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Research Papers

Poromechanics Solutions to Plane Strain and Axisymmetric Mandel-Type Problems in Dual-Porosity and Dual-Permeability Medium

[+] Author and Article Information
Vinh X. Nguyen

Mewbourne School of Petroleum and Geological Engineering, PoroMechanics Institute, University of Oklahoma, Sarkeys Energy Center, Suite P119, 100 East Boyd Street, Norman, OK 73019nxvinh@ou.edu

Younane N. Abousleiman1

Mewbourne School of Petroleum and Geological Engineering, ConocoPhillips School of Geology and Geophysics, School of Civil Engineering and Environmental Science, PoroMechanics Institute, University of Oklahoma, Sarkeys Energy Center, Suite P119, 100 East Boyd Street, Norman, Oklahoma 73019yabousle@ou.edu

1

Corresponding author.

J. Appl. Mech 77(1), 011002 (Sep 23, 2009) (18 pages) doi:10.1115/1.3172146 History: Received September 05, 2008; Revised May 24, 2009; Published September 23, 2009

The two-dimensional Mandel-type problem geometry is well-known to bio-geomechanicians for testing rocks, cartilages, and bones with solutions in Cartesian coordinates for rectangular specimens or polar coordinates for cylindrical and disk samples. To date, all existing solutions are only applicable to single-porosity and single-permeability models, which could fall short when the porous material exhibits multiporosity and/or multipermeability characteristics, such as secondary porosity or fracture. This paper extends the plane strain and axisymmetric Mandel-type solutions from single-to dual-porosity and dual-permeability poromechanics. The solutions are presented in explicit analytical forms and account for arbitrary time-dependent external loading conditions, e.g., cyclic and ramping. The derived analytical solutions and results exhibit general behaviors characterized by two time scales. Stresses, pore pressures, and displacements are plotted for various time scale ratios to illustrate the interplaying effects of permeability and stiffness contrast of both porous regions, in addition to the interporosity exchange, on the overall responses of the system. Also, examples with realistic loading conditions for laboratory testing or field simulation such as cyclic and ramping are provided to demonstrate the engineering applications of the presented dual-poroelastic formulation and solutions.

Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

The Mandel’s problem geometry and setup for rectangular strip specimens

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Figure 2

The axisymmetric Mandel-type problem geometry and setup for cylindrical or circular disk samples

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Figure 3

Pore-pressure profile for strip geometry at various times

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Figure 4

Pore-pressure profile for cylindrical geometry at various times

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Figure 5

Vertical stress profile for strip geometry at various times

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Figure 6

Vertical stress profile for cylindrical geometry at various times

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Figure 7

Pore-pressure histories in the center of the rectangular strip

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Figure 8

Pore-pressure histories in the center of the cylinder

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Figure 9

Vertical displacement histories at the top of the sample for both strip and cylindrical geometries

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Figure 10

Lateral displacement histories on the outside of the sample for both strip and cylindrical geometries

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Figure 11

Vertical stress histories at x=0 and x=a for rectangular strip

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Figure 12

Axial stress histories at r=0 and r=R for cylinder

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Figure 13

Pore-pressure histories at the cylinder’s center r=0 for different interporosity flow coefficients while keeping constant permeability contrast kII/kI=105 and stiffness ratio KI/KII=50

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Figure 14

Vertical displacement histories at the top of the cylinder for different interporosity flow coefficients while keeping constant permeability contrast kII/kI=105 and stiffness ratio KI/KII=50

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Figure 15

Pore-pressure histories at the cylinder’s center r=0 for different permeability ratios while keeping constant stiffness contrast KI/KII=50 and interporosity flow Γ=1.67×10−7

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Figure 16

Vertical displacement histories at the top of the cylinder for different permeability ratios while keeping constant stiffness contrast KI/KII=50 and interporosity flow Γ=1.67×10−7

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Figure 17

Pore-pressure histories at the cylinder’s center r=0 for different bulk modulus ratios while keeping constant permeability contrast kII/kI=105 and interporosity flow Γ=1.67×10−7

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Figure 18

Vertical displacement histories at the top of the cylinder for different bulk modulus ratios while keeping constant permeability contrast kII/kI=105 and interporosity flow Γ=1.67×10−7

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Figure 19

Pore-pressures histories at the cylinder’s center r=0 under cyclic loading

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Figure 20

Pore-pressures histories at the cylinder’s center r=0 under different linear ramp loading rates

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Figure 21

Pore-pressure fluctuations at the cylinder’s center (r=0) through times under combined cyclic and linear ramp loadings (the cyclic loading period is T=2 s and ramping characteristic time is to=10 s)

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