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Ye Zhang 1 , Jianying Jiao 1 , Juraj Irsa 2 , Dongdong Wang 1 yzhang9@uwyo

MODFLOW and More, June 2-5, Golden, CO. Direct Method of Parameter Estimation for Steady-State Flow in Heterogeneous Aquifers with Unknown Boundary Conditions. Ye Zhang 1 , Jianying Jiao 1 , Juraj Irsa 2 , Dongdong Wang 1 yzhang9@uwyo.edu

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Ye Zhang 1 , Jianying Jiao 1 , Juraj Irsa 2 , Dongdong Wang 1 yzhang9@uwyo

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  1. MODFLOW and More, June 2-5, Golden, CO Direct Method of Parameter Estimation for Steady-State Flow in Heterogeneous Aquifers with Unknown Boundary Conditions Ye Zhang1, Jianying Jiao1, Juraj Irsa2, Dongdong Wang1 yzhang9@uwyo.edu 1Dept. of Geology & Geophysics, University of Wyoming, Laramie, WY 2 Optimum Engineering Solutions, Edwardsville, IL http://faculty.gg.uwyo.edu/yzhang/

  2. Aquifer Inversion: Non-Uniqueness Observations: 3 heads 1 flow rate Both solutions yield zero obj  both flow fields are perfectly calibrated to the observed data  Results are non-unique. obj=0 obj=0 Irsa, J., and Y. Zhang (2012) Water Resour. Res., 48, W09526, doi:10.1029/2011WR011756.

  3. New Inverse Method: Objective • Simultaneously estimate aquifer: • hydraulic conductivities (Ks), • source/sink strengths (Ns), • boundary conditions (BC), • while accounting for subsurface static & dynamic data uncertainty.

  4. New Inverse Method: Theory • Aquifer flow equation (below, unconfined) + a set of BC (not shown): • Global continuity of hydraulic head and Darcy flux at grid cell boundaries: Fundamental solution of the flow equation • Local Conditioning by measurement data: Depending on the goal of inversion, one or more data equations are needed. T • Parameter constraint equations (not used yet)

  5. Confined Aquifer: N=0 • No recharge, no wells • 40 hydraulic head measurements + 1 subsurface flow rate measurement • K varies over 2 orders of magnitude • only K1 is estimated: K2, K3, K4 obtained from prior info EQs Irsa & Zhang (2012) WRR True Field (FDM) Inversion (40+1) BC Recovery

  6. Confined Aquifer: Source/Sinks Inversion grid* Measurements: 62 fluid pressures + well rates (Q1, Q2) FDM (LGR at wells) * Inverse grid has 31 cells without LGR Jiao & Zhang, in prep.

  7. Unconfined Aquifer: Uniform N Jiao & Zhang, in review. True heads (<50x50x20) Inverted heads (31 cells + LGR)

  8. Inversion of Keq: Structured field Inversion for Keq: Random field Equivalent K True Keq • Given: • Random Gaussian ln(K) • 25 (5x5) observations on heads + 1 Qy (0%) • Inversion result: 30x30 grid, K=2.4 True: K(x,y) True Keq • Given: • Random Gaussian ln(K) • 15 (3x5) observations on heads +1 Qy (0%) • Inversion result: 30x30 grid, K=2.6 True equivalent K: 37.0 True equivalent K: K=2.7 • Given: • 18 (3x6) observed heads • 1 Qy along y=0.5 • error (3%, std=13) • Result: 2x2 grid K=39.6 • Given: • 18 (3x6) observed heads • 1 Qy along y=0.5 • error (3%, std=13) • Result: 10x10 grid K=1.6 • 30x30 grid K=1.8 True Keq Inverted: Keq • Given: • Random Gaussian ln(K) • 15 (3x5) observations on heads +1 Qy (0%) • Inversion result: 30x30 grid, K=2.2

  9. Reservoir Inversion: Effective Parameter Estimation

  10. Stochastic Inversion

  11. Stochastic Inversion hydrofacies Static Data Integration 12 wells sampled head & fluxes Dynamic Data Integration

  12. Stochastic Inversion Wang, Zhang, & Irsa, in prep.

  13. Fractured Aquifer Zhang, Zhang, & Wang, in prep.

  14. Summary • A new inverse theory for modeling steady-state single-phase flow in confined and unconfined aquifers; • Simultaneous estimation of heterogeneous parameters (multiple Ks, multiple Ns) and BC; • Given sufficient and adequate measurement data, solutions (parameters, BC, flow fields) are “unique”; • Stable outcomes with increasing measurement errors (not shown); • Computationally efficient; • Can account for subsurface static data uncertainty; • If underlying parameter (K, N) heterogeneity is unknown, yield equivalent or average parameter values; • Is not sensitive to high K contrast (Kmax/Kmin up to 106tested for 2D);

  15. Extra

  16. Inversion: Work Breakdown • Irsa & Zhang (2012): 2D confined aquifer inversion without source/sink: jointly estimate a single k & BC; • (2) Zhang (2013): 1D Unconfined aquifer inversion with source/sink: jointly estimate k1, k2, k3, N1, N2, N3, & BC, using pressure data and as few as flow rate from a single well; Estimate equivalent k* & average source/sink terms; • (3) Ongoing work: • 3D confined inversion without source/sink  parameter sensitivity; • 2D unconfined inversion with source/sink; estimate equivalent k* tensor & average source/sink; • 2D stochastic confined inversion (without source/sink): accounting for static data uncertainty; • 2D fractured aquifer inversion (without source/sink): strongly variable heterogeneity

  17. Confined Aquifer: Tensor Inversion (N=0) Inversion: 0% error (31 cells) True Model (<50x50 cells) Inversion: ±0.2% err (31 cells) FDM Solution: h(x,y) 31 heads + 10 q sampled Also successful: simultaneous inversion of diagonal K* (Kx/Ky=100) & nonzero N;

  18. 3D Inversion (Confined Aquifer)

  19. 3D Inversion (Confined Aquifer) Irsa & Zhang (2012) under review • Heterogeneous Aquifer: • 2 zones (Kmax/Kmin=1,000) • Linear flow • Homogeneous Aquifer • Non-linear flow Data: 12 heads & one qz (no flow rate measurement); FDM FDM One flow rate measurement One flow rate measurement inversion inversion FDM FDM

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