Ensemble Kalman filter assimilation of transient groundwater flow data via stochastic moment equations

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Ensemble Kalman filter assimilation of transient groundwater flow data via stochastic moment equations Alberto Guadagnini (1,), Marco Panzeri (1), Monica Riva (1,), Shlomo P. Neuman () (1) Department of Civil and Environmental Engineering, Politecnico di Milano, Milan, Italy () Department of Hydrology and Water Resources, University of Arizona, Tucson, Arizona, USA

Objective: Inverse Modeling Data Assimilation History Matching Data Assimilation History Matching Inverse Modeling Prior model: Posterior model: x WCT P1 x x x x x x x x x x x t WCT P1 x x x x x x x x x x x x x x t Producer P1 Injector well W1 Observations WCT P1 x x x x x x x x x x x t x x x t The inverse problem aims at determining the unknown model parameters by making use of observed state data.

Inverse modeling / parameter estimation 3 x 4 35 3 5 15 1 5 13 Impervious boundary 1 1 14 Reference field: exponential variogram (Sill =.; Integral scale = 4.) Guadagnini A., Panzeri M., Riva M., Neuman S.P. 3 15 5 4 16 6 7 17 1 3 4 8 18 Impervious boundary x 1 9 1 11 19 from Panzeri et al. [13] ref 4 - -4 h Pumping well (flow rate = 3) 9 measurements σ =.1 h measurement locations 1 observation times σ =.1 1..8.4. -.4 -.8 ( ) E ( ) he from Panzeri et al. [13].1 5 15 3 5 14 1 4 3 8 9 6 t.5 1. 7

Ensemble Kalman Filter (EnKF) 4 Start with the Kalman Filter (KF) [Kalman, 196] First developed to integrate noisy measured data in a physical model characterized by (a) linear dynamics and (b) a Gaussian distribution of system variables and measurement errors. Two-step implementation scheme: (1) mean and covariances of system state variables are propagated/advanced in time until new measurements of these states (and/or other) variables are available (forward step); () measured values and the associated error measurement variances are employed to evaluate updated mean and covariances of system state variables (updating step). t = t t = t 1 t = t f(y ) f(y 1 y ) f(y 1 d 1 ) f(y y 1 ) f(y d ) Guadagnini A., Panzeri M., Riva M., Neuman S.P. f(d 1 y 1 ) f(d y )

Ensemble Kalman Filter (EnKF) 5 Start with the Kalman Filter (KF) [Kalman, 196] First developed to integrate noisy measured data in a physical model characterized by (a) linear dynamics and (b) a Gaussian distribution of system variables and measurement errors. Two-step implementation scheme: (1) mean and covariances of system state variables are propagated/advanced in time until new measurements of these states (and/or other) variables are available (forward step); () measured values and the associated error measurement variances are employed to evaluate updated mean and covariances of system state variables (updating step). t = t t = t 1 t = t f(y ) f(y 1 y ) f(y 1 d 1 ) f(y y 1 ) f(y d ) Guadagnini A., Panzeri M., Riva M., Neuman S.P. f(d 1 y 1 ) f(d y )

Ensemble Kalman Filter (EnKF) 6 Advantages: EnKF is associated with dynamic systems; observed data are obtained in time and employed for sequential updates of the model. no sensitivity analysis (gradient-based methods) relatively affordable computational cost sequential update of model variables/states well-suited for real time model estimation Ensemble of models (conceptual uncertainty quantification) t = t t = t 1 t = t f(y ) f(y 1 y ) f(y 1 d 1 ) f(y y 1 ) f(y d ) Guadagnini A., Panzeri M., Riva M., Neuman S.P. f(d 1 y 1 ) f(d y )

EnKF (traditional MC approach) 7 Data assimilation of transient groundwater flow data in complex geologic media via Ensemble Kalman filter First and second order statistical moments of parameters and state variables are traditionally approximated via Monte Carlo approach Difficulties a. A good approximation of the covariance function requires a large number of realizations (computationally intensive) b. When the collection ( ensemble ) of realizations is small, the spurious correlations in the empirical covariance matrix may lead to incorrect parameters updates (Filter Inbreeding)

EnKF coupled with MEs 8 Data assimilation of transient groundwater flow data via Ensemble Kalman filter Key point coupling EnKF with the solution of stochastic Moment Differential Equations (MEs) of transient groundwater flow Tartakovsky and Neuman [1998a, b, c]; e et al. [4]; Hernandez et al. [3, 6]; Riva et al. [9, 1]; Bianchi Janetti et al. [1]; Panzeri et al. [13a, b] Advantages a) MEs provide theoretical insights in the nature of the solution b) Solution of MEs can be achieved (in principle) by adopting a coarser grid than the one required by MC (computational efficiency) c) Alleviate Filter Inbreeding

Brief history of groundwater flow MEs 9 Neuman and Orr [1993]: Steady-state groundwater flow MEs (nonlocal and localized format) Tartakovsky and Neuman [1998a, b, c]: Transient exact and recursive approximations (including effective parameters for uniform mean flow) Guadagnini and Neuman [1999a, b]; e et al. [4]: Numerical solution of steady-state and transient MEs. x x 8 6 4 4 6 8 1 1 14 16 18 x1 8 6 4 4 6 8 1 1 14 16 18 x1 x 1 6. 5. 4. 3.. 1.. -1. -. -3. -4. -5. -6. -7.. 1.5 1..5. -.5-1. -1.5 -. -.5-3. -3.5 x from Guadagnini and Neuman [9]

Brief history of groundwater flow MEs 1 Winter et al. [; 3]: Flow statistics in composite media p 1 (k) P 1 (x) p (k) p(b) Riva et al. [1]; Guadagnini et al. [3]; Neuman et al. [4, 7]; Riva et al. [9]: Analytical solutions well flow; stochastic interpretation of pumping tests + field applications 3 G 1 G Hernandez et al. [3, 6]; Riva et al. [9, 1]; Bianchi Janetti et al. [1]: Inverse modeling (steady-state and transient) + field applications 1-1 -3.1 1 1 from Riva et al. [1]

Brief history of groundwater flow MEs 11 Riva et al. [11]: MEs & Model discrimination criteria Panzeri et al. [13a, b]: MEs & EnKF Neuman et al. [8]; Riva and Guadagnini [9]: MEs and hierarchical media. Guadagnini and Neuman [1]; Morales Casique et al. [6a, b]; Riva et al. [6] + others: MEs for transport; well capture zones.

Extended MEs 1 Stochastic equations developed by e et al. [4] are extended to account for random initial conditions Zero-order mean flow equations: Second-order mean flow equations: ( ) ( x, t ) ( ) ( + q ) = ( ) h S( x) x, t f x, t t q ( ) ( ) ( ) ( ) ( x, t = K x h x, t ) + B.C. & I.C. G ( ) ( x, t ) ( ) ( ) h S( x) + q x, t = t σ x q G q r ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( x, t = K x h x, t + x, t + ) ( x, t) + B.C. & I.C.

Extended MEs 13 Stochastic equations developed by e et al. [4] are extended to account for random initial conditions Equations for the residual flux ( ) u ( yx,, s) () ( ) ( ) ( ) ( ) ( S y [ ( x,, )] = ( ) [, q ) y KG y yu y s KG x y C x y y, s ] s + B.C. & I.C. ( ) ( ) ( y ) ( ) u ( yx,, s) = K x h, s r ( ) ( ) x, y, s = u ( x, y, s) () y Second-order head (cross)covariance equations: () Ch ( yx,,,) st ( ) S( y) = K y C yxst + u yxt h ys s + B.C. & I.C. () () y [ G( ) y h (,,, )] y [ (,, ) y (, ) ]

EnKF: flowchart 14 MONTE CARLO MOMENT EQUATIONS Initial conditions: h f, T j j = 1,, NMC h f, T C u + h ( ) ( ) ( ) h u C h f, T

EnKF: flowchart 15 MONTE CARLO MOMENT EQUATIONS Initial conditions: h f, T j j = 1,, NMC h f, T C u + h ( ) ( ) ( ) h u C h f, T Forecast Step NMC deterministic flow problems Stochastic Moment Equations h f, Tk j j = 1,, NMC h f, Tk C u + h ( ) ( ) ( ) h u C h f, T k

EnKF: flowchart 16 MONTE CARLO MOMENT EQUATIONS Initial conditions: h f, T j j = 1,, NMC h f, T C u + h ( ) ( ) ( ) h u C h f, T Forecast Step NMC deterministic flow problems Stochastic Moment Equations h f, Tk j j = 1,, NMC h f, Tk C u + h ( ) ( ) ( ) h u C h f, T k EnKF Updating Step: Observed data T d k Tk Σ dd

EnKF: flowchart 17 MONTE CARLO MOMENT EQUATIONS Initial conditions: h f, T j j = 1,, NMC h f, T C u + h ( ) ( ) ( ) h u C h f, T Forecast Step NMC deterministic flow problems Stochastic Moment Equations h f, Tk j j = 1,, NMC h f, Tk C u + h ( ) ( ) ( ) h u C h f, T k EnKF Updating Step: Observed data T d k Tk Σ dd h f, Tk j j = 1,, NMC h ut, k C u + h ( ) ( ) ( ) h u C h ut, k

Objectives of first analyses 18 Comparison of the two methodologies: 1. EnKF coupled with the solution of the stochastic moment equations (MEs) of transient groundwater flow;. Traditional MC-based EnKF with diverse numbers of model realizations (NMC = 1; 5; 1,; 1,; 1,). a. Accuracy of the estimated mean and variance of logconductivity and head fields b. Effect of magnitude of measurement error c. Effect of transient v. pseudo-steady state data d. Analysis of the occurrence of filter inbreeding (systematic underestimation of estimation variance with the assimilation) e. Computational efficiency

Inverse modeling / parameter estimation 19 x Constant head = 1 4 35 3 5 15 1 5 13 Impervious boundary 1 1 14 3 15 5 4 16 1 3 4 Reference field: exponential variogram (Sill =.; Integral scale = 4.) 6 7 17 8 18 Impervious boundary x 1 Guadagnini A., Panzeri M., Riva M., Neuman S.P. 9 1 11 19 Constant head = from Panzeri et al. [13] ref 4 - -4 h Pumping well (flow rate = 3) 9 measurements σ =.1 h measurement locations 1 observation times σ =.1 1..8.4. -.4 ( ) E ( ) he from Panzeri et al. [13] -.8 5 15 3 5.1 14 1 4 3 8 9 6 t.5 1. 7

Tests performed x 4. Impervious boundary I = 4. 4 3 σ =.5 σ = 1. σ =. a) b) c) 4 4 3 3 Constant head =.8 3.. 1... 1.. 3. 4. Impervious boundary Constant head =. I = 1. 1 4 3 1 1 3 4 1 3 4 1 4 3 1 1 3 4 1 3 4 1 4 3 1 1 3 4 1 3 4 x 1 4 4 4 I =. 3 3 3 1 1 1 1 3 4 1 3 4 1 3 4 from Panzeri et al. [14]

Objectives of first analyses 1 Comparison of the two methodologies: 1. EnKF coupled with the solution of the stochastic moment equations (MEs) of transient groundwater flow;. Traditional MC-based EnKF with different number of model realizations (NMC = 1; 5; 1,; 1,; 1,). a. Accuracy of the estimated mean and variance of logconductivity and head fields b. Effect of magnitude of measurement error c. Effect of transient v. pseudo-steady state data d. Analysis of the occurrence of filter inbreeding (systematic underestimation of estimation variance with the assimilation) e. Computational efficiency

Inverse modeling / parameter estimation Reference Estimated mean fields T k = T k = 5 T k = 1 T k = T k = 3 T k = 4 T k = 8 1.5 1..95 4. 3.. 1.. -1. -. -3. -4. E.9.85.8.75 4 6 8 T k N 1 ( ) = ( ) ( ) E T x x k i i ref N i = 1 T k from Panzeri et al. [14]

Inverse modeling / parameter estimation 3 Estimated variance fields T k = T k = 5 T k = 1 T k = σ.5. 1.5. 1.8 T k = 3 T k = 4 T k = 8 1..5 V 1.6 1.4 1.. 1..8 4 6 8 More than 9% of the estimates lie inside the ± σ confidence intervals at each time T K from Panzeri et al. [14] T k N 1 ( ) σ ( ) V T = x k i N i= 1 T k

Objectives of first analyses 4 Comparison of the two methodologies: 1. EnKF coupled with the solution of the stochastic moment equations (MEs) of transient groundwater flow;. Traditional MC-based EnKF with different number of model realizations (NMC = 1; 5; 1,; 1,; 1,). a. Accuracy of the estimated mean and variance of logconductivity and head fields b. Effect of magnitude of measurement error c. Effect of transient v. pseudo-steady state data d. Analysis of the occurrence of filter inbreeding (systematic underestimation of estimation variance with the assimilation) e. Computational efficiency

Synthetic example: increasing number of observation times 5 1. E Test Case 3 (TC3): Adding 1 additional observation times to the conditioning data set 1.5 1..95.9.85.8.75.7 from Panzeri et al. [13] TC1 TC3 4 6 8 T k h V.8.4. -.4 -.8. 1.8 1.6 1.4 1. 1..8 5 15 3 5 T k TC1 TC3 4 6 8 T k

Objectives of first analyses 6 Comparison of the two methodologies: 1. EnKF coupled with the solution of the stochastic moment equations (MEs) of transient groundwater flow;. Traditional MC-based EnKF with diverse numbers of model realizations (NMC = 1; 5; 1,; 1,; 1,). a. Accuracy of the estimated mean and variance of logconductivity and head fields b. Effect of magnitude of measurement error c. Effect of transient v. pseudo-steady state data d. Analysis of the occurrence of filter inbreeding (systematic underestimation of estimation variance with the assimilation) e. Computational efficiency

Results: estimated mean fields 7 Estimated fields of ME ( x ) at the latest assimilation step ( t =1.) NMC =1, NMC = 1, True field ref, 4 - -4 4... -. -4. NMC = 1, NMC = 5 NMC = 1 from Panzeri et al. [14]

Results: estimated variance fields 8 Estimated fields of ME σ ( x ) at the latest assimilation step ( t =1.) NMC =1, NMC =1, σ ( x) 4. 1.5 1. - -4.5. NMC = 1, NMC = 5 NMC = 1 from Panzeri et al. [14]

Results 9 1.1 1.5 E 1.9 V 1.8.5.7..4.6.8 1..4.6.8 1 t t ( ) ( ) N 1 ( ) = ( i) ( i) ref N i= 1 E t x x Filter Inbreeding (see also later) t 1 N σ i N i= 1 V t = x ME NMC =1 NMC = 5 NMC =1, NMC =1, NMC =1, t from Panzeri et al. [14]

Objectives of first analyses 3 Comparison of the two methodologies: 1. EnKF coupled with the solution of the stochastic moment equations (MEs) of transient groundwater flow;. Traditional MC-based EnKF with different number of model realizations (NMC = 1; 5; 1,; 1,; 1,). a. Accuracy of the estimated mean and variance of logconductivity and head fields b. Effect of magnitude of measurement error c. Effect of transient v. pseudo-steady state data d. Analysis of the occurrence of filter inbreeding (systematic underestimation of estimation variance with the assimilation) e. Computational efficiency

Filter inbreeding 31 The EnKF leads to systematically underestimated error during the assimilation. It is caused by the juxtaposition of diverse effects: i. updating of the collection of model realizations through a Kalman gain calculated from the same sample; ii. iii. influence of the finiteness of the sample size on the Kalman gain (spurious covariances in the sample covariance matrix); nonlinear relationship between hydraulic heads and logtransmissivities.

Filter inbreeding 3 1.9 1.5 P σ.8.7 V E 1.6.5.5..4.6.8 1 t..4.6.8 1 t N 1 P t H x x x { } i t i i ref ( ) = ( ) ( ) ( ) σ σ N i= 1 H { } : Heaviside step function t ME NMC =1 NMC = 5 NMC =1, NMC =1, NMC =1, from Panzeri et al. [14]

ME NMC =1, NMC = 1, NMC = 5 NMC = 1 σ =.5 I = 4. ME NMC =1, NMC = 1, NMC = 5 NMC = 1 σ =. I =. Spatial location (black dots) of the reference values of that lie outside the corresponding confidence intervals of width equal to ± two standard deviations about their mean obtained after the final updating time step. Also related to Filter inbreeding. from Panzeri et al. [14]

Field scale application data analyzed 34 Corrected drawdown.14.1.1.8.6.4. Pumping at B4 B1 B B3 Guadagnini A., Panzeri M., Riva M., Neuman S.P.. 1 1 1 1 1 Time [s] Heterogeneous fluvial aquifer in the Neckar river valley near Tübingen, Germany. The aquifer consists mainly of sandy gravel material. Prior works at the site include: a) Geostatistical analyses b) Modeling of tracer tests c) Stochastic delineation of wellprotecion zones

Field scale application ME-EnKF modeling 35 s c s c [ m] [ m].16.1.8.4. 1 1 1 1 t [ s] Pumping well: B4.16 c).1.8.4 a) B5 Pumping well: B. 1 1 1 1 t s [ ] B1 B3 B4 B5 B3 B1 B4. 1 1 1 1 t s Measurement errors are considered Gaussian with zero mean and standard deviation of 1-3 m. Conditioning data set is composed by 15 selected drawdown measurements for each well taken at 4 observation wells (e.g., B1, B3, B4, and B5 when pumping from B)..8.6.4. b) Pumping well: B3 [ ] B3 B1 B4 B5 Assimilation of drawdown data by considering pumping test in B & in B3. The initial log-transmissivity mean field is set to a constant value of = -3.86 (T is in m /s) (from previous analyses stochastic well testing). The initial covariance is stationary and set according to an exponential variogram function with sill 1.5 and integral scale.5 m (from previous analyses on vertically integrated conductivity data).

Field application prior field (pumping from B) 36 Corrected drawdown [m] Corrected drawdown [m].16.1.8.4..16.1.8.4. Observation: B1 1 1 1 1 1 1 Time [s] Observation: B4 Time [s] 1 1 1 1 1 1 Guadagnini A., Panzeri M., Riva M., Neuman S.P. Corrected drawdown [m] Corrected drawdown [m].16.1.8.4..16.1.8.4. Observation: B3 1 1 1 1 1 1 Time [s] Observation: B5 Time [s] 1 1 1 1 1 1

Field application ME- & MC-EnKF modeling V 1.6 1..8.4. N 1 i N i= 1 ( ) σ ( ) V t = x Mean field variance Mean field variance t [s] MEs NMC = 5 4 t x 197. 19. 187.4 18.6 177.8 173. 168. 163.4 B4 B4 37 158.6 158.6 163.4 168. 173. 177.8 18.6 187.4 19. 197. x1 x 1 B1 B5 B B5 B B1 B3 B3 T k = T k = 1 T k = 1 T k = 15 4 m T k = 5-5 -3-1 1.5 1 1.5

Field scale application ME-EnKF modeling 197. 19. 38 B5 B5 187.4 18.6 B4 B4 177.8 173. B B 168. B3 B3 B1 163.4 B1 158.6 158.6 163.4 168. 173. 177.8 18.6 187.4 19. 197. Mean field variance Mean field variance x x1 x 1 T k = T k = 1 T k = 1 T k = 15 4 m T k = 5-5 -3-1 1.5 1 1.5

Field scale application ME-EnKF modeling 1.6 1.5 1.4 V 1.3 1. 1.1 1...5 1. 1.5. t [ s] N 1 i N i= 1 ( ) σ ( ) V t = x t

Field application Validation (pumping from B4) 4 Corrected drawdown [m].16.1.8.4. Validation: B1 1 1 1 1 1 1 Time [s] Corrected drawdown [m].16.1.8.4. Validation: B 1 1 1 1 1 1 Time [s] Corrected drawdown [m].16.1.8.4. Validation: B3 Time [s] 1 1 1 1 1 1 Guadagnini A., Panzeri M., Riva M., Neuman S.P. Corrected drawdown [m].16.1.8.4. Validation: B5 Time [s] 1 1 1 1 1 1

Field application Validation (pumping from B4) 41 s B1 [ m].1.1.8.6.4.. 1 1 1 1 Prior After assimilation of B After assimilation of B and B3.1.1.8.6.4.. 1 1 1 1 t [ s] [ s].1.1.8.6.4.. 1 1 1 1 t t [ s] s B [ m].14.1.1.8.6.4.. Prior After assimilation of B After assimilation of B and B3.14.14.1.1.1.1.8.8.6.6.4.4.... 1 1 1 1 1 1 1 1 1 1 1 1 t [ s] [ s] t t [ s]

Field scale application Validation and reciprocity 4 Corrected drawdown / Q Corrected drawdown [m] 5 15 1 5.16.1.8.4. 1 1 1 1 1 1 Time [s] Pump from B4 Observation: B5 Assimilation set: B B3 B B3 B4 Time [s] 1 1 1 1 1 1 Corrected drawdown / Q Corrected drawdown [m] 5 15 1 5.6.4.. 1 1 1 1 1 1 Time [s] Pump from B3 Observation: B5 Assimilation set: B B4 B5 B3 B5 Time [s] 1 1 1 1 1 1

Field scale application Comparisons 43 Elevation [m ASL] 36 35.5 35 34.5 34 33.5 33 3.5 3 31.5 31 1.E-5 1.E-4 1.E-3 1.E- 1.E-1 K [m/s] B4 31 flowmeter K values from all B and F wells were locally upscaled to obtain T G =. 1 [m /s] Variance (ln T) = 1.5 Integral scale =.5 m - -4-6 -8 B1 B B3 B4 B5 multiscale geostatistical analysis Riva et al. [8] ME-EnKF Riva et al. [8] Multiscale geostatistical analysis based on impeller flowmeter data and stochastic type-curves interpretation of pumping tests Estimated = ln T Estimated = ln T - -4-6 -8 Assimilation set: B Assimilation set: B + B4 B1 B B3 B4 B5

Conclusions 44 Coupling (nonlocal) conditional MEs of groundwater flow with EnKF obviates the need for: (a) computationally intensive MC simulations; (b) batch inverse solution Combining EnKF and MEs renders the estimation of variogram parameters less critical than in batch inverse solution (not shown in details here) Increasing the log-conductivity error measurement variance by one order of magnitude brought about only a minor deterioration in the quality of the parameter estimates Assimilation of early time hydraulic head measurements resulted in an increased rate of improvement of parameter estimation without underestimating their variance.

Conclusions 45 Traditional MC-based EnKF yields reliable results when NMC 1, Filter inbreeding affects the MC simulation performed with a low number of model realization. The ME-based assimilation is not affected by filter inbreeding in the considered case. Results from field scale application are promising.

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