An explicit one-dimensional consolidation solution with semi-permeable drainage boundary for unsaturated soil

Xudong Zhao, Charles W.W. Ng, Shuai Zhang, Junjun Ni, Chao Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

13 Citations (Scopus)


Existing solutions for analyzing one-dimensional (1-D) consolidation of unsaturated soil are only derived to cater to two extreme drainage conditions (fully drained and undrained). This study presents a new explicit solution for 1-D consolidation of unsaturated soil with semi-permeable drainage boundary. Based on the assumptions of two independent stress variables and the governing equations proposed by Fredlund, the eigenfunction expansion method is adopted to develop an explicit analytical solution to calculate excess pore-water and pore-air pressures in an unsaturated soil when it is subjected to external loads. The developed general solutions are expressed in terms of depth, z, and time, t. For the semi-permeable drainage boundary, eigenvalues and eigenfunctions in the space domain are developed. The technique of Laplace transform is used to solve the coupled ordinary differential equations in the time domain. The newly derived explicit solution is verified with the existing semi-analytical method in the literature, and an excellent agreement is obtained. Compared with the semi-analytical solution, the newly derived analytical solution is more straightforward and explicit so that this solution is relatively easier to be implemented into a computer program to carry out a preliminary assessment of 1-D consolidation of unsaturated soil.

Original languageEnglish
Pages (from-to)2241-2253
Number of pages13
JournalInternational Journal for Numerical and Analytical Methods in Geomechanics
Issue number16
Publication statusPublished - 1 Nov 2020


  • 1-D consolidation
  • eigenfunction expansion method
  • explicit analytical solution
  • semi-permeable boundary
  • unsaturated soil

ASJC Scopus subject areas

  • Computational Mechanics
  • Materials Science(all)
  • Geotechnical Engineering and Engineering Geology
  • Mechanics of Materials


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