drift frequency corrections to these new modes are also discussed.

Neoclassical flows and transport in nonaxisymmetric toroidal plasmas

The moment equation approach to neoclassical transport theory has been generalized to nonaxisymmetric toroidal systems under the assumption of the existence of magnetic surfaces. In particular, the parallel plasma flows and bootstrap current are calculated in both the Pfirsch–Schlüter and banana regimes. It is found that both parallel plasma flows and the bootstrap current can be reduced as the toroidal bumpiness increases in an otherwise axisymmetric system.

The use of moment equations for calculating the mean square response of a linear system to non-stationary random excitation

The moment equations approach is used to calculate the mean square response of a linear system to non-stationary random excitation which is expressed as a product of a deterministic envelope function and a Gaussian stationary non-white noise. The moment equations are derived by performing single integrations in the time domain and are solved numerically by digital computer. Numerical examples are given for the response of single and two degree-of-freedom systems which are excited by noise with an exponentially decaying harmonic correlation function. It is shown that an overshoot, in the sense that the transient response exceeds its stationary value, may occur even in the case of an exponential envelope function, but that the response does not exhibit overshoot when the natural frequency of the system is almost coincident with the dominant frequency of the input.