M. Harrison, M. Riva, M. Mousavi Nezhad, A. Guadagnini
{"title":"Estimation of auto-covariance of log hydraulic conductivity from Generalized Sub-Gaussian porosity and particle size random fields","authors":"M. Harrison, M. Riva, M. Mousavi Nezhad, A. Guadagnini","doi":"10.1098/rspa.2023.0476","DOIUrl":null,"url":null,"abstract":"We derive analytical formulations relating the spatial covariance ( <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> </mml:math> </jats:inline-formula> ) of (log-transformed) hydraulic conductivities to auto- and cross-covariances of porosity ( <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ϕ</mml:mi> </mml:math> </jats:inline-formula> ) and representative soil particle sizes within the framework of the classical Terzaghi model. The latter provides an empirical relationship which is widely used to obtain conductivity estimates. We frame the study within recent stochastic approaches and conceptualize appropriate transformations of <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ϕ</mml:mi> </mml:math> </jats:inline-formula> and representative soil particle size as Generalized Sub-Gaussian (GSG) spatially cross-correlated random processes. Consistency of the theoretical framework against sample distributions of <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ϕ</mml:mi> </mml:math> </jats:inline-formula> and particle size is assessed through the analysis of field data. A perturbation-based approach yields workable expressions of <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> </mml:math> </jats:inline-formula> upon truncating the otherwise exact analytical solution at given orders of approximations. Our analytical (truncated) log-conductivity covariance is in agreement with its Monte Carlo-based counterpart. A Global Sensitivity Analysis relying on classical Sobol indices quantifies the relative importance of all parameters embedded in the formulation of <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> </mml:math> </jats:inline-formula> . We show that parameters driving the GSG nature of the distribution of (transformed) porosity are key to the main features of <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> </mml:math> </jats:inline-formula> . We also document the relevance of properly capturing emergences of possible cross-correlations between <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ϕ</mml:mi> </mml:math> </jats:inline-formula> and representative particle size to reconstruct conductivity fields.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"5 1","pages":""},"PeriodicalIF":2.9000,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","FirstCategoryId":"103","ListUrlMain":"https://doi.org/10.1098/rspa.2023.0476","RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}
引用次数: 0
Abstract
We derive analytical formulations relating the spatial covariance ( CY ) of (log-transformed) hydraulic conductivities to auto- and cross-covariances of porosity ( ϕ ) and representative soil particle sizes within the framework of the classical Terzaghi model. The latter provides an empirical relationship which is widely used to obtain conductivity estimates. We frame the study within recent stochastic approaches and conceptualize appropriate transformations of ϕ and representative soil particle size as Generalized Sub-Gaussian (GSG) spatially cross-correlated random processes. Consistency of the theoretical framework against sample distributions of ϕ and particle size is assessed through the analysis of field data. A perturbation-based approach yields workable expressions of CY upon truncating the otherwise exact analytical solution at given orders of approximations. Our analytical (truncated) log-conductivity covariance is in agreement with its Monte Carlo-based counterpart. A Global Sensitivity Analysis relying on classical Sobol indices quantifies the relative importance of all parameters embedded in the formulation of CY . We show that parameters driving the GSG nature of the distribution of (transformed) porosity are key to the main features of CY . We also document the relevance of properly capturing emergences of possible cross-correlations between ϕ and representative particle size to reconstruct conductivity fields.
在经典特尔扎吉模型的框架内,我们推导出了(对数变换)水力传导性的空间协方差(C Y)与孔隙度(j)和代表性土壤颗粒尺寸的自协方差和交叉协方差的相关分析公式。后者提供了一种经验关系,被广泛用于估算导流系数。我们将这项研究纳入最新的随机方法,并将 ϕ 和代表性土壤粒径的适当变换概念化为广义子高斯(GSG)空间交叉相关随机过程。通过分析实地数据,评估了理论框架与 ϕ 和粒径样本分布的一致性。采用基于扰动的方法,在给定近似阶数下截断原本精确的分析解,即可得到可行的 C Y 表达式。我们的分析(截断)对数电导协方差与基于蒙特卡洛的协方差一致。全局敏感性分析依赖于经典的索布尔指数,量化了 C Y 公式中所有参数的相对重要性。我们表明,驱动(转换)孔隙度分布 GSG 性质的参数是 C Y 主要特征的关键。我们还证明了正确捕捉 ϕ 与代表性粒度之间可能存在的交叉相关性对重建电导场的重要性。
期刊介绍:
Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.