Science and Products
Algorithms for model parameter estimation and state estimation using the Kalman Filter for forecasting, filtering, and fixed-lag smoothing applied to a state-space model for one-dimensional vertical infiltration
The algorithms in this data release implement a State-Space Model (SSM) of vertical infiltration through the unsaturated zone and recharge to the water table. These algorithms build on previous investigations available at https://doi.org/10.1029/2020WR029110 and https://doi.org/10.1111/gwat.13206. The SSM is defined by observed states (i.e., the water-table altitude) and unobserved states (i.e., f
Algorithms for model parameter estimation and state estimation applied to a state-space model for one-dimensional vertical infiltration incorporating snowmelt rate as a system input
The algorithms and input data included in this data release are used to interpret time-series
data (water-table altitude, precipitation, snowmelt, and potential evapotranspiration) over an
observation period to estimate model parameters of a State-Space Model (SSM) of vertical
infiltration to the groundwater table. The SSM model is coupled with a Kalman Filter (KF)
to estimate system states (w
Algorithms for model parameter estimation, state estimation, and forecasting applied to a State-Space model coupled with the Kalman Filter for one-dimensional vertical infiltration to fractured rock aquifers
The algorithms and input data included in this data release are used to interpret time-series data (water- table altitude, precipitation, and potential evapotranspiration) over an observation period to estimate model parameters of a State-Space (SS) model of vertical infiltration to a fractured-rock aquifer. The SS model is coupled with a Kalman Filter (KF) to estimate system states (water-table a
Water Level Altitude in Bedrock Wells and Meteorological Data at the Masser Groundwater Recharge Site between February 1 and December 31, 1999
This product contains time-series data for groundwater level altitudes in bedrock boreholes and meteorological information from a site administered by the U.S. Department of Agriculture (USDA), Agricultural Research Service (ARS). The site, referred to as the Masser Groundwater Recharge Site, is located in the Valley and Ridge Physiographic Province of eastern-central Pennsylvania, USA, approximat
Concentrations of Chlorinated Ethene Compounds in Rock Core Collected from the Mudstone Underlying the former Naval Air Warfare Center, West Trenton, New Jersey
These data sets present results from the analyses of chlorinated ethene compounds in samples of rock core collected from boreholes in the mudstone underlying the former Naval Air Warfare Center, West Trenton, New Jersey.
Organic and total carbon analyses of rock core collected from boreholes 83BR, 84BR, 85BR, 86BR, 87BR, 88BR, and 89BR in the mudstone underlying the former Naval Air Warfare Center, West Trenton, New Jersey
These data sets present results for analyses of the fraction of total and organic carbon in samples of rock core collected from boreholes 83BR, 84BR, 85BR, 86BR, 87BR, 88BR, and 89BR in the Lockatong Formation (mudstone) underlying the former Naval Air Warfare Center, West Trenton, New Jersey.
Biogeochemical analyses of water samples collected in the mudstone aquifer underlying the Naval Air Warfare Center, West Trenton, NJ (2008-2013)
This data set presents results from the analyses of groundwater water samples collected from monitoring wells and monitoring intervals in bedrock wells in the mudstone aquifer underlying the former Naval Air Warfare Center (NAWC), West Trenton, NJ. The water samples were collected between 2008 and 2013 and were analyzed for field parameters, inorganic and organic constituents, and the abundances o
Data from Mercury Intrusion Porosimetry conducted on samples of a mudstone underlying the Naval Air Warfare Center, West Trenton, NJ
This data set presents the results of conducting Mercury Intrusion Porosimetry on core samples of the mudstone underlying the Naval Air Warfare Center, West Trenton, NJ. The core samples were selected from the continuous core collected from boreholes 83BR-89BR.
Lithologic characterization of cores from boreholes 83BR-89BR collected from the mudstone aquifer underlying the Naval Air Warfare Center, West Trenton, NJ
This data set presents the lithologic interpretation of rock cores from boreholes 83BR-89BR collected from the mudstone aquifer underlying the Naval Air Warfare Center (NAWC), West Trenton, NJ. Continuous core from these boreholes was collected and visually interpreted to identify characteristics of the depositional environment of the mudstone. Three types of mudstone were identified: a black-fis
Groundwater tracing experiments conducted in the mudstone aquifer underlying the former Naval Air Warfare Center, West Trenton, NJ (2007-2008)
This data set presents results from two groundwater tracing experiments conducted in the mudstone aquifer underlying the former Naval Air Warfare Center (NAWC), West Trenton, NJ. In each test, a bromide solution was introduced into a hydraulically isolated section of borehole 36BR (denoted as 36BR-A); the hydraulically isolated section of the borehole isolated specific bedding plane parting fractu
Filter Total Items: 68
Benefits and cautions in data assimilation strategies: An example of modeling groundwater recharge
Assimilating recent observations improves model outcomes for real-time assessments of groundwater processes. This is demonstrated in estimating time-varying recharge to a shallow fractured-rock aquifer in response to precipitation. Results from estimating the time-varying water-table altitude (h) and recharge, and their error covariances, are compared for forecasting, filtering, and fixed-lag smoo
Authors
Allen M. Shapiro, Frederick Day-Lewis
Mapping areas of groundwater susceptible to transient contamination events from rapid infiltration into shallow fractured-rock aquifers in agricultural regions of the conterminous United States
Current time-invariant groundwater vulnerability assessments may not capture intermittent contamination events in landscape areas that experience rapid infiltration following precipitation or snowmelt. Occurrences of rapid infiltration and intermittent degradation of groundwater quality are frequently reported in fractured-rock aquifers. This investigation identifies landscape areas underlain by f
Authors
Allen M. Shapiro, James A. Falcone
Application of recursive estimation to heat tracing for groundwater/surface-water exchange
We present and demonstrate a recursive-estimation framework to infer groundwater/surface-water exchange based on temperature time series collected at different vertical depths below the sediment/water interface. We formulate the heat-transport problem as a state-space model (SSM), in which the spatial derivatives in the convection/conduction equation are approximated using finite differences. The
Authors
W. Anderson McAliley, Frederick Day-Lewis, David Rey, Martin Briggs, Allen M. Shapiro, Dale Werkema
Incorporating snowmelt into daily estimates of recharge using a state-space model of infiltration
A state-space model (SSM) of infiltration estimates daily groundwater recharge using time-series of groundwater-level altitude and meteorological inputs (liquid precipitation, snowmelt, and evapotranspiration). The model includes diffuse and preferential flow through the unsaturated zone, where preferential flow is a function of liquid precipitation and snowmelt rates and a threshold rate, above w
Authors
Allen M. Shapiro, Frederick Day-Lewis, William M. Kappel, John H. Williams
Reframing groundwater hydrology as a data-driven science
No abstract available.
Authors
Allen M. Shapiro, F.D. Day-Lewis
Contributing areas to domestic wells in dipping sedimentary rocks under extreme recharge events
We use particle tracking to determine contributing areas (CAs) to wells for transient flow models that simulate cyclic domestic pumping and extreme recharge events in a small synthetic watershed underlain by dipping sedimentary rocks. The CAs consist of strike-oriented bands at locations where the water table intersects high-hydraulic conductivity beds, and from which groundwater flows to the pump
Authors
Claire R. Tiedeman, Allen M. Shapiro
Estimating and forecasting time-varying groundwater recharge in fractured rock: A state-space formulation with preferential and diffuse flow to the water table
Rapid infiltration following precipitation may result in groundwater contamination from surface contaminants or pathogens. In fractured rock, contaminants can migrate rapidly to points of groundwater withdrawals. In contrast to the temporal availability of groundwater quality chemical indicators, meteorological and groundwater level observations are available in real-time to estimate time-varying
Authors
Allen M. Shapiro, Frederick Day-Lewis
The complex spatial distribution of trichloroethene and the probability of NAPL occurrence in the rock matrix of a mudstone aquifer
Methanol extractions for chloroethene analyses are conducted on rock samples from seven closely spaced coreholes in a mudstone aquifer that was subject to releases of the nonaqueous phase liquid (NAPL) form of trichloroethene (TCE) between the 1950's and 1990's. Although TCE concentration in the rock matrix over the length of coreholes is dictated by proximity to subhorizontal bedding planefractur
Authors
Allen M. Shapiro, Daniel J. Goode, Thomas E. Imbrigiotta, Michelle M. Lorah, Claire R. Tiedeman
Variability of organic carbon content and the retention and release of trichloroethene in the rock matrix of a mudstone aquifer
Contaminants diffusing from fractures into the immobile porosity of the rock matrix are subject to prolonged residence times. Organic contaminants can adsorb onto organic carbonaceous materials in the matrix extending contaminant retention. An investigation of spatial variability of the fraction of organic carbon (foc) is conducted on samples of rock core from seven closely spaced boreholes in a m
Authors
Allen M. Shapiro, Rebecca J. Brenneis
Bioremediation in fractured rock: 1. Modeling to inform design, monitoring, and expectations
Field characterization of a trichloroethene (TCE) source area in fractured mudstones produced a detailed understanding of the geology, contaminant distribution in fractures and the rock matrix, and hydraulic and transport properties. Groundwater flow and chemical transport modeling that synthesized the field characterization information proved critical for designing bioremediation of the source ar
Authors
Claire R. Tiedeman, Allen M. Shapiro, Paul A. Hsieh, Thomas E. Imbrigiotta, Daniel J. Goode, Pierre Lacombe, Mary F. DeFlaun, Scott R. Drew, Carole D. Johnson, John H. Williams, Gary P. Curtis
Bioremediation in fractured rock: 2. Mobilization of chloroethene compounds from the rock matrix
A mass balance is formulated to evaluate the mobilization of chlorinated ethene compounds (CE) from the rock matrix of a fractured mudstone aquifer under pre- and postbioremediation conditions. The analysis relies on a sparse number of monitoring locations and is constrained by a detailed description of the groundwater flow regime. Groundwater flow modeling developed under the site characterizatio
Authors
Allen M. Shapiro, Claire R. Tiedeman, Thomas E. Imbrigiotta, Daniel J. Goode, Paul A. Hsieh, Pierre Lacombe, Mary F. DeFlaun, Scott R. Drew, Gary P. Curtis
Porosity and pore size distribution in a sedimentary rock: Implications for the distribution of chlorinated solvents
Characterizing properties of the rock matrix that control retention and release of chlorinated solvents is essential in evaluating the extent of contamination and the application of remediation technologies in fractured rock. Core samples from seven closely spaced boreholes in a mudstone subject to trichloroethene (TCE) contamination were analyzed using Mercury Intrusion Porosimetry to investigate
Authors
Allen M. Shapiro, Chrsitopher E. Evans, Erin C. Hayes
Non-USGS Publications**
Pinder, G. F. and Shapiro, A. 1982. Physics of Flow in Geothermal Systems, in Recent Trends in Hydrogeology. ed. T. N. Narasimhan. Geological Society of America, Boulder, CO. p. 25-30. https://doi.org/10.1130/SPE189-p25.
Pinder, G. F. and Shapiro, A. 1979. A new collocation method for the solution of the convection-dominated transport equation. Water Resources Research 15(5): 1177-1182. https://doi.org/10.1029/WR015i005p01177.
Pinder, G. F. and Shapiro, A. 1980. Reply to comment on "A new collocation method for the solution of the convection-dominated transport equation". Water Resources Research 16(6): 1137. https://doi.org/10.1029/WR016i006p01137.
Shapiro, A. and Pinder, G. F. 1981. Analysis of an upstream weighted collocation approximation to the transport equation. Journal of Computational Physics 39(1): 46-71. https://doi.org/10.1016/0021-9991(81)90136-4.
Andersson, J. and Shapiro, A. M. 1983. Stochastic analysis of one-dimensional steady state unsaturated flow: A Comparison of Monte Carlo and Perturbation Methods. Water Resources Research 19(1): 121-133. 10.1029/WR019i001p00121.
Shapiro, A. M. and Andersson, J. 1983. Steady state fluid response in fractured rock: A boundary element solution for a coupled, discrete fracture continuum model. Water Resources Research 19(4): 959-969. 10.1029/WR019i004p00959.
Andersson, J., Shapiro, A. M. and Bear, J. 1984. A Stochastic Model of a Fractured Rock Conditioned by Measured Information. Water Resources Research 20(1): 79-88. 10.1029/WR020i001p00079.
Bear, J. and Shapiro, A. M. 1984. On the shape of the non-steady interface intersecting discontinuities in permeability. Advances in Water Resources 7(3): 106-112. https://doi.org/10.1016/0309-1708(84)90037-X.
Bear, J., Shamir, U., Gamliel, A. and Shapiro, A. M. 1985. Motion of the seawater interface in a coastal aquifer by the method of successive steady states. Journal of Hydrology 76(1): 119-132. https://doi.org/10.1016/0022-1694(85)90093-9.
**Disclaimer: The views expressed in Non-USGS publications are those of the author and do not represent the views of the USGS, Department of the Interior, or the U.S. Government.
Science and Products
- Data
Algorithms for model parameter estimation and state estimation using the Kalman Filter for forecasting, filtering, and fixed-lag smoothing applied to a state-space model for one-dimensional vertical infiltration
The algorithms in this data release implement a State-Space Model (SSM) of vertical infiltration through the unsaturated zone and recharge to the water table. These algorithms build on previous investigations available at https://doi.org/10.1029/2020WR029110 and https://doi.org/10.1111/gwat.13206. The SSM is defined by observed states (i.e., the water-table altitude) and unobserved states (i.e., fAlgorithms for model parameter estimation and state estimation applied to a state-space model for one-dimensional vertical infiltration incorporating snowmelt rate as a system input
The algorithms and input data included in this data release are used to interpret time-series data (water-table altitude, precipitation, snowmelt, and potential evapotranspiration) over an observation period to estimate model parameters of a State-Space Model (SSM) of vertical infiltration to the groundwater table. The SSM model is coupled with a Kalman Filter (KF) to estimate system states (wAlgorithms for model parameter estimation, state estimation, and forecasting applied to a State-Space model coupled with the Kalman Filter for one-dimensional vertical infiltration to fractured rock aquifers
The algorithms and input data included in this data release are used to interpret time-series data (water- table altitude, precipitation, and potential evapotranspiration) over an observation period to estimate model parameters of a State-Space (SS) model of vertical infiltration to a fractured-rock aquifer. The SS model is coupled with a Kalman Filter (KF) to estimate system states (water-table a
Water Level Altitude in Bedrock Wells and Meteorological Data at the Masser Groundwater Recharge Site between February 1 and December 31, 1999
This product contains time-series data for groundwater level altitudes in bedrock boreholes and meteorological information from a site administered by the U.S. Department of Agriculture (USDA), Agricultural Research Service (ARS). The site, referred to as the Masser Groundwater Recharge Site, is located in the Valley and Ridge Physiographic Province of eastern-central Pennsylvania, USA, approximatConcentrations of Chlorinated Ethene Compounds in Rock Core Collected from the Mudstone Underlying the former Naval Air Warfare Center, West Trenton, New Jersey
These data sets present results from the analyses of chlorinated ethene compounds in samples of rock core collected from boreholes in the mudstone underlying the former Naval Air Warfare Center, West Trenton, New Jersey.Organic and total carbon analyses of rock core collected from boreholes 83BR, 84BR, 85BR, 86BR, 87BR, 88BR, and 89BR in the mudstone underlying the former Naval Air Warfare Center, West Trenton, New Jersey
These data sets present results for analyses of the fraction of total and organic carbon in samples of rock core collected from boreholes 83BR, 84BR, 85BR, 86BR, 87BR, 88BR, and 89BR in the Lockatong Formation (mudstone) underlying the former Naval Air Warfare Center, West Trenton, New Jersey.Biogeochemical analyses of water samples collected in the mudstone aquifer underlying the Naval Air Warfare Center, West Trenton, NJ (2008-2013)
This data set presents results from the analyses of groundwater water samples collected from monitoring wells and monitoring intervals in bedrock wells in the mudstone aquifer underlying the former Naval Air Warfare Center (NAWC), West Trenton, NJ. The water samples were collected between 2008 and 2013 and were analyzed for field parameters, inorganic and organic constituents, and the abundances oData from Mercury Intrusion Porosimetry conducted on samples of a mudstone underlying the Naval Air Warfare Center, West Trenton, NJ
This data set presents the results of conducting Mercury Intrusion Porosimetry on core samples of the mudstone underlying the Naval Air Warfare Center, West Trenton, NJ. The core samples were selected from the continuous core collected from boreholes 83BR-89BR.Lithologic characterization of cores from boreholes 83BR-89BR collected from the mudstone aquifer underlying the Naval Air Warfare Center, West Trenton, NJ
This data set presents the lithologic interpretation of rock cores from boreholes 83BR-89BR collected from the mudstone aquifer underlying the Naval Air Warfare Center (NAWC), West Trenton, NJ. Continuous core from these boreholes was collected and visually interpreted to identify characteristics of the depositional environment of the mudstone. Three types of mudstone were identified: a black-fisGroundwater tracing experiments conducted in the mudstone aquifer underlying the former Naval Air Warfare Center, West Trenton, NJ (2007-2008)
This data set presents results from two groundwater tracing experiments conducted in the mudstone aquifer underlying the former Naval Air Warfare Center (NAWC), West Trenton, NJ. In each test, a bromide solution was introduced into a hydraulically isolated section of borehole 36BR (denoted as 36BR-A); the hydraulically isolated section of the borehole isolated specific bedding plane parting fractu - Publications
Filter Total Items: 68
Benefits and cautions in data assimilation strategies: An example of modeling groundwater recharge
Assimilating recent observations improves model outcomes for real-time assessments of groundwater processes. This is demonstrated in estimating time-varying recharge to a shallow fractured-rock aquifer in response to precipitation. Results from estimating the time-varying water-table altitude (h) and recharge, and their error covariances, are compared for forecasting, filtering, and fixed-lag smooAuthorsAllen M. Shapiro, Frederick Day-LewisMapping areas of groundwater susceptible to transient contamination events from rapid infiltration into shallow fractured-rock aquifers in agricultural regions of the conterminous United States
Current time-invariant groundwater vulnerability assessments may not capture intermittent contamination events in landscape areas that experience rapid infiltration following precipitation or snowmelt. Occurrences of rapid infiltration and intermittent degradation of groundwater quality are frequently reported in fractured-rock aquifers. This investigation identifies landscape areas underlain by fAuthorsAllen M. Shapiro, James A. FalconeApplication of recursive estimation to heat tracing for groundwater/surface-water exchange
We present and demonstrate a recursive-estimation framework to infer groundwater/surface-water exchange based on temperature time series collected at different vertical depths below the sediment/water interface. We formulate the heat-transport problem as a state-space model (SSM), in which the spatial derivatives in the convection/conduction equation are approximated using finite differences. TheAuthorsW. Anderson McAliley, Frederick Day-Lewis, David Rey, Martin Briggs, Allen M. Shapiro, Dale WerkemaIncorporating snowmelt into daily estimates of recharge using a state-space model of infiltration
A state-space model (SSM) of infiltration estimates daily groundwater recharge using time-series of groundwater-level altitude and meteorological inputs (liquid precipitation, snowmelt, and evapotranspiration). The model includes diffuse and preferential flow through the unsaturated zone, where preferential flow is a function of liquid precipitation and snowmelt rates and a threshold rate, above wAuthorsAllen M. Shapiro, Frederick Day-Lewis, William M. Kappel, John H. WilliamsReframing groundwater hydrology as a data-driven science
No abstract available.AuthorsAllen M. Shapiro, F.D. Day-LewisContributing areas to domestic wells in dipping sedimentary rocks under extreme recharge events
We use particle tracking to determine contributing areas (CAs) to wells for transient flow models that simulate cyclic domestic pumping and extreme recharge events in a small synthetic watershed underlain by dipping sedimentary rocks. The CAs consist of strike-oriented bands at locations where the water table intersects high-hydraulic conductivity beds, and from which groundwater flows to the pumpAuthorsClaire R. Tiedeman, Allen M. ShapiroEstimating and forecasting time-varying groundwater recharge in fractured rock: A state-space formulation with preferential and diffuse flow to the water table
Rapid infiltration following precipitation may result in groundwater contamination from surface contaminants or pathogens. In fractured rock, contaminants can migrate rapidly to points of groundwater withdrawals. In contrast to the temporal availability of groundwater quality chemical indicators, meteorological and groundwater level observations are available in real-time to estimate time-varyingAuthorsAllen M. Shapiro, Frederick Day-LewisThe complex spatial distribution of trichloroethene and the probability of NAPL occurrence in the rock matrix of a mudstone aquifer
Methanol extractions for chloroethene analyses are conducted on rock samples from seven closely spaced coreholes in a mudstone aquifer that was subject to releases of the nonaqueous phase liquid (NAPL) form of trichloroethene (TCE) between the 1950's and 1990's. Although TCE concentration in the rock matrix over the length of coreholes is dictated by proximity to subhorizontal bedding planefracturAuthorsAllen M. Shapiro, Daniel J. Goode, Thomas E. Imbrigiotta, Michelle M. Lorah, Claire R. TiedemanVariability of organic carbon content and the retention and release of trichloroethene in the rock matrix of a mudstone aquifer
Contaminants diffusing from fractures into the immobile porosity of the rock matrix are subject to prolonged residence times. Organic contaminants can adsorb onto organic carbonaceous materials in the matrix extending contaminant retention. An investigation of spatial variability of the fraction of organic carbon (foc) is conducted on samples of rock core from seven closely spaced boreholes in a mAuthorsAllen M. Shapiro, Rebecca J. BrenneisBioremediation in fractured rock: 1. Modeling to inform design, monitoring, and expectations
Field characterization of a trichloroethene (TCE) source area in fractured mudstones produced a detailed understanding of the geology, contaminant distribution in fractures and the rock matrix, and hydraulic and transport properties. Groundwater flow and chemical transport modeling that synthesized the field characterization information proved critical for designing bioremediation of the source arAuthorsClaire R. Tiedeman, Allen M. Shapiro, Paul A. Hsieh, Thomas E. Imbrigiotta, Daniel J. Goode, Pierre Lacombe, Mary F. DeFlaun, Scott R. Drew, Carole D. Johnson, John H. Williams, Gary P. CurtisBioremediation in fractured rock: 2. Mobilization of chloroethene compounds from the rock matrix
A mass balance is formulated to evaluate the mobilization of chlorinated ethene compounds (CE) from the rock matrix of a fractured mudstone aquifer under pre- and postbioremediation conditions. The analysis relies on a sparse number of monitoring locations and is constrained by a detailed description of the groundwater flow regime. Groundwater flow modeling developed under the site characterizatioAuthorsAllen M. Shapiro, Claire R. Tiedeman, Thomas E. Imbrigiotta, Daniel J. Goode, Paul A. Hsieh, Pierre Lacombe, Mary F. DeFlaun, Scott R. Drew, Gary P. CurtisPorosity and pore size distribution in a sedimentary rock: Implications for the distribution of chlorinated solvents
Characterizing properties of the rock matrix that control retention and release of chlorinated solvents is essential in evaluating the extent of contamination and the application of remediation technologies in fractured rock. Core samples from seven closely spaced boreholes in a mudstone subject to trichloroethene (TCE) contamination were analyzed using Mercury Intrusion Porosimetry to investigateAuthorsAllen M. Shapiro, Chrsitopher E. Evans, Erin C. HayesNon-USGS Publications**
Pinder, G. F. and Shapiro, A. 1982. Physics of Flow in Geothermal Systems, in Recent Trends in Hydrogeology. ed. T. N. Narasimhan. Geological Society of America, Boulder, CO. p. 25-30. https://doi.org/10.1130/SPE189-p25.Pinder, G. F. and Shapiro, A. 1979. A new collocation method for the solution of the convection-dominated transport equation. Water Resources Research 15(5): 1177-1182. https://doi.org/10.1029/WR015i005p01177.Pinder, G. F. and Shapiro, A. 1980. Reply to comment on "A new collocation method for the solution of the convection-dominated transport equation". Water Resources Research 16(6): 1137. https://doi.org/10.1029/WR016i006p01137.Shapiro, A. and Pinder, G. F. 1981. Analysis of an upstream weighted collocation approximation to the transport equation. Journal of Computational Physics 39(1): 46-71. https://doi.org/10.1016/0021-9991(81)90136-4.Andersson, J. and Shapiro, A. M. 1983. Stochastic analysis of one-dimensional steady state unsaturated flow: A Comparison of Monte Carlo and Perturbation Methods. Water Resources Research 19(1): 121-133. 10.1029/WR019i001p00121.Shapiro, A. M. and Andersson, J. 1983. Steady state fluid response in fractured rock: A boundary element solution for a coupled, discrete fracture continuum model. Water Resources Research 19(4): 959-969. 10.1029/WR019i004p00959.Andersson, J., Shapiro, A. M. and Bear, J. 1984. A Stochastic Model of a Fractured Rock Conditioned by Measured Information. Water Resources Research 20(1): 79-88. 10.1029/WR020i001p00079.Bear, J. and Shapiro, A. M. 1984. On the shape of the non-steady interface intersecting discontinuities in permeability. Advances in Water Resources 7(3): 106-112. https://doi.org/10.1016/0309-1708(84)90037-X.Bear, J., Shamir, U., Gamliel, A. and Shapiro, A. M. 1985. Motion of the seawater interface in a coastal aquifer by the method of successive steady states. Journal of Hydrology 76(1): 119-132. https://doi.org/10.1016/0022-1694(85)90093-9.**Disclaimer: The views expressed in Non-USGS publications are those of the author and do not represent the views of the USGS, Department of the Interior, or the U.S. Government.