Heterogeneity: Why important? What to do?

Transcription

1 Heterogeneity: Why important? What to do? Jesus Carrera Institut Jaume Almera (IJA) for Earth Sciences Higher Council for Scientific Research (CSIC) Barcelona, Spain 1

2 Transport Effect of space and time variability How to characterize. 2

3 Solute transport Advection: v = q/φ (q proportional to K) Dispersion: Proportional to: α q Reactions Mass Conservation c c c f r φ = ( D ) q + t Porosity Dispersion coeff. Water flux Reactions 3

4 Solution of ADE c ( x,t) c 1/ 2 ( 2π ) = + 0 ( ) 18 u e 1 x Vt y + z exp + V d 2 2tDL 2tDT Initial Pulse At early times, little displacement, significant dilution and spreading Later on, dilution and spreading continue, but displacement becomes apparent 4

5 Dispersion and advection scales Longitud (-) Longitud LD LA = = vt 2αvt Tiempo (-) Tiempo c max M M = = φ 2 πdt / φ φ LD Dispersion dominates small distances and times, but later advection becomes dominant. Everything is controlled by D, K and 5 φ

6 Field data Does all the above work? Is transport defined by K, porosity and dispersivity? 6

7 Scale dependence of Hyd. Conductivity (FEBEX) log K (m s -1 ) PULSE HORNER CROSS-HOLE GLOBAL MOD. Fractures Rock Matrix One by one test Inflow SCALE OF INFLUENCE (m) 7

8 Scale dependence of dispersivity Data from tracer tests and pollution plumes worldwide Dispersivity grows with test scale (Lallemand-Barres y Peandecerf, 1978 ) 8

9 Kinematic porosity function of residence time 0 log b (m) Dolomita 5 6 log time (min) Hard rocks Creta pendiente 0.5 Efective porosity in fractured rocks appears to increase with residence time 9

10 Transport predictions are awful!!! Good calibration Good calibration El Cabril (UPC, 1999) 10

11 Reaction rates Observed reaction rates are 2-3 orders of magnitude slower than expected from measurements Physical surface area (log m 2 /kg water) Bruton (1986) White (1986) Liu (1987) Hurd (1973) Herman & Lorah (1987) Delany (1985) Velbel (1985) Claassen & White (1973) Paces (1983) Gislason & Eugster (1987) (White & Peterson,1990) Reactive surface area (log m 2 /kg water) 11

12 The problem MVT deposit outflow Expected Diss/Precip Results Conservative species Resident water Dissolution (Corbera et al, 2003 ) Precip. Brine inflow BUT, Do we know how to simulate mixing? 12

13 1st conclusion: ADE full of problems Scale dependence of K Scale dependence of dispersivity Time (and space, and direction) dependence of porosity Poor prediction of space distribution and time evolution of c and reaction THE CULPRIT: Variability Heterogeneity of K (hydraulic conductivity), and minerals, and... (BUT ESPECIALLY K) Temporal fluctuations of water flux 13

14 Random functions Random variable: concept used for describing variables with an uncertain value e.g.: result from throwing a dice z(ξ) z(ξ) z(ξ 1 )=3, z(ξ 2 )=5, z(ξ 3 )=2, z(ξ 4 )=4, ξ 1, ξ 2, ξ 3, are realizations Random functions: Functions whose values are random variables; or Random entities whose realizations are functions: z(x,ξ) z(x,ξ 1 ) z(x,ξ2 ) z(x,ξ 5 ) 14

15 Random functions need spatial correlation Pdf is not sufficient The uncorrelated field will behave as a homogeneous field (can be treated as constant) Stochastics needed when spatial (and/or temporal) structure (often characterized by correlation function) In passing, this is why recharge is usually viewed as constant (a view that may need to be revised after Howard s presentation) 15

16 Dealing with heterogeneity Deterministic (but uncertain) methods Try to describe reality from geology Stochastic methods Describe reality as one realization of a random function. In practice, work with an ensemble of realizations in the hope that some of them are similar to reality or, rather, that jointly describe the actual uncertainty. 16

17 Stochastic methods Motivation: Deal with variability Procedure: 1) Characterize (stochastically) medium variability of K, minerals, surfaces,... 2) Generate n spatially variable fields 3) Solve them 4) Compute output statistics Solve analytically or numerically 17

18 Methods to describe spatial variability Traditional (stationary) geostatistics Heterogeneity defined by pdf and variogram (or correlation function) Tools: kriging, conditional simulation Sophisticated simulation methods (multipoint geostatistics, Markov chain, etc) Genetic models Very powerful free codes (GSLIB, GEMS) available atstanford web page 18

19 Kriging vs conditional simulation Estimated Y field 1.0 Y Reference X Y field Kriging is an interpolation method. Does not reproduce small scale heterogeneity. 6 1 Y X

20 Traditional (stationary) geostatistics Problem: requires many data or highly uncertain and does not look real 20

21 PluriGaussian Model Problem: requires many trials to be realistic 21

22 Genetic models Try to simulate geological processes. Quite powerful for sedimentary formations 22

23 Genetic models Problem: lots of data, assumptions and CPU, hard to condition to real data 23

24 Conditioning on hydraulic data Problem statement: Find K(x) that Honors K(x i ) point measurements of K Reproduces h m (x i,t), head observations Inverse modelling 24

25 Parametrization Express model properties, which may vary in space and time, as a function of a few unknown scalars (MODEL PARAMETERS) Y Zonation p 1 p 2 p 3 N q 0 i i i= 1 q( x,t) = q ( x,t) + p α ( x,t) x Y Interpolation Y Small scale variability p 1 p 2 p 3 p 1 p 2 p 3 x x 25

26 Optimum parameters defined as minimum of objective function h(p) p 2 h(p 1 ) h(p 0 ) p 0 h(p) t Contour lines of objective function: p 2 p 1 h(p 2 ) t Sum (h-h m ) 2 p 1 26

27 Generic optimization method 1) Initialization: Read input data, i=0, p 0 h(p) h(p 2 ) 2) Solve the simulation problem, h(p i ) 3) Compute an updating vector, d. 4) Update parameters, p i+1 = p i + d. p 2 h(p 1 ) h(p 0 ) t 5) If convergence, stop. Otherwise, i=i+1 and go to 2 p 0 Contour lines of objective function p 2 p 1 p 27 1

28 Step 1: Random Field Example: Stochastic inversion Step 2: Simulate drawdowns from three pump tests to be used as data 28

29 A synthetic example of hydraulic tomography B B Data: Drawdowns at 13 obs. Points, caused by three pumping tests Point measurements of T at 13 points O-1 B-1 B-2 O-2 Estimate assuming: B O-9 B-3 O-4 O-7 O-6 O-3 O-10 O-8 O-5 Known variogram Varying number of pilot points, whose prior estimates come from kriging 29

30 Method Given: - Local T data (e.g. Pulses, sp. capacity..) - Cross hole drawdowns - Conceptual model (i.e., B.C. s, geost,...) Find T field such that: - reproduces cross hole drawdowns - reproduces local T data Approach: - maximize likelihood - adjoint state method 30

31 10 Conditioning on hydraulic data O-1 O-9 B-1 B-3 O-4 O-7 O-3 O-6 B-2 O-10 O-2 O-8 O-5 Joint interpretation 10 of 3 pumping tests Drawdown (m) Observation point B Time (s) Observation point O-5 N p = 41; RMSE d = 0.8 N p =241; RMSE d = 0.6 Measured drawdown O-1 O-9 B-1 B-3 O-4 O-7 O-3 Drawdown (m) O B-2 O Time(s) O-2 O-8 O-5 Observation point O-10 Observation point O-8 N p = 41; RMSE d = 1.29 N p =241; RMSE d = 1.19 Measured drawdown Perfect fit Drawdown (m) 1 Drawdown (m) N p = 41; RMSE d = 2.1 N p =241; RMSE d = 1.1 Measured drawdown N p = 41; RMSE d = 1.9 N p =241; RMSE d = 0.9 Measured drawdown Time(s)

32 Hydraulic data insufficient for unequivocal identification λ=.001 Opt. λ λ = 100 N p =41 N p =65 μ= 10 2 μ= 10 1 Few pilot points do not allow identifying narrow features Small weight to prior information leads to unstable results, but provides food for thought N p =97 N p =241 μ= 10 1 Too much weight to prior information leads to stable but boring results Many parameters (p.p.) OK if properly regularized μ=

33 Blind prediction of tracer tests Each simulation reproduces: Spatial variability Tailing Local concentrations 33

34 Conditioning on geophysical data - Problem statement: Find K(x) that Honors K(x i ) point measurements of K Correlates with geophysical image Methods: External drift Cokriging (cosimulation) Qualitative conditioning Any hope to include basic geological (genetic) info? 34

35 In practice, qualitative use 35

36 Fractured rock site (hydro calibration) Descenso (m) Descenso (m) Descenso (m) SR4-1 (punto de bombeo) Falla Sur - S10 Fractura SR4-2 (Unidad alterada) Tiempo (s) S10 (Fractura S10) Tiempo (s) North Fault 474 Fault South Fault S ` Dike Sr4 F.S10 M I N E Sr1 S5 F.SR1-3 PM 27 Dike 285 Fault Sr3 S5 (Frac.285, 27' Dique) Tiempo (s) S14 Sr3 Dike Sr2 Y Z X Descenso (m) Descenso (m) Descenso (m) PM (Mina) Tiempo (s) SR1-1 (Bandas fracturadas) SR1-2 (Dique 27) SR1-3 (SR1-3 Fractura) SR1-4 (Unidad alterada) Tiempo (s) 36

37 Qualitative integration of geophysics facilitates reasonable long term prediction Descenso (m) Descenso (m) 10 S14: Matriz Externa S10: S10 Fractura SR4-3: Unidad alterada 5 10 SR4-2: Matriz Mina 5 10 SR4-1:Falla Sud - S10 Frac Time (s) PM: Mina (Punto Bombeo) Tiempo (s) Descenso (m) Descenso (m) S14 Filon Sr3 F. Norte Filon 27 Filon 27- Sr2 PM F. 474 M I N A Sr3 Sr1 S5 S10 Sr4 F. Sur 15 SR1-1: 27 Dique SR1-2: Matriz Mina SR1-3: Unidad alterada 5 F. 285 Y X Z Descenso (m) 10 SR2-1: Falla Norte 5 10 SR2-2: Unidad alterada SR3-1: Falla Norte SR3-2: Matriz Mina SR3-3: Unidad alterada S5: 285 Falla - 27' Dique Tiempo (s) Tiempo (s) 37

38 Conditioning on geological data - Geostatistics: Facies simulations Honors point textural information Correlates with known patterns Methods: Traditional geostatistics Indicator simulation Multiple Point Geostatistics (training imag) Transition Probabilities Any hope to include basic geological (genetic) info? (i.e., nonstationarity) 38

39 SUMMARY Spatial variability needed to understand observations (at least, transport observations) Very active research field Work still needed on: Refinement in geophysics (shall we just keep working with images?) Methodological developments in geostatistics (process- Vs. Structure- imitating) Joint inversion Use history of the Earth (i.e., geology) info. 39