Document Details
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Document Type |
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Article In Conference |
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Document Title |
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Study of the Factors Affecting Joint Conditional Simulation of Groundwater Flow and Solute Transport in Heterogeneous Aquifers دراسة العوامل المؤثرة على النمذجة الازدواجية المشترطة لحركة المياه الجوفية ومسار الملوثات في الخزانات الجوفية غير المتجانسة |
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Subject |
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earth sciences, hydrogeology and environmental sciences |
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Document Language |
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Arabic |
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Abstract |
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Aquifers are inherently heterogeneous at various observation scales. Characterizing the
heterogeneity at a scale of our interest, generally requires information of hydrologic properties at
every point in the aquifer. Such a detailed hydraulic property distribution in aquifers requires
numerous measurements, considerable time, and great expense, and is generally considered
impractical and infeasible. The alternative is to utilize a small number of samples to estimate the
variability of parameters in a statistical framework. That is, the spatial variation of a hydraul ic
property is characterized by its probability distribution estimated from samples.
Recent analyses of heterogeneity showed that, although the hydraulic conductivity values
vary significantly in space, the variation is not entirely random, but correlated in space. Such a
correlated nature implies that the parameter values are not statistically independent in space and
they must be treated as a stochastic process, instead of a single random variable. Stochastic
process is defined briefly as an infinite collection of random variables. Yeh (1992) provided an
overview of several stochastic approaches developed in the last few years for modeling water
flow and solute transport in heterogeneous aquifers, they classified them into two main
categories: homogeneous or effective parameters and heterogeneous approaches. Most of these
models are known to be valid only if the spatial heterogeneity of the soil is moderate and are
limited to relativeJy simplified analytical models. The effective parameter approach assumes that
the heterogeneous geologic formation can be homogenized to obtain effective parameters with
which one can predict the ensemble behavior of the flow and transport processes. Examples of
such studies include those by, Dagan (1982 a and b a985a).
The heterogeneous approach is designed to consider the nature of spatial variability of
hydrologic properties of the aquifer with limited amount of data. Methods in this approach
generally consist of geostatistics, Monte Carlo simulation, and conditional simulation.
Geostatistics (kriging-cokriging) is a mathematical interpolation and extrapolation tool, which
uses the spatial Statistics of the data set to estimate the property at unsampled locations.
Although hydraulic head and transmissivity fields derived from cokriging have been found to be
reasonable, there is no guarantee these estimates satisfy the principle of conservation of mass
Harter and Yeh (1993); and Yeh et al. (19C ia). Monte Carlo simulation is the most intuitive
approach for dealing with spatial variability in a stochastic sense. Although it belongs to the
heterogeneous approach since hydraulic prop rty at every point in the aquifer is specified, it is,
in principle, equivalent to the effective paran.eter approach. 80th Monte Carlo simulation and
the effective parameter approach derive the mean and variance of the hydraulic head, but Monte
Carlo simulation requires fewer assumptions, and it can predict shape of frequency distribution
of the output variables. Typical examples of studies using this approach can be found in Freeze
(1975).
Conditional simulation is an approach that combines geostatistics and Monte Carlo
simulation. Unlike Monte Carlo simulation, it provides only a subset of all possible realizations
i:i
of the hydrologic property, which consists of the values of the properties at sample locations and
confirms with a predefined spatial statistics of the hydrologic property. In this context,
realizations that do not agree with measured values at the sampled locations are discarded.
Because the conditional simulation includes the data values at the sampled location and all
possible values at the unsampled locations, the conditional simulation is considered the most
rational approach for dealing with uncertainties in heterogeneous geologic formations, Yeh
(1992). The complete theory of conditional simulation is given by Matheron (1973) and Joumel
and Huijbregts (1978).
The objective of this research is to characterize aquifer heterogeneity based on limited
data sets of transmissivities and/or hydraulic head in such a way that the obtained fields honors
the values of these properties at the pre-sampled locations (conditional simulation). There are
several factors control the process of conditioning fields. In this study, the influence of degree of
heterogeneity, correlation scale, sampling location, size of the data set and hydraulic gradient
were investigated and interpreted as a travel time distribution of solute particle released at a prespecified
location.
Two dimensional depth-averaged saturated steady groundwater flow equation was
adopted in this study to illustrate the stochastic conditional simulation approach. Transmissivity
is considered as spatially heterogeneous parameter that is unknown, except at data locations and
is modeled statistically as a second order stationary random field. Uncertainty in the
transmissivity then propagates through the model and results in uncertainty in the hydraulic head.
Assuming head and log-transmissivity to be spatial stochastic processes, they further decompose
as the sum of mean and perturbation parts about the mean. Randomness is introduced and the
model is linearized by a first-order small perturbation expansion. Following up the mathematics
results in a linearized governing perturbation flow equation.
The basic simulation process can be summarized in three main steps. First, use the
perturbation equation and the associated linearized solution to obtain the spectral representation.
Second, use the spectral representation to get covariances and cross-covariances functions of
both transmissivity and hydraulic head. Third, generate multiple realizations and condition them
to data. Conditioning on data was accomplished by the standard cokriging geostatistical
procedure. In fact the implicit linearity in the perturbation solution works well with low variance
values (less than one) of log-transmissivity. However for large variances, the conditioned logtransmissivity
is substituted into the flow equation and the equation solved, many of the
generated fields of this case violate continuity conditions. One option to avoid this problem is to
use iterative conditioning approach.
In the iterative approach, the transmissivity field is conditioned on both head and
transmissivity measurements. This transmissivity field was then used, along with the boundary
conditions, to solve for a new head fields. The re~ulting fields satisfy the continuity conditions
but the head do not agree with the measured heac. Consequently the new head at the previous
data locations is used to again condition the transmissivity field, and the process is repeated until
the head fields are close to the measured values. Unfortunately the solution to the actual flow
equation is very different from the linearized equation for high variances. Thus the iteration can
only try to improve the head differences, but it can never completely remove them. This
procedure significantly improves head difference after few iterations.
fro
Since we are dealing with a large number of random fields (realizations), random numbe
generator is·utilized here as a tool to generate hundreds of realizations. There are several randon
number generators capable of generating spatially correlated random fields .. In this study thl
spectral approach utilizing Fast Fourier Transform (FFT) random field generator (RFG) is usee
due primarily to its computational speed advantage and ease of incorporating an existin!
subroutine into the algorithm. Random fields of transmissivity values (realizations) are generate(
using spectral random field generator, [Gutjahr (l989)J, assuming an exponential covarianc(
function of random variable.
The iterative conditional approach was applied to a hypothetical case to study five pre
specified factors affecting the performance of the proposed approach. Each factor was studie(
separately. Before the beginning of any step, a standard reference case was decided upon. Thi
case served as a reference for comparison with other developed cases or scenarios. Each facto
studied was divided into scenarios. Eleven different scenarios based on each parameter valUl
variations were studied and results were obtained. Several small and big computer codes wen
written to perform all runs of different scenarios. For each scenario, 100 different transmissivit~
fields were generated (realizations) from the same process.
The proposed iterative conditional simulation approach was implemented successfully ir
characterizing the heterogeneity of a medium using limited number of transmissivity and heat
data. The proposed factors affecting the conditioning process were studied separately and tht
following specific conclusions were made:
1. The proposed approach worked well when the variance of generated
transmissivities was less than one. When higher variances are used a problem of
particle trap occurred.
2. At low Ln[TJ variances, the flow paths of real izations appear converging to an
ensemble mean surrounding the real path flow.
3. In general the travel time of released particle increases as the variance of Ln[T]
increases.
4. Correlation scale proved to be an important factor. Small correlation scale value
produced wider spectrum and longer travel time than large correlntion scale value.
5 No significant impact in the flow paths occurred when menn flow gradient
changed. However, the travel time distribution has changed dramatically.
6. An interesting result obtained when data size has changed. Few data points
scenario produced wider spectrum of flow paths than when more data points are
used.
7. The mean Ln[TJ of the 100 conditioned fields of the standard case IS
overestimated slightly in some scenarios and underestimmed in others. |
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Conference Name |
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the second annual meeting for scientific research |
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Duration |
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From : 27 محرم AH - To : 28 محرم AH
From : 30 مارس AD - To : 31 مارس AD |
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Publishing Year |
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1424 AH
2003 AD |
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Number Of Pages |
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7 |
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Article Type |
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Article |
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Added Date |
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Wednesday, January 14, 2009 |
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Researchers
| خالد سعيد بالخير | Balkhair, N/A N/A | Researcher | | |
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