CHARACTERIZATION OF SUBSURFACE HETEROGENEITY: INTEGRATION OF SOFT AND HARD
INFORMATION USING MULTI-DIMENSIONAL COUPLED MARKOV CHAIN APPROACH
E. Park (Research Associates)
Department of Civil Engineering,
Section Hydrology and Ecology
Characterization of the subsurface heterogeneity is the most important step in many subsurface studies, yet hardly
attainable due to intrinsic limitations in subsurface accessibility. In this study, the recent 2D Coupled Markov Chain
(2DCMC) model (Elfeki and Dekking, 2001) is extended to a three-dimensional model. The newly developed
stochastic technique of 3D Coupled Markov Chain (3DCMC) is purely based on Markovian properties and
conditioning on field data, based on explicit formulae. This model is theoretically sound, easy to implement, and
computationally cheap. Several real and hypothetical examples are given to show the effectiveness of the developed
model.
Introduction
The heterogeneity of the subsurface has been a long-existing troublesome topic from the very beginning of subsurface
hydrology. The saturated/unsaturated subsurface flow through the geologic media is governed by many components.
Out of these components, the properties of the subsurface material is the most important factor that influences the
subsurface flow yet the most difficult to quantify because of its spatial structure that cannot be described by
deterministic methods. One can easily experience this feature in most fields by observing huge variation of its
properties from point to point (Gelhar, 1993). Because of the extreme difficulties in predicting subsurface
characteristics, the results of the subsurface flow and transport phenomenon, which is highly sensitive to the permeability
or the hydraulic conductivity input, are also very difficult to simulate realistically.
The traditional deterministic approach in which the subsurface properties are represented by a unique value
throughout the entire modeled domain, or represented by a multi-layered system in which each layer can be
characterized by a distinct parameter, is not realistic in most geologic settings. In reality, their properties vary in a
discrete or continuous manner on a multiplicity of scales from one location to another. Moreover, there is uncertainty
due to the fact that parameters are measured only at some sampled locations such as selected well locations and depth
intervals which are often sparse. A stochastic process can be defined mathematically as a collection of random
variables. In a statistical framework, the seemingly uncorrelated subsurface properties can be quantified in terms of
statistical variables. As a tool for quantitative prediction, the purpose of the stochastic simulation is to describe the
subsurface properties on the basis of this statistical inference, whether the real nature is stochastic or deterministic
(Gelhar, 1993). Groundwater flow and transport are therefore more realistically modeled via the stochastic approach.
The characterization of the subsurface is the most important step for every subsurface flow and transport simulation
because the existing heterogeneity of the subsurface governs the preferential movement of the fluids and
contaminants. Currently, many subsurface characterizing tools are available, e.g. borehole, seismic, CPT, GPR, etc.
Because the subsurface is inherently not fully accessible with all these techniques, there exist huge amounts of
uncertainties that need to be resolved to draw a plausible picture of the targeted subsurface. Geostatistical stochastic
simulations are currently the most promising solutions to these uncertainty problems and they have been used for
many case studies. Indicator-based simulation methods are well suited for the geological characterization for the
heterogeneities due to their conceptual (categorical and nonparametric) agreement with real geology (Carle and Fogg,
1996). They have been intensively used in many fields of geology such as soil, stratigraphy, hydrogeology,
sedimentology, etc. In most indicator-based simulations, conventionally, indicator variograms have been extensively
used.
Recently, Carle and Fogg (1996) and Carle et al. (1998) developed a new type of sequential indicator simulation
(SIS) algorithm that communicates with Markovian transition probability instead of the indicator variograms. The
approaches of Carle and Fogg (1998) brought many improvements in terms of asymmetry, which could not be
modeled by conventional approaches because the semivariograms are intrinsically symmetric (Carle et al., 1998).
Their approach has the advantageous aspect of adaptability to effectively utilize many soft and hard data in the stage
of building the transition matrix. Also descriptive subjectivity of geologists can be easily associated at this stage.
Although their transition probabilities are built in Markovian framework, the application of these probabilities to
conventional simulation methods that does not depend on the Markovian properties, can lose some part of its merits.
Also the conditioning should be performed under Markovian framework to be theoretically more rigorous. Elfeki and
Dekking (2001) developed a two dimensional conditional indicator simulation algorithm using Markovian transition
probabilities under a Markovian framework. This model is a direct branch of one-dimensional Markov chain model
developed by Krumbein (1967). The coupled Markov chain (CMC) model is theoretically sound and easy to
implement. Also conditioning is easy by using explicit formulae. But the developed CMC model is only confined to
two-dimensional problems although many practical subsurface characterization problems are 3D. The first objective
of this study is to improve the resolving ability of 2D CMC using active conditioning, which will be effectively
applied for 3D simulations. The second objective is to extend 2D CMC to 3D problems for making the model more
practical.
This research project started in September 2002 and it is due to finish in August 2004
Results in 2002
We developed a 3D coupled Markov chain theories and its corresponding software called CMC3D as an extension of
the previous work of Elfeki and Dekking (2001). We also provide improved scheme of 2D conditioning that uses the
concept of angle tolerance to account for conditioning to the nearest data that is off the line of propagation scheme.
Monte Carlo simulation using the developed model shows that our model is stable and robust.
Research plan for 2003
The followings are the points to be developed or improved regarding our model and software:
1) Developing error minimizing sequences
2) Enhancing data adaptability by incorporating data from various sources (soft geologic data, i.e. geophysical, GPR,
CPT, seismic data)
3) Integrating flow simulator (FDM or FEM)
4) Integrating solute transport simulator (RWPT or MOC)
5) Integrating with spatial database (GIS)
6) Need to verify the Model through various 3D application using hard data as well as soft data and cross-validation
techniques
7) Optimum estimation of horizontal transition probabilities
Publications in 2002
E. Park, A. Elfeki, M. Dekking, annual progress report “Characterization of subsurface heterogeneity: Integration of
soft and hard information using multi-dimensional Coupled Markov chain approach”, Delft, The Netherlands, 2002.
Supervised M.Sc. thesis
-none
Figure 1. Simulation results of 3D CMC model using hypothetical input data
(a) outside
(b) inside