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Determining Spatial Distributions of Permeability

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Abstract

Conventional laboratory experimentation provides an apparent value of hydraulic permeability which, at best, is representative of the entire sample. Nuclear magnetic resonance imaging (MRI) provides unique opportunities to probe spatial distributions of permeability at a much finer scale (Seto et al., Transp Porous Media 42:351–388, 2001). We advance the methodology for determining spatial distributions of permeability and provide, for the first time, laboratory determinations of permeability distributions with complete three-dimensional (3D) spatial resolution. We investigate new experimental designs that mitigate a possible lack of identifiability and provide for more accurate estimates of permeability. We demonstrate the application of MRI experiments and analyses that provide substantial improvements in the determination of the porosity distribution, an essential step for obtaining reliable measurements of spatially resolved velocity distributions. We investigate the use of global optimization to solve the associated inverse problem for determining permeability distributions from the measured velocity distributions. Our methodology is demonstrated with experimental data on sandstone and trabecular bone samples.

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Abbreviations

A c :

Cross-sectional area

\({B_i^m}\) :

ith B-spline basis function with the order of m

c :

B-spline coefficient

c :

Column vector of B-spline coefficients

F :

Velocity values calculated from mathematical model

G :

Gram matrix

g :

Gravity

H :

Matrix composed of Gram matrix elements consisting regularization term

J :

Performance index

K :

Permeability tensor

k(z):

Permeability at position z

k app :

Apparent permeability

k :

Wave vector for spin position

L :

Length of sample

N id :

Dimension of the spline in z id direction

N v :

Number of imaging voxels

n :

Normal vector

P :

Normalized displacement distribution function

p :

Pressure

Q :

Volumetric flow rate

q :

Wave vector for spin displacement

S :

NMR signal intensity

t :

Time

U :

Uniform random number

V :

Volume of the sample

v app :

Apparent velocity

v :

Darcy velocity

W :

Weighting matrix

x :

Extended partition of B-spline basis function

Y :

Velocity values measured by experiments

Z :

Position displacement

z :

Position vector

s :

Position vector

\({\fancyscript{T}}\) :

“Temperature” parameter for simulated annealing algorithm

z id :

Position parallel to the dimension id

Γ:

Index representing a set of flow experiments

Δ :

Observation time

δ :

Pulse width of pulsed-gradient-field

λ :

Regularization parameter

μ :

Viscosity

ρ :

Bulk fluid density

ζ:

Joint displacement density (or velocity) function

\({\phi}\) :

Porosity

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Authors and Affiliations

  1. Department of Chemical and Biological Engineering, Colorado State University, Fort Collins, CO, 80523, USA

    Jinsoo Uh & A. Ted Watson

  2. Advanced Imaging Research Center, University of Texas Southwestern Medical Center, Dallas, TX, 75390, USA

    Jinsoo Uh

Authors
  1. Jinsoo Uh
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  2. A. Ted Watson
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Correspondence to Jinsoo Uh.

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Uh, J., Watson, A.T. Determining Spatial Distributions of Permeability. Transp Porous Med 86, 385–414 (2011). https://doi.org/10.1007/s11242-010-9627-3

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