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Dynamic Data Integration and Stochastic Inversion of a Two-Dimensional Confined Aquifer Dongdong Wang1, [email protected]; Ye Zhang1, [email protected]; 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 SIS Realization 1 10 10 10 8 6 75 45 0 0 Rows 50 50 60 Rows 40 Rows 40 50 60 70 70 70 70 80 80 80 80 90 90 90 90 100 10 20 30 40 No50 flow60 70 80 90 40 50 60 70 80 90 10 Columns Simulated Annealing Realization 1 100 1. Variogram Modeling Based on Static Hydrofacies Data 50 60 70 80 90 10 100 20 20 20 30 30 30 40 40 40 50 50 60 70 70 70 80 80 80 90 90 20 30 40 50 60 70 80 90 100 100 experimental variogram 40 50 60 70 80 90 100 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 100 10 Columns Grid Coarsening Realization 1 30 50 60 10 20 Columns Simulated Annealing Realization 60 10 0.3 0.16 40 10 100 0.18 30 10 90 0.35 20 20 30 40 50 60 70 80 90 100 10 20 Columns Grid Coarsening Realization 30 30 40 50 60 70 80 90 100 Columns Grid Coarsening Realization 60 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) 0.2 0.15 0.1 0 5 10 15 20 25 30 lag distance 35 40 45 10 10 20 20 20 30 30 30 40 40 40 50 Rows 80 10 50 Rows 0.12 0.25 Rows 0.14 50 60 60 70 70 70 80 80 80 90 90 90 100 10 20 30 40 50 60 70 80 90 100 100 10 20 30 40 50 60 70 80 Boundary Indices 50 55 60 65 70 75 80 85 90 95 46 90 100 10 20 30 40 50 60 70 80 90 Observed (error free) heads vs. Recovered heads 100 450 true head boundary 105 400 41 estimated head boundary of a given realization 115 31 120 26 125 21 130 16 135 11 140 6 145 Hydraulic Heads 110 36 350 300 250 200 150 100 0 1 195 190 185 180 175 170 165 160 155 150 20 40 60 80 100 120 140 160 180 Boundary Indices (d) 25% 12 8 Frequency Continuity Equations: Relative Esimate Error Percentage 10 6 4 h: hydraulic head x, z: Cartesian coordinates ai: approximated parameter k: hydraulic conductivity δ: weight for continuity equations 0 0.85 0.9 0.95 1 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 200 100 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 Grid Size 2X2 4X4 8X8 16X16 32X32 64X64 128X128 256X256 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 50 60 100 Grid Size 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. Columns Simulated Annealing Realization 30 Modify the type size, column widths and banner headings as needed. vertical variogram horizontal variogram Grid Size 100 100 100 Rows 20 30 30 50 60 200 60 4 Original Matrix 60 40 Static Data Integration 70 90 15 Columns 60 Original Matrix 105 2 60 40 50 lag distance 12 120 30 This will be enlarged 200% to fit within the allotted 47" x 68” space. 30 14 30 AGU Poster Template 20 16 30 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. 10 135 30 X 0 18 20 10 0.08 150 20 100 0.2 10 20 20 40 Z SIS Realization 60 SIS Realization 30 (b) 20 0.1 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 Rows 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 Rows Matlab shows your figure, "Edit"->"Copy Figure"->paste the figure into ppt Rows Poster Number: 18 AGU Hydrology Days 2013 0 0 10 20 30 40 Realization Number 50 60 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