3-spatial continuity analysis - ulisboa · pdf filespatial continuity analysis classes...

3-Spatial Continuity Analysis

CERENA

Instituto Superior Técnico

1

Spatial Continuity Analysis

of petrophysical properties or lithofacies of a

petroleum reservoir

Characterization of spatial patterns of the main properties

that characterize the quality of a reservoir: lithofacies,

porosity, permeability, acoustic impedances,....

5000 m.

Spatial continuity patterns statistical tools to quantify the

spatial continuity patterns

randomness and spatial anisotropy


With spatial continuity analysis one intend to reach two main

objectives:

i) Structural Analysis – To understand and quantify the main patterns of

the spatial phenomenon – main directions of continuity, anisotropic

behaviour of internal properties,...

ii) To build a spatial correlation model representative of the entire

area which is the basis of geostatistical estimation and simulation.

Spatial Continuity Analysis Spatial Continuity Analysis based in structural elements

Bi-phasic set: two lithogroups

a1

a2

0

1

a1 a2

A

r dxxIA

)(1 Spatial Continuity

Analysis based in a

circle of radius r

Multi-point statistics - multivariate statistics between z(x), z(x+1),

z(x+2),... z(x+h).

When the spatial phenomenon is known through a full image where the values v(x)

are known in the entire space (expert images, outcrops,…).


z(x) z(x+h)

Bi-point statistics – bi-variate statistics of z(x) and z(x+h)

When the spatial phenomenon is known through a limited and discrete set of sample values.

A

z(x) property z located at x

x=(x,y,z) x+h=(x+h,y+h,z+h)

h is the vector that separates the two points


v(x)

v(x+h)

Bi-point Statistics

• Spatial Continuity is evaluated through the correlation between all pairs of

values separated by a vector h.

• The lower the value of h, the higher is the correlation between values z(x),

located in x , and values z(x+h) located in x+h.


Representation of z(x) e

z(x+h) on a bi-plot

Pairs of values separated by h = 4 m.



11



The set of bi-plots can be summarized in one diagram (ρ(h),h):

ρ(h)

h

correlogram

1.0


Estimation of Semi-variogram and Spatial Covariance

• Measures that summarize the dispersion of bi-plots between z(x) and z(x+h)

• Tools to quantify the spatial continuity of the phenomenon.

Semi-variogram: The mean of the square differences between z(x)

and z(x+h) for different h values

)()()(2

1)(

2)(

1

hxZxZhN

hhN

i

Z(x)

Z(x+h)

z(xi)

z(xi+h)

z(xi)- z(xi+h)


Non-centred Covariance: The mean of the products z(x).z(x+h)

)().()(

1)('

)(

1

hxZxZhN

hChN

i

Centred Covariance: The mean of the products z(x).z(x+h),

normalized by the arithmetic means of the points located at x and x+h

respectively.

h)+m(x).m(x - )().()(

1)(

)(

1

hxZxZhN

hChN

i

with the arithmetic means of the points located at x and x+h:

N

i

ixZN

xm1

)(1

)(

N

i

i hxZN

hxm1

)(1

)(


Correlogram: normalized covariance

2

)(

2

)( .)(

hxx

hCh

with:

2

1

2

)(1

hN

i

x xmxzhN

2

1

2

)(1

hN

i

hx hxmhxzhN

• Covariance C(h) and Correlogram (h) are measures of

similarity

•Variogram (h) is a measure of dissimilarity


Variogram Representation

...capturing different spatial

behaviours in different

directions


The variogram is calculated by the mean of the square differences

between the pairs of points separated by a vector h.

)()()(2

1)(

2)(

1

1

hN

i

ii xZxZhN

h

Regular grid of samples

Estimation of experimental variograms


Irregularly Spaced Data

Classes of angles and distances

Irregularly spaced data implies that tolerances of angles

(d) and distances (hdh) have to be defined.

7o

110 m.

x

h + h

-

h

+

-

x+h



x1 x4

x2

x3

x5

x6 x7

x8

X1 X2 X3 X4 X5 X6 X7 X8

X1 0 600 800 150 330 220 800 950

X2 0 200 500 340 600 850 700

X3 0 850 400 700 800 520

X4 0 200 60 480 650

X5 0 150 450 480

X6 0 230 430

X7 0 440

X8 0

X1 X2 X3 X4 X5 X6 X7 X8

X1 0 1 1 2 - 2 2 -

X2 0 - 3 2 3 3 2

X3 0 1 1 1 3 2

X4 0 1 - 2 -

X5 0 1 3 -

X6 0 1 -

X7 0 1

X8 0

Angles

Direction 1: 0º ± 30º

Direction 2: 90º ± 30º

Direction 3: 45º ±30º

Distances (m)



Classes Distância N. de Pares de pontos Variograma

0 200 2

200 400 2

400 600 2

600 800 2

800 1000 1

Classes Distância N. de Pares de pontos Variogram

0 200 1

200 400 2

400 600 2

600 800 2

800 1000 -

X1 X2 X3 X4 X5 X6 X7 X8

X1 0 600 800 150 330 220 800 950

X2 0 200 500 340 600 850 700

X3 0 850 400 700 800 520

X4 0 200 60 480 650

X5 0 150 450 480

X6 0 230 430

X7 0 440

X8 0

Variograms

Direction 1

Direction 2


• In presence of a given scarcity of data, the increase of tolerances of angles

and distances has a single goal: to obtain more consistent statistics for the

direction and distance h.

• In apparently isotropic spatial phenomena with a lack of data it is usual to

calculate just one variogram for all directions – omnidirectional variogram

(with a angle tolerance of 180º).

• When the phenomenon is clearly anisotropic, increasing of the tolerance of

angles can result in smoothing of the ranges (measure of maximum distance

up to which spatial correlation can be considered to exist) for different

directions. This means that the lower ranges are overestimated and the higher

ranges are underestimated.


(h)

h1

(h)

h2

dh2= 2.dh1

Large tolerances of distance can lead to high values of variogram near the origin.


h=1

h=2

h=4

Spatial Representativity of the variogram

Columns of

values

without pairs.

Representativ

e area of the

variogram for

distance h.

In the calculation of the variogram

the values h should not be greater

than approximately 1/2 of the

dimension of the field A in the

direction of h

Practice II – Experimental Variograms geoVAR e geoMOD of geoMS

Spatial Continuity Models Variogram models

INTRODUÇÃO À

GEOESTATÍSTICA

h

(h)

Samples

Experimental

variogram

Reality

Real variogram

?

Objective: Infer the real variogram based on the experimental variogram

Variograms Models

Variograms Models

Method:

Interpolate the experimetal points by a

smooth curve: the variogram model (h)

The variogram model (h) must be a function

of a small number of parameters.

The variogram model (h) must be representative of

the spatial pattern of the unknown reality, i.e., must

be a good estimator of the real variogram.

This is an important and crucial step of a geostatistical reserves

estimation study: as the reality is unknown, normally a

multidisciplinary team – geologists, geophysists, petroleum

engineers., ..- is involved in order to squeeze all the knowlege about

the orebody into the variogram model.

h

(h)

Variograms Models: Spherical Model

ah

ahC

a

h

a

hC

h

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

2

1

2

3.

3

5000 m.

a

C

The range a is defined as the distance to which the model reaches the sill

a=1000 m.

a=2000 m.

Variograms Models: Exponential Model

a=2000 m.

The range a is defined as the distance to which the model reaches 95% of

the sill

(h)=1-exp (-3h/a)

a

a=1000 m.

Variograms Models: Spherical and Exponential

a=1000 m.

a=2000 m.

Variograms Models: Gaussian Model

(h)=1-exp (h/a)2

a

a=1000 m.

a=2000 m.

Variograms Models: Nested Structures

(h)=Esf(a=500m.) (h)=Esf(a=2000m.)

(h)= 0.5 Esf(a=500m.) +

0.5 Esf(a=2000m.)

Variograms Models: Nested Structures

(h)= 0.3 Esf(a=500m.) +

0.7 Esf(a=2000m.)

(h)= 0.5 Esf(a=500m.) +

0.5 Esf(a=2000m.)

Variograms Models : Nugget Effect

C0

Nugget effect is a structure (constant) that has to be added

to the other structures:

(h)= C0 + C1(h)+C2(h) + …

measures the small scale variability

C0 = C

C0 = 0.5C

Variograms Models: Nugget Effect

Variograms Models: Anisotropy Models

N/S

E/W

a=1000m

a=500m

Variograms Models: Geometric Anisotropy

N/S

E/W

N/S

E/W

Variograms Models : Geometric Anisotropy

ry =a/ay=1 rx =a/ax=2 a is the highest range 1000 m.

dy dy´=dy . ry =dy

dx dx´=dx . rx =2. dx

dy

dy´

dx´

dx

a=500(dx)=a=1000(dx´)

Structural transformations

x0

x1 x2 x0

x0

x1

x2

Transformation of the spatial referential for a

better estimation of the spatial continuity –

variograms, covariances, ..

Automatic or visual modeling?

Isotropic or anisotropic models?

How many structures?

How good is the model?

The objective is to capture the main spatial patterns of the

mineralization, not to build a variogram that better fits the

experimental points.

Estimation of the Variogram

Estimation of the Variogram: Nugget Effect

High local mean and variance

These pairs can originate high spikes of variogram values

Estimation of the Variogram: Proportional Efect

Estimation of the Variogram: Proportional Efect

2)(

)()(

hm

hhr

N

i

i

N

i

i

r

hN

hm

hhN

h

1

21 )(

)(

)(

2

2

1

2

1)(

hixix

hixix

hNh

N

i

r

Relative variogram

m(h) – mean of a ll pair of values

Ni(h) – Number of sample pairs for each region.

Denominator reduces the influence of very large values of x(i)-x(i+h)

Local relative variogram

Pairwise relative variogram

0C

hCh

N

i

hixixhN

h1

2

11)(

hixix

hNh

N

i

1

1)(

Correlogram

Rodogram

Madogram

)()0()( hCCh

γ(h)

C(h)

C(0)

Variogram and spatial

covariances

Practice III- Variogram Models geoVAR and geoMOD of geoMS

3-spatial continuity analysis - ulisboa · pdf filespatial continuity analysis classes...

Documents