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