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Analytical Solution for One‐Dimensional Steady‐ and Transient‐State Flow in Vertical Heterogeneous Unsaturated Soils.

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Title: Analytical Solution for One‐Dimensional Steady‐ and Transient‐State Flow in Vertical Heterogeneous Unsaturated Soils.
Authors: Liu, Shuang1 (AUTHOR), McCartney, John S.2 (AUTHOR), Xiao, Yang1 (AUTHOR) xiaoy@cqu.edu.cn
Source: Water Resources Research. Dec2025, Vol. 61 Issue 12, p1-22. 22p.
Subjects: One-dimensional flow, Hydraulic conductivity, Equilibrium, Waterlogging (Soils), Engineering models, Heterogeneity, Slope stability
Abstract: Exact solutions for one‐dimensional steady‐state and transient liquid flow toward a water table in heterogeneous unsaturated soils are critical in predicting saturation profiles in several real‐world applications including interpretation of climate change effects on the subsurface and impacts on slope stability. In this study, vertical heterogeneity in saturated hydraulic conductivity with depth is characterized by an exponential decay function. A steady‐state solution is derived based on Darcy's law and the water table depth, and two transient‐state solutions are obtained from Richards' equation using the Laplace transform and the modified Bessel equation under common upper boundary conditions, that is, flow rate and pressure head, following the initial steady‐state flow condition and a water table at a specified depth. The transient and steady‐state solutions are compared with numerical solutions obtained from a multi‐layered configuration and an analytical solution for homogeneous soils, demonstrating their reliability and efficacy. Various hypothetical heterogeneous soils with differing parameters are employed to assess the flow behaviors, illustrating that vertical heterogeneity impacts the pressure head profiles. More pronounced heterogeneous soils exhibit lower pressure head and effective saturation during flow compared to homogeneous soils, which depend on the air‐entry value and the water table depth, irrespective of the upper boundary conditions. The solution has been employed to predict volumetric water content profiles in the Loess Plateau of China and evaluate the effect of infiltration on the stability of a shallow landslide in Japan based on the infinite slope model and the suction stress concept. Key Points: Exact solutions are derived for 1D flow in heterogeneous unsaturated soils, considering depth‐dependent saturated hydraulic conductivityThe novel flow analytical solutions can be employed to assess shallow landslide mobilization, nitrate transport, and other surficial issuesThe analytical solution is a reliable benchmark for numerical models that handle seepage problems while considering spatial variability [ABSTRACT FROM AUTHOR]
Copyright of Water Resources Research is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 190516128
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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Analytical Solution for One‐Dimensional Steady‐ and Transient‐State Flow in Vertical Heterogeneous Unsaturated Soils.
– Name: Author
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  Data: <searchLink fieldCode="AR" term="%22Liu%2C+Shuang%22">Liu, Shuang</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22McCartney%2C+John+S%2E%22">McCartney, John S.</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Xiao%2C+Yang%22">Xiao, Yang</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> xiaoy@cqu.edu.cn</i>
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  Data: <searchLink fieldCode="JN" term="%22Water+Resources+Research%22">Water Resources Research</searchLink>. Dec2025, Vol. 61 Issue 12, p1-22. 22p.
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  Data: <searchLink fieldCode="DE" term="%22One-dimensional+flow%22">One-dimensional flow</searchLink><br /><searchLink fieldCode="DE" term="%22Hydraulic+conductivity%22">Hydraulic conductivity</searchLink><br /><searchLink fieldCode="DE" term="%22Equilibrium%22">Equilibrium</searchLink><br /><searchLink fieldCode="DE" term="%22Waterlogging+%28Soils%29%22">Waterlogging (Soils)</searchLink><br /><searchLink fieldCode="DE" term="%22Engineering+models%22">Engineering models</searchLink><br /><searchLink fieldCode="DE" term="%22Heterogeneity%22">Heterogeneity</searchLink><br /><searchLink fieldCode="DE" term="%22Slope+stability%22">Slope stability</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Exact solutions for one‐dimensional steady‐state and transient liquid flow toward a water table in heterogeneous unsaturated soils are critical in predicting saturation profiles in several real‐world applications including interpretation of climate change effects on the subsurface and impacts on slope stability. In this study, vertical heterogeneity in saturated hydraulic conductivity with depth is characterized by an exponential decay function. A steady‐state solution is derived based on Darcy's law and the water table depth, and two transient‐state solutions are obtained from Richards' equation using the Laplace transform and the modified Bessel equation under common upper boundary conditions, that is, flow rate and pressure head, following the initial steady‐state flow condition and a water table at a specified depth. The transient and steady‐state solutions are compared with numerical solutions obtained from a multi‐layered configuration and an analytical solution for homogeneous soils, demonstrating their reliability and efficacy. Various hypothetical heterogeneous soils with differing parameters are employed to assess the flow behaviors, illustrating that vertical heterogeneity impacts the pressure head profiles. More pronounced heterogeneous soils exhibit lower pressure head and effective saturation during flow compared to homogeneous soils, which depend on the air‐entry value and the water table depth, irrespective of the upper boundary conditions. The solution has been employed to predict volumetric water content profiles in the Loess Plateau of China and evaluate the effect of infiltration on the stability of a shallow landslide in Japan based on the infinite slope model and the suction stress concept. Key Points: Exact solutions are derived for 1D flow in heterogeneous unsaturated soils, considering depth‐dependent saturated hydraulic conductivityThe novel flow analytical solutions can be employed to assess shallow landslide mobilization, nitrate transport, and other surficial issuesThe analytical solution is a reliable benchmark for numerical models that handle seepage problems while considering spatial variability [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Water Resources Research is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1029/2024WR039104
    Languages:
      – Code: eng
        Text: English
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      Pagination:
        PageCount: 22
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    Subjects:
      – SubjectFull: One-dimensional flow
        Type: general
      – SubjectFull: Hydraulic conductivity
        Type: general
      – SubjectFull: Equilibrium
        Type: general
      – SubjectFull: Waterlogging (Soils)
        Type: general
      – SubjectFull: Engineering models
        Type: general
      – SubjectFull: Heterogeneity
        Type: general
      – SubjectFull: Slope stability
        Type: general
    Titles:
      – TitleFull: Analytical Solution for One‐Dimensional Steady‐ and Transient‐State Flow in Vertical Heterogeneous Unsaturated Soils.
        Type: main
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          Name:
            NameFull: Liu, Shuang
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            NameFull: McCartney, John S.
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            NameFull: Xiao, Yang
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          Dates:
            – D: 01
              M: 12
              Text: Dec2025
              Type: published
              Y: 2025
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            – TitleFull: Water Resources Research
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