Geostatistical Reservoir Modeling
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Transcript of Geostatistical Reservoir Modeling
Geostatistical Reservoir Modeling
MSIYJose R. Villa©2007
Geostatiscal reservoir modeling(GSLIB, SGeMS, SReM)
Reservoir simulation + IPM(ECLIPSE100, PRiSMa)
Well location, type and trajectory optimization(PRiSMa-O)
Uncertainty assessment(EED)
Contents
1. Introduction
2. Statistical concepts
3. Spatial continuity
4. Estimation– Kriging
5. Simulation– Sequential Gaussian simulation (SGS)
6. Facies modeling– Sequential indicator simulation (SIS)– Fluvial modeling
7. Porosity and permeability modeling
Contents
1. Introduction
2. Statistical concepts
3. Spatial continuity
4. Estimation1. Kriging
5. Simulation– Sequential Gaussian simulation (SGS)
6. Facies modeling– Sequential indicator simulation (SIS)– Fluvial modeling
7. Porosity and permeability modeling
Workflow for Reservoir Modeling
Caers, J., Introduction to Geostatistics for Reservoir Characterization, Stanford University, 2006 (modified)
Data for Reservoir Modeling
Deutsch, C., Geostatiscal Reservoir Modeling, Oxford University Press, 2002
Geostatistics
• Branch of applied statistics that places emphasis on– The geological context of the data– The spatial relationship of the data– Integration of data with different support volume and scales
• Indispensable part of reservoir management since reservoir models are required for planning, economics and decision making process
• Provides a numerical description of reservoir heterogeneity for:– Estimation of reserves– Field management and optimization– Uncertainty assessment
Reservoir Modeling (1)
Reservoir Modeling (2.1)
Reservoir Modeling (2.2)
Modeling Scale
Caers, J., Petroluem Geostatistics, Society of Petoroluem Engineers, 2005
Model Building
Caers, J., Petroluem Geostatistics, Society of Petoroluem Engineers, 2005
Uncertainty
Caers, J., Petroluem Geostatistics, Society of Petoroluem Engineers, 2005
Role of Geostatistics
• Tool for building a reservoir model using all available reservoir and well data allowing a realistic representation of geology
• Links the static and dynamic reservoir model• Framework for uncertainty assessment
Application of Geostatistics
• Data integration– Well and seismic data– Co-variable (porosity and permeability)– Tendencies
• High-resolution reservoir models for flow simulation
• Uncertainty assessment– Volumetrics– Recoverable reserves
Further Readings
Geostatistical Reservoir ModelingClayton DeutschOxford University Press
GSLIB: A Geostatistical Software LibraryClayton Deutsch and Andre JournelOxford University Press
Introduction to Applied GeostatisticsEd Isaaks and Mohan SrivastavaOxford University Press
Petroleum GeostatisticsJef CaersSociety of Petroleum Engineers
Contents
1. Introduction
2. Statistical concepts
3. Spatial continuity
4. Estimation– Kriging
5. Simulation– Sequential Gaussian simulation (SGS)
6. Facies modeling– Sequential indicator simulation (SIS)– Fluvial modeling
7. Porosity and permeability modeling
Exploratory Data Analysis
• Definitions– Population– Sample
• Data preparation and quality check
• Decision of stationarity
Probability Distributions
• Variable: a measure (, k, i) which can assume any of the prescribed set of values– Continuous (z): , k– Categorical (ik=0,1; k=category): facies
• Random variable Z: a random variable whose outcome (z) is unknown but its frequency of outcome is quantified by a random function
• Random function: describes/models the variability or uncertainty of unknown true values (outcomes), either as cumulative (CDF) or density (PDF) distribution model – RF is location and information dependent
Probability Distributions – cdf
( ) ( ) Prob{ }ZF z F z Z z
( ) [0,1]
( ) 0, ( ) 1
Prob{ ( , )} Prob{ } Prob{ }
Prob{ ( , )} ( ) ( )
F z
F F
Z a b Z b Z a
Z a b F b F a
Probability Distributions – pdf
0
( ) ( )( ) '( ) lim
dz
F z dz F zf z F z
dz
( ) 0
( ) ( ) Prob{ }
( ) 1
z
f z
f x dx F z Z z
f z dz
Expected Value and Variance
2222 }{}}{{}var{
)(}{
mZEZEZEZ
dzzzfmZE
• Expected value: the actual mean of a population
• Variance: measure of spread of a random variable from the expected value
Mean and variance (1)
• Sample
• Population
n
ii
n
ii
zzn
s
zn
z
1
22
1
1
1
}{
}{22 mZE
ZEm
Mean and variance (2)
Distributions
• Parametric– Uniform– Exponential– Normal standard– Normal– Lognormal
• Non-parametric– Experimental, inferred by data (histograms)
Uniform Distribution
1 z [a,b]
( )0 z [a,b]
0 z a
( ) Prob{ } ( ) z [a,b]
1 z
z
f z b a
z aF z Z z f z dz
b ab
2 2 2 2
2
2 2 2
{ }2
1 1{ }
3
=Var{ } { }12
b
a
a bm E Z
E Z z dz a ab bb a
b aZ E Z m
Exponential Distribution
0 0 0
1( ) z>0
1 1( ) ( ) 1 z>0
z
a
zz z x x z
a a a
f z ea
F z f x dx e dx ae ea a
2 2 2
0 0
2 2 2 2 2 2
{ }
1{ } 2 2
=Var{ } { } 2
z z
a a
m E Z a
E Z z e dz ze dz aa
Z E Z m a a a
Normal Standard Distribution
2
2
0 0
1( ) z ( , )
2
1( ) ( ) z ( , )
2
za
z z xa
f z e
F z f x dx e dx
2
{ } 0
=Var{ } 1
m E Z
Z
0
1Z N
Normal Distribution
2
2
2
2
2
2
1( ) z ( , )
2
1( ) ( ) z ( , )
2
z m
x mz
f z e
F z f z dz e dx
2 2
{ }
=Var{ }
m E Z m
Z
2
0
1
mZ N
Z mY N
Lognormal Distribution
2
2
2
2 2
{ }
=Var{ } 1
m E Z e
Z m e
2
2
0 logm
Z N
Y Ln Z N
2
2
2
2
- ln z
2
- ln xz 2
0 0
1( ) e z 0
2
1 e( ) ( ) z 0
2
m
mz
f zz
F z f x dx dxx
Histograms
Quantiles and Probability Intervals (1)
• Quantile: is a z-value that corresponds to a fixed cumulative frequency. For example, the 0.5 quantile (median or q(0.5)) is the z-value that separates the data into two equally halves. – Lower quantile: q(0.25)– Upper quantile: q(0.75)– Interquantile range: IQR = q(0.75) - q(0.25)
Quantiles and Probability Intervals (2)
( )
( ( )) Prob{ ( )} [0,1]
q p z
F q p Z q p p
p=
q(p)
Q-Q Plots (1)
• Tool for comparing two different distributions
• Plot of matching p-quantile values q1(p) vs. q2(p) from the two different distributions
Q-Q Plots (2)
Q-Q Plots (3)
data 1
data
2
data 1
data
2
data 1
data
22 1
2 22 1
2 1pdf pdf
m m
Monte Carlo Simulation
• Generate a set of uniform random numbers p (random number generator)• Retrieve for each such number p, the quantile q of the cdf• The set of values qp are called “samples drawn from the distribution F• The histogram of these qp values match the cdf F
zp
Data Transformation (1)
• Transform distribution of a dataset into another distribution
• Applications– Well-log porosity to core porosity if the latter is
deemed more reliable– Simulation results into a specific target distribution– Transformation into known analytical models
(Gaussian or normal score transform)
Data Transformation (2)
Deutsch, C., Geostatiscal Reservoir Modeling, Oxford University Press, 2002
Data Transformation (3)
2121
( )2
z m
f z e
Deutsch, C., Geostatiscal Reservoir Modeling, Oxford University Press, 2002
Contents
1. Introduction
2. Statistical concepts
3. Spatial continuity
4. Estimation– Kriging
5. Simulation– Sequential Gaussian simulation (SGS)
6. Facies modeling– Sequential indicator simulation (SIS)– Fluvial modeling
7. Porosity and permeability modeling
Spatial Correlation
Distancia (ft) Permeabilidad (md)50 11.75100 4.09150 3.05
. .
. .
. .6400 12.02
Contents
1. Introduction
2. Statistical concepts
3. Spatial continuity
4. Estimation– Kriging
5. Simulation– Sequential Gaussian simulation (SGS)
6. Facies modeling– Sequential indicator simulation (SIS)– Fluvial modeling
7. Porosity and permeability modeling
Contents
1. Introduction
2. Statistical concepts
3. Spatial continuity
4. Estimation– Kriging
5. Simulation– Sequential Gaussian simulation (SGS)
6. Facies modeling– Sequential indicator simulation (SIS)– Fluvial modeling
7. Porosity and permeability modeling
Contents
1. Introduction
2. Statistical concepts
3. Spatial continuity
4. Estimation– Kriging
5. Simulation– Sequential Gaussian simulation (SGS)
6. Facies modeling– Sequential indicator simulation (SIS)– Fluvial modeling
7. Porosity and permeability modeling
Contents
1. Introduction
2. Statistical concepts
3. Spatial continuity
4. Estimation– Kriging
5. Simulation– Sequential Gaussian simulation (SGS)
6. Facies modeling– Sequential indicator simulation (SIS)– Fluvial modeling
7. Porosity and permeability modeling