Random field models for hydraulic conductivity
in ground water flow
Special Session on Random Fields and Long Range Dependence
AMS Regional Meeting
Michigan State University
15 March 2015
Mark M. Meerschaert
Department of Statistics and Probability
Michigan State University
mcubed@stt.msu.edu
http://www.stt.msu.edu/users/mcubed
Partially supported by NSF grant EAR-1344280.
Abstract
A standard model in ground water hydrology uses random fields
to interpolate sparse data on hydraulic conductivity. The resulting random field is used to parameterize a partial differential
equation model for flow and transport, which is then solved by
numerical methods. This talk will review the state of the art in
such models, and discuss open problems.
Acknowledgments
Boris Baeumer, Maths & Stats, U Otago, Dunedin, New Zealand
David A. Benson, Geology & Geo Eng, Colorado School of Mines
Hermine Biermé, MAP5 Université René Descartes, Paris
Geoffrey Bohling, Kansas Geological Survey, Lawrence, Kansas
Mine Dogan, Geology and Geophysics, University of Wyoming
David Hyndman, Geological Sciences, Michigan State University
Tomasz J. Kozubowski, Math and Stat, Univ. of Nevada, Reno
Chae Young Lim, Statistics and Probability, Michigan State U
Silong Lu, Tetra Tech, Inc., Atlanta, GA
Fred J. Molz, Environmental Eng & Science, Clemson University
Farzad Sabzikar, Statistics and Probability, Michigan State U
Hans-Peter Scheffler, Mathematics, Uni Siegen, Germany
Remke Van Dam, Institute for Future Environments, QUT
Equations for groundwater flow and transport
Step 1: Measure hydraulic conductivity K (x), a tensor field that
codes how easily water can flow through a porous medium.
Step 2: Compute the hydraulic head (pressure) h(x, t) by numerically solving the groundwater flow equation
∂h(x, t)
= a∇ · K (x)∇h(x, t) + f (x, t),
∂t
where f (x, t) is a source/sink term.
Step 3: Compute the velocity field v (x, t) using Darcy’s Law
v (x, t) = −bK (x)∇h(x, t).
Groundwater flow and transport (continued)
Step 4: Compute solute concentration C(x, t) by numerically
solving the advection-dispersion equation
∂C(x, t)
= −∇ · v (x, t)C(x, t) + ∇ · D (x, t)∇C(x, t) + F (x, t),
∂t
where F (x, t) is a source/sink term and D (x, t) is the dispersivity
tensor (normal covariance matrix).
Typical simplifying assumptions:
1. Steady state hydraulic head h(x)
2. Steady state velocity field v (x)
3. Scalar dispersion D (x) = λkv (x)kI
4. Scalar hydraulic conductivity K (x) = K(x)I
5. Gaussian ln K field with isotropic exponential correlation
Data requirements
Consider a bounded (rectangular) domain in R2 or R3
Establish (or assume) boundary conditions for hydraulic head h
Measure hydraulic conductivity K at some points (expensive!)
Extrapolate K field to all (> 106) grid points in the domain
Typically, we have just a few vertical columns of K data
Each vertical column contains 100 to 500 data points
Data source: The MADE site
MAcroDispersion Experimental Site (MADE) in Columbus MS.
300
LEGEND
MADE−2 test boundary
MADE−3 source trench
N
2D GPR lines
3D GPR cubes
HRK profiles
200
northing [m]
ICA cube
100
121108A
111108C
MLS cube
111108B
111108A
0
0
100
easting [m]
200
Hydraulic conductivity data distribution
probability density
Histogram of measured ln K data (one column, n = 561).
0.6
(a)
0.4
0.2
0
−6
−4
−2
0
ln (K [m/day])
2
Hydraulic conductivity data correlation
Autocorrelation function for one column of ln K data (n = 561).
ACF
1
(a)
0.5
0
d= 0
0
0.2
0.4
0.6
distance [m]
0.8
1
Fractional difference (fractal filter)
Apply a fractional difference filter to the data Xn = ln K(x, y, zn)
to obtain an uncorrelated sequence
Zn =
∞
X
j=0
(−1)j
!
d
Xn−j
j
where the fractional binomial coefficients
!
Γ(d + 1)
d
=
j
j!Γ(j − d + 1)
for some d > 0. Then Xn is a fractionally integrated white noise.
If d = 1 then Zn = Xn − Xn−1, so that Xn = Z1 + · · · + Zn.
If Zn is Gaussian and 0.5 < d < 1.5, then Xn is an fBm with
Hurst index H = d − 0.5 sampled at equally spaced points.
Filtering out the data correlation
A fractional difference with d = 0.9 removes the correlation.
ACF
1
(b)
0.5
0
d= 0.9
0
0.2
0.4
0.6
distance [m]
0.8
1
Distribution of the filtered data
probability density
A fractional difference with d = 0.9 reveals that the underlying
distribution has heavier tails and a sharper peak than the best
fitting Gaussian.
3
(b)
2
1
0
−1.5
−1
−0.5
0
0.5
fractionally differenced ln K
1
1.5
Our model for ln K
We simulate an anisotropic Gaussian ln K field in each layer,
conditional on the observed data (solid black lines) and sample
one column (white dotted line) for model validation.
Simulated column of ln K data
probability density
The simulated ln K data in a single column (white dotted line)
is a Gaussian mixture, resembling the ln K data.
3
(d)
2
1
0
−1.5
−1
−0.5
0
0.5
fractionally differenced ln K
1
1.5
Operator scaling random field model
We simulate an anisotropic Gaussian random field
Xψ (x) = Re
Z
eix·k − 1 ψ(k)−H−q/2 B(dk).

d
X
Rd
with 0 < H < 1, q = a1 + · · · + ad,
ϕ(k) = 
j=1
1/2
Dj |k · θj |2/aj 
and B(dk) is independently scattered complex-valued Gaussian.
Then each xi 7→ Xψ (x1, . . . , xd) is an fBm with Hi = H/ai.
d
Operator scaling: Xψ (cE x) = cH Xψ (x) where E = diag(a1, . . . , ad).
Here cE = exp(E ln c) where exp(A) = I + A + A2/2! + · · ·
Fitting to the K data
Ground penetrating radar identifies the layer boundaries.
Simulate ln K field with H = 0.4, θ1 = (1, 0)′, θ2 = (0, 1)′,
D1 = 10, D2 = 1, a1 = a2 = 1 (mild anisotropy).
Adjust ln K field in each layer to observed mean and variance.
Condition on the observed ln K data by random bridging: Apply
kriging to the data and the realization, then subtract.
The resulting correlation in the x direction is consistent with the
data (4 m spacing), insufficient to judge H1 6= H2.
Simulation method (spectral synthesis)
Define
X̄ψ (xℓ) =
X
eixℓ·kj ψ(kj )−H−q/2 B(∆kj ).
j
where the kj are evenly spaced grid points in the small subrectangles ∆kj of a large annular rectangle in Rd.
Simulate B(∆kj ), iid as Z1 + iZ2, Z1, Z2 ∼ N (0, σ 2|∆kj |/2).
Then X̄ψ (xℓ) is the discrete FT of ψ(kj )−H−q/2 B(∆kj ).
h
i
Use FFT to compute, then Xψ (xℓ) ≈ Re X̄ψ (xℓ) − X̄ψ (0) .
In dimension d = 1 this is the spectral synthesis method.
Parameter estimation
Given spatial data wℓ on a regular grid of points xℓ ∈ Rd
Assume wℓ = X̄ψ (xℓ) =
P
ixℓ·kj ψ(k )−H−q/2 B(∆k )
j
j
je
Invert the discrete FT to get yj = ψ(kj )−H−q/2 B(∆kj )
Solve the nonlinear regression log |yj | = log |ψ(kj )−H−q/2|+log |εj |
Here εj = B(∆kj ) are iid N (0, σ 2|∆kj |)
Note E(log |εj |) → −∞ as ∆k → 0.
Consistent, asymptotically normal estimates of H1, H2, θ1, θ2, D1, D2.
Open question
We can also define the anisotropic random field
X ϕ (x ) =
Z
y ∈Rd
h
i
H−q/2
H−q/2
ϕ(x − y )
− ϕ(0 − y )
B(dy )
where B(dy ) is an independently scattered Gaussian random
measure, and ϕ(cE x) = c ϕ(x) is an operator scaling filter.
Idea: ψ(k) is the FT of ϕ(x), B(dk) is the FT of B(dy ).
Again xi 7→ Xϕ(x1, . . . , xd) is an fBm with Hi = H/ai.
d
Again Xϕ(cE x) = cH Xϕ(x) where E = diag(a1, . . . , ad).
Are these two processes the same? [Yes when d = 1 or isotropic.]
Problem: Explicit computation of the covariance function?
Open problems
• Infinitely divisible noise B(dk)
• tempered power law (Matérn) filters in Rd
• Simulation methods with conditioning
• Sample path properties [Xiao]
• Operator scaling vector fields [Didier et al.]
References
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and the Effect on Solute Mixing and Dispersion, Water Resources Research, Vol. 42
(2006), No. 1, W01415 (18 pp.), doi:10.1029/2004WR003755.
2. H. Biermé, M.M. Meerschaert, H.P. Scheffler, Operator Scaling Stable Random Fields,
Stochastic Processes and their Applications, Vol. 117 (2007), pp. 312–332.
3. G. Didier and V. Pipiras, Integral representations and properties of operator fractional
Brownian motions, Bernoulli, 17(1), pp. 1–33 (2011).
4. Mine Dogan, Remke L. Van Dam, Gaisheng Liu, Mark M. Meerschaert, James J. Butler
Jr., Geoffrey C. Bohling, David A. Benson, and David W. Hyndman, Predicting flow
and transport in highly heterogeneous alluvial aquifers, Geophysical Research Letters,
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doi:10.1016/j.jmva.2008.10.002.
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velocity, and the order of the fractional dispersion derivative in a highly heterogeneous
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He, Xinya Li, and Luanjing Guo, Examining the Influence of Heterogeneous Porosity
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scaling random fields, Journal of Multivariate Analysis, Vol. 123 (2014), pp. 172–183,
DOI: 10.1016/j.jmva.2013.09.010.
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pp. 1–4, doi:10.1029/2003GL019320.
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Hydraulic Conductivity Fields: Gaussian or Not? Water Resources Research, Vol. 49
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continuity for anisotropic Gaussian random fields, Transactions of the American Mathematical Society, Vol. 365 (2013), No. 2, pp. 1081–1107.
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