Stochastic methods for aquifer protection and management

Transcription

1 Stochastic methods for aquifer protection and management Dr Adrian Butler Department of Civil & Environmental Engineering G-WADI 07 International Workshop on Groundwater Modeling for Arid and Semi-arid Areas Lanzhou, China, Slide 1

2 Arid zone water resources Shibam, Yemen Slide 2

3 Why is groundwater protection important? groundwater is a vital water resource difficult to treat once contaminated out of sight, out of mind many potential sources for contamination Slide 3

4 Sources of groundwater contamination Evaporation Transpiration Ploughing Pesticides and Fertiliser Application Manure Spreading Industrial Storage/ Contaminated Land Uncovered Road Salt Oil Storage Tanks Leaking Sewers Petrol Station Landfills Urban Run Off Soil Public Water Supply Unsaturated Zone Unsaturated Zone Saturated Zone Septic Systems Water Table Soil Cone of Depression Groundwater Flow Regional Groundwater Flow Groundwater Flow Groundwater Flow Saturated Zone Impermeable Layer Slide 4

5 Groundwater protection Prevention rather than cure Implemented using Aquifer protection policies Groundwater source protection Slide 5

6 Application of stochastic modelling methods to Source Protection Zones Delineation of Source Protection Zones Using Statistical Methods FRITZ STAUFFER 1, ALBERTO GUADAGNINI 2, ADRIAN BUTLER 3, HARRIE- JAN HENDRICKS FRANSSEN 1, NEELTJE VAN DE WIEL 4, MAHMOUD BAKR 3, MONICA RIVA 2 and LAURA GUADAGNINI 2 1 Federal Institute of Technology (ETH), Zurich, Switzerland; 2 Politecnico di Milano, D.I.I.A.R., Milan, Italy; 3 of Science, Technology and Medicine,, U.K.; 4 Delft University of Technology, The Netherlands Water Resources Management (2005) 19: European research project Stochastic Analysis of Well Head Protection and Risk Assessment (W-SAHaRA), Contract No. EVK1-CT Slide 6

7 Effect of a pumping well on groundwater flow Slide 7

8 Definitions commonly used in well capture zones TERM Capture Zone Well catchment Isochrone Source Protection Zone (SPZ) Well head protection area (WHPA) DEFINITION The area around the well from which water is captured within a certain time t. Sometimes the time dependent aspect is emphasised by using the word time-related capture zone. The capture zone for infinite time. It encompasses the entire area over which the well draws the water pumped from it. Another word that can refer to both a capture zones and a well catchment is the zone of contribution (ZOC). The boundary of a capture zone. It is a contour line of equal travel time to the well. The area around a pumping well, which is associated with a specified level of protection. Its delineation is a political consideration based on a risk analysis, which may or may not involve an actual capture zone delineation. Slide 8

9 Groundwater Source Protection Source protection zones (SPZ) abstraction borehole III II I In UK: Zone I 50 day Zone II 400 day Zone III Catchment Slide 9

10 Example Source Protection Zones Slide 10

11 Simplified SPZ analysis W Q r b n c R Unconfined aquifer Slide 11

12 Simplified SPZ analysis W Q r t t = bn c W 2 R ln 2 R r 2 b n c R Unconfined aquifer Slide 12

13 Time-dependent calculation b = 10 m W = 1 mm/yr Q = 10 ML/d n c = time [d] distance [m] Slide 13

14 Sensitivity t t = bn c W 2 R ln 2 R r time [d] distance [m] Slide 14

15 Requirements for SPZ calculation Geometry of aquifer Thickness Boundaries Groundwater heads Flow field Well pumping rate Groundwater recharge Infiltration from rivers/streams Hydraulic conductivities Kinematic porosities Numerical Model Slide 15

16 Typical SPZ calculation Calculate flow field (numerically) Use particle tracking to determine travel times and trajectories Problems Groundwater flow field Neglects dispersion Deterministic V x Δt V y Δt Slide 16

17 Advection dispersion equation (n ec) * = D c (qc ) + Qc t where D is the dispersion tensor [m 2 s -1 ]: D L 0 0 D = 0 D T D T where D L = a L v and D T = a T v and a L and a T are the longitudinal and transverse dispersivities of the porous medium [m]. Slide 17

18 Macrodispersion (effects of heterogeneity) The advection-dispersion equation implies that for a uniform flow field the dispersion coefficient (i.e. the dispersivity) is a constant governed by the geometrical structure of the porous medium. However, in practice dispersivities are found to be scale dependent (i.e. the larger the area/volume this parameter is measured the greater its value). This has been markedly demonstrated by various authors (e.g. Gelhar, 1986). Slide 18

19 Examples of heterogeneity Slide 19

20 Examples of heterogeneity Distribution of hydraulic conductivity along a cross section through a glacial drift aquifer. (expressed as negative log value) (Sudicky, WRR, 22,1986) Slide 20

21 Solution If aquifer structure can be represented in detail, the scaledependence of dispersion can be represented but it is practically impossible to build up a detailed deterministic description. A stochastic approach can be used in which aquifer properties can be represented as random functions, but still preserving known data points. Numerical methods can be used in repeated simulations, sampling from the statistical distributions of properties, using a Monte Carlo approach. Such an approach would therefore incorporate: a) scale-dependent dispersion b) model uncertainty due to sparse input data Slide 21

22 Geostatistical Analysis Generally distributed spatial data are available on aquifer properties Use estimation techniques to build up a detailed picture of the distribution of properties over the whole aquifer Kriging is probably the most commonly used estimation technique in groundwater modelling Kriging is effectively a weighted estimator that uses information about the relationship between known data points to build up a best linear unbiased estimate of the field. It involves producing a statistical model of the field using a semi-variogram that describes the spatial inter-relation between field data points. Slide 22

23 Geostatisitics Let z(x) be a random space variable m ( x) = E[ z( x)] var[ z( x )] = E{[ z( x ) m( x )] 2 } C( x1,x2 ) = E{[ z( x1 ) m( x1 )][ z( x2 ) m( x2 )]} 2γ ( x1,x2 ) = var[ z( x1 ) z( x2 )] Slide 23

24 Stationarity (weak) 1. E[z(x)] = m (constant) 2. C(h) = E{[z(x) m][z(x+h) m]} covariance only depends on separation distance and not location Then, if C(0) = σ 2 variance i.e. var[z(x)] 0 C(h)dh = σ 2 I where I is the integral scale Slide 24

25 Hydraulic conductivity 1. Assume distribution of K(x) for a specific formation is approx. stationary log normal dist. [i.e. Y(x) = ln(k(x)) ] 2. Thus the mean E(Y(x)) = K g (Geometric mean) 3. The variance is σ Y 2 4. It is often assumed that the two-point covariance is an exponential relationship. Hence C Y (x 1,x 2 )=C Y (h)=σ Y2 exp(-h/i Y ) (where h = x 1 -x 2 ) Slide 25

26 Stochastic Approaches The stochastic approach views the media properties as random functions For contamination transport, hydraulic conductivity is considered as the random variable This leads to the formulation of stochastic groundwater flow equations Analytical solutions using spectral/perturbation methods Moment analyses Numerical methods using a Monte Carlo approach Slide 26

27 Numerical Stochastic Methods Generate spatially correlated hydraulic conductivity/transmissivity fields Combine random field generation with geostatistical field estimation Resulting solution can be conditioned on hydraulic head and contaminant concentration measurements Slide 27

28 Numerical Stochastic Approach View K (T) as spatiallycorrelated log-normally distributed random variable (Y=ln(K), μ Y, σ Y 2,I Y ) γ(x) Establish field structure and variable parameters x Slide 28

29 Numerical Stochastic Approach (2) Produce random K field using generation method (e.g. Turning bands, Fourier transform, Sequential Gaussian) Slide 29

30 Numerical Stochastic Approach (3) Solve for flow, simulate transport using particle tracking to generate well capture zone Well Slide 30

31 Numerical Stochastic Approach (4) Numerical methods can be used in repeated simulations, sampling from the statistical distributions of properties, using a Monte Carlo approach. γ(x) Well x Well Slide 31

32 Numerical Stochastic Approach (5) Probabilistic well capture zone obtained from many realisations Slide 32

33 Incorporation of hydraulic conductivity measurements van Leeuwen, M., te Stroot, C.B.M., Butler, A.P. and Tompkins, J.A., Stochastic determination of well capture zones, Water Resour. Res. 34, (1998), van Leeuwen, M., Butler, A.P., te Stroot, C.B.M. and Tompkins, J.A., Stochastic determination of well capture zones conditioned on regular grids of transmissivity measurements, Water Resour. Res. 36, (2000), Slide 33

34 Model simulation setup Governing flow equation x T( h h x ) + T( x ) = x y y A( x ) h T(x) A(x) hydraulic head transmissivity sources/sinks Slide 34

35 Aquifer Properties Treat transmissivity T(x) as a random space function (RSF) with following properties: Log-normal distribution Transmissivity fluctuation Y = ln(t/t G ) T G is the geometric mean Two point covariance is: C Y (h)=σ Y 2 exp( h /I Y ) where and h is the lag separation vector I Y is the integral scale σ Y2 is the variance Slide 35

36 Model grid Set up simulation grid using dimensionless variables: 2πq 2πq = x y 0 = y Q Q 2 2πq 2πq t 0 = t IY 0 = IY neq Q Δh where: q0 = TG J J L TG TG = b x 0 = Mean regional head gradient Mean transmissivity per unit aquifer thickness and Q well location x,y = 0,0 Slide 36

37 Model grid structure (not to scale) Well Slide 37

38 Unconditional result Slide 38

39 Uncertainty measurement Calculate uncertainty according to the areal integral of the resulting capture zone probability distribution (capd) proximity to P(x') = 0.5 (i.e. locations with greatest uncertainty) ω u c = f ( x ) dx x c where: f c ( x ) = 0. 5 P( x ) 0. 5 Slide 39

40 Effect of conditioning Assume that we now have a set of measured hydraulic conductivity/transmissivity values. Use these to re-calculate the probabilistic well capture zone. ω c c = f ( x ) dx x c Uncertainty change: Ω c = ω u c ω ω u c c c Slide 40

41 Conditioning: the value of data Use of transmissivity data to reduce model uncertainty of well capture zone Theoretical studies a) Regular grids b) Single points Implications for site sampling strategies Slide 41

42 Regular grid Conditioning Conditioning density α c = 2 Δx c δ where Δx = grid node separation = 0.2 and δ c = condition node separation Consider: a) effect of conditioning density on 'true' capture zone determination b) uncertainty reduction Slide 42

43 Effect of conditioning grid density on uncertainty Ω c For σ Y2 = 0.5, I Y = 1.0 and n = 50 Slide 43

44 Effect of integral scale on uncertainty Ω c Slide 44

45 Single point conditioning Investigate uncertainty contribution at each grid node for a single 'truth' (i.e. reference) field Plot uncertainty reduction as a function of position and transmissivity averaged over 100 reference fields Consider time-dependence Slide 45

46 True well capture zone Reference transmissivity field True capture zone 1.0 Well y' x' Y values σ Y 2 = 2.0 I' Y = 1.0 Slide 46

47 Uncertainty reduction due to single transmissivity measurement True capture zone Map of Ω c field for reference transmissivity field y' 0.0 Well x' Slide 47

48 Relationship with transmissivity value Graph of Ω c as a function of Y x' = -1 x' = 0 x' = 1 x' = Ωc Y Slide 48

49 As a function of time Graph of Ω c as a function of t' t'=0.5 t'=1.0 t' = 2.0 t' = Ωc x' Slide 49

50 Implications of K/T Conditioning on Well Capture Zones Heterogeneity has a major impact on uncertainty Conditioning can reduce uncertainty, but this is dependent on the aquifer's geostatistical properties Large amounts of conditioning data may only give marginal improvements in highly heterogeneous systems Uncertainty reduction from single point transmissivity measurement depends on both location and magnitude (and may result in increased uncertainty for T values >> T G ) Development of sampling strategies for source protection Slide 50

51 Worth of head/transmissivity data in well capture zone estimation Bakr, M.I. and Butler, A.P., Worth of head data in well capture zone design; deterministic and stochastic analysis. Journal of Hydrology, (2004), 290, Slide 51

52 Head conditioning methods Sequential self-calibrated method Multiple realisations of hydraulic conductivity field Solve groundwater flow problem Compare results against observed heads Perturb K values fit observed heads Representer method Follow stages 1 and 2 as above Calculate objective function (OF) from head values Minimise OF using adjoint of flow equation and state variables Slide 52

53 Problem setup Mean Y= 0 σ Y2 = 2.0 I = 2.0 q w =-0.16 Exponential covariance 5000 realizations Constant head boundary, 10 (L) Pumping well No flow boundary Constant head boundary, 10.5 (L) No flow boundary Slide 53

54 Deterministic Analysis Slide 54

55 Effect of Calibration on Heads meas 35 meas meas Reference head field versus calibrated. Left maps are for variance {Y}=0.5 and right maps are for variance of {Y}=2. Solid lines are the reference contours and the dashed lines are the calibrated contours. Red filling is the reference capture zone and +" is the mean capture zone for the calibrated cases. Slide 55

56 Calibration using h & T measurements 35 T measurements 186 T measurements h & 35 T measurements 186 h & 186 T measurements Reference versus calibrated head and mean well capture zone for variance of {Y}=2 ( post-pumping calibration). Solid lines are the reference contours and the dashed lines are the calibrated contours. "+" is the reference capture zone and "o" is mean capture zone for the calibrated cases. Slide 56

57 Effect of pumping rate m 80 m 40 m ωu ωc α = ω u α ω = i CAPr CAP r CAPi CAP Pumping rate (m 3 /day) Effect on uncertainty reduction α due to changes in pumping rate and sampling density. Slide 57

58 Summary For low variability of Y it is possible to delineate mean well capture zones to a good accuracy using head data only. Higher variability of Y results in a less accurate delineation of well capture zones. Combining h and Y measurements in calibrating the spatially variable Y leads to an improved delineation of the mean well capture zones, especially for highly variable fields. It is observed that, although calibration using Y alone did not produce hydraulic gradients comparable to those for the reference case, it was more successful in producing the reference well capture zone. Thus indicating the greater importance of Y measurements over h data in well capture zone design for moderate pumping rate. Higher abstraction rates increase the importance of h data in calibrating Y. They also demonstrate the improvement in the calibrated capture zone that increasing h measurement densities provide. Slide 58

59 Stochastic analysis Slide 59

60 Effect of number of h measurements 40-T capture zone probabilities for: a) unconditional Y. b) Conditioned Y using 9 head measurements c) Conditioned Y using 48 head measurements d) Conditioned Y using 176 head measurements. Contours shown are 2.5%, 50% and 97.5%. Filling red is the 40-T reference capture zone. 10 (a) (b) 12 (c) (d) Slide 60

61 Effect of Y versus h measurements T well capture zone probabilities for conditional Y on 48 Y measurements (left) and 48 Y measurements and 48 Y and h measurements (right). Circles indicates measurements locations. Slide 61

62 Effect of Y versus h measurements, cont. 2.50% 97.50% 95% confidence interval Unconditional h measurements h measurements Y measurements h and Y measurements h measurements Area of 95% confidence interval for different number of h and/or Y measurements. Slide 62

63 Summary Uncertainty of well capture zones decrease with increasing measurement density. This increase stagnates beyond a threshold value of conditioning density. In the scenarios considered, a combination of 48 h and Y data could reduce the area of the 95% confidence interval in well capture zone location by over 50% (compared to a 40% reduction using either h or Y data). This performance was comparable to that obtained through the calibration on three and a half times the number of head observations alone. Slide 63

64 Effects of uncertainty in recharge Spatially varying recharge (known mean) Uncertainty in mean value Synthetic problem Recharge h = 10 m 10 km No flow Well No flow Unconfined aquifer h = 0 m 10 km Slide 64

65 Synthetic study 9000 (A) 9000 (B) 9000 (C) y (m) 5000 y (m) 5000 y (m) WELL 3000 WELL 3000 WELL x (m) x (m) x (m) Ensemble averaged well capture zones with capture probabilities for (a) spatial variable recharge with σ R = 79 mm/year, (b) uncertain mean recharge with σ R =79 mm/year and (c) no uncertainty of recharge. In all cases σ Y2 =0.2 and I R =2000 m. Slide 65

66 Results Table 5 Effect of uncertainty in the mean recharge, as compared to the effect of an equivalent uncertainty in recharge due to spatial variability, on the capture zone uncertainty expressed by AESD(CZ). Simulations results are given for three different degrees of transmissivity heterogeneity. As reference also the capture zone uncertainty is given in case the recharges are perfectly known. σ R =32 mm/year (mean) σ R =32 mm/year (spatial) σ R =79 mm/year (mean) σ R =79 mm/year (spatial) Without recharge uncertainty σ 2 Y = σ 2 Y = σ 2 Y = Slide 66

67 Summary Impact of spatially varying recharge on SPZ probabilities appears not to be too dramatic However, the impact of uncertainty in the mean is. This has important implications for arid-zone regions where recharge determination is problematic and, therefore, often highly uncertain. Slide 67

68 Effects of uncertainty in boundary conditions No flow Recharge h = 0 m h = 5 ± Δhm 5 km Well Unconfined aquifer No flow 5 km Slide 68

69 Results (A) (B) Ensemble averaged well capture zones with capture probabilities for (a) reference field, (b) unconditional simulations, (c) conditioning to 10 transmissivity and head data and no uncertainty in the boundary conditions, (d) conditioning to 10 transmissivity and head data and uncertainty in the boundary conditions. y (m) y (m) WELL x (m) WELL (C) y (m) y (m) WELL x (m) WELL (D) x (m) x (m) Slide 69

70 Semi-analytical analyses Calculate velocity co-variance from knowledge of spatial statistics for transmissivity field Apply these to particles tracked from deterministic flow field Construct longitudinal and transverse second moments of particle displacement Slide 70

71 Method comparison y [m] 800 y [m] x [m] x [m] Cumulative probability distribution for well catchment for unconfined problem. Left: semi-analytical solution. Right: MC results. Isolines Γ(0.023), Γ(0.5) and Γ(0.977) are shown. Slide 71

72 A methodology for stochastic SPZs Identify groundwater abstraction and potential sources of pollution No Delineate capture zones Have well capture/ catchment zones been demarcated? Yes Method of derivation Simple analytical solution (e.g. Bear and Jacobs) Existing numerical model (2/3D) e. g. MODFLOW, MODPATH Is there a need to quantify model uncertainty? Yes Collect and/or collate data No STOP Develop new/modified numerical model (2D/3D) Slide 72

73 Stochastic SPZs (cont.) Probabilistic analysis: Identify sources of uncertainty (e.g. conceptual model, hydrogeological parameters, boundary conditions) Undertake preliminary unconditional simulation Incorporate data for conditional simulation Select method 2 Select type and location of additional input measurements 1 Monte Carlo analysis Moment Equation analysis Yes Reduce uncertainty No STOP Slide 73

74 Challenges for Arid-zone groundwater resources Abstraction wells are vital resources. Need to ensure these are protected against contamination. Achieved by delineation of SPZs. However, information required to calculate these are uncertain. Consequently, incorporation of uncertainty into SPZ calculation is required A range of methods and tools available to allow this to be achieved. Slide 74