Dynamic Data Integration and Stochastic Inversion of a Two-Dimensional Confined Aquifer
Dongdong Wang1, dwang12@uwyo.edu; Ye Zhang1, yzhang9@uwyo.edu; Juraj Irsa1
1Department of Geology & Geophysics, University of Wyoming, 1000 E. University Avenue, Laramie, WY 82071, USA
Ground Truth Model
With Sampling Wells
No flow
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experimental variogram
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The modified matrix, while improving the computation performance
(Fig. 5), also ensures that the estimated k is correct (Table 1).
Table 1 The estimated conductivity difference
between original coefficient matrix and modified coefficient matrix
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Note:
variogram model(nested model) fitted
1. Each presenter is provided with a 4-foot-high by 6-foot-wide poster board. Poster boards have a 2.5 cm (1-inch)
frame. Dimensions of the useable work area are 1.2 meters high by 1.8 meters wide (4′ high x 6′ wide).
Fig. 2 Experimental
based on 12 sampling wells
in horizontal
(left) the
and vertical
(right) directions.
2. variograms
The presentation
must
cover
material
as cited in the abstract.
2. Sequential Indicator
Simulation
3. Place
the(SIS)
title of your paper and your paper number prominently at the top of the poster board to allow
Following Chambers etviewers
al. (2000), this
study encoded
the true
hydrofacies
by 1 (sandy
facies)abstract’s
and
to identify
your
paper.
Indicate
1) the
presentation number, 2) title, and 3) authors’ names.
0 (clay facies). Based on the indicator histogram and variogram models, SIS is conducted using the
Highlight
GSLIB (Duetsch 4.
and Journel
2000 ). the authors’ names, e-mails, and address information in case the viewer is interested in contacting
Columns
Columns
Columns
3. Simulated Annealingyou for more information.
Fig. 3 Stochastic Simulation results via SIS based on the sandy facies (white) and the clay facies (black). Three of the sixty SIS
5. Prepare
all diagrams
charts
neatly
and legibly
in a size sufficient to be read at a distance of 2
The concept of compactness
in topology
is adopted or
to design
simulated
annealing
algorithm beforehand
realizations and the corresponding inverted models are presented here, including simulated annealing results and coarsening results.
applicable to our problem.
meters. Paragraph and figure caption text should be ATDynamic
LEAST 24-point
font (0.9 cm height) and headers AT
Data Integration Results
Dynamic Data Integration
LEAST 36 point font (1.2 cm height). Use creativity by using
font sizes
styles,
perhaps
even
The 60different
coarsened realizations
(Fig. and
3, bottom
row) are
inverted based
on color.
600 heads and120 fluxes sample
the 12contained
wells. Hydraulic
k of the
hydrofacies
(black
zones(e.g.,
in Fig. 3) is estimated. The
Based on the inversion
proposed by colors
Irsa and Zhang
(2012), stochastic inversionfor
haseach
been line form
6. method
Use different
and textures/symbols
or bar
inconductivity
your graph
orclay
chart.
A serif
font
k of the sandy facies (white zones in Fig. 3) is obtained by a prior information equation, i.e., kclay/ksand ratio
developed. Since this research tried to address large-scale problem, computation efficiency is key. To
easiersolver
for reading
text,(Paige
andanda non-serif
font
(e.g.,
Arial
or Helvetica)
headers
and
figure
is assumed
known.
Both
the estimated
conductivityfor
distribution
and the
inverted
hydraulic head boundary
efficiently solve inversionTimes)
equations,isaoften
linear iterative
was utilized,main
i.e., LSQR
condition were compared to the true values, i.e., the solution of Fig. 1. (Fig. 4)
Saunder 1982). Coefficient
matrix is modified to improve computation performance, including
labels.
(a)
(b)
isomorphism, linear transform and inverse equation simplification.
7. Organize the paper on the poster board so it is clear, orderly, and self-explanatory. You have complete freedom
Approximation Function:
in displaying your information in figures, tables, text, photographs, etc.
8. Use squares, rectangles, circles, etc., to group similar ideas. Avoid cluttering your poster with too much text.
Label different elements as I, II, III; or 1, 2, 3; or A, B, C, making it easier for a viewer to follow your display.
9. Include the background of your research followed by results and conclusions. A successful poster presentation
depends on
how well you convey information to an interested audience.
Simplified
10. Please do not laminate your poster to ensure that it can
be recycled.
(c)
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true head boundary
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estimated head boundary
of a given realization
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Frequency
Continuity Equations:
Relative Esimate Error Percentage
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h: hydraulic head x, z: Cartesian coordinates ai: approximated parameter
k: hydraulic conductivity
δ: weight for continuity equations
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K1 Recovery (True K1 = 1.0)
1.05
1.1
Difference
1.8E-5
5.05E-5
9.5E-5
1.499E-4
1.2605E-4
1.2824E-4
2.974E-2
2.980E-2
CONCLUSIONS
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1. The estimated k was accurate within ±10% of the true values;
for boundary condition estimation, the accuracy was within ±
15% of the true values.
2. The inverse method was proven applicable for integrating static
and dynamic data within a stochastic framework.
3. For inverting large problems, however, careful scaling analysis
is needed to improve the performance of the iterative solver.
REFERENCES
Chambers, Richard L, Jeffery M. Yarus, and Kirk B. Hird (2000),
Petroleum geostatistics for nongeostatististicians, The Leading
Edge, 19(5), 474 – 479,
doi: 10.1190/​1.1438630.
Deutsch, C.V., and A.G. Journel, GSLIB, Geostatistical Software
Library and User’s Guide, Oxford University Press, New York,
New York, 1992.
Irsa, Juraj, and Ye Zhang (2012), A direct method of parameter
estimation for steady state flow in heterogeneous aquifer with
unknown boundary conditions, Water Resource Research, 48,
W09526, doi:10.1029/2011WR011756.
Paige, Christopher C., and Michael A. Saunders (1982), LSQR: an
algorithm for sparse linear equations and sparse least squares, ACM
Transactions on Mathematical Software, 8(1), 43-71.
20%
ACKNOWLEDGMENTS
15%
10%
This work was funded by the Center for Fundamentals of
Subsurface Flow at the School of Energy Resources at the
University of Wyoming.
5%
2
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Estimated Hydraulic Conductivity
Original
Modified
0.999952
0.999970
0.9999327
0.9998822
0.999908232
1.000004039
0.999899788
1.000049760
0.99989413
1.00002018
0.99989194
1.00002018
0.99988
1.02962
0.999858
1.02966
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Fig. 5 Computation efficiency comparison between original coefficient matrix and
modified coefficient matrix. (a) shows the converge iteration cost and (b) shows the
converge time cost with increasing grid discretization.
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vertical variogram
horizontal variogram
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AGU Poster Template
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Fig. 1 The ground truth model (Chambers, Yarus and Hird 2000) with a lateral flow boundary
conditions . All quantities assume a consistent set of units. High hydraulic head is 300 units, low head
is 200 units and the horizontally normal flow path length is 100 units. At each well, 100 hydrofacies, 50
hydraulic heads and 10 fluxes are sampled following a regular pattern.
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Irsa and Zhang (2012) proposed a new direct inverse method which
successfully modeled state variables in an aquifer where boundary
conditions are unknown. The following works were completed by
them:
1. The new inverse method directly incorporated noisy observed data
(hydraulic heads and flow rates) at the measurement points in a
single step, without solving a boundary value problem.
2. This method effectively solved small-scale aquifer inversion
problems, where hydraulic conductivity spatial distributions are
know a priori, which results in a deterministic inverse
formulation.
3. The stability and computational efficiency of this method were
both proven for the inverse problems at small-scale.
However, for large-scale problems in real applications, the hydrofacies
distributions are usually unknown and need to be estimated. The
computational efficiency of this method also needs to be explored
when it comes to large-scale problems. This research will try to
address stochastic large-scale aquifer inversion problems without
knowledge of boundary conditions, and thus, proving the applicability
of the new method mentioned above with necessary modifications and
adaptions for stochastic inversion with high efficiency on the twodimensional confined aquifer. To achieve this goal, hybrid data
integration will be required for inverse modeling, including dynamic
data, such as hydraulic heads and fluxes, and static data, such as
hydraulic conductivities. This study conducted a stochastic static and
dynamic data integration follows these steps:
1. Develop a synthetic model to obtain observed data, including
static hydrofaices conductivity and error-free dynamic hydraulic
data.
2. Generate variogram models based on the static data from the12
sampling wells.
3. Conduct simulated annealing to smooth the images generated by
SIS.
4. Conduct aquifer parameter inversion on each smoothed
realization.
5. Improve computation performance (speed, stability and accuracy)
of the inverse solver.
Sixty realizations were generated by SIS and simulated annealing was used to smooth the resultant images.
Image coarsening was adopted to help increase computation speeds on the condition that large-scale
hydrofacies continuity was preserved.
10
300
(a)
Static Data Integration Results
Time Cost (hours)
Hydrofacies and error-free hydraulic data observed at 12 sampling wells were obtained from a
synthetic model (Fig.1). This model adopted Finite Difference Method (FDM) and MODFLOW
2000 was used as an analysis platform to conduct this FDM model.
COMPUTATION EFFICIENCY
Iterations (105)
Synthetic Model
experimental variogram
INTRODUCTION
RESULTS
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Much work has been done in developing and applying inverse
methods to aquifer simulation problems. A new direct inversion
method was proposed by Irsa and Zhang (2012) which proved that
boundary condition is not required for estimating parameters in
small-scale aquifer problems. The scope of this paper is to
investigate the applicability of this new method for large simulation
problems and to incorporate uncertainty measures in the inversion
outcomes. The problem considered is a two-dimensional inverse
model (50×50 grid) of steady-state flow for a heterogeneous ground
truth model (100 ×100 grid) with two hydrofacies types (Chambers
et al. 2000). From the ground truth model, twelve wells were
sampled for facies types, based on which experimental indicator
histogram and directional variograms were computed. These
parameters and models were used by Sequential Indicator Simulation
(SIS) to generate 60 realizations of hydrofacies patterns, which were
conditioned to the facies measurements at wells. These realizations
were further smoothed with Simulated Annealing, before they were
conditioned with the direct inversion method to the dynamic data,
i.e., observed heads and groundwater fluxes sampled at the same
well locations. A set of realizations of estimated conductivities, flow
fields, and boundary conditions have been created, which center on
the “true” solutions from solving the forward problem with the
ground truth model. For conductivity estimation, the estimated k was
accurate within ±10% of the true values; for boundary condition
estimation, the accuracy was within ± 15% of the true values. The
inversion system of equations was solved with LSQR (Paige and
Saunders 1982) for which we have adopted a matrix scaling
preprocessor which speeds up the convergence of the solver by 50
times.
METHODS
experimental variogram
ABSTRACT
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Matlab shows your figure, "Edit"->"Copy
Figure"->paste the figure into ppt
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Poster Number: 18
AGU Hydrology Days 2013
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Fig. 4 (a) Boundary indices shown for the true model
(c) Estimated hydraulic conductivity distribution
(b) Inverted hydraulic head boundary conditions (d) Relative error of estimated boundary hydraulic heads
Funding Number: SER 1217-20756-2013