Theoretical analysis of non‐ G aussian heterogeneity effects on subsurface flow and transport
Abstract
Abstract Much of the stochastic groundwater literature is devoted to the analysis of flow and transport in Gaussian or multi‐Gaussian log hydraulic conductivity (or transmissivity) fields, ( x being a position vector), characterized by one or (less frequently) a multiplicity of spatial correlation scales. Yet Y and many other variables and their (spatial or temporal) increments, Δ Y , are known to be generally non‐Gaussian. One common manifestation of non‐Gaussianity is that whereas frequency distributions of Y often exhibit mild peaks and light tails, those of increments are generally symmetric with peaks that grow sharper, and tails that become heavier, as separation scale or lag between pairs of Y values decreases. A statistical model that captures these disparate, scale‐dependent distributions of Y and in a unified and consistent manner has been recently proposed by us. This new “generalized sub‐Gaussian (GSG)” model has the form where is (generally, but not necessarily) a multiscale Gaussian random field and is a nonnegative subordinator independent of G . The purpose of this paper is to explore analytically, in an elementary manner, lead‐order effects that non‐Gaussian heterogeneity described by the GSG model have on the stochastic description of flow and transport. Recognizing that perturbation expansionmore »
- Authors:
-
- Dipartimento di Ingegneria Civile e Ambientale Politecnico di Milano Milano Italy, Department of Hydrology and Atmospheric Sciences University of Arizona Tucson Arizona USA
- Department of Hydrology and Atmospheric Sciences University of Arizona Tucson Arizona USA
- Publication Date:
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1402144
- Resource Type:
- Publisher's Accepted Manuscript
- Journal Name:
- Water Resources Research
- Additional Journal Information:
- Journal Name: Water Resources Research Journal Volume: 53 Journal Issue: 4; Journal ID: ISSN 0043-1397
- Publisher:
- American Geophysical Union (AGU)
- Country of Publication:
- United States
- Language:
- English
Citation Formats
Riva, Monica, Guadagnini, Alberto, and Neuman, Shlomo P. Theoretical analysis of non‐ G aussian heterogeneity effects on subsurface flow and transport. United States: N. p., 2017.
Web. doi:10.1002/2016WR019353.
Riva, Monica, Guadagnini, Alberto, & Neuman, Shlomo P. Theoretical analysis of non‐ G aussian heterogeneity effects on subsurface flow and transport. United States. https://doi.org/10.1002/2016WR019353
Riva, Monica, Guadagnini, Alberto, and Neuman, Shlomo P. Tue .
"Theoretical analysis of non‐ G aussian heterogeneity effects on subsurface flow and transport". United States. https://doi.org/10.1002/2016WR019353.
@article{osti_1402144,
title = {Theoretical analysis of non‐ G aussian heterogeneity effects on subsurface flow and transport},
author = {Riva, Monica and Guadagnini, Alberto and Neuman, Shlomo P.},
abstractNote = {Abstract Much of the stochastic groundwater literature is devoted to the analysis of flow and transport in Gaussian or multi‐Gaussian log hydraulic conductivity (or transmissivity) fields, ( x being a position vector), characterized by one or (less frequently) a multiplicity of spatial correlation scales. Yet Y and many other variables and their (spatial or temporal) increments, Δ Y , are known to be generally non‐Gaussian. One common manifestation of non‐Gaussianity is that whereas frequency distributions of Y often exhibit mild peaks and light tails, those of increments are generally symmetric with peaks that grow sharper, and tails that become heavier, as separation scale or lag between pairs of Y values decreases. A statistical model that captures these disparate, scale‐dependent distributions of Y and in a unified and consistent manner has been recently proposed by us. This new “generalized sub‐Gaussian (GSG)” model has the form where is (generally, but not necessarily) a multiscale Gaussian random field and is a nonnegative subordinator independent of G . The purpose of this paper is to explore analytically, in an elementary manner, lead‐order effects that non‐Gaussian heterogeneity described by the GSG model have on the stochastic description of flow and transport. Recognizing that perturbation expansion of hydraulic conductivity diverges when Y is sub‐Gaussian, we render the expansion convergent by truncating Y 's domain of definition. We then demonstrate theoretically and illustrate by way of numerical examples that, as the domain of truncation expands, (a) the variance of truncated Y (denoted by Y t ) approaches that of Y and (b) the pdf (and thereby moments) of Y t increments approach those of Y increments and, as a consequence, the variogram of approaches that of . This in turn guarantees that perturbing to second order in (the standard deviation of Y t ) yields results which approach those we obtain upon perturbing to second order in even as the corresponding series diverges. Our analysis is rendered mathematically tractable by considering mean‐uniform steady state flow in an unbounded, two‐dimensional domain of mildly heterogeneous Y with a single‐scale function G having an isotropic exponential covariance. Results consist of expressions for (a) lead‐order autocovariance and cross‐covariance functions of hydraulic head, velocity, and advective particle displacement and (b) analogues of preasymptotic as well as asymptotic Fickian dispersion coefficients. We compare these theoretically and graphically with corresponding expressions developed in the literature for Gaussian Y . We find the former to differ from the latter by a factor k = ( denoting ensemble expectation) and the GSG covariance of longitudinal velocity to contain an additional nugget term depending on this same factor. In the limit as Y becomes Gaussian, k reduces to one and the nugget term drops out.},
doi = {10.1002/2016WR019353},
journal = {Water Resources Research},
number = 4,
volume = 53,
place = {United States},
year = {Tue Apr 11 00:00:00 EDT 2017},
month = {Tue Apr 11 00:00:00 EDT 2017}
}
https://doi.org/10.1002/2016WR019353
Works referenced in this record:
Anisotropic statistical scaling of soil and sediment texture in a stratified deep vadose zone near Maricopa, Arizona
journal, February 2014
- Guadagnini, A.; Neuman, S. P.; Schaap, M. G.
- Geoderma, Vol. 214-215
Inverse sequential simulation: A new approach for the characterization of hydraulic conductivities demonstrated on a non‐ G aussian field
journal, April 2015
- Xu, Teng; Gómez‐Hernández, J. Jaime
- Water Resources Research, Vol. 51, Issue 4
Best practice stochastic facies modeling from a channel-fill turbidite sandstone analog (the Quarry outcrop, Eocene Ainsa basin, northeast Spain)
journal, July 2006
- Falivene, Oriol; Arbues, Pau; Gardiner, Andy
- AAPG Bulletin, Vol. 90, Issue 7
Higher Order Models using Entropy, Markov Random Fields and Sequential Simulation
book, January 2005
- Daly, Colin
- Geostatistics Banff 2004
Comparison of three geostatistical methods for hydrofacies simulation: a test on alluvial sediments
journal, December 2011
- dell’Arciprete, Diana; Bersezio, Riccardo; Felletti, Fabrizio
- Hydrogeology Journal, Vol. 20, Issue 2
Scaling of random fields by means of truncated power variograms and associated spectra
journal, May 1997
- Di Federico, Vittorio; Neuman, Shlomo P.
- Water Resources Research, Vol. 33, Issue 5
Anisotropy, lacunarity, and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms
journal, October 1999
- Di Federico, Vittorio; Neuman, Shlomo P.; Tartakovsky, Daniel M.
- Water Resources Research, Vol. 35, Issue 10
A Comparison of Methods for the Stochastic Simulation of Rock Fractures
journal, September 2007
- Dowd, Peter A.; Xu, Chaoshui; Mardia, Kanti V.
- Mathematical Geology, Vol. 39, Issue 7
River and floodplain process simulation for subsurface characterization
journal, September 1998
- Gross, Lori Juergens; Small, Mitchell J.
- Water Resources Research, Vol. 34, Issue 9
Numerical investigation of apparent multifractality of samples from processes subordinated to truncated fBm: NUMERICAL INVESTIGATION OF APPARENT MULTIFRACTALITY
journal, December 2011
- Guadagnini, Alberto; Neuman, Shlomo P.; Riva, Monica
- Hydrological Processes, Vol. 26, Issue 19
Sub-Gaussian model of processes with heavy-tailed distributions applied to air permeabilities of fractured tuff
journal, March 2012
- Riva, Monica; Neuman, Shlomo P.; Guadagnini, Alberto
- Stochastic Environmental Research and Risk Assessment, Vol. 27, Issue 1
The Pros and Cons of the Truncated Gaussian Method
book, January 1994
- Galli, A.; Beucher, H.; Le Loc’h, G.
- Geostatistical Simulations
Three-dimensional high resolution fluvio-glacial aquifer analog – Part 2: Geostatistical modeling
journal, July 2011
- Comunian, A.; Renard, P.; Straubhaar, J.
- Journal of Hydrology, Vol. 405, Issue 1-2
A strategy to overcome adverse effects of autoregressive updating of streamflow forecasts
journal, January 2015
- Li, M.; Wang, Q. J.; Bennett, J. C.
- Hydrology and Earth System Sciences, Vol. 19, Issue 1
The Direct Sampling method to perform multiple-point geostatistical simulations: PERFORMING MULTIPLE-POINTS SIMULATIONS
journal, November 2010
- Mariethoz, Gregoire; Renard, Philippe; Straubhaar, Julien
- Water Resources Research, Vol. 46, Issue 11
Recent Advances in Statistical and Scaling Analysis of Earth and Environmental Variables
book, January 2013
- Neuman, Shlomo P.; Guadagnini, Alberto; Riva, Monica
- Advances in Hydrogeology
Hierarchical object-based stochastic modeling of fluvial reservoirs
journal, October 1996
- Deutsch, Clayton V.; Wang, Libing
- Mathematical Geology, Vol. 28, Issue 7
Filter-Based Classification of Training Image Patterns for Spatial Simulation
journal, January 2006
- Zhang, Tuanfeng; Switzer, Paul; Journel, Andre
- Mathematical Geology, Vol. 38, Issue 1
Flow and Transport in Bimodal Heterogeneous Formations
journal, October 1995
- Rubin, Yoram
- Water Resources Research, Vol. 31, Issue 10
Anisotropic statistical scaling of vadose zone hydraulic property estimates near Maricopa, Arizona: STATISTICAL SCALING OF VADOSE ZONE HYDRAULIC PROPERTIES
journal, December 2013
- Guadagnini, A.; Neuman, S. P.; Schaap, M. G.
- Water Resources Research, Vol. 49, Issue 12
Using the Snesim program for multiple-point statistical simulation
journal, December 2006
- Liu, Yuhong
- Computers & Geosciences, Vol. 32, Issue 10
Truncated Plurigaussian Simulations to Characterize Aquifer Heterogeneity
journal, January 2009
- Mariethoz, Grégoire; Renard, Philippe; Cornaton, Fabien
- Ground Water, Vol. 47, Issue 1
Evidence for Non-Gaussian Scaling Behavior in Heterogeneous Sedimentary Formations
journal, May 1996
- Painter, Scott
- Water Resources Research, Vol. 32, Issue 5
Extended power-law scaling of air permeabilities measured on a block of tuff
journal, January 2012
- Siena, M.; Guadagnini, A.; Riva, M.
- Hydrology and Earth System Sciences, Vol. 16, Issue 1
Facies Modeling Using a Markov Mesh Model Specification
journal, July 2011
- Stien, Marita; Kolbjørnsen, Odd
- Mathematical Geosciences, Vol. 43, Issue 6
Numerical modeling of macrodispersion in heterogeneous media: a comparison of multi-Gaussian and non-multi-Gaussian models
journal, March 1998
- Wen, Xian-Huan; Gómez-Hernández, J. Jaime
- Journal of Contaminant Hydrology, Vol. 30, Issue 1-2
Updating multipoint simulations using the ensemble Kalman filter
journal, February 2013
- Hu, L. Y.; Zhao, Y.; Liu, Y.
- Computers & Geosciences, Vol. 51
Solute transport in heterogeneous porous formations
journal, August 1984
- Dagan, Gedeon
- Journal of Fluid Mechanics, Vol. 145, Issue -1
To be or not to be multi-Gaussian? A reflection on stochastic hydrogeology
journal, February 1998
- Gómez-Hernández, J. Jaime; Wen, Xian-Huan
- Advances in Water Resources, Vol. 21, Issue 1
Theory and generation of conditional, scalable sub-Gaussian random fields: CONDITIONAL SCALABLE SUB-GAUSSIAN RANDOM FIELDS
journal, March 2016
- Panzeri, M.; Riva, M.; Guadagnini, A.
- Water Resources Research, Vol. 52, Issue 3
Using Multiple Grids in Markov Mesh Facies Modeling
journal, November 2013
- Kolbjørnsen, Odd; Stien, Marita; Kjønsberg, Heidi
- Mathematical Geosciences, Vol. 46, Issue 2
Transition probability-based indicator geostatistics
journal, May 1996
- Carle, Steven F.; Fogg, Graham E.
- Mathematical Geology, Vol. 28, Issue 4
FLUVSIM: a program for object-based stochastic modeling of fluvial depositional systems
journal, May 2002
- Deutsch, C. V.; Tran, T. T.
- Computers & Geosciences, Vol. 28, Issue 4
Simulation and analysis of scalable non-Gaussian statistically anisotropic random functions
journal, December 2015
- Riva, Monica; Panzeri, Marco; Guadagnini, Alberto
- Journal of Hydrology, Vol. 531
Transport in heterogeneous sediments with multimodal conductivity and hierarchical organization across scales
journal, July 2004
- Dai, Zhenxue; Ritzi, Robert W.; Huang, Chaocheng
- Journal of Hydrology, Vol. 294, Issue 1-3
Correction to “Stochastic modeling of macrodispersion in heterogeneous porous media” by Yoram Rubin
journal, October 1990
- Rubin, Yoram
- Water Resources Research, Vol. 26, Issue 10
A SGeMS code for pattern simulation of continuous and categorical variables: FILTERSIM
journal, December 2008
- Wu, Jianbing; Boucher, Alexandre; Zhang, Tuanfeng
- Computers & Geosciences, Vol. 34, Issue 12
A simulation model of alluvial stratigraphy
journal, October 1979
- Bridge, John S.; Leeder, Michael R.
- Sedimentology, Vol. 26, Issue 5
Comment on “Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations” by Scott Painter
journal, April 1997
- Liu, Hui Hai; Molz, Fred J.
- Water Resources Research, Vol. 33, Issue 4
Stochastic modeling of groundwater flow by unconditional and conditional probabilities: 2. The solute transport
journal, August 1982
- Dagan, G.
- Water Resources Research, Vol. 18, Issue 4
Flow in multiscale log conductivity fields with truncated power variograms
journal, May 1998
- Di Federico, Vittorio; Neuman, Shlomo P.
- Water Resources Research, Vol. 34, Issue 5
Conditional Simulation with Patterns
journal, April 2007
- Arpat, G. Burc; Caers, Jef
- Mathematical Geology, Vol. 39, Issue 2
Uncertainty assessment and data worth in groundwater flow and mass transport modeling using a blocking Markov chain Monte Carlo method
journal, January 2009
- Fu, Jianlin; Jaime Gómez-Hernández, J.
- Journal of Hydrology, Vol. 364, Issue 3-4
Relationship between juxtaposed, overlapping, and fractal representations of multimodal spatial variability: RELATIONSHIP BETWEEN REPRESENTATIONS OF VARIABILITY
journal, August 2003
- Neuman, Shlomo P.
- Water Resources Research, Vol. 39, Issue 8
Integration of Markov mesh models and data assimilation techniques in complex reservoirs
journal, November 2015
- Panzeri, M.; Della Rossa, E. L.; Dovera, L.
- Computational Geosciences, Vol. 20, Issue 3
Statistical scaling of pore-scale Lagrangian velocities in natural porous media
journal, August 2014
- Siena, M.; Guadagnini, A.; Riva, M.
- Physical Review E, Vol. 90, Issue 2
Effects of non-Gaussian copula-based hydraulic conductivity fields on macrodispersion: EFFECTS OF SPATIAL DEPENDENCE OF K ON MACRODISPERSION
journal, July 2012
- Haslauer, C. P.; Guthke, P.; Bárdossy, A.
- Water Resources Research, Vol. 48, Issue 7
Impact of two geostatistical hydro-facies simulation strategies on head statistics under non-uniform groundwater flow
journal, January 2014
- Perulero Serrano, Raul; Guadagnini, Laura; Riva, Monica
- Journal of Hydrology, Vol. 508
New scaling model for variables and increments with heavy-tailed distributions: SCALING MODEL FOR VARIABLES WITH HEAVY-TAILED DISTRIBUTIONS
journal, June 2015
- Riva, Monica; Neuman, Shlomo P.; Guadagnini, Alberto
- Water Resources Research, Vol. 51, Issue 6
Transport in multiscale log conductivity fields with truncated power variograms
journal, May 1998
- Di Federico, Vittorio; Neuman, Shlomo P.
- Water Resources Research, Vol. 34, Issue 5
Anisotropic Scaling of Berea Sandstone Log Air Permeability Statistics
journal, January 2013
- Riva, Monica; Neuman, Shlomo P.; Guadagnini, Alberto
- Vadose Zone Journal, Vol. 12, Issue 3
Stochastic modeling of macrodispersion in heterogeneous porous media
journal, January 1990
- Rubin, Yoram
- Water Resources Research, Vol. 26, Issue 1