Geohydrology ii (1)

Geohydrology IIGeohydrology IIWeek onWeek on

Subsurface Characterization Subsurface Characterization and and

Transport ProcessesTransport ProcessesBy By

Dr. Ir. Amro ElfekiDr. Ir. Amro Elfeki

Time TableTime Table

• 16/02/2004 16/02/2004 Characterization of Subsurface Heterogeneity Characterization of Subsurface Heterogeneity

by Elfeki. 10:30-12:30, Room 4.99by Elfeki. 10:30-12:30, Room 4.99

• 17/02/200417/02/2004Transport Processes in Porous Media Transport Processes in Porous Media

by Elfeki.8:30 -10:30. Room 4.99by Elfeki.8:30 -10:30. Room 4.99

• 20/02/200420/02/2004Incorporating Uncertainty in Groundwater Flow and Incorporating Uncertainty in Groundwater Flow and Transport Models.Transport Models.

by Elfeki.8:30 -10:30. Room 3.99by Elfeki.8:30 -10:30. Room 3.99

Lecture (1)Lecture (1)

Characterization of Subsurface Heterogeneity

Lecture (1)Lecture (1)

1. How can we model heterogeneity? 2. How can we extend the information from boreholes?3. How heterogeneity influences flow and transport?

Layout of the LectureLayout of the Lecture• Field Observations.Field Observations.

• Some methods of Modelling Heterogeneity Some methods of Modelling Heterogeneity ( Site Characterization).( Site Characterization).

• Gaussian Random Fields.Gaussian Random Fields.

• Markovian Random Fields.Markovian Random Fields.

• Applications.Applications.

Trench Trench

Hanford Formation, southeast Washingtonby Blume, et al. (2002).

Observation: Layers of coarse and fine sediments.

OutcropsOutcrops

Outcrop with Cross-Bedding: by Skinner and Porter. (1995).

Observation: Large-scale stratification with fine scale cross-bedding

Space Series Data Space Series Data

Space series from Mount Simon sandstone aquifer: Gelhar, (1996).

Laboratory Measurements: Permeability and porosity.

Observation: Variability of hydrological parameters.

Subsurface HeterogeneitySubsurface Heterogeneity

• One can easily experience the heterogeneity from One can easily experience the heterogeneity from most fields by observing huge variation of its most fields by observing huge variation of its properties from point to point properties from point to point (Gelhar, 1993) (Gelhar, 1993)

• The heterogeneity of subsurface has been a long-The heterogeneity of subsurface has been a long-existing troublesome topic from the very beginning existing troublesome topic from the very beginning of the subsurface hydrology of the subsurface hydrology (Anderson, 1983)(Anderson, 1983)

Deterministic, Random, or Stochastic?Deterministic, Random, or Stochastic?

Purely random?

No Regularity

Pure Random Process

Purely deterministic?

Deterministic Regularity

Pure Deterministic Process

Something in between?

Stochastic Regularity

Stochastic Process

Deterministic ApproachDeterministic Approach

0 200 400 600 800 1000 1200 1400 1600-15

-10

-5

0

From boreholes, geologists construct the geological cross-section utilizing geological experience and technical background from practitioners. The produced image is considered as a deterministic one.

The final user of that image may rely on it as a subjectively certain picture of the subsurface.

Stochastic ApproachStochastic Approach

• The word stochastic has its origin in the Greek The word stochastic has its origin in the Greek adjective adjective στoχαστικoςστoχαστικoς which means skilful at aiming which means skilful at aiming or guessing. or guessing.

• Mathematical models that employ stochastic Mathematical models that employ stochastic methods occupy an intermediate position in the methods occupy an intermediate position in the spectrum of dynamic models. spectrum of dynamic models.

Pure Deterministic

Pure Random

Stochastic Model

Why do we need the Stochastic Approach?Why do we need the Stochastic Approach?

-The uncertainty due to -The uncertainty due to the lack of information the lack of information about the subsurface about the subsurface structure which is known structure which is known only at sparse sampled only at sparse sampled locations.locations.

-The erratic nature ofThe erratic nature of the subsurface the subsurface parameters observed parameters observed at field scale.at field scale.

Scales of Natural Variability Scales of Natural Variability (The Hierarchical Structure of Heteogeneity)(The Hierarchical Structure of Heteogeneity)

1. Microscopic Scale: micrometer to mmNavier-Stokes Eqn.2. Macroscopic Scale:Soil samples 1-10 cm(REV) Darcy’s law.3. Megascopic Scale:Geological units with different REV Characteristics, 1m- 1 Km. 4.Gigascopic Scale: Regional geological features (Aquifer Size and depositional environments (1-10 Km)

GeostatisticsGeostatistics

Kriging (stochastic Kriging (stochastic interpolation)interpolation)

Gaussian Random Gaussian Random FieldField

Non-Gaussian Random Non-Gaussian Random FieldField

Simulation of Sedimentary Depositional

Process

a priori knowledge

sedimentary history

geometry of sedimentary structure

Site Specific Information

a priori knowledge

well logs

geophysical data

Koltermann and Gorelick (1996)

Facts about Each MethodFacts about Each Method

• DescriptiveDescriptive– Quantification is difficultQuantification is difficult

• Process-ImitatingProcess-Imitating– Conditioning is difficult, too sensitive to Conditioning is difficult, too sensitive to

initial condition, and computationally initial condition, and computationally demandingdemanding

• Structure-ImitatingStructure-Imitating– Lateral variability data is hard to getLateral variability data is hard to get– Produce multiple, equally probable imagesProduce multiple, equally probable images

Gaussian Random FieldsGaussian Random Fields

0 20 40 60 80 100 120 140 160 180 200

-40

-20

0

-3 .3 -2 .3 -1 .3 -0 .3 0.7 1.7 2.7

Y=Log (K )

Random Field:is a collection of random variables.Gaussian: the random variables follow a Gaussian distribution.

Spatial Auto-CorrelationSpatial Auto-Correlation It is a measure of the spatial correlation structure of a process.

2

( )

1

( ), ( )( )

1( ( ), ( )) ( ) - ( ) -( )

ZZZ

n

i i i j i i j ij

Cov Z Z

Cov Z Z Z Z Z Zn

s

x + s xs

x + s x + sx xs

The auto-correlation function has the following properties:

)(- = )(0)(

1)0(

ZZ ss

ZZ

ZZ

ZZ

= =

ij

X

Y

0

ZZ

ij

1

p

2 3

Calculation of Auto-Correlation Function Calculation of Auto-Correlation Function (1D)(1D)

σ

sCov = sρ

Z - iZ Z - siZsn

sCov

Z

xxZZ

sn

i=x

xx

x

2

)(

1

)()(

)()()(

1)(

Some Auto-Correlation of SeriesSome Auto-Correlation of Series

Auto-correlation ModelsAuto-correlation Models

Auto-correlation models

2

)(

)(

1)(

s

s

es

es

s s

0 5 1 0 1 5 2 0 2 50 5 1 0 1 5 2 0 2 50 5 1 0 1 5 2 0 2 5

Correlation Scale (Range)Correlation Scale (Range)The correlation scale is defined as the distance over which the process is autocorrelated in space.

It is calculated as the distance at which the autocorrelation function tends to zero. Some authors suggest a threshold value taken as e-1.

Simulated Image using CMC

Markov Random FieldsMarkov Random Fields

Geological Map (Adams and Gelhar, 1992)

Random Field:

is a collection an integer random variables.

Markov:

the relation between the values have some probabilistic dependence.

SIS Model Simulation

Bierkens, 1996

Indicator Random FieldsIndicator Random Fields

Schelluinen study area, The Netherlands

LU Decomposition Method (1)LU Decomposition Method (1)

The algorithm for generating random fields with a given covariance structure based on the covariance matrix of the system is as follows:

1) Build the covariance matrix C of the system. The elements of C are denoted by,

)( Z, Z = Cov c jiij

and if i=j the covariances becomes the variances.

...),(............),(

),(..),(

21

2

212

1212

2

1

p

i

Zp

Z

Z

pZ

ZZCov

ZZCovZZCovZZCov

C

ij

X

Y

0

ZZ

1

p

2 3

LU Decomposition Method (2)LU Decomposition Method (2)

In case of stationary random field, the elements of the covariance matrix are given as: 2 ( )ij ijz = c s σ2

Z is the variance of the process Z, ρ(sij) is the auto-correlation function, and sij is the distance vector between point i and point j.

)y - y( + )x - x( = s 2ji

2jiij

For pairs of values Zi and Zj with i = 1,...p and j = 1,...p determine (xi - xj) and (yi - yj) where, (xi,yi) are the coordinates of point Zi and point (xj,yj) are the corresponding coordinates for point Zj. The distance sij between two points is,

ij

X

Y

0

ZZ

1

p

2 3

Auto-Correlation Matrix Auto-Correlation Matrix

1..........1.....1

..1

1

21

112

2

p

p

Z

C1R

ρij = cij/σ2Z is the autocorrelation coefficient between point i and point j.

ij

X

Y

0

ZZ

1

p

2 3

Properties of Auto-correlation MatrixProperties of Auto-correlation Matrix

1) All the diagonal elements are equal to one, i.e. correlation between the point and itself is perfect (complete correlation).

2) If ρij = 0, this means no correlation between i and j.

3) All the off-diagonal elements are called autocorrelation coefficients and they are less than one.

4) The autocorrelation matrix is symmetric, i.e., ρij = ρji.

5) According to the stationarity assumption: ρ12 = ρ23 =...= ρp-1p, ρ13 = ρ24 =...= ρp-2p, and so on.

LU Decomposition Method (3)LU Decomposition Method (3)

2) One has to decompose the covariance matrix by the Cholesky factorization method (Square-Root method),

U L= Cwhere, L is a unique lower triangular matrix, U is a unique upper triangular matrix, and U is LT , i.e., U is the transpose of L.

pj<i ,0 = l

pi<j<1 ,ll - cl1 = l

pi1 ,l - c = l

pi1 ,cc = l

ij

jkik1-j

=1kijjj

ij

2ik

1-i=1kii

1/2ii

11

i1i1

LU Decomposition Method (4)LU Decomposition Method (4)

3) Generation of normally distributed p-dimensional sequence of independent random numbers with zero mean and unit standard deviation N(0,1) which can be expressed as,

where, ε is vector of normally distributed random numbers, and εi is the i-th random number drawn from N(0,1).

T21 },...,,{ p

4) Multiplication of the independent random vector ε by the triangular matrix U to get a vector of auto-correlated random numbers. This vector can be expressed by matrix multiplication convention as,

ε U= Xwhere, X is a vector of multi-variate normal random Np(0,I), 0 is zero mean vector (p*1), and I is the identity matrix (p*p).

X + μ= Z

Example of LU Decomposition (Two-points)Example of LU Decomposition (Two-points)

22

122

111

2

12

2

1

22

22

22

2

22

221

2

21

22221

21

21

22

222

21

221121

211

2221

21

1

10

10,

10

)())((2)(

121exp

12

1),(

)())((2)(121exp

121),(

XZ

XZ

XX

ZZZZ

--

π = ZZf

ZZZZ-

- π

= ZZf

UL

C

O O

z 1 z 2

Numerical Example of LU Decomposition(1)Numerical Example of LU Decomposition(1)

i0 1 i+1i-1 N2

2

2

{ 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 }

{ 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 }10

2.451.1 0.83 0.55 0.28 0.0 0.0 0.0 0.0 0.0 0.00.83 1.1 0.83 0.55 0.28 0.0 0.0 0.0 0.0 0.00.55 0.83 1.1 0.83 0.55 0.28 0.0 0

1

z

z

i jZ

mx m

x x

μ

C

.0 0.0 0.00.28 0.55 0.83 1.1 0.83 0.55 0.28 0.0 0.0 0.00.0 0.28 0.55 0.83 1.1 0.83 0.55 0.28 0.0 0.00.0 0.0 0.28 0.55 0.83 1.1 0.83 0.55 0.28 0.00.0 0.0 0.0 0.28 0.55 0.83 1.1 0.83 0.55 0.280.0 0.0 0.0 0.0 0.28 0.55 0.83 1.1 0.83 0.550.0 0.0 0.0 0.0 0.0 0.28 0.55 0.83 1.1 0.830.0 0.0 0.0 0.0 0.0 0.0 0.28 0.55 0.83 1.1

1.05 0.00 0.00 0.00 0.0 0.0 0.0 0.0 0.0 0.00.79 0.69 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.52 0.60 0.68 0.00 0.00 0.00 0.00 0.00 0.00 0.

L

000.27 0.49 0.58 0.67 0.00 0.00 0.00 0.00 0.00 0.000.00 0.41 0.45 0.55 0.66 0.00 0.00 0.00 0.00 0.000.00 0.00 0.41 0.46 0.59 0.60 0.00 0.00 0.00 0.000.00 0.00 0.00 0.42 0.49 0.58 0.60 0.00 0.00 0.000.00 0.00 0.00 0.0 0.43 0.49 0.57 0.60 0.00 0.000.00 0.00 0.00 0.0 0.00 0.47 0.47 0.56 0.59 0.000.0 0.0 0.0 0.0 0.0 0.0 0.47 0.48 0.58 0.56

Numerical Example of LU Decomposition(2)Numerical Example of LU Decomposition(2)

.

.

.

.

.

.

.162.1241.0397.0

.

.

.

.

.

.

.562.0841.0203.0

60.060.060.060.060.060.060.060.060.060.0

.

.

.

.

.

.

.562.0841.0203.0

639.0543.0721.0876.0480.0787.0548.0201.0998.0193.0

.

56.058.048.047.000.000.000.000.000.000.000.059.056.047.047.000.000.000.000.000.000.000.060.057.049.043.000.000.000.000.000.000.000.060.058.049.042.000.000.000.000.000.000.000.060.059.046.041.000.000.000.000.000.000.000.066.055.045.041.000.000.000.000.000.000.000.067.058.049.027.000.000.000.000.000.000.000.068.060.052.000.000.000.000.000.000.000.000.069.079.00.00.00.00.00.00.000.000.000.005.1

LεμZ

Lε

Auto-correlation of a Stationary Medium (1)Auto-correlation of a Stationary Medium (1)

myxm 25.0,1

0 5 10 15 20 25 30 35 40 45 50-25

-20

-15

-10

-5

0

0 5 10 15 20 25 30 35 40 45 50-25

-20

-15

-10

-5

0

-5.5

-3.5

-1.5

0.5

2.5

4.5

0

0.01

0.1

1

10

Auto-correlation of a stationary Medium (1’)Auto-correlation of a stationary Medium (1’)

0 1 0 2 0 3 0 4 0 5 0

0

0 . 4

0 . 8

1 . 2

myxm 25.0,1

2/1

2

22

exp)(

var.exp.2

yx ss

ianceCoIstorpicD

s

0 1 0 2 0 3 0 4 0 5 0

0

0 . 4

0 . 8

1 . 2

Single Realization Ensemble=70

Auto-correlation of a stationary Medium (2)Auto-correlation of a stationary Medium (2)

myxmm yx 25.0,5,25

0 5 10 15 20 25 30 35 40 45 50-25

-20

-15

-10

-5

0

0 5 10 15 20 25 30 35 40 45 50-25

-20

-15

-10

-5

0

- 7

- 3 . 5

- 1 . 5

0 . 5

2 . 5

4 . 5

0

0 . 0 1

0 . 1

1

1 0

Auto-correlation of a Stationary Medium (2’)Auto-correlation of a Stationary Medium (2’)

Single Realization

Ensemble=10

myxmm yx 25.0,5,25

2/1

2

2

2

2

exp)(

var.exp.2

y

y

x

x ss

ianceCoAnistorpicD

s

0 1 0 2 0 3 0 4 0 5 0

0

0 . 4

0 . 8

1 . 2

0 1 0 2 0 3 0 4 0 5 0

0

0 . 4

0 . 8

1 . 2

Markov ChainsMarkov Chains

“…“…which may be regarded as a sequence or which may be regarded as a sequence or chain of discrete state in time (or space) in chain of discrete state in time (or space) in which the probability of the transition from which the probability of the transition from one state to another in the next step in the one state to another in the next step in the

chain depends on the previous state…”chain depends on the previous state…”

((Harbaugh and Bonham-Carter, 1969Harbaugh and Bonham-Carter, 1969 ) )

History of Markov Chains Application in History of Markov Chains Application in Subsurface CharacterizationSubsurface Characterization

• Early Work (Traditional Markov Chain)Early Work (Traditional Markov Chain)– Krumbein, 1-D (1967)Krumbein, 1-D (1967)– Harbaugh and Bonham-Carter, 1-D (1969)Harbaugh and Bonham-Carter, 1-D (1969)

• Recent WorkRecent Work– Elfeki, 2-D (1996) – unconditional CMCElfeki, 2-D (1996) – unconditional CMC– Elfeki and Dekking, 2-D (2001) – conditional CMCElfeki and Dekking, 2-D (2001) – conditional CMC– Carle and Fogg, 3-D Model (1996) – MC + SIS +SACarle and Fogg, 3-D Model (1996) – MC + SIS +SA

Theory of One-dimensional Markov ChainTheory of One-dimensional Markov ChainSS S

i0 1 i+ 1i-1 N2

l k q

,:)Pr()Pr(

p S Z | S ZS Z ,..., S Z ,S Z ,S Z | S Z

lkl1-iki

p0r3-in2-il1-iki

................

..

1

21

11211

nnn

lk

n

pp

pp

ppp

p 1,...,0

1

ppn

klklk

w p kN

lkN

)(lim 1,0

...,

1

1

n

kkk

klkl

n

l

ww

n ,1 k ,w p wMarginal prob.

Transition prob.

A B C DA 0 0 0 0B 0 0 0 0C 0 0 0 0D 0 0 0 0

Transition ProbabilityTransition Probability

Tally MatrixTally Matrix to to

Transition Transition Probability Probability

MatrixMatrix A B C D

A 0.6 0.1 0.2 0.1B 0.375 0.625 0 0C 0.1 0.1 0.8 0D 0.5 0 0 0.5

p

x

A B C DA 6 1 2 1B 3 5 0 0C 1 1 8 0D 1 0 0 1

Transition ProbabilityTransition Probability

• Properties of Transition Properties of Transition Probability MatrixProbability Matrix

AB

limCD

n

n

ww

pww

0.3659 0.1951 0.3659 0.0732w

1

1n

lkk

p

n np p

Row-sum

n-step transition

Marginal probability equals to volume proportion of each lithology

n x

Example on One-dimensional Markov ChainExample on One-dimensional Markov Chain

................

..

1

21

11211

nnn

lk

n

pp

pp

ppp

p

Estimation of transition prob.

Estimation of marginal prob.

w p kN

lkN

)(lim

Coupled Markov ChainCoupled Markov Chain

i0 1 i+ 1i-1 N2

S lS f

i0 1 i+ 1i-1 N2

S kS q

X-Chain

Y-Chain

lkfqqifik1+il1+i p S Y , S X | S Y ,S X ,)Pr(

},...,,,...,,,....{. 2121, nnqkfllkfq ssssssppp Transition probabilities

kllk www .Marginal probabilities

Coupled Chain on a LatticeCoupled Chain on a Lattice Dark Grey (Boundary Cells)Light Grey (Previously Generated Cells)W hite (Unknown Cells)

i-1 ,j i,ji,j-1

1 ,1

N x ,N y

N x ,1

1 ,N y

N x ,j

nkp . p

p . p SZSZSZ p

p . p C

SYSYSXSXCS Z ,S Z | S Z

SYSYSXSXS Z ,S Z | S Z

SYSYS Z | S Z

SXSXS Z | S Z

f

vmf

hlf

vmk

hlk

mjiljikjiklm

n

f

vmf

hlf

mjkjlikimjiljikji

mjkjlikimjiljikji

mjkjmjikji

likiljikji

,...1.),|Pr(:

)Pr()Pr()(Pr

)Pr()Pr()(Pr

)Pr()(Pr

)Pr()(Pr

1,,1,,

1

1

111,,1,

111,,1,

11,,

1,1,

Coupled Chain Prob. In Two-State ModelCoupled Chain Prob. In Two-State ModelTwo-State Model

1

1 22

1

2 11

1

2 22

1

1 11

2

1 11

2

1 22

2

1 21

2

1 12

2

2 11

2

2 22

2

2 21

2

2 12

1

1 21

1

1 12

1

2 12

1

2 21

P(ij,lk) = P(i,l) P (j,k) , i,j,l,k = 1,2 h v

The Transition P robab ilities o f The C oupled C hain

k= l k<> lP (11,11) P (11,22) P (11,21) P (11,12) P (12,11) P (12,22) P (12,21) P (12,12)P(21,11) P (21,22) P (21,21) P (21,12) P (22,11) P (22,22) P (22,21) P (22,12)

ExampleExample of Two-State Modelof Two-State ModelNumerical Exam ple of Two-State Model

The Transition Probabilities of The Coupled Chain

k=l k<>lP(11,11) =0.250 P(11,22) =0.250 P(11,21) =0.250 P(11,12) =0.250 P(12,11) =0.250 P(12,22) =0.250 P(12,21) =0.250 P(12,12) =0.250P(21,11) =0.125 P(21,22) =0.375 P(21,21) =0.375 P(21,12) =0.125 P(22,11) =0.125 P(22,22) =0.375 P(22,21) =0.375 P(22,12) =0.125

P(ij,lk) = P(i,l) P(j,k) , i,j,l,k = 1,2 h v

The Horizontal Chain The Vertical Chain 1 2 1 2 1 0.50 0.50 1 0.50 0.50 2 0.25 0.75 2 0.50 0.50

Normalized Transition Probabilities of The Coupled Chain

k=l k<>lP(11,11) =0.500 P(11,22) =0.500 P(11,21) =0.000 P(11,12) =0.000 P(12,11) =0.500 P(12,22) =0.500 P(12,21) =0.000 P(12,12) =0.000P(21,11) =0.250 P(21,22) =0.750 P(21,21) =0.000 P(21,12) =0.000 P(22,11) =0.250 P(22,22) =0.750 P(22,21) =0.000 P(22,12) =0.000

P(ij,kk) = P(i,f) P(j,f)

P(i,k) P(j,k)i,j,k =1,2

f

0 1 0 0 2 00 30 0 40 0 50 0 600 700 8 00 9 00 1 000-5 00

-4 00

-3 00

-2 00

-1 00

0

(a) G eo lo gica l P a ttern w ith Tw o P ha se M ater ia ls .

0 10 20 30 40 5 0 60 70 80 90 100-5 0

-4 0

-3 0

-2 0

-1 0

0

0 5 0 10 0 15 0 200-5 0

0

1 2 3 4

0 10 20 30 40 5 0 60 70 80 90 100-5 0

-4 0

-3 0

-2 0

-1 0

0

(a )

(b )

(c )

0 10 20 30 40 5 0 60 70 80 90 100-5 0

-4 0

-3 0

-2 0

-1 0

0

(d )

0 100 200 300 400 500 600 700 800 900 1000-4 00

-3 00

-2 00

-1 00

0

(b ) R e la tive ly T hin S tra tified F o rm ation .

0 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00-50 0

-40 0

-30 0

-20 0

-10 0

0

(a ) G eo lo g ica l S ys tem w ith O n e U n it H a s B ig D im en sion s.

0 20 0 40 0 60 0 80 0 10 00 12 00-60 0

-40 0

-20 0

0

(b ) L arge Sca le G eo lo g ica l U n its .

0 200 4 00 6 00 8 00 100 0 12 00 1 40 0

-4 0 0

-2 0 0

0

(a ) Inc lined B ed d in g w ith -4 5 d eg rees .

0 10 0 20 0 3 00 40 0 50 0 60 0 7 00 80 0 900 100 0-500

-400

-300

-200

-100

0

(b ) Inc lin ed B ed d ing w ith -7 degrees.

0 20 0 40 0 6 00 80 0 10 00 12 00 14 0 0 1 600 18 00 2000

-400

-200

0

(c) Inc lin ed B edd ing w ith 26 .5 deg rees .0 200 400 600 80 0 100 0 12 00 140 0

-4 00

-2 00

0

(c ) B ed R o ck w ith Se dim en tary D ep osit.

Unconditional CMCUnconditional CMC

Coupled Markov Chain (application 1)Coupled Markov Chain (application 1) Two-dimensional Cross-sectional Panel of the Fluvial Succession of the Medial

Area of the Tόrtola Fluvial System, Spain

Length of The Section (m) = 648. Depth of The Section (m) = 115.Sampling interval in X-axis (m) = 9. Sampling interval in Y-axis (m) = 2.5

Horizontal Transition Probability Matrix State 1 2 3 4 5 6 7 8 1 0.893 0.009 0.005 0.000 0.000 0.000 0.000 0.093 2 0.000 0.796 0.011 0.000 0.000 0.000 0.000 0.194 3 0.000 0.000 0.989 0.000 0.000 0.000 0.000 0.011 4 0.006 0.000 0.013 0.885 0.000 0.000 0.000 0.096 5 0.074 0.000 0.000 0.074 0.593 0.037 0.000 0.222 6 0.000 0.013 0.000 0.000 0.000 0.946 0.000 0.040 7 0.040 0.000 0.000 0.000 0.000 0.000 0.940 0.020 8 0.007 0.006 0.002 0.007 0.005 0.005 0.001 0.968

Vertical Transition Probability Matrix State 1 2 3 4 5 6 7 8 1 0.591 0.000 0.000 0.000 0.014 0.000 0.042 0.353

2 0.011 0.753 0.097 0.000 0.000 0.000 0.000 0.1403 0.032 0.000 0.623 0.000 0.000 0.238 0.000 0.1073 0.000 0.025 0.000 0.662 0.013 0.000 0.000 0.2995 0.111 0.000 0.000 0.074 0.519 0.000 0.000 0.2966 0.000 0.000 0.026 0.032 0.006 0.084 0.000 0.8517 0.120 0.000 0.000 0.100 0.000 0.000 0.360 0.420

8 0.029 0.008 0.039 0.017 0.003 0.031 0.010 0.863

1 2 3 4 5 6 7 8

Coupled Markov Chain (application 1 cont.)Coupled Markov Chain (application 1 cont.)

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Stochastic Simulations

SchematizedOutcrop

Outcrop

Conditional Simulation Conditional Simulation

From a practical point of view, it is desirable that the random fields not only

reproduce the spatial structure of the field

but also

honour the measured data and their locations.

This requires an implementation of some kind of conditioning, so that the generated realizations are constrained to the available field measurements.

Representation of a Conditional SimulationRepresentation of a Conditional Simulation

Methods of ConditioningMethods of Conditioning

D irec t M eth od s"M etrica l M eth od s "

In d irec t M eth od s"K rig in g M eth od "

M eth od s o f C on d it ion in g

Graphical Illustration of Conditioning by Graphical Illustration of Conditioning by KrigingKriging

(1) A kriged map is generated from the data:

(2) Unconditional simulation is generated from the data:

(3) Allocation pseudo measurements.

(4) Kriged map is generated from the pseudo measurements:

(5) A pseudo error =kriged map(step 4) - the unconditional simulation in step 2.

(6) Conditional simulation = the pseudo error in step 5 + the kriged map in step 1:

( - )cs kd us kus Z Z Z Z

kdZ

usZ

kusZ

cs Z

Conditioning in One-dimensional Markov Conditioning in One-dimensional Markov ChainChain

i0 1 i+ 1i-1 N2

S kS l S q

1

11

1

1 11

Pr ( )

Pr( , , )Pr ( )

Pr ( , )Pr ( | , ) Pr (

Pr ( )

i i Nk l q

i i Nl k qi i Nk l q

i Nl q

N i i iq l ki i Nk l q

| , SS SZ Z Z S S SZ Z Z | , S S SZ Z Z

S SZ Z S S SZ Z Z Z | , S S SZ Z Z

1

11

1

1

, )Pr ( , )

Pr ( | ).Pr ( , )Pr ( )

Pr ( , )

Pr ( | )Pr ( )

il k

i Nl q

N i i iq k l ki i Nk l q

i Nl q

N iq ki i Nk l q

S SZ S SZ Z

S S S SZ Z Z Z | , S S SZ Z Z S SZ Z

S SZ Z | , S S SZ Z Z

1 1.Pr ( | ). Pr ( )i i ik l l S S SZ Z Z

1 1Pr ( | ). Pr( )N i iq l l S S SZ Z Z ( )

1 ( 1)

( )

( 1)

Pr ( )N i

kq lki i Nk l q N i

lq

N ilk kq

lk q N ilq

p p | , S S SZ Z Z

p

p pp

p

Coupled Markov Chain “CMC”and CS_CMC Coupled Markov Chain “CMC”and CS_CMC in 2Din 2D

Dark Grey (Boundary Cells)Light Grey (Previously Generated Cells)W hite (Unknown Cells)

i-1 ,j i,ji,j-1

1 ,1

N x ,N y

N x ,1

1 ,N y

N x ,j

nkp . p

p . p SZSZSZ p

f

vmf

hlf

vmk

hlk

mjiljikjiklm ,...1.),|Pr(: 1,,1,,

.,...1,

),,|Pr(:

)(

)(

,1,,1,,

nkp . p . p

p .p . p

SZSZSZSZp

f

vmf

iNhfq

hlf

vmk

iNhkq

hlk

qjNmjiljikjiqklm

x

x

x

Coupled Markov Chain (application)Coupled Markov Chain (application) Two-dimensional Cross-sectional Panel of the Fluvial Succession of the Medial

Area of the Tόrtola Fluvial System, Spain

Length of The Section (m) = 648. Depth of The Section (m) = 115.Sampling interval in X-axis (m) = 9. Sampling interval in Y-axis (m) = 2.5

Horizontal Transition Probability Matrix State 1 2 3 4 5 6 7 8 1 0.893 0.009 0.005 0.000 0.000 0.000 0.000 0.093 2 0.000 0.796 0.011 0.000 0.000 0.000 0.000 0.194 3 0.000 0.000 0.989 0.000 0.000 0.000 0.000 0.011 4 0.006 0.000 0.013 0.885 0.000 0.000 0.000 0.096 5 0.074 0.000 0.000 0.074 0.593 0.037 0.000 0.222 6 0.000 0.013 0.000 0.000 0.000 0.946 0.000 0.040 7 0.040 0.000 0.000 0.000 0.000 0.000 0.940 0.020 8 0.007 0.006 0.002 0.007 0.005 0.005 0.001 0.968

Vertical Transition Probability Matrix State 1 2 3 4 5 6 7 8 1 0.591 0.000 0.000 0.000 0.014 0.000 0.042 0.353

2 0.011 0.753 0.097 0.000 0.000 0.000 0.000 0.1403 0.032 0.000 0.623 0.000 0.000 0.238 0.000 0.1073 0.000 0.025 0.000 0.662 0.013 0.000 0.000 0.2995 0.111 0.000 0.000 0.074 0.519 0.000 0.000 0.2966 0.000 0.000 0.026 0.032 0.006 0.084 0.000 0.8517 0.120 0.000 0.000 0.100 0.000 0.000 0.360 0.420

8 0.029 0.008 0.039 0.017 0.003 0.031 0.010 0.863

1 2 3 4 5 6 7 8

Geological Application of CS_CMCGeological Application of CS_CMC

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3D CMC model single realization3D CMC model single realization