Post on 19-Mar-2018
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
Spatial Continuity Analysis
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,…).
Spatial Continuity Analysis
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
Spatial Continuity Analysis
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.
Spatial Continuity Analysis
Representation of z(x) e
z(x+h) on a bi-plot
Pairs of values separated by h = 4 m.
Pairs of values separated by h = 8 m.
Spatial Continuity Analysis
11
Pairs of values separated by h = 16 m.
Pairs of values separated by h = 48 m.
The set of bi-plots can be summarized in one diagram (ρ(h),h):
ρ(h)
h
correlogram
1.0
Spatial Continuity Analysis
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)
Spatial Continuity Analysis
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
)(
Spatial Continuity Analysis
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
Spatial Continuity Analysis
Variogram Representation
...capturing different spatial
behaviours in different
directions
Spatial Continuity Analysis
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
Spatial Continuity Analysis
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
Estimation of experimental variograms
Spatial Continuity Analysis
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)
Estimation of experimental variograms
Spatial Continuity Analysis
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
Spatial Continuity Analysis
• 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.
Spatial Continuity Analysis
(h)
h1
(h)
h2
dh2= 2.dh1
Large tolerances of distance can lead to high values of variogram near the origin.
Spatial Continuity Analysis
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