Groundwater.

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Groundwater. Notes on geostatistics
Monica Riva, Alberto Guadagnini Politecnico di
Milano, Italy Key reference de Marsily, G.
(1986), Quantitative Hydrogeology. Academic
Press, New York, 440 pp
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Modelling flow and transport in heterogenous
media motivation and general idea
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Understanding the role of heterogeneity
Jan 2000 editorial "It's the Heterogeneity!
(Wood, W.W., Its the Heterogeneity!, Editorial,
Ground Water, 38(1), 1, 2000) heterogeneity of
chemical, biological, and flow conditions should
be a major concern in any remediation
scenario. Many in the groundwater community
either failed to "get" the message or were forced
by political considerations to provide rapid,
untested, site-specific active remediation
technology. "It's the heterogeneity," and it is
the Editor's guess that the natural system is so
complex that it will be many years before one can
effectively deal with heterogeneity on societally
important scales. Panel of experts
(DOE/RL-97-49, April 1997) As flow and transport
are poorly understood, previous and ongoing
computer modelling efforts are inadequate and
based on unrealistic and sometimes optimistic
assumptions, which render their output unreliable.
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Flow and Transport in Multiscale Fields
(conceptual)
?
Field laboratory-derive conductivities
dispersivities appear to vary continuously with
the scale of observation (conductivity support,
plume travel distance). Anomalous
transport. Recent theories attempt to link such
scale-dependence to multiscale structure of Y
ln K. Predict observed effect of domain size on
apparent variance and integral scale of
Y. Predict observed supra linear growth rate of
dispersivity with mean travel distance
(time). Major challenge develop more
powerful/general stochastic theories/models for
multiscale random media, and back them with
lab/field observation.
?
?
?
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Neuman S.P., On advective transport in fractal
permeability and velocity fields, Water Res.
Res., 31(6), 1455-1460, 1995.
Shed some light Conceptual difficulty Data
deduced by means of deterministic Fickian models
from laboratory and field tracer tests in a
variety of porous and fractured media, under
varied flow and transport regimes. Linear
regression aLa ? 0.017 s1.5 Supra-linear growth
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Natural Variability. Geostatistics revisited

• Introduction Few field findings about spatial
  variability
• Regionalized variables
• Interpolation methods
• Simulation methods

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AVRA VALLEY Clifton and Neuman, 1982 Clifton,
P.M., and S.P. Neuman, Effects of Kriging and
Inverse Modeling on Conditional Simulation of the
Avra Valley Aquifer in southern Arizona, Water
Resour. Res., 18(4), 1215-1234, 1982. Regional
Scale
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Columbus Air Force Adams and Gelhar,
1992 Aquifer Scale
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Mt. Simon aquifer Bakr, 1976 Local Scale
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• Summary Variability is present at all scales
• But, what happens if we ignore it? We will see in
  this class that this would lead to interpretation
  problems in both groundwater flow and solute
  transport phenomena
• Examples in transport - Scale effects in
  dispersion
• - New processes arising
• Heterogeneous parameters ALL (T, K, , S, v
  (q), BC, ...)
• Most relevant one T (2D), or K (3D), as they
  have been shown to vary orders of magnitude in an
  apparently homogeneous aquifer

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Variability in T and/or K
Summary of data from many different places in the
world. Careful though! Data are not always
obtained with rigorous procedures, and moreover,
as we will see throughout the course, data depend
on interpretation method and scale of
regularization Data given in terms of mean and
variance (dispersion around the mean value)
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Variability in T and/or K Almost always slnT (or
slnK ) lt 2 (and in most cases lt1) This can be
questioned, but OK by now Correlation scales
(very important concept later!!)
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• But, what is the correct treatment for natural
  heterogeneity?
• First of all, what do we know?
• - real data at (few) selected points
• - Statistical parameters
• - A huge uncertainty related to the lack of
  data in most part of the aquifer. If parameter
  continuous (of course they are), then the number
  of locations without data is infinity
• Note The value of K at any point DOES EXIST. The
  problem is we do not know it (we could if we
  measured it, but we could never be exhaustive
  anyway)
• Stochastic approach K at any given point is
  RANDOM, coming from a predefined (maybe known,
  maybe not) pdf, and spatially correlated ------
  REGIONALIZED VARIABLE

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Regionalized Variables

• T(x,?) is a Spatial Random Function iif
• If ? ?0 then T(x,?0) is a spatial function
  (continuity?, differentiability?)
• If x x0 then T(x0) (actually T(x0, ?)) is a
  random function
• Thus, as a random function, T(x0) has a
  univariate distribution (log-normal according to
  Law, 1944 Freeze, 1975)

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Hoeksema and Kitanidis, 1985
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Hoeksema Kitanidis, 1985 Log-T normal, log-K
normal Both consolidated and unconsolidated
deposits
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Now we look at T(x), so we are interested in the
multivariate distribution of T(x1), T(x2), ...
T(xn) Most frequent hypothesis Y(Y(x1),
Y(x2), ... Y(xn))(ln T(x1), ln T(x2), ... ln
T(xn)) Is multinormal with But most
important NO INDEPENDENCE
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What if independent? and then we are in
classical statistics But here we are not, so we
need some way to characterize dependency of one
variable at some point with the SAME variable at
a DIFFERENT point. This is the concept of the
SEMIVARIOGRAM (or VARIOGRAM)
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Classification of SRF

• Second order stationary
• EZ(x)const
• C(x, y) is not a function of location (only of
  separation distance, h)
• Particular case isotropic RSF C(h) C(h)
• Anisotropic covariance different correlation
  scales along different directions
• Most important property if multinormal
  distribution, first and second order moments are
  enough to fully characterize the SRF multivariate
  distribution

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Relaxing the stationary assumption
1. The assumption of second-order stationarity
with finite variance, C(0), might not be satisfied
(The experimental variance tends to increase
with domain size)
2. Less stringent assumption INTRINSIC HYPOTHESIS
The variance of the first-order increments is
finite AND these increments are themselves
second-order stationary. Very simple example
hydraulic heads ARE non intrinsic SRF
EY(x h) Y(x) m(h) varY(x h) Y(x)
?(h)
Independent of x only function of h
Usually m(h) 0 if not, just define a new
function, Y(x) m(x), which satisfies this
consition
Definition of variogram, ?(h)
EY(x h) Y(x) 0 ?(h) (1/2) varY(x h)
Y(x) (1/2) E(Y(x h) Y(x))2
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Variogram v. Covariance
1. The variogram is the mean quadratic increment
of Y between two points separated by h.
2. Compare the INTRINSIC HYPOTHESIS with
SECOND-ORDER STATIONARITY
EY(x) m constant ?(h) (1/2) E(Y(x h)
Y(x))2 (1/2) ( EY(x h)2
EY(x)2 2 m2 2 EY(x h) Y(x) 2 m2)
C(0) C(h)
variogram
covariance
h
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The variogram
The definition of the Semi-Variogram is usually
given by the following probabilistic formula
When dealing with real data the
semi-variogram is estimated by the Experimental
Semi-Variogram. For a given separation vector,
h, there is a set of observation pairs that are
approximately separated by this distance. Let the
number of pairs in this set be N(h). The
experimental semi-variogram is given by
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Some comments on the variogram
If Z(x) and Z(xh) are totally independent,
then If Z(x) and Z(xh) are totally
dependent, then One particular case is when x
xh. Therefore, by definition
In the stationary case
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Variogram Models

• DEFINITIONS
• Nugget
• Sill
• Range
• Integral distance or correlation scale

• Models
• Pure Nugget
• Spherical
• Exponential
• Gaussian
• Power

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• Correlation scales Larger in T than in K.
  Larger in horizontal than in vertical. Fraction
  of the domain of interest

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Additional comments

• Second order stationary
• EZ(x)constant
• ?(h) is not a function of location
• Particular case isotropic RSF ?(h) ?(h)
• Anisotropic variograms two types of anisotropy
  depending on correlation scale or sill value
• Important property ?(h) ?2 C(h)
• Most important property if multinormal
  distribution, first and second order moments are
  enough to fully characterize the SRF multivariate
  distribution

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Estimation vs. Simulation

• Problem Few data available, maybe we know mean,
  variance and variogram
• Alternatives
• (1) Estimation (interpolation) problems KRIGING
• Kriging BLUE
• Extremely smooth
• Many possible krigings Alternative cokriging

http//www-sst.unil.ch/research/variowin/
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The kriging equations - 1
We want to predict the value, Z(x0), at an
unsampled location, x0, using a weighted average
of the observed values at N neighboring
locations, Z(x1), Z(x2), ..., Z(xN). Let
Z(x0) represent the predicted value a weighted
average estimator be written as
The associated estimation error is
In general, we do not know the (constant) mean,
m, in the intrinsic hypothesis. We impose the
additional condition of equivalence between the
mathematical expectation of Z and Z0.
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The kriging equations - 2
Unknown mathematical expectation of the process Z.
This condition allows obtaining an unbiased
estimator.
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The kriging equations - 3
We wish to determine the set of weights. IMPOSE
the condition
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The kriging equations - 4
We then use the definition of variogram
THEN
Which I will use into
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The kriging equations - 5
By substitution
Noting that
We finally obtain
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The kriging equations - 6
This is a constrained optimization problem. To
solve it we use the method of Lagrange
Multipliers from the calculus of variation. The
Lagrangian objective function is
To minimize this we must take the partial
derivative of the Lagrangian with respect to each
of the weights and with respect to the Lagrange
multiplier, and set the resulting expressions
equal to zero, yielding a system of linear
equations
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The kriging equations - 7
Minimize this
and get (N1) linear equations with (N1)
unknowns
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The kriging equations - 8
The complete system can be written as A ? b
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The kriging equations - 9
We finally get the Variance of the Estimation
Error
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Estimation vs. Simulation (ii)

• (2) Simulations try to reproduce the look of
  the heterogeneous variable
• Important when extreme values are important
• Many (actually infinite) solutions, all of them
  equilikely (and with probability 0 to be
  correct)
• For each potential application we are interested
  in one or the other

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Estimation. 1
AVRA VALLEY. Regional Scale - Clifton, P.M., and
S.P. Neuman, Effects of Kriging and Inverse
Modeling on Conditional Simulation of the Avra
Valley Aquifer in southern Arizona, Water Resour.
Res., 18(4), 1215-1234, 1982.
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Estimation. 2
AVRA VALLEY. Regional Scale - Clifton, P.M., and
S.P. Neuman, Effects of Kriging and Inverse
Modeling on Conditional Simulation of the Avra
Valley Aquifer in southern Arizona, Water Resour.
Res., 18(4), 1215-1234, 1982.
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Estimation. 3
AVRA VALLEY. Regional Scale - Clifton, P.M., and
S.P. Neuman, Effects of Kriging and Inverse
Modeling on Conditional Simulation of the Avra
Valley Aquifer in southern Arizona, Water Resour.
Res., 18(4), 1215-1234, 1982.
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Estimation. 4
AVRA VALLEY. Regional Scale - Clifton, P.M., and
S.P. Neuman, Effects of Kriging and Inverse
Modeling on Conditional Simulation of the Avra
Valley Aquifer in southern Arizona, Water Resour.
Res., 18(4), 1215-1234, 1982.
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Estimation. 5
AVRA VALLEY. Regional Scale - Clifton, P.M., and
S.P. Neuman, Effects of Kriging and Inverse
Modeling on Conditional Simulation of the Avra
Valley Aquifer in southern Arizona, Water Resour.
Res., 18(4), 1215-1234, 1982.
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Monte Carlo approach
CONDITIONAL CROSS-CORRELATED FIELDS Y lnT

h1
Statistical CONDITIONAL moments, first and second
order
h2
. . .
. . .
h2000
2000 simulations
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NUMERICAL ANALYSIS - MONTE CARLO
? Evaluation of key statistics of medium
parameters (K, porosity, ) ? Synthetic
generation of an ensemble of equally likely
fields ? Solution of flow/transport problems on
each one of these ? Ensemble statistics
? Simple to understand ? Applicable to a wide
range of linear and nonlinear problems ? High
heterogeneities ? Conditioning
? Heavy calculations ? Fine computational grids ?
Reliable convergence criteria (?)
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Problems reliable assessment of convergence
Ballio and Guadagnini 2004
Hydraulic head variance
Number of Monte Carlo simulations