Stochastic delineation of capture zones: classical versus Bayesian approach

A Bayesian approach to characterize the predictive uncertainty in the delineation of time-related well capture zones in heterogeneous formations is presented and compared with the classical or non-Bayesian approach. The transmissivity field is modelled as a random space function and conditioned on distributed measurements of the transmissivity. In conventional geostatistical methods the mean value of the log transmissivity and the functional form of the covariance and its parameters are estimated from the available measurements, and then entered into the prediction equations as if they are the true values. However, this classical approach accounts only for the uncertainty that stems from the lack of ability to exactly predict the transmissivity at unmeasured locations. In reality, the number of measurements used to infer the statistical properties of the transmissvity field is often limited, which introduces error in the estimation of the structural parameters. The method presented accounts for the uncertainty that originates from the imperfect knowledge of the parameters by treating them as random variables. In particular, we use Bayesian methods of inference so as to make proper allowance for the uncertainty associated with estimating the unknown values of the parameters. The classical and Bayesian approach to stochastic capture zone delineation are detailed and applied to a hypothetical flow field. Two different sampling densities on a regular grid are considered to evaluate the effect of data density in both methods. Results indicate that the predictions of the Bayesian approach are more conservative.     

Stochastic differential dynamic programming for multi-reservoir system control

: As with all dynamic programming formulations, differential dynamic programming (DDP) successfully exploits the sequential decision structure of multi-reservoir optimization problems, overcomes difficulties with the nonconvexity of energy production functions for hydropower systems, and provides optimal feedback release policies. DDP is particularly well suited to optimizing large-scale multi-reservoir systems due to its relative insensitivity to state-space dimensionality. This advantage of DDP encourages expansion of the state vector to include additional multi-lag hydrologic information and/or future inflow forecasts in developing optimal reservoir release policies. Unfortunately, attempts at extending DDP to the stochastic case have not been entirely successful. A modified stochastic DDP algorithm is presented which overcomes difficulties in previous formulations. Application of the algorithm to a four-reservoir hydropower system demonstrates its capabilities as an efficient approach to solving stochastic multi-reservoir optimization problems. The algorithm is also applied to a single reservoir problem with inclusion of multi-lag hydrologic information in the state vector. Results provide evidence of significant benefits in direct inclusion of expanded hydrologic state information in optimal feedback release policies.     

An algorithm for generating stochastic cloud fields from radar profile statistics

An algorithm is described for generating stochastic three-dimensional (3D) cloud fields from time–height fields derived from vertically pointing radar. This model is designed to generate cloud fields that match the statistics of the input fields as closely as possible. The major assumptions of the algorithm are that the statistics of the fields are translationally invariant in the horizontal and independent of horizontal direction; however, the statistics do depend on height. The algorithm outputs 2D or 3D stochastic fields of liquid water content (LWC) and (optionally) effective radius. The algorithm is a generalization of the Fourier filtering methods often used for stochastic cloud models. The Fourier filtering procedure generates Gaussian stochastic fields from a “Gaussian” cross-correlation matrix, which is a function of a pair of heights and the horizontal distance (or “lag”). The Gaussian fields are nonlinearly transformed to give the correct LWC histogram for each height. The “Gaussian” cross-correlation matrix is specially chosen so that, after the nonlinear transformation, the cross-correlation matrix of the cloud mask fields approximately matches that derived from the input LWC fields. The cloud mask correlation function is chosen because the clear/cloud boundaries are thought to be important for 3D radiative transfer effects in cumulus.The stochastic cloud generation algorithm is tested with 3 months of boundary layer cumulus cloud data from an 8.6-mm wavelength radar on the island of Nauru. Winds from a 915-MHz wind profiler are used to convert the radar fields from time to horizontal distance. Tests are performed comparing the statistics of 744 radar-derived input fields to the statistics of 100 2D and 3D stochastic output fields. The single-point statistics as a function of height agree nearly perfectly. The input and stochastic cloud mask cross-correlation matrices agree fairly well. The cloud fractions agree to within 0.005 (the total cloud fraction is 18%). The cumulative distributions of optical depth, cloud thickness, cloud width, and intercloud gap length agree reasonably well. In the future, this stochastic cloud field generation algorithm will be used to study domain-averaged 3D radiative transfer effects in cumulus clouds.     

Stochastic analysis of immiscible displacement of the fluids with arbitrary viscosities and its dependence on support scale of hydrological data

Stochastic analysis is commonly used to address uncertainty in the modeling of flow and transport in porous media. In the stochastic approach, the properties of porous media are treated as random functions with statistics obtained from field measurements. Several studies indicate that hydrological properties depend on the scale of measurements or support scales, but most stochastic analysis does not address the effects of support scale on stochastic predictions of subsurface processes. In this work we propose a new approach to study the scale dependence of stochastic predictions. We present a stochastic analysis of immiscible fluid–fluid displacement in randomly heterogeneous porous media. While existing solutions are applicable only to systems in which the viscosity of one phase is negligible compare with the viscosity of the other (water–air systems for example), our solutions can be applied to the immiscible displacement of fluids having arbitrarily viscosities such as NAPL–water and water–oil. Treating intrinsic permeability as a random field with statistics dependant on the permeability support scale (scale of measurements) we obtained, for one-dimensional systems, analytical solutions for the first moments characterizing unbiased predictions (estimates) of system variables, such as the pressure and fluid–fluid interface position, and we also obtained second moments, which characterize the uncertainties associated with such predictions. Next we obtained empirically scale dependent exponential correlation function of the intrinsic permeability that allowed us to study solutions of stochastic equations as a function of the support scale. We found that the first and second moments converge to asymptotic values as the support scale decreases. In our examples, the statistical moments reached asymptotic values for support scale that were approximately 1/10000 of the flow domain size. We show that analytical moment solutions compare well with the results of Monte Carlo simulations for moderately heterogeneous porous media, and that they can be used to study the effects of heterogeneity on the dynamics and stability of immiscible flow.     

河道油藏边界面的最优化随机模拟

以河道的观测深度为硬数据，利用贝叶斯理论通过随机建模的方法生成描述河道的方向线和河道几何参数。选这些参数的最大概率作为河道的最优化模型，利用所生成的最优化参数计算出河道砂体的边界面。对大家所公认的横截面为抛物线形状的河道给出了计算的方法，并以此为例实现了这一计算方法。     

Simulating Concentration Fluctuation Time Series with Intermittent Zero Periods and Level Dependent Derivatives

Intermittent concentration fluctuation time series were produced with a stochastic numerical model derived from the assumption that the concentration fluctuations at a fixed receptor in a point-source plume can be modelled as a first order Markov process. The time derivative of concentration was assumed to be level-dependent and constrained by a stationary lognormal probability density function. The input parameters required to reconstruct the intermittent time series are the intermittency factor , the conditional fluctuation intensity i _p ² , and the time scale T _c . A clipped lognormal probability distribution was used to describe the fluctuation time series. Good agreement between the stochastic simulation and experimental water-channel data was demonstrated by comparing the time derivative of concentration and the upcrossing rates over a range of intermittency factors = 0.7 to 0.01 and fluctuation intensities i _w ² = 2.2 to 7.5.     

Estimation or simulation? That is the question

The issue of smoothing in kriging has been addressed either by estimation or simulation. The solution via estimation calls for postprocessing kriging estimates in order to correct the smoothing effect. Stochastic simulation provides equiprobable images presenting no smoothing and reproducing the covariance model. Consequently, these images reproduce both the sample histogram and the sample semivariogram. However, there is still a problem, which is the lack of local accuracy of simulated images. In this paper, a postprocessing algorithm for correcting the smoothing effect of ordinary kriging estimates is compared with sequential Gaussian simulation realizations. Based on samples drawn from exhaustive data sets, the postprocessing algorithm is shown to be superior to any individual simulation realization yet, at the expense of providing one deterministic estimate of the random function.     

A probabilistic/deterministic approach for the prediction of the sediment transport rate

In this paper we present a probabilistic/deterministic model for the evaluation of the sediment transport rate in a stream. Starting from Einstein’s theory, the approach was obtained by trying to overcome some of the intrinsic limitations. The approach is based on two distinct probability functions, one relevant to the detachment of grains and the second relevant to the length of particle jumps. The sediment transport rate is obtained by integrating the distribution of the ranges of the particle jumps multiplied by the average particle velocity. The relationship for the average ranges of particle jumps is an opportune combination of the Einstein and Yalin expressions. The final formulation was calibrated by means of a large number of experimental data and also by comparison with some of the most widely-used empirical formulas. The results show a better agreement between theory and experiments than do the other theories analyzed.