1. Groundwater. Notes on geostatistics Monica Riva, Alberto Guadagnini Politecnico di Milano, Italy Key reference: de Marsily, G. (1986), Quantitative Hydrogeology. Academic Press, New York, 440 pp

2. In practice: random spatial variability of hydrogeologic medium properties, and stochastic nature of corresponding flow (hydraulic head, fluid flux and velocity) and transport (solute concentration, solute flux and velocity) variables, are often ignored. Instead, the common approach has been to analyse flow and transport in multiscale, randomly heterogeneous soils and rocks deterministically. Yet with increasing frequency, the popular deterministic approach to hydrogeologic analysis is proving to be inadequate.    Modelling flow and transport in heterogenous media motivation and general idea

3. Understanding the role of heterogeneity Jan 2000 editorial "It's the Heterogeneity!“ (Wood, W.W., It’s the Heterogeneity!, Editorial, Ground Water, 38(1), 1, 2000): heterogeneity of chemical, biological, and flow conditions should be a major concern in any remediation scenario. Many in the groundwater community either failed to "get" the message or were forced by political considerations to provide rapid, untested, site- specific active remediation technology. "It's the heterogeneity," and it is the Editor's guess that the natural system is so complex that it will be many years before one can effectively deal with heterogeneity on societally important scales. Panel of experts (DOE/RL-97-49, April 1997): As flow and transport are poorly understood, previous and ongoing computer modelling efforts are inadequate and based on unrealistic and sometimes optimistic assumptions, which render their output unreliable.

4. Flow and Transport in Multiscale Fields (conceptual) conductivities & dispersivities varyscale of observation Field & laboratory-derive conductivities & dispersivities appear to vary continuously with the scale of observation (conductivity support, plume travel distance). Anomalous transport. theorieslinkscale-dependencemultiscale structure Recent theories attempt to link such scale-dependence to multiscale structure of Y = ln K. Predicteffect of domain size Predict observed effect of domain size on apparent variance and integral scale of Y. Predictsupra linear growth rate of dispersivity Predict observed supra linear growth rate of dispersivity with mean travel distance (time). Major challenge Major challenge: develop more powerful/general stochastic theories/models for multiscale random media, and back them with lab/field observation.

5. Shed some light Conceptual difficulty: Data deduced by means of deterministic Fickian models from laboratory and field tracer tests in a variety of porous and fractured media, under varied flow and transport regimes. Linear regression: a La  0.017 s 1.5 Supra-linear growth Neuman S.P., On advective transport in fractal permeability and velocity fields, Water Res. Res., 31(6), 1455-1460, 1995.

6. Natural Variability. Geostatistics revisited Introduction: Few field findings about spatial variability Regionalized variables Interpolation methods Simulation methods

7. AVRA VALLEY Clifton and Neuman, 1982 Clifton, P.M., and S.P. Neuman, Effects of Kriging and Inverse Modeling on Conditional Simulation of the Avra Valley Aquifer in southern Arizona, Water Resour. Res., 18(4), 1215-1234, 1982. Regional Scale

8. Columbus Air Force [Adams and Gelhar, 1992] Aquifer Scale

9. Mt. Simon aquifer Bakr, 1976 Local Scale

10. Summary: Variability is present at all scales But, what happens if we ignore it? We will see in this class that this would lead to interpretation problems in both groundwater flow and solute transport phenomena Examples in transport: - Scale effects in dispersion - New processes arising Heterogeneous parameters: ALL (T, K,, S, v (q), BC,...) Most relevant one: T (2D), or K (3D), as they have been shown to vary orders of magnitude in an apparently homogeneous aquifer

11. Variability in T and/or K Summary of data from many different places in the world. Careful though! Data are not always obtained with rigorous procedures, and moreover, as we will see throughout the course, data depend on interpretation method and scale of regularization Data given in terms of mean and variance (dispersion around the mean value)

12. Variability in T and/or K Almost always σ lnT (or σ lnK ) < 2 (and in most cases <1) This can be questioned, but OK by now Correlation scales (very important concept later!!)

13. But, what is the correct treatment for natural heterogeneity? First of all, what do we know? - real data at (few) selected points - Statistical parameters - A huge uncertainty related to the lack of data in most part of the aquifer. If parameter continuous (of course they are), then the number of locations without data is infinity Note: The value of K at any point DOES EXIST. The problem is we do not know it (we could if we measured it, but we could never be exhaustive anyway) Stochastic approach: K at any given point is RANDOM, coming from a predefined (maybe known, maybe not) pdf, and spatially correlated ------ REGIONALIZED VARIABLE

14. Regionalized Variables T(x,ω) is a Spatial Random Function iif: -If ω = ω 0 then T(x,ω 0 ) is a spatial function (continuity?, differentiability?) -If x = x 0 then T(x 0 ) (actually T(x 0, ω)) is a random function Thus, as a random function, T(x 0 ) has a univariate distribution (log-normal according to Law, 1944; Freeze, 1975)

15. Hoeksema and Kitanidis, 1985

16. Hoeksema & Kitanidis, 1985 Log-T normal, log-K normal Both consolidated and unconsolidated deposits

17. Now we look at T(x), so we are interested in the multivariate distribution of T(x 1 ), T(x 2 ),... T(x n ): Most frequent hypothesis: Y=(Y(x 1 ), Y(x 2 ),... Y(x n ))=(ln T(x 1 ), ln T(x 2 ),... ln T(x n )) Is multinormal with But most important: NO INDEPENDENCE

18. What if independent? and then we are in classical statistics But here we are not, so we need some way to characterize dependency of one variable at some point with the SAME variable at a DIFFERENT point. This is the concept of the SEMIVARIOGRAM (or VARIOGRAM)

19. Classification of SRF Second order stationary E[Z(x)]=const C(x, y) is not a function of location ( only of separation distance, h ) Particular case: isotropic RSF; C(h) = C(h) Anisotropic covariance: different correlation scales along different directions Most important property: if multinormal distribution, first and second order moments are enough to fully characterize the SRF multivariate distribution

21. Relaxing the stationary assumption 1. The assumption of second-order stationarity with finite variance, C(0), might not be satisfied (The experimental variance tends to increase with domain size) 2. Less stringent assumption: INTRINSIC HYPOTHESIS The variance of the first-order increments is finite AND these increments are themselves second-order stationary. Very simple example: hydraulic heads ARE non intrinsic SRF E[Y(x + h) – Y(x)] = m(h) var[Y(x + h) – Y(x)] =  (h) Independent of x; only function of h Usually: m(h) = 0; if not, just define a new function, Y(x) – m(x), which satisfies this consition Definition of variogram,  (h) E[Y(x + h) – Y(x)] = 0  (h) = (1/2) var[Y(x + h) – Y(x)] = (1/2) E[(Y(x + h) – Y(x)) 2 ]

22. Variogram v. Covariance 1. The variogram is the mean quadratic increment of Y between two points separated by h. 2. Compare the INTRINSIC HYPOTHESIS with SECOND-ORDER STATIONARITY E[Y(x)] = m = constant  (h) = (1/2) E[(Y(x + h) – Y(x)) 2 ] = = (1/2) ( E[Y(x + h) 2 ] + E[Y(x) 2 ] – 2 m 2 – 2 E[Y(x + h) Y(x)] + 2 m 2 ) = = C(0) – C(h) covariance variogram h

23. The variogram The definition of the Semi-Variogram is usually given by the following probabilistic formula When dealing with real data the semi-variogram is estimated by the Experimental Semi-Variogram. For a given separation vector, h, there is a set of observation pairs that are approximately separated by this distance. Let the number of pairs in this set be N(h). The experimental semi-variogram is given by:

25. Some comments on the variogram If Z(x) and Z(x+h) are totally independent, then If Z(x) and Z(x+h) are totally dependent, then One particular case is when x = x+h. Therefore, by definition In the stationary case:

26. Variogram Models DEFINITIONS: Nugget Sill Range Integral distance or correlation scale Models: Pure Nugget Spherical Exponential Gaussian Power

28. Correlation scales: Larger in T than in K. Larger in horizontal than in vertical. Fraction of the domain of interest

29. Additional comments Second order stationary E[Z(x)]=constant  (h) is not a function of location Particular case: isotropic RSF  (h) =  (h) Anisotropic variograms: two types of anisotropy depending on correlation scale or sill value Important property:  (h) =  2 – C(h) Most important property: if multinormal distribution, first and second order moments are enough to fully characterize the SRF multivariate distribution

30. Estimation vs. Simulation Problem: Few data available, maybe we know mean, variance and variogram Alternatives: (1) Estimation (interpolation) problems: KRIGING Kriging – BLUE Extremely smooth Many possible krigings Alternative: cokriging http://www-sst.unil.ch/research/variowin/

31. The kriging equations - 1 We want to predict the value, Z(x 0 ), at an unsampled location, x 0, using a weighted average of the observed values at N neighboring locations, {Z(x 1 ), Z(x 2 ),..., Z(x N )}. Let Z*(x 0 ) represent the predicted value; a weighted average estimator be written as The associated estimation error is In general, we do not know the (constant) mean, m, in the intrinsic hypothesis. We impose the additional condition of equivalence between the mathematical expectation of Z* and Z 0.

32. The kriging equations - 2 This condition allows obtaining an unbiased estimator. Unknown mathematical expectation of the process Z.

33. The kriging equations - 3 We wish to determine the set of weights. IMPOSE the condition

34. The kriging equations - 4 We then use the definition of variogram THEN: Which I will use into:

35. The kriging equations - 5 By substitution We finally obtain: Noting that:

36. The kriging equations - 6 This is a constrained optimization problem. To solve it we use the method of Lagrange Multipliers from the calculus of variation. The Lagrangian objective function is To minimize this we must take the partial derivative of the Lagrangian with respect to each of the weights and with respect to the Lagrange multiplier, and set the resulting expressions equal to zero, yielding a system of linear equations

37. The kriging equations - 7 Minimize this: and get (N+1) linear equations with (N+1) unknowns

38. The kriging equations - 8 The complete system can be written as: A = b

39. The kriging equations - 9 We finally get the Variance of the Estimation Error Hint: just replace into

40. Estimation vs. Simulation (ii) (2) Simulations: try to reproduce the “look” of the heterogeneous variable Important when extreme values are important Many (actually infinite) solutions, all of them equilikely (and with probability = 0 to be correct) For each potential application we are interested in one or the other

41. Estimation. 1 AVRA VALLEY. Regional Scale - Clifton, P.M., and S.P. Neuman, Effects of Kriging and Inverse Modeling on Conditional Simulation of the Avra Valley Aquifer in southern Arizona, Water Resour. Res., 18(4), 1215-1234, 1982.

42. Estimation. 2 AVRA VALLEY. Regional Scale - Clifton, P.M., and S.P. Neuman, Effects of Kriging and Inverse Modeling on Conditional Simulation of the Avra Valley Aquifer in southern Arizona, Water Resour. Res., 18(4), 1215-1234, 1982.

43. Estimation. 3 AVRA VALLEY. Regional Scale - Clifton, P.M., and S.P. Neuman, Effects of Kriging and Inverse Modeling on Conditional Simulation of the Avra Valley Aquifer in southern Arizona, Water Resour. Res., 18(4), 1215-1234, 1982.

44. Estimation. 4 AVRA VALLEY. Regional Scale - Clifton, P.M., and S.P. Neuman, Effects of Kriging and Inverse Modeling on Conditional Simulation of the Avra Valley Aquifer in southern Arizona, Water Resour. Res., 18(4), 1215-1234, 1982.

45. Estimation. 5 AVRA VALLEY. Regional Scale - Clifton, P.M., and S.P. Neuman, Effects of Kriging and Inverse Modeling on Conditional Simulation of the Avra Valley Aquifer in southern Arizona, Water Resour. Res., 18(4), 1215-1234, 1982.

46. Monte Carlo approach h1h1h1h1 h2h2h2h2 h 2000...... 2000 simulations Statistical CONDITIONAL moments, first and second order CONDITIONAL CROSS-CORRELATED FIELDS Y = lnT

47. NUMERICAL ANALYSIS - MONTE CARLO Simple to understand Applicable to a wide range of linear and nonlinear problems High heterogeneities Conditioning  Heavy calculations  Fine computational grids  Reliable convergence criteria (?)  Evaluation of key statistics of medium parameters (K, porosity, …)  Synthetic generation of an ensemble of equally likely fields  Solution of flow/transport problems on each one of these  Ensemble statistics

48. Hydraulic head variance Number of Monte Carlo simulations Problems: reliable assessment of convergence – Ballio and Guadagnini [2004]