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Reducing Concentration Uncertainty in Geological Structures by Conditioning on Boreholes Using The Coupled Markov Chain Approach Amro Elfeki Section Hydrology, Dept. of Water Management, TU Delft, The Netherlands.

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Outlines Motivation of this research. Methodology: Markov Chain in One-dimension. Markov Chain in Multi-dimensions: Coupled Markov Chain (CMC). Application of CMC at the Schelluinen study area (Bierkens, 94). Comparison between: CMC (Elfeki and Dekking, 2001) and SIS (Sequential Indicator Simulation, Gomez-Hernandez and Srivastava, 1990). Flow and Transport Models in a Monte-Carlo Framework. Geostatistical Results. Transport Results. Conclusions.

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Motivation and Issues Motivation of this research: Test the applicability of CMC model on field data at many sites. Compare CMC with SIS (well-known model in geostatistics). Incorporating CMC model in flow and transport models to study uncertainty in concentration fields. Deviate from the literature: - Non-Gaussian stochastic fields: (Markovian fields), - Statistically heterogeneous fields, and - Non-uniformity of the flow field (in the mean) due to boundary conditions.

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Geological and Parameter Uncertainties Unconditional CMC Elfeki, Uffink and Barends, 1998 Geological Uncertainty: Geological configuration. Parameter Uncertainty: Conductivity value of each unit.

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Application of CMC at MADE Site Elfeki, 2003 (in review) Real field situation: Data is in the form of boreholes. Geological prediction is needed at unsampled locations.

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One-dimensional Markov Chain

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Coupled Markov Chain CMC in 2D (Elfeki and Dekking, 2001)

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CMC vs. Conventional Methods CMCConventional Methods Based on conditional probability (transition matrix). Based on variogram or autocovariance. Marginal Probability.Sill. Asymmetry can be described. Asymmetry is impossible to describe. A model of spatial dependence is not necessary. A model of spatial dependence is needed for implementation. Compute only the one- step transition and the model takes care of the n-step transition probability. Need to compute many lags for the variogram or auto-correlations. (unreliable at large lags)

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Schelluinen study area, The Netherlands Soil Coding Soil description 1 Channel deposits (sand) 2 Natural levee deposits (fine sand, sandy clay, silty clay) 3Crevasse splay deposits (fine sand, sandy clay, silty clay) 4Flood basin deposits (clay, humic clay) 5Organic deposits (peaty clay, peat) 6Subsoil (sand) Data from Bierkens, 1994

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Parameter Estimation and Procedure

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Horizontal transition probability matrix of 1650 m section calculated over sampling intervals of 25 m. Soil123456 10.9790.0040.0010.0060.0090.001 20.0200.9650.0010.0080.0060.000 30.0030.0020.9660.0130.0160.000 4 0.0010.0090.9830.0070.000 50.001 0.0060.0070.9840.001 60.000 0.0010.0000.0020.997 Vertical transition probability matrix 1650 m section calculated over sampling intervals of 0.25 m. Soil123456 10.9450.0000.0090.0000.0090.037 20.0710.7960.0210.0410.0710.000 3 0.7970.0860.0890.028 40.0030.0130.0410.7140.2220.007 50.0040.0120.0470.1190.7680.050 60.000 1.000 Transition Probabilities (1650 x10 m)

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Transition Probabilities (240 x10 m) Horizontal transition probability matrix Vertical transition probability matrix State3456 3456 30.9790.0100.0110.000 30.9690.0270.0040.000 40.0110.9740.0150.00040.0080.7240.2680.000 50.0080.1200.9770.00350.0250.1390.7910.045 6 0.0100.0000.0070.983 6 0.000 1.000 Sampling intervals Dx = 2 m Dy= 0.25 m 0.9660.0130.0160.000 0.0090.9830.0070.000 0.0060.0070.9840.001 0.0000.0020.997 0.7970.0860.0890.028 0.0410.7140.2220.007 0.0470.1190.7680.050 0.000 1.000 Horizontal Transition Probability from 1650x10 Vertical Transition Probability from 1650x10

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Parameter Numerical Value Time step5 [day] Longitudinal dispersivity0.1 [m] Transverse dispersivity0.01 [m] Effective porosity0.30 [-] Injected tracer mass1000 [grams] Head difference at the site1.0 [m] Monte-Carlo Runs50 MC Number of particles10,000 [particles] Physical and Simulation Parameters Numerical simulation values used in the numerical tracer experiment. Soil Properties at the core scale from Bierkens, 1996 (Table 1). Soil Coding Soil typeWiWi 6Fine & loamy sand0.120.601.764.400.09 5Peat0.39-2.001.70.302.99 3Sand & silty clay0.19-4.973.490.15.86 4Clay & humic clay0.30-7.002.490.0110.1 Convergence: ~14000 Iterations Accuracy 0.00001

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Flow Model is the hydraulic head, V x and V y are pore velocities, is the hydraulic conductivity, and is the effective porosity. Hydrodynamic Condition: Non-uniform Flow in the Mean due to Boundary Conditions.

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Transport Model Governing equation of solute transport : C is concentration V x and V y are pore velocities, and D xx, D yy, D xy, D yx are pore-scale dispersion coefficients is effective molecular diffusion, is delta function, is longitudinal dispersivity, and is lateral dispersivity.

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The displacement is a normally distributed random variable, whose mean is the advective movement and whose deviation from the mean is the dispersive movement. instantaneous injection + uniform flow Random Walk Method

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Application of SIS at the Site Geological Section Deterministic and Stochastic Zones In SIS Model Bierkens, 1996

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Comparison between CMC and SIS (1) Conditioning on half of the drillings SIS Model Simulation CMC Model Simulation Geological Section Bierkens, 1996

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Comparison between CMC and SIS (2) Conditioning on all drillings SIS Model Simulation CMC Model Simulation Geological Section Bierkens, 1996

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Monte-Carlo on CMC

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Effect of Conditioning on 240 x10m Sec. 31 boreholes 25 boreholes 9 boreholes 2 boreholes

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Ensemble Indicator Function

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Effect of Conditioning on Variogram Measure of Variability

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Effect of Conditioning on S. R. Plume mg/lit T= 82 years # drillings 2 3 5 9 25 31

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Effect of Conditioning Single Realiz. C max Practical convergence is reached after about 21 boreholes

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First Moment (Single Realization) Trend is reached at 3 boreholes Convergence at 9 boreholes

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Second Moment (Single Realization) Trend is reached at 3 boreholes Convergence at 5 and 25 boreholes Convergence at 9 boreholes

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Breakthrough Curve (Single Realization) Convergence at 25 boreholes

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Conditioning on 2 boreholes (Ensemble ) mg/lit T = 4.1 years T = 82.2 years T = 136.9 years

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Conditioning on 5 boreholes (Ensemble) mg/lit

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Conditioning on 9 boreholes (Ensemble)

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Conditioning on 21 boreholes(Ensemble)

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Conditioning on 31 boreholes(Ensemble)

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Effect of Conditioning on Ensemble C max

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Conclusions 1. CMC model proved to be a valuable tool in predicting heterogeneous geological structures which lead to reducing uncertainty in concentration distributions of contaminant plumes. 2. Comparison between SIS and CMC have shown more or less similar results in terms of the geological configuration of the confining layers. However, CMC has more merits over the SIS: -some parts of the confining layers are treated determistically in SIS method which is not the case in CMC method. -non-stationarity in the confining layers is treated straightforwardly by CMC, it is inherited in the method, while in SIS model subdivision into three sub-layers has to be performed. -three variogram models with different parameters have been used in SIS, while direct transition probabilities were used in CMC. 3. Convergence to actual concentration is of oscillatory type, due to the fact that some layers are connected in one scenario and disconnected in another scenario.

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Conclusions 4. In non-Gaussian fields, single realization concentration fields and the ensemble concentration fields are non-Gaussian in space with peak skewed to the left. 5. Reproduction of peak concentration, plume spatial moments and breakthrough curves in a single realization requires many conditioning boreholes (20-31 boreholes). However, reproduction of plume shapes require less boreholes (5 boreholes). 6. Ensemble concentration and ensemble variance have the same pattern. Ensemble variance is peaked at the location of the peak ensemble concentration and decreases when one goes far from the peak concentration. This supports early work by Rubin (1991). However, in Rubins case the maximum concentration was in the center of the plume which is attributed to Gaussian fields. The non- centered peak concentration, in this study, is attributed to the non- Gaussian fields.

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Conclusions 7. Coefficient of variation of max concentration [CV(Cmax)] decreases significantly when conditioning on more than 5 boreholes. 5. Reproduction of peak concentration, plume spatial moments and breakthrough curves in a single realization requires many conditioning boreholes (20-31 boreholes). However, reproduction of plume shapes require less boreholes (5 boreholes). 6. Ensemble concentration and ensemble variance have the same pattern. Ensemble variance is peaked at the location of the peak ensemble concentration and decreases when one goes far from the peak concentration. This supports early work by Rubin (1991). However, in Rubins case the maximum concentration was in the center of the plume which is attributed to Gaussian fields. The non- centered peak concentration, in this study, is attributed to the non- Gaussian fields.

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