Sparse Randomized Maximum Likelihood (SpRML) for subsurface flow model calibration and uncertainty quantification

Despite their apparent high dimensionality, spatially distributed hydraulic properties of geologic formations can often be compactly (sparsely) described in a properly designed basis. Hence, the estimation of high-dimensional subsurface flow properties from dynamic performance and monitoring data can be formulated and solved as a sparse reconstruction inverse problem. Recent advances in statistical signal processing, formalized under the compressed sensing paradigm, provide important guidelines on formulating and solving sparse inverse problems, primarily for linear models and using a deterministic framework. Given the uncertainty in describing subsurface physical properties, even after integration of the dynamic data, it is important to develop a practical sparse Bayesian inversion approach to enable uncertainty quantification. In this paper, we use sparse geologic dictionaries to compactly represent uncertain subsurface flow properties and develop a practical sparse Bayesian method for effective data integration and uncertainty quantification. The multi-Gaussian assumption that is widely used in classical probabilistic inverse theory is not appropriate for representing sparse prior models. Following the results presented by the compressed sensing paradigm, the Laplace (or double exponential) probability distribution is found to be more suitable for representing sparse parameters. However, combining Laplace priors with the frequently used Gaussian likelihood functions leads to neither a Laplace nor a Gaussian posterior distribution, which complicates the analytical characterization of the posterior. Here, we first express the form of the Maximum A-Posteriori (MAP) estimate for Laplace priors and then use the Monte-Carlo-based Randomize Maximum Likelihood (RML) method to generate approximate samples from the posterior distribution. The proposed Sparse RML (SpRML) approximate sampling approach can be used to assess the uncertainty in the calibrated model with a relatively modest computational complexity. We demonstrate the suitability and effectiveness of the SpRML formulation using a series of numerical experiments of two-phase flow systems in both Gaussian and non-Gaussian property distributions in petroleum reservoirs and successfully apply the method to an adapted version of the PUNQ-S3 benchmark reservoir model.     

GEO卫星Hill方程的分析型解法

提出并实现了一种求解静地轨道GEO(geostationary orbit)卫星Hill方程摄动解的分析型方法。根据地固坐标系下GEO卫星运动的特点,在其定点处把摄动力进行泰勒展开,通过选择特定的参考轨道获得了GEO卫星的地球扁率摄动解。在此基础上,成功地将拉普拉斯变换运用于GEO卫星的Hill方程,得到了一组可用于求解摄动分析解的递推积分公式。通过逐次趋近的方法,利用这组积分公式可以有效地实现由低阶解推求高阶解。     

总强度磁异常各阶垂向导数换算新方法

磁异常垂直导数换算在磁性目标解释推断过程中具有重要意义。分析了总强度磁异常各阶垂向导数频率转换因子的滤波特性,指出常规傅立叶变换法在求解总强度磁异常各阶垂向导数时,观测数据中高频噪声会显著放大,甚至会淹没掉真实信息。从理论上证明了总强度磁异常沿垂直方向的积分和各阶导数均为调和函数,在此基础上,提出联合采用空间域和频率域运算求解其沿垂直方向的各阶导数。该方法的基本思路是：利用频率域转换关系,计算平面上总强度磁异常沿垂直方向的积分值;选择三点二阶中心差分或双三次样条曲线函数法计算总强度磁异常垂向积分值沿水平方向的二阶导数;最后将二阶水平导数代入拉普拉斯方程求解出总强度磁异常的垂向一阶导数;以总强度磁异常及其垂向一阶导数沿水平方向的二阶导数为基础,结合拉普拉斯方程,可进一步求解出总强度磁异常任意阶垂向导数。同时,为了研究提出方法的有效性,采用球体磁场模型进行验证,并首次推导了地磁场方向和磁化强度方向不一致时,球体总强度磁异常沿垂直方向的一阶和二阶导数表达式。经研究表明：提出的方法换算得到的垂向导数结果精度明显优于常规傅立叶变换法换算结果,且具有较强的抗噪能力,尤其是在计算高阶导数时效果更加明显。     

Accuracy of formation factors for three-dimensional pore-scale images of geo-materials estimated by renormalization technique

The formation factor, the dimensionless electric resistivity of porous rock/sediment saturated with conductive fluid, is an important quantity in geophysical exploration for petroleum reservoirs and groundwater aquifers. The renormalization technique is a promising approximation method for the quick estimation of the formation factors from large three-dimensional images of porous geo-materials obtained by X-ray microtomography. In the present study, we applied the renormalization technique to various pore-scale image sets of real geo-materials (sandstones, pumice, lava, and sandy sediments). The purpose is to explore the factors controlling the estimation accuracy of the formation factor. The results revealed that the accuracy increases with increasing (i) porosity, (ii) degree of the pore elongation along the direction of the applied electric field, and (iii) size of the initial subsystem with which the renormalization step starts and with decreasing (iv) pore or grain size. Most importantly, a high degree of elongation of the pore structure along the applied field ensures good accuracy even if the porosity is low, the initial subsystem is small, and the pore or grain size is large.     

Wellbore flow‐rate solution for a constant‐head test in two‐zone finite confined aquifers

The solution describing the wellbore flow rate in a constant‐head test integrated with an optimization approach is commonly used to analyze observed wellbore flow‐rate data for estimating the hydrogeological parameters of low‐permeability aquifers. To our knowledge, the wellbore flow‐rate solution for the constant‐head test in a two‐zone finite‐extent confined aquifer has never been reported so far in the literature. This article is first to develop a mathematical model for describing the head distribution in the two‐zone aquifer. The Laplace domain solutions for the head distributions and wellbore flow rate in a two‐zone finite confined aquifer are derived using the Laplace transform, and their corresponding time domain solutions are then obtained using the Bromwich integral method and residue theorem. These new solutions are expressed in terms of an infinite series with Bessel functions and not straightforward to calculate numerically. A large‐time solution for the wellbore flow rate is therefore developed by employing the relationship of small Laplace variable versus large time variable and L'Hospital's rule. The result shows that the large‐time solution is identical to the steady‐state solution obtained after applying the Tauberian theorem into the Laplace domain solution. This large‐time solution can reduce to the Thiem equation in the case of no skin. Finally, the newly developed solution is used to investigate the effects of outer boundary distance and conductivity ratio on the wellbore flow rate. Copyright © 2011 John Wiley & Sons, Ltd.     

Exact analytical solutions for two-dimensional advection-dispersion equation in cylindrical coordinates subject to third-type inlet boundary condition

Exact analytical solutions for two-dimensional advection-dispersion equation (ADE) in cylindrical coordinates subject to the third-type inlet boundary condition are presented in this study. The finite Hankel transform technique in combination with the Laplace transform method is adopted to solve the two-dimensional ADE in cylindrical coordinates. Solutions are derived for both continuous input and instantaneous slug input. The developed analytical solutions are compared with the solutions for first-type inlet boundary condition to illustrate the influence of the inlet condition on the two-dimensional solute transport in a porous medium system with a radial geometry. Results show significant discrepancies between the breakthrough curves obtained from analytical solutions for the first-type and third-type inlet boundary conditions for large longitudinal dispersion coefficients. The developed solutions conserve the solute mass and are efficient tools for simultaneous determination of the longitudinal and transverse dispersion coefficients from a laboratory-scale radial column experiment or an in situ infiltration test with a tracer.     

相位相关法辅助的重复纹理区低空影像匹配

针对重复纹理区低空影像匹配中存在的多义性易造成同名点过少或匹配失败等问题利用相位相关法计算了像对中两影像傅立叶变换复共轭的互功率谱预测同名点在平移旋转和尺度上的遍历范围并以尺度遍历范围自适应估计Harris－Laplace尺度空间和检测角点然后通过相关系数法和核线约束匹配同名点实验验证了该方法对于重复纹理区低空影像匹配的可靠性和实用性可获得数量足够且分布均匀的同名点     

Axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent due to groundwater drawdown

Axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent induced by groundwater drawdown was studied based on Biot’s axisymmetric consolidation theory. Laplace and Hankel transforms were employed to solve the governing equation. Explicit analytical solutions are obtained in the Laplace–Hankel transform domain when groundwater drawdown is induced by a constant pumping well. Based on the solutions, numerical computations were performed to study the influences of the compressibility of the fluid constituent on the consolidation behavior of the soil layer.     

Laplace变换解双层地基固结问题

连云港地区的软土是海相淤泥和淤泥质土.在连云港软土地区修筑高速公路时,通常采用水泥土搅拌桩地基处理方法使路基强度和变形满足工程的要求.该类软土压缩沉降量大,排水固结缓慢,地基稳定性差,造成高速公路建设中及建成后工程问题频繁出现.从江苏海相软土的工程实例出发,将经水泥土搅拌桩处理后的高速公路路基简化为双层地基,借助Laplace变换求解了二级加荷下的双层地基一维固结问题,为克服对固结方程进行Laplace逆变换的困难,在Laplace逆变换中运用Stehfest算法进行数值求解,该双层地基固结沉降计算结果与实际观测数据基本一致.     

分布热源作用下裂隙岩体渗流-传热的拉氏变换-格林函数半解析计算方法

以裂隙岩体高放射性核废物地下处置库性能评估为目标,提出了分布热源作用下单裂隙岩体渗流-传热的简化概念模型、控制微分方程和拉氏变换-格林函数半解析法,为进一步采用半解析法计算分布热源作用下多裂隙岩体的渗流-传热问题奠定了基础。针对单裂隙岩体的渗流-传热问题,建立考虑岩石内热源和二维热传导的控制微分方程,利用拉氏变换域微分方程的基本解建立格林函数积分方程,采用解析法处理其中的奇点,通过数值积分和拉氏数值逆变换求解,计算任意时刻裂隙水和岩石的温度分布。通过算例,与基于岩石一维热传导假定的解析解进行了对比,并计算分析了分布热源作用下单裂隙岩体的渗流-传热特征及其对裂隙开度、岩石热传导系数和热流集度的敏感度。算例表明,（1）就裂隙水温度而言,由于考虑了岩石的二维热传导,拉氏变换-格林函数半解析解小于基于岩石一维热传导假定的解析解;（2）裂隙水温度和岩石温度对裂隙开度和热流集度的敏感度较大,对岩石热传导系数的敏感度较小。