[]_Geostatistics_in_12_Lessons.doc

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Preface to Geostatistics in 12 Lessons

Introduction

This web page is a set of companion notes to accompany the twelve lectures presented in the summer

of 1777 8or #andmark raphics in !ustin Teas The lectures are intended to be an informal training

seminar for those employees involved in the development, documentation, and testing of software that

implement geostatistics

Key Tasks

There are some key tasks that will be accomplished by the end of the summer These include"

o un informal training seminars with members of the testing, documentation, and

development groups

o 9evelop a web-based training resource for testing, documentation, and development

groups

o 9evelop a glossary of geostatistical terms and key concepts

o :ork with the testing manager to develop a test plan for the testing of geostatistical

components

o 9evelop workflow specifications for shared earth modeling

o 9ocument procedure and assumptions * underlying techni$ues in the eostat 9;

0raining #e.inars

The training seminars will be presented in 12 lectures #ectures will be presented on Tuesday, and

Thursday at 1" am until noon The lectures will be presented in varying depth 5n depth seminars

will be held on Tuesday and Thursday, and light seminars will be held on :ednesday The :ednesday

seminars will be a high-level overview of the Tuesday and Thursday seminars sers of the web site will have the

opportunity to submit their $ui66es by email for marking or allow for self-marking

Glossar! of 0er.s

! glossary of terms will be provided to assist the user :ords that appear in the training resource that

also appear in the glossary will be hot linked

0esting

8amiliarity with geostatistical concepts is essential in any testing strategy 8amiliarity will come as a

result of the seminars and the training resource 'ommon mistakes in the construction of a

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geostatistical model and items that indicate problems will be outlined 5n depth pages will tackle theory

and provide tools for proving algorithms by hand

orflo Diagra.sThe training resource will provide a diagrammatic illustration of a workflow for

reservoir modeling The training resource will be hot linked to the workflow model =ot linking the

workflow diagram will allow the user to go to specific lectures instead of having to browse through the

entire training resource

Docu.entation of ssu.ptions / Procedures

The construction of a geostatistical model re$uires sometimes confusing procedures and assumptions

The training resource and the seminars will clearly state and eplain all assumptions and procedures

5n discussion with


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o eservoir planning

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4arts of this website are patterned after the book Geostatistical Reservoir Modeling, a currently

unpublished book authored by 9r 'layton 9eutsch 9r 9eutsch is aware and has granted permission

for the use of material

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Lecture 1: Purpose / Motivation for Geostatistics

Bualitative easoning

eservoir 4lanning

ncertainty 4ortfolio &anagement

The #ife 'ycle of a 4roect

#ecture 1 Bui6

Introduction

5n view of #andmarkEs core business, ie reservoir characteri6ation, geostatistics can be defined as a

collection of tools for $uantifying geological information leading to the construction of 39 numerical

geological models to be used for assessment and prediction of reservoir performance

eostatistics deals with spatially distributed and spatially correlated phenomena eostatistics allows

$uantification of spatial correlation and uses this to infer geological $uantities from reservoir data at

locations where there are no well data (through interpolation and etrapolation) 5n addition, the main

benefits from geostatistics are" (1) modeling of reservoir heterogeneity, (2) integrating different types

of data of perhaps different support and different degrees of reliability, and (3) assessing and

$uantifying uncertainty in the reservoir model

This course was not developed as a cookbook recipe for geostatistics


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$ualified decision making ! distribution of uncertainty is generated, and using a loss function the risk

is assessed and an optimal estimate is determined, the estimate that incurs the least loss 9ifferent loss

functions can be used for pessimistic and optimistic estimates

8igure 13, !n illustration showing the concept of risk $ualified decision making Dote that the loss function is scenariospecific, and that the histogram of possible costs are in addition to those costs if the estimate were correct

ortfolio Management

4ortfolio management re$uires that the best possible decisions be made in the face of uncertainty

ome of these decisions include"

67ploration License Bidding:using limited seismic and well data, decide which 6ones to

focus on and * or commit resources to

67ploration Drilling:given a few wells (1-3), decide whether or not the field warrants further

investigation

Drilling Ca.paign: decide how many wells to drill, placement of wells, and timing of

enhanced oil recovery tactics

Develop.ent Planning:decide how large support facilities must be, negotiate pipeline or salesagreements and contractor commitments

Mature field:decide on the value of infill drilling or the implementation of enhanced oil

recovery schemes such as flooding and steam inection

*andon.ent / #ale:timing of environmentally and economically sound closing of facilities

These decisions are being made with less data and greater uncertainty for proects that are marginally

profitable ound estimates backed by rigorous mathematical methods secures investors and fosters

good economic relations

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Lecture 1: Purpose / Motivation for Geostatistics, The Quiz

Question )

#ist three maor benefits that geostatistics offers, and describe what each mean and eplain why they

are important

Question *

9ecision making in presence of uncertainty is important :hat are the two steps for risk-$ualified

decision makingK

Question +

5n general terms, eplain the link between spatial variability (heterogeneity) and uncertainty

Question

5n your own words describe the information effect and how it relates to uncertainty

Question -

eostatistics is useful at every point in the life cycle of a reservoir, but where is it most useful and

whyK

solutions

Lecture 1: Purpose / Motivation for Geostatistics, The Quiz

Question )

#ist three maor benefits that geostatistics offers, and describe what each mean and eplain why they

are important

Buantification of uncertainty" summari6es our lack of knowledge for better decision making

igorous mathematics" means that there are sound mathematical laws applied for repeatability

9ata 5ntegration" data of many types can be integrated using geostatistical tools

Question *

9ecision making in presence of uncertainty is important :hat are the two steps for risk-$ualified

decision makingK

Buantification of uncertainty and then $uantification of risk

Question +

5n general terms, eplain the link between spatial variability (heterogeneity) and uncertainty

1+
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!s spatial variability increases heterogeneity increases and hence uncertainty increases

Question

5n your own words describe the information effect and how it relates to uncertainty

The information effect is the result of increased available information which leads to less uncertainty

Question -

eostatistics is useful at every point in the life cycle of a reservoir, but where is it most useful and

whyK

eostatistics is most important in the early stages of the life cycle because it makes intelligent use of

limited data and allows for decision making that is tempered with a knowledge and understanding of

the uncertainty inherent in the numerical-geological model

Auly /, 1777

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Lecture 2: Basic Concepts

tatistical Tools

=istograms

4robability 9istribution

'ategorical Cariables

'omparing =istograms 9ata Transformation

&onte 'arlo imulation

%ootstrap

eostatistical, and .ther ;ey 'oncepts

Dumerical 8acies &odeling

'ell %ased &odeling

.bect %ased &odeling

#ecture 2


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and

(23)

Median

The midpoint of the ranked (ie sorted from smallest to largest) data 5f there were 2 data, the median

would be the 13th value 5t also represents the th percentile in a cumulative histogram

Mode

The mode is the most commonly occurring data value in the data set

8ariance

The variance is a measure of spread 5t can be thought of as the average s$uared-distance of the data

from the mean 5t can be found using the e$uation below"

(2+)

#tandard Deviation

The standard deviation is the s$uare root of the variance 5t is sometimes the preferred measure of

spread because it has the same units as the mean whereas the variance has s$uared units

(2)

Coefficient of #eness

The coefficient of skewness is the average cubed difference between the data values and the mean 5f a

distribution has many small values and a long tail of high values then the skewness is positive, and the

distribution is said to be positively skewed 'onversely, if the distribution has a long tail of small

values and many large values then it is negatively skewed 5f the skewness is 6ero then the distribution

is symmetric 8or most purposes we will only be concerned with the sign of the coefficient and not its

value

(2/)

Coefficient of 8ariation

The coefficient of variation is the ratio of the variance and the mean :hile the standard deviation and

the variance are measures of absolute variation from the mean the coefficient of variation is a relativemeasure of variation and gives the standard deviation as a percentage of the mean 5t is much more

fre$uently used than the coefficient of skewness ! coefficient of variation ('C) greater than 1 often

indicates the presence of some high erratic values (outliers)

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8ig 27 &onte 'arlo simulation consists of drawing a normally distributed number and recording the appropriate valuefrom the cdf

&onte-'arlo simulation is the foundation of all stochastic simulation techni$ues &uch care should be

taken to ensure that the parent cdf is a representative distribution, as any biases will be translated intothe results during the transformation

Bootstra#

The bootstrap is a method of statistical resampling that allows uncertainty in the data to be assessed

from the the data themselves The procedure is as follows"

1 draw nvalues from the original data set with replacement

2 calculate the re$uired statistic The re$uired statistic could be any of the common summary

statistics 8or eample we could calculate the uncertainty in the mean from the first set of n

values3 repeat 1times to build up a distribution of uncertainty about the statistic of interest 8or the

eample above we would find the mean of the n values 1 times yielding a distribution of

uncertainty about the mean

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8ig 21 The bootstrap is used to determine the uncertainty in the data itself This diagram shows how the uncertainty in

the mean is found 8irst randomly draw nvalues from the data set and calculate the mean epeat this many times, and thedistribution of the mean $uantifies the uncertainty about the mean

Geostatistical, and 2t$er Key Conce#ts

Petrop-!sical Properties

There are three principle petrophysical properties discussed in this course" (1) lithofacies type, (2)

porosity, and (3) permeability =ard datameasurements are the lithofacies assignments porosity and

permeability measurements taken from core (perhaps log) !ll other data types including well logs and

seismic data are called soft dataand must be calibrated to the hard data H9eutsch, 177?I

Modeling #cale

5t is not possible nor optimal to model the reservoir at the scale of the hard core data The core data

must be scaled to some intermediate resolution (typical geological modeling cell si6e" 1 ft N 1 ft N

3 ft) &odels are built to the intermediate scale and then possibly further scaled to coarser resolutions

for flow simulation

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>ni?ueness; #.oot-ing; and @eterogeneit!

'onventional mapping algorithms were devised to create smooth maps that reveal large scale geologic

trendsG for fluid flow problems, however the etreme high and low values have been diluted and will

often have a large impact on the flow response (eg time to breakthrough would be systematically

under-estimated during a water flood) These algorithms remove the inherent variability of the

reservoirG they remove the heterogeneity within the reservoir 8urthermore, they only provide one

uni$ue representation of the reservoir

nalogue Data

There are rarely enough data to provide reliable statistics, especially hori6ontal measures of continuity

8or this reason analogue data from outcrops and similar more densely drilled reservoirs are used to

help infer spatial statistics that are impossible to calculate from the available subsurface reservoir data

D!na.ic Reservoir C-anges

eostatistical models provide static descriptions of the petrophysical properties Time dependent

processes such as changes in pressure and fluid saturation are best modeled with flow simulatorsbecause they take into account physical laws such as conservation of mass and so on

Data 0!pes

The following list represents the most common types of data used in the modeling of a reservoir"

'ore data ( andby lithofacies)

:ell log data (stratigraphic surfaces, faults, measurements of petrophysical

properties)

eismic derived structural data (surface grids * faults, velocities)

:ell test and production data (interpreted , thickness, channel widths,

connected flow paths, barriers) e$uence stratigraphic interpretations * layering (definition the continuity and

the trends within each layer of the reservoir)

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Lecture !: Geo"oica" Princip"es for $eservoir Mo%e"in

eservoir Types and


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sheets

%9rrentl ver im.ortant

approaches

Car*onate Reservoirs

%y definition carbonate (limestone) rocks are those have greater than O carbonate material Thecarbonate material is either derived from organisms that secrete carbonate as skeletal material or as

fecal matter, or precipitated out of solution #imestone is chemically unstable and is easily converted

to dolomite when hydrothermal fluids rich in magnesium pass through it #imestone is also easily

metamorphised into other rock types such as marble &ost (7O) of carbonate reservoirs can be

modeled using cell based indicator simulation to model limestone * dolomite conversion

9olomiti6ation often has a directional trend 8luid flow in rock is almost always directional and the

magnesium re$uired for dolomit6ation is carried by hydrothermal fluids The fluid flows through the

rock and magnesium replaces calcium creating dolomite The trends can be seen with seismic

(dolomiti6ed limestone has different acoustic properties than limestone) %ecause there are at least two

rock types (limestone and dolostone) we must use estimation methods that make use of multiple

variables Trends such as the conversion of limestone to dolostone may also show up in the geologic

contour maps from the wells

5n other cases, reefs may sometimes be modeled as obects, and there may be areal trends associated

with these as well (! change in sea level may cause a reef to die out and another to form further in or

out)

Table 32 Table for 'arbonate reservoirs

Reservoir0!pe

C-aracteristic #-apes 67a.ples / .portance Modeling0ec-ni?ue

'arbonate


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8igure 3+, ! location map of a sample data set

The assessment of the sampling scheme was correct, there is a northerly bias in the sampling scheme

5t is useful to draw a contour map of the data ! contour map helps gain some insight to the nature of

the data, and can sometimes reveals important trends The map below shows that most of the sampling

occurred in areas of high potential The map in 8igure 3 does not reveal any trends but illustrates thevalue of a contour map

8igure 3, ! contour map using the sample data set The accuracy of the map is not critical 5ts purpose is to simplyillustrate trends

The contour map illustrates that any areas of high potential (red areas) are heavily sampledG a biased

sampling procedure The contour map also illustrates that we may want to etend the map in the east

direction

5t is common practice to use the 'artesian coordinate system and corner-point grids for geological

modeling The corner-point grid system is illustrated in 8igure 3/

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8ig 3/ The standard grid system used for geological modeling

Dotice that theZdimension $in 8igure 3/ is not the same as the dimension ain the areal grid, but the

XY dimension for both the areal and vertical grids are the same 8or the sake of computational

efficiency the stacked areal grids are aligned with Zais, but for fleibility theZais need not be of

the same dimensions as the areal grid This techni$ue proves valuable for"

1 modeling the hydrocarbon bearing formation as a stack of stratigraphic layers" 5t is intuitively

obvious that a model should be built layer by layer with each layer derived from a homogenous

depositional environment !lthough each depositional environment occurred over a large span

of time in our contet the depositional environment actually occurred for only a brief period of

geological time and for our purposes can be classified as a homogenous depositional

environment

2 volume calculations" The model must conform to the stratigraphic thickness as closely as

possible &odeling the formation as a @sugar cube@ model leads to poor estimates

3 flow calculations" 8low nets must have e$uipotential across facies ! @sugar cube@ modelwould yield erroneous results

This permits modeling the geology in stratigraphic layers The stratigraphic layers are modeled as 2-9

surface maps with a thickness and are then stacked for the final model Thus having a non regular grid

in theZdirection allows for conformity to thickness permitting accurate volume calculations, also

allows for flow nets (must be e$uipotential across any face)

eological events are rarely oriented with longitude and latitude There is usually some a6imuth, dip,

or plunge to the formation 5f the angle between the formation and the coordinate ais is large there

will be error a'in the cell dimensions as indicated by 8igure 30 !lso, it is confusing to have to deal

with the angles associated with a6imuth, dip, and plunge, so we remove them and model in some moreeasily understood coordinate system

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8igure 3? 5llustrates the process of rotating the coordinate ais to be aligned with the maor ais of the reservoir 8irst theais are rotated about the 6 ais to accommodate the the a6imuth of there reservoir, second ais are rotated about the y aisto accommodate dip in the reservoir

The rotations can be removed using the the transforms indicated in &atri 33 and 3+

(33)

(3+)

ometimes fluvial channels can be difficult to model because they deviate significantly 5n this case it

is possible to straighten the channel using the transform in


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8igure 3?, Transforming a twisty channel into a straight channel

"tratigra#$ic Coordinates

eservoirs often consist of stratigraphic layers separated by a surfaces that correspond to somese$uence of geologic time events, much like growth rings in a tree The bounding surfaces that

differentiate the strata are the result of periods of deposition or periods of deposition followed by

erosion The surfaces are named according to these geologic events"

Proportional:The strata conform to the eisting top and base The strata may vary in thickness

due to differential compaction, lateral earth pressures, different sedimentation rates, but there is

no significant onlap or erosion (9eutsch, 1777)

0runcation" The strata conform to an eisting base but have been eroded on top The

stratigraphic elevation in this case is the distance up from the base of the layer

)nlap" The strata conform the eisting top (no erosion) but have @filled the eistingtopography so that a base correlation grid is re$uired

Co.*ination" The strata neither conform to either the eisting top or bottom surfaces Two

additional grids are re$uired

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8igure 37 5llustrates proportional, truncation, onlap, and combination type correlation surfaces

The stratigraphic layers must me be moved so that they conform to a regular grid This is done by

transforming the 6 coordinate to a relative elevation using"

(3)

8igure 31 shows how the strata is moved to a regular grid Dote that features remain intact, ust theelevation has been altered to a relative elevation

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8igure 31 5llustrates the result of transferring the 6 coordinate to a regular grid using formula 37

5t is important to note that these are coordinate transformsG the data is transformed to a modeling spaceand transformed back to reality There are no property or distance changes here, ust the movement

from reality to some virtual space then back to reality

Cell #iAe

The cell si6e used in the model is a serious issue 5f the cell si6e is to small an enormous number of

cells will be re$uired to populate the model Too many cells holds the conse$uence of having a model

that is too difficult to manipulate and very taing on the '4> 5f the cells are too large then important

geological features will be removed from the model !s processing power increases model si6e is of

lesser and lesser importance, but with todayEs computers models that range from 1 million cells to million cells are appropriate

&or' ("o)

The specific process employed for 3-9 model building will depend on the data available, the time

available, the type of reservoir, and the skills of the people available 5n general, the following maor

steps are re$uired"

1 9etermine the areal and vertical etent of the model and the geological modeling cell si6e

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f) enerate 3-9 permeability models

g) &erge and translate back to real coordinates

+ Cerify the model

'ombine 6ones into a single model

8igure 311 illustrates the modeling concepts discussed in this lecture"

8ig 311 ! flow chart showing the geostatistical work flow for #ecture 3

Auly 7, 1777

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Question *

:hat does a distribution look like when the coefficient of skewness isG positiveK, negativeK, 6eroK 'an

you provide sets petrophysical properties that would present" a positive coefficient of skewnessK a

negative coefficient of skewnessK a low coefficient of skewnessK

, a) permeability, b) watersaturation, c) porosity

Question +

9efine the coefficient of variation :hat does it mean when the coefficient of variation is greater than

oneK

there are a lot of outliers

Question

4rovide 2 geophysical properties that are positively correlated, negatively correlated, and not

correlated

+ve: porosity and permeability

-ve: impedance and porosity

not: permeability and the weather

Question -

=ow would you interpret a $uantile-$uantile plot whose plotted points deviated from the reference

lineG in a parallel fashionK at an angle from the +th$uantileK

a parallel deviation indicates a dierence in the mean,and a change in slope indicates a dierence in

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Auly /, 1777

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Lecture : -ata .na"*sis

9ata !nalysis

5nference

9eclustering

Trends

econciliation of 9ata oft 9ata 9eclustering

:ork 8low

#ecture + Bui6

Data 5nalysis

9ata analysis is the gathering, display, and summary of the data 9ata analysis is an important step for

building reliable numerical models 5mportant features of the data are reali6ed, erroneous data, and

outliers are revealed The issues addressed by a 9ata !nalysis are"

>nderstanding and cleaning the data

lobal resources assessment" first order * back-of-the-envelop calculations of volumetrics,

facies*6one summary statistics, used to estimate and confirm magnitude of epected results

5dentification of geological populations

5dentification of geological trends

econciliation of different data types" eg transform log-derived porosity measurements to

match the core-derived data

5nference of representative statistics * distributions

'alibration of soft data

Inference

5n recent years the growth of eostatistics has made itself felt more in the petroleum industry than any

other, and an important feature of this growth is the shift in philosophy from deterministic response to

stochastic inference tochastic inference concerns generali6ations based on sample data, and beyond

sample data The inference process aims at estimating the parameters of the random function model

from sample information available over the study area The use of sample statistics as estimates of the

population parameters re$uires that the samples be volume * areally representative of the underlying

population ampling schemes can be devised to ensure statistical representativity, but they are rarely

used in reality 5t is up to the geoscientist to repair the effect of biased sampling, integrate data of

different types, cope with trends, etc, and in general ensure that the data truly is representative of the

population

)utliers and 6rroneous Data


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earch the location map for the outliers 9o they appear to be all in the same locationK 9o they

appear to be inappropriateK

how a cross plot of the local averages versus the data (every point is mapped versus the

average of the surrounding data)

8igure +a, a mean that differs from the mode significantly, a maimum that is significantly higher than the mean, or evena posting that sticks out indicates the presence of outliers

8igure +1 shows some of the things that indicate an outlier ! mean that deviates significantly from

the median, a maimum that deviates significantly from the mean or the median, or a single posting

way out in the middle of nowhere There are three possible solutions for coping with outliers (1) we

can decide to leave them as they are, (2) we can remove them from the data set, (3) or we alter the

value to something more appropriate to the surrounding data


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8igure +b, hould two distributions be separatedK 5t depends on the study

9ecisions made in the geostatistical study must be backed up by sound practice and good udgement

The strategies indicated only serve as an aid in ustifying your decisions you must also document your

decisionsG you know when you may need to ustify them

Declustering

nfortunately this

sampling practice leads to location biased sampling 8igure +1, the location map of the sample data

illustrates location biased sampling The areas of low potential are not as well represented as the areas

of high potential

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8igure +1, Dote that all areas are not sampled unbiasedly ome are areas are heavily sampled and others are poorlysampled

9eclustering corrects the distribution for the effect of location-biased sampling 9eclustering assigns a

weight to each data and calculates the summary statistics using each weighted data

8igure +2, The single data in !rea 1 informs a much larger area than the data in !rea 2

8igure +2 illustrates location-biased sampling The single data in !rea 1 informs a larger area than the

data of !rea 2 5ntuitively one would weight each of the data in !rea 2 by one fifth and the data in

!rea 1 by one 'alculating the weights this way is effective but a more efficient way is to overlay a

grid and weight each data relative to the number of data in the cell area defined by the grid using

8unction +1 below"

(+1)

where /i(c)is the weight, niis the number of data appearing in the cell, and 1+is total number of cells

with data 8igure +3 shows how geometric declustering, or cell declustering works


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8igure +3 The declustering weights for three different cells has been calculated, all other declustering weights are found inthe same way

The declustering algorithm can be summari6ed by"

1 choose an initial declustering cell si6e that includes about one datum per cell and calculate the

mean The goal to finding the optimal declustering cell si6e is to find the lowest mean for datasets where the high potential areas have been over sampled or the largest mean for data sets

where the low potential areas have been over sampled, and there are two parameters that can

effect the declustered mean (1) the declustering cell si6e and the location of the grid origin To

ensure that the optimal declustering cell si6e is chosen several cell si6es and grid origin

locations should be used to calculate the mean The cell si6e that yields the lowest * largest

mean is chosen as the optimal declustering cell si6e 8igure ++ shows the declustered mean

and declustering cell si6e using 2+ different cell si6es between 1 and 2 units and grid offsets

The posted declustered mean is the average of the mean calculated from grid offsets

2 8igure ++, The scatter plot shows the declusteredmean for a variety of grid si6es The minimum declustered mean is the lowest mean The declustering cell si6e is a cell yielding a declustered mean of 2+2

3 >sing the optimal declustering cell si6e decluster the data ecall that step 1 only determines

the optimal declustering cell si6e, it does not decluster the data 8igure + shows histograms

before and after declustering Dotice that the low values (the poorly represented values) are

now more represented and the high value (the overly represented values) are less represented in

the declustered histogram !lso notice that the declustered mean and variance are lower than

the clustered statistics 8igure +/ is a map showing the magnitude of weight applied to each

data Dote the clustered data are under weighted, the sparse data are over weighted, and the

well spaced data are not weighted

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+ 8igure+, =istograms of the data before and after declustering 4oints to note are the clustered mean is larger than the

declustered mean, which is the epected result, and the clustered variance is larger than the declustered variance,also an epected result

8igure +/, ! map of the magnitude of weight Dotice that sparsely sampled areas are up weighted, clustered areasare down weighted, and well spaced are not weighted

#oft Data Declustering

eometric declustering is the preferred method of declustering, but what should be done when there

are too few data to decluster, or when there is systematic preferential drillingK .ne method is to use a

soft data set to develop a distribution for the hard data in locations where there is no hard data, only

soft data 9ata is weighted by the value of the secondary variable at its location oft data declustering

re$uires"

ridded secondary variable, that is defined over the area of study

'alibration cross-plot

&athematical manipulation

0


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'onsider a reservoir that has a porosity known to be negatively correlated with depth, such as that

indicated in 8igure +0 The wells were preferentially drilled along the crest of the structure, and a

model of the reservoir is to be built 'learly the well data will not represent the reservoir unbiasedly,

but seismic data has been collected over the entire area of interest

8igure +0, ! location map showing the locations of well and seismic data The well data is not representative of the entirereservoir and there is very little data to decluster, thus soft data declustering is used

The protocol for soft data declustering is"

&ap the secondary variable over the area of interest

9evelop a bivariate relation between the primary and secondary data, such as that in 8igure

+?a The calibration is critical to this operation and great care must be eercised in developing

it

construct a representative distribution by accumulating all of the conditional distributions

8igure +?b illustrates the concept of adding distributions to create a single distribution

8igure +?a, ! scatter plot of the hard porosity data versus the seismic depth data The inferred correlation between the hardand soft data is shown The relation is inferred from analogue data and the data themselves

?


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Trends

Cirtually all natural phenomena ehibit trends ravity worksG vertical profiles of permeability and

porosity fine upward within each successive strata (9eutsch, 1777) ince almost all natural

phenomena ehibit a trend it is not always appropriate to model using a stationary C educing the

si6e of the study area to a si6e where the assumption of stationarity is appropriate or reducing the

assumption of stationarity to the search radius are two methods for coping with trends >niversal

kriging, an adaptation of ordinary kriging system produces good local estimates in the presence of

trend >niversal kriging can also be used to calculate a trend automatically, but its use should be

tempered with good udgement and sound reasoning instead of ust accepting the result The best

method for coping with a trend is to determine the trend (as a deterministic process) subtract it fromthe observed local values and estimate the residuals and add the trend back in for the final estimate

(&ohan, 17?7) .ften it is possible to infer areal or vertical trends in the distribution of rock types

and*or petrophysical properties, and inect this deterministic information into the model

8igure +?a, Trend mapping is important for at least two reasons" (1) it is wise to inect all deterministic features and atrend is a deterministic feature, and estimation re$uires stationarity and a trend implies that there is no stationary mean

Thus, remove the trend and estimate the residual

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&or' ("o)

Auly 1, 1777

/1


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Lecture : -ata .na"*sis, The Quiz

Question )

%riefly eplain the principle of declusteringK :hy is declustering important in geostatistical oil

reservoir characteri6ationK

Question *

hould all outliers be automatically be removedK :hy or why notK

Question +

:hat tools you use to split data sets into different faciesK =ow would you proceedK

Question

:hy bother using soft data to decluster hard dataK

Question -

:hen we speak of inference in the contet of data analysis what are we striving forK 5s it importantK

:hyK

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Lecture : -ata .na"*sis, The Quiz +o"utions

Question )

%riefly eplain the principle of declusteringK :hy is declustering important in geostatistical oil

reservoir characteri6ationK

:eighting the data so that each is volume representative of the volume of interest 5t is important

because we re$uire that the data used is as representative of the reservoir as possible arbage in

garbage out

Question *

hould all outliers be automatically be removedK :hy or why notK

Do, not all outliers are bad Fou must refer to the location of the data 5s it surrounded by other high

ranking dataK 5s it sitting out in the middle of nowhereK 5s it surrounded by data that would imply thatthere should be a low * high value instead of what is thereK

Question +

:hat tools you use to split data sets into different faciesK =ow would you proceedK

=istograms, scatterplots, $uantile-$uantile plots, location (base maps), contour maps tart with a

location map and a contour map eparate those data that appear to show trends or directional

continuity look at the histogram, do the trends and the peaks (indicating different distributions)

correspond to those locations on the mapsK once separated, look at the scatterplots !re the data setswell correlatedK

Question

:hy bother using soft data to decluster hard dataK

:e know that the data is not representative and it must be declustered 5f we ignore the problem then

we are building models that are not accurate, and there is no way to tell how inaccurate oft data

declustering may seem crude but it is much better than ignoring the problem

Question -

:hen we speak of inference in the contet of data analysis what are we striving forK 5s it importantK

:hyK

:e are striving for summary statistics that are as representative as possible 5t is very important The

entire estimation * simulation process relies on an accurate representation of the reservoir

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Lecture : +patia" -ata .na"*sis

1 Cariograms

2 Cariogram 5nterpretation and &odeling

3 #ecture Bui6

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8igure 2 Two variograms and the corresponding maps The variogram on the right shows no spatial correlation and theresulting map is random The variogram on the right is very continuous showing etensive spatial correlation and therelevant map shows good spatial correlation

8igure 2 shows a map that was made using the variogram 5&2a The variogram indicates that the data

have no correlation at any distance, and hence image a is a random map 5mage 2$was made using

variogram 2$ Cariogram 2$indicates that the data are well correlated at long distances and image

2$shows some correlation at long distance 8igure 3 shows a close up the two images in 8igure2 Dotice that in figure 3$the colors gradually change from blue to green to orange then red, and

that this is not the case in figure 3$ 5n 8igure 3athe colors change randomly with no correlation

from one piel to the net ! piel in image 3ais not well correlated to a neighboring piel, whereas

in image 3$neighboring piels are well correlated

8igure 3 ! close up of an area on each map shows that map a using the variogram having no spatial correlation israndom whereas the map on the right which used a variogram that is continuos and thus the map shows spatial correlation

'orrelation is the characteristic of having linear interdependence between random variables or between

sets of numbers %etween what variables is the variogram measuring correlationK 5n the variograms

presented so far, correlation is being measured between the same variable, but separated by a distance

approimately e$ual toh 8igure + shows conceptually how an eperimental variogram is calculated

The lag distance or distance h is decided upon by the practitioner The two variables are (1) the data at

the head of the vector, and (2) the data at the tail of the vector The data tail of the vector (the circled

end in figure +) is calledz(u)(the random variable at location u)and the data at the head of the vector

is called z(u+h)(the random variable at location u+h) tarting with the smallest lag distance the

algorithm visits each data and determines if there are any data approimately one lag away 5f there

are, the algorithm computes variogram value for one lag !fter each data has been visited, the

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algorithm doubles the lag distance and repeats the calculation 5n this way the eperimental variogram

$uantifies the spatial correlation of the data

8igure + The variogram is not calculated from one single point over varying distances -, rather it moves from point to

point and calculates the variogram for each distance -at each data location

Com#onents of t$e 0ariogram

There are a few parameters that define some important properties of the variogram"

1 ill" the sill is e$ual to the variance of the data (if the data are normal score the

sill will be one)2 ange" the range is the distance at which the variogram reaches the sill

3 Dugget


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8igure a The components of the variogramG the sill is the variance of the variable under study, the range is the distanceat which the variogram plateaus, the nugget effect is the short scale variability

The nugget effect is a measure of short scale variability, any error in the measurement value or the

location assigned to the measurement contributes to the nugget effect The range shows the etent of

correlation, and the sill indicates the maimum variability, or the variance of the data 8igure b

shows what happens when we change two parameters the nugget effect and the range ecall that the

sill is a fied value 5t is the variance of the data 5mages a $ and c in 8igure b shows the effect of

different ranges ! variogram with no range is shown in image a image $has an intermediate range,

and image chas a long range 5mages d eandfshow the effect of increasing nugget effect 5mage d

shows the effect of no nugget effect, or no short scale variability, image e shows an intermediate

amount of nugget effect, and imagefshows pure nugget effect, or complete short scale variability

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/?


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8igure b ome variogram eamples to show the effect of different parameters

Qualitative "#atial Data 5nalysis

5n probabilistic notation the variogram is written"

(1)

:hich says the variogram is the epected value of the s$uared difference of Z(u) andZ(u#h)& The

semivariogram is defined"

(2)

To be precise the semivariogram is one half the variogram 5n this lecture we will assume that the

variogram and the semivariogram are synonymous

%efore the variogram is calculated some data preparation must be performed 9ata must be free from

outliers and systematic trends, as well, since geostatistical simulation re$uires normally distributed

data, the data must be transformed into normal space


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8igure / !n illustration of the the lag, lag tolerance, a6imuth, a6imuth tolerance and bandwidth parameters for variogrammodeling


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8igure 7a

There is some discussion as to whether or not the sill is or is not e$ual to the variance The concern is

that the variance is susceptible to outliers, this is why it is important weed out outliers before the

variogram analysis another concern is the use of declustering weights using declustering weights

reduces the variance so which variance do we useK for now we will use the apriori variance the final

issue concerns the dispersion variance 5 will leave the issue of dispersion variance for the reader to

investigate

5nisotro#y

5f a petrophysical property has a range of correlation that is dependent on direction then the

petrophysical property is said to ehibit geometric anisotropy if the petrophysical property reaches the

sill in one direction and not in another it is said to ehibit 6onal anisotropy 5n other words, a

variogram ehibits 6onal anisotropy when the variogram does not reach the epected sill&ost

reservoir data ehibit both geometric and 6onal anisotropy 8igure 7 first geometric anisotropy,

second 6onal anisotropy, and lastly both forms of anisotropy

figure 7b

Sonal anisotropy can be the result of two different reservoir features" (1) layering, the hori6ontal

variogram does not reach the epected sill because there are layer like trends that eist and variogram

is not reaching full variabilityG and (2) areal trends, the vertical variogram does not reach the epected

sill due to a significant difference in the average value in each well

Cyclicity

eological phenomenon often formed in repeating cycles, that is similar depositional environments

occurring over and over ! variogram will show this feature as cyclicity as the variogram measures the

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spatial correlation it will pass through regions that bear positive then negative correlation while still

trending to no correlation ! cyclic variogram can be seen in 8igure 1

8igure 1 ray scale image of an


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(arge "cale Trends

virtually all geological processes impart a trend in the petrophysical property distribution

9olomiti6ation is the result of hydrothermal fluid flow, upward fining of clastics, and so on, are largescale trends 8igure 11 shows how large scale trends affect the histogram Trending causes the

variogram to climb up and beyond the sill of the variogram

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0ariogram modeling

!ll directional variograms must be considered simultaneously to understand the 39 spatial correlation

1 'ompute and plot eperimental variograms in what are believed to be the principal directions

of continuity based on a-priori geological knowledge

2 4lace a hori6ontal line representing the theoretical sill >se the value of the eperimental(stationary) variance for continuous variables (1 if the data has been transformed to normal

score) and p(l Pp) for categorical variables where p is the global proportion of the category of

interest 5n general, variograms are systematically fit to the theoretical sill and the whole

variance below the sill must be eplained in the following steps

3 5f the eperimental variogram clearly rises above the theoretical sill, then it is very likely that

there eists a trend in the data The trend should be removed as detailed above, before

proceeding to interpretation of the eperimental variogram

+ 5nterpretation"

o #-ort=scale variance: the nugget effect is a discontinuity in the variogram at the origin

corresponding to short scale variability 5t must be chosen as to be e$ual in all

directionsG pick from the directional eperimental variogram ehibiting the smallestnugget !t times, one may chose to lower it or even set it to

o nter.ediate=scale variance:geometric anisotropy corresponds to a phenomenon with

different correlation ranges in different directions


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8igure 13

2 The spherical model The spherical model is the most common variogram model type The

spherical model is mathematically defined by formula 1+, and 8igure 1+ shows a spherical

type model

(+)

8igure 1+

3 The eponential model The eponential model is similar to the spherical model but it

approaches the sill asymptotically 5t is mathematically defined by formula and shown as a

variogram in 8igure 1

()

8igure 1

0/


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8igure 1?

/ The last model type is known as the dampened hole effect because it includes a damping

function in its mathematical formula (formula 7) The model variogram is shown in 8igure

17

(7)

8igure 17

!4am#les

The variogram models in the principal directions (maor hori6ontal, minor hori6ontal, and vertical)

must be consistent, ie, same nugget effect and same number and type of structures This is re$uired so

that we can compute variogram values at angles not aligned with the principle ais in off-diagonal

directions and between eperimental values The responsibility for a licit variogram model is left to

practitioner, current software does not help very much

%asic idea is to eplain the total variability by a set of nested structures where each nested structure

each having different range parameters in different directions"

(1)

where his the distance, h, hy, h6are the direction specific distance parameters, and a, ay, a6, are the

directional range parameters derived from the variogram models The range parameters a , ay, a6can

approach or positive infinity

=ow do we ensure a legitimate modelK

0?


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1 pick a single (lowest) isotropic nugget effect

2 choose the same number of variogram structures for all directions based on most comple

direction

3 ensure that the same sill parameter is used for all variogram structures in all directions

+ allow a different range parameter in each direction

model a 6onal anisotropy by setting a very large range parameter in one or more of the

principal directions

8igure 2 shows some simple 19 eperimental variograms, the respective models and parameters

8or each eample the sill is 1

8igure 2, The top eample shows a 19 eample with two nested structures 4oints to note are (1) the sum of the sill

contributions is e$ual to one, (2) the total range of the variogram is the sum of the component ranges The middle eampleshows a power model variogram, note that neither the eperimental or the the model variogram reach a sill The bottomvariogram is a simple eponential model

&ost beginning practitioners over model the eperimental variogram That is most beginners apply to

many nested structures and try very hard to catch each point in the model There should never be more

than three nested structure (not including the nugget effect), and the model need only be accuratewithin the range of its use eferring to the middle variogram of 8igure 2, if the variogram is not

needed beyond 12 units there is no need to model beyond 12 units 8igure 21 shows some more

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comple eamples Dote that they are all 29 eamples &odeling 39 variograms is a simple etension

of the same principles used to model 29 5n short only model what you need to

8igure 21, The top left and bottom right eamples model 6onal anisotropy The bottom left eample shows a model ofgeometric anisotropy Th top right eample is $uite comple, re$uiring the need of a hole effect model to correctly modelthe spatial correlation

6ork &lo1

&odeling the spatial correlation is the most difficult and important step in the geostatistical modeling

process reat care should be taken

?


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Lecture : +patia" -ata .na"*sis, The Quiz

Question )

The scatterplot is an ecellent visual tool to display correlation The correlation coefficient is an

intuitive measure of two point correlation 8or every point on a variogram plot ((-) versus -) there is

a scatterplot of z(u) and z(u-) values 9raw a scatterplot and give an approimate correlation

coefficient for the three points labeled !, %, and ' on the above figure

Question *

!n analytical variogram model is fit to eperimental points ive three reasons why such variogram

modeling is necessary

Question +

:hat is the difference between the variogramandsemivariogramK :hat is the difference between an

eperimental and a model semivariogramK

Question

'omplete the shorthand notation for the following variogram model"

Question -

=ow does geometric anisotropy differ from 6onal anisotropyK :hat is happening in an eperimental

variogram ehibits 6onal anisotropyK

?2


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Question -

=ow does geometric anisotropy differ from 6onal anisotropyK :hat is happening in an eperimental

variogram ehibits 6onal anisotropyK

a) eometric" the range in one direction differs than the range in another Sonal" anges are the same

but one direction does not reach the sill

b) The eperimental variogram in the direction that is reaching the sill is achieving full variability, and

the other is not The could be a result of either an areal trend, or layering in the reservoir

?+


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8igure /2 The kriging weights must consider redudancy of the data, the closeness of the data, and the

direction and magnitude of continuity

There is one other goal when estimating the unknown attribute" minimi6e the error variance 5f the

error variance is minimi6ed then the estimate will be the best estimate The error variance is theepected value of the difference between the known and the estimate and is defined by"

(/+)

where6N(u) is the estimator, andz(u) is the true value .ne obvious $uestion raised by this e$uation is

how can we determine the error if we do not know the true valueK True, we do not know the true value,

but we can choose weights that do minim6e the error To minimi6e the estimation variance take the

partial derivative of the error variance (e$uation /+) and set to , but before taking the derivative

e$uation /+ is epanded"

(/)

The result is an e$uation that refers to the covariance between the data points %(u;u), and the data

and estimator%(u;u) ! first this may seem like a problem because we have not dicussed the

covariance between the data and the estimator, but we did discuss the variogram, and the variogram

and the covariance are related ecall that the variogram is defined by"

and note that the covariance is defined by (the covariance is not the s$uared difference whereas the

variogram is)"

The link between the variogram and the covariance is"

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so the variogram and the covariance are linked by"

(//)

where (-) is the variogram, '() is the variance of the data, and '(-) is the covariance

This makes it possible to perform kriging in terms of the variogram instead of the covariance

'ontinuing with the derivation of the kriging e$uations, we know that formula / must be minimi6ed

by taking the partial derivative with respect to the weights and set to 6ero"

setting to 6ero

(/0)

The result of the derivation in terms of the variogram is the same because both the variogram and the

covariance measure spatial correlation, mathematically"

(/?)

and the system of e$uatiuons in terms of the variogram is"

(/7)

This is known as simple kriging There are other types of kriging but they all use the same fundamental

concepts derived here

??


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Discussion

There are a couple of motivations behind deriving kriging e$uaitons in terms of the covariance"

1 5ts easier olving the kriging e$uations in terms of the variogram re$uires that the mean be

carried throughout the derivation 5t is easier to simplify in terms of covariance

2 5t is possible to have the variance at hL be 6ero with a variogram, this makes the matri veryunstable The covariance is defined as the epected vaue of the difference, not the s$uared

difference therefore the value of the covariance at hL is always large and hence the main

diagonal in the matri will always be large &atrices that have small main diagonal elements

such as when using the variogram are difficult for solution algorithms to solve due to truncation

errors and so on

3 it is eay to convert the variogram to covariance

Im#limenting Kriging

.nce again, consider the problem of estimating the value of an attribute at any unsampled location u,

denoted 6N(u), using only sample data collected over the study area A, denoted byz(un) as illustrated in

8igure /3 8igure /3 shows the estimator (the cube), and the data (z(un)) To perform kriging ust fill

in the matrices 8or eample, filling in the left hand matri, entry 1,1, consider the variogram between

points 1 and 1 The distance between a point and itself is , and thus the first entry would be the nugget

effect entry number 1,2, consider the distance h between points 1 and 2, read the appropriate

variogram measure and enter it into the matri repeat for the all of the variogram entreis and solve for

the weights n

8igure /3

?7


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8igure /+

8igure /

The estimate is then calculated as"

(/1)

The result is an estimate of the true value and the error associated with the estimate, as 8igure //

illustrates

8igure //

;riging provides the best estimate but there are some issues"

T$e ros and Cons of Kriging

ros7 Cons7

The best linear unbiased estimator

Smooths

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(/10)

The se$uential simulation workflow is as follows"

1 Transform the original S data to a standard normal distribution (all work will be done in

@normal@ space) :e will see later why this is necessary

2 o to a location u and perform kriging to obtain kriged estimate and the corresponding kriging

variance"

9raw a random residual (u) that follows a normal distribution with mean of and varianceof 2;(u)

!dd the kriged estimate and residual to get simulated value"

Dote that ZN (u) could be e$uivalently obtained by drawing from a normal distribution with

meanZN(u) and variance 2;(u)

!dd F? (u) to the set of data to ensure that the covariance with this value and all futurepredictions is correct !s stated above, this is the key idea of se$uential simulation, that is, to

consider previously simulated values as data so that we reproduce the covariance between all of

the simulated values

Cisit all locations in random order (to avoid artefacts of limited search)

%ack-transform all data values and simulated values when model is populated

'reate another e$uiprobable reali6ation by repeating with different random number seed

(9eutsch, 1777)

6$y a Gaussian 8%ormal9 Distriution:

The key mathematical properties that make se$uential aussian simulation work are not limited to the

aussian distribution The covariance reproduction of kriging holds regardless of the data distribution,

the correction of variance by adding a random residual works regardless of shape of the residual

distribution (the mean must be 6ero and the variance e$ual to the variance that must be re-introduced),

and the covariance reproduction property of kriging holds when data are added se$uentially from any

distribution There is one very good reason why the aussian distribution is used" the use of any other

distribution does not lead to a correct distribution of simulated values The mean may be correct, thevariance is correct, the variogram of the values taken all together is correct, but the @shape@ will not be

5n @aussian space@ all distributions are aussian H13I There is a second, less important, reason why

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the aussian distribution is used" the central limit theorem tells us that the se$uential addition of

random residuals to obtain simulated values leads to a aussian distribution The construction of

kriging estimates is additive %y construction, the residuals are independent and there are many of

them The only caveat of the central limit theorem that we could avoid is the use of the same shape of

distribution, that is, we may avoid multivariate aussianity if the shape of the residual distribution was

changed at different locations The challenge would be to determine what the shape should be

(9eutsch, 1777)

T$e ros and Cons of "imulation

ros7 Cons7

Honors the histogram ulti!le images

Honors the variogram

"once!tually difficult to

understand

#uantifies globaluncertainty

$ot locally accurate

a%es avaiable multi!lereali&ations

&acies "imulation

8acies are considered an indicator (categorical) variable and simulating facies re$uires indicator

simulation

7+


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Lecture 6: Geostatistica !a""i#$ %o#ce"ts& he uiz

9uestion 1

;riging is often referred to as a $est linear 9n$iased estimatorat an unsampled location


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9uestion 3

:hat is &onte 'arlo simulationK =ow is &onte 'arlo simulation performed from a cdf :(x);

draw a random number read the appropriate $uantile

9uestion 4

:hy is decl9steringused prior to stochastic simulation of porosityK

:e use the distribution in se$uential aussian simulation so the distribution must be as accurate as

possible

9uestion "

:hat are the features of a simulated reali6ation that make it preferable to a ;riging map for oil

reservoir evaluationK :hat features are not appropriate for aussian simulationK

models the heterogeneity whereas the kriging map smooths out the heterogeneity

features that are connected like permeability are not appropriate for simulation

7/


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Lecture : +tructura" Mo%e"in

Celocity >ncertainty

urface %ased &odeling

urface 8lapping 9istribution

8ault =andling #ecture 0 Bui6

Introduction

5t is estimated that hydrocarbon reserves recoverable through improved reservoir management eceed

new reserves that can be added through eploration 5ncreasingly, it is being recogni6ed that 39

seismic data analysis is a critical reservoir management technology and plays a key role in reservoir

detection, delineation, characteri6ation, and monitoring =owever, 39 seismic alone is inade$uate for

many applications due to (1) limited resolution, and (2) the indirect and*or weak relationships between

critical reservoir parameters such as permeability, porosity, and water saturation !s a result, it isgenerally recogni6ed by reservoir scientists that proper reservoir description and monitoring re$uire

full integration of 39 seismic with engineering, geological (including geochemical and geostatistical),

petrophysical, and borehole geophysical methods

eismic is very good at many things such as resolving large scale structural features, recovering

information from 7777O of the volume of interest, but it is not good at resolving fine scale features

such as resolving petrophysical properties, this task is left to geostatistics .ther more intimate

petrophysical property sensing tools such as logs offer fine scale measures of petrophysical properties

but offer no insight into what lies beyond the tools range, this task is left to geostatistics 'ore data is

an even more finely scaled measure of petrophysical properties, but it range of insight is even less than

that of log data eostatistics

eostatistics uses the coarse-scale structural information offered by seismic, the mid-scale information

offered by electric logs and the fine scale information of core data to generate high resolution models

of oil reservoirs ! reservoir model is built starting with the large scale features, the structural features,

first This lecture will discuss the uncertainty in interpreted seismic surfaces, and in the event there is

no reliable seismic data, how to simulate the surfaces that would define structure and fault handling

0elocity 'ncertainty

The geophysical branch of the eploration science is primarily concerned with defining subsurface

geological features through the use of seismic techni$ues, or the study of energy wave transmissionsthrough rock eismic techni$ues can be used to make subsurface maps similar to those developed by

standard geological methods

The three main rock properties that the geophysicist studies are 1) elastic characteristics, 2) magnetic

properties, and 3) density !lthough studies of density and magnetic properties provide useful

information, elastic characteristics are considerably more important since they govern the transmission

of energy waves through rock 5t is this elastic characteristic that is studied in seismic surveys The

word seismic pertains to earth vibrations which result from either earth$uakes or artificially induced

disturbances

eflection seismic surveys record the seismic waves that return or reflect from subsurface formation

interfaces after a seismic shock wave has been created on the surface %y measuring the time re$uired

70
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for different waves to be reflected from different formations, the geophysicist can identify structural

variations of the formations 8igure 01 illustrates this process in a typical land survey operation

8igure 01, a typical land survey operation

The obective of seismic work is to develop maps that indicate structures which might form traps for

oil or gas from the data provided on the record cross sections The geophysicist makes maps bycalibrating the seismic attribute to core data, well log data, and analogue data !ny feature that causes

a change in propagation of sound in rock shows up in the seismic survey 'hanges in dip, different

rock types, possible faults, and other geological features that are some of the features indicated in the

sections These features are not immediately evident in the seismic data everal steps are necessary to

convert seismic data into useful structural and stratigraphic information


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8igure 0+ The parametric surface after @undulation@ has been added

The model is built surface by surface and each surface is deposited on top of the eisting surfaces (if

there are any, and there wont be at first) using the following protocol"

1 The central location of the new surface (x+ +) is selected stochastically The distribution used

for selection of the location is derived from the distribution of possible locations given the

thickness of the reservoir !t the beginning of the simulation all surfaces have the same

probability of selection, but as the simulation continues the reservoir builds up thickness and

there fewer permissible surfaces that will comply to the selection and conditioning criteria

2 The length of the surface Xis randomly selected from a triangular pdf with the minimum and

maimum parameters being user selected

3 The inner and outer widths, the height of the surface and the orientation of the surface are

selected from a triangular distribution with parameters provided by the user

+ The surface is @dropped@ onto the reservoir !ny eisting surfaces will truncate the new

surface 8igure 0 shows the dropping principle in action

8igure 0 The dropping principle used in the simulation of surfaces

'ondition the surface to the data !ll of the surfaces are dropped to the same datum as

indicated in figure 0 There are two solutions if the surface does not conform to the

intersections provided by the well data (1) raise the surface to meet the intersection, and (2)

lower the surface to meet the intersection 5f the surface is raised it could be reected if it too

short and will not be truncated by eisting surfaces, instead, the surface is lowered to the

intersection as in 8igure 0/

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8igure 0/ 'onditioning of the surface to a single well data

/ epeat until the reservoir is fully populated

8igure 00 shows an eample of simple parametric surface simulation

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"urface &la##ing

urface flapping is an acronym for surface uncertainty 8igure 0? shows surface flapping The pink

vertical lines are well data that the surface must be conditioned to The light blue hori6ontal line is the

gas oil contact and the pink hori6ontal line is the oil water contact The dark blue lines are top surface

reali6ations and the green lines are bottom surface reali6ations There is uncertainty in the true location

of the top surface and the bottom surfaces everywhere ecept at the wells The blue and green lines

illustrate the etent of uncertainty about these surfaces The green lines do not flap as wildly as the

blue lines There is sound reasoning for this urface uncertainty cannot be assessed independently,

once the uncertainty in the top surface with respect to the present-day surface has been established allother remaining surfaces will have less uncertainty The uncertainty in remaining surfaces is accounted

for in the uncertainty with respect to the distance between layers .ne could imagine considering each

of the surface uncertainties modeled independently but in doing so negative volumes could be created

(the surface lines could cross) !lso the distribution of thickness would be ignored 5n those locations

where the top and bottom surfaces cross there might be at minimum 6ero thickness, but the seismic

derived distribution of thickness might suggest that there is a very low possibility of 6ero thickness

This is why we model the top surface uncertainty first conditioned to the uncertainty of the surface

with respect to the present-day surface and all remaining surfaces are modeled conditional to the

uncertainty of the thickness between surfaces !n important point that must be pointed out is velocity

uncertainty and how this uncertainty relates to depth * surface determination is not considered here,

this is a simplified model meant only to illustrate surface uncertainty with respect to the well data The

uncertainty in the surfaces can be modeled using se$uential aussian simulation

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8igure 0? ! diagram of surface uncertainty

!ssessing the uncertainty in surfaces is important for the determination of pore volume and hence

predicted oil in place volumes 8or eample consider the calculation of the gross pore volume"

The net-to-gross ratio and net porosity are inferred from the well data, available seismic data and

geological interpretations There are uncertainties eisting in the determination of the net-to-gross

ratio and the net porosity due to limited well data and uncertainty in the calibration of soft seismic and

geological data >ncertainties in all factors propagate to uncertainty in the final calculation of pore

volume The uncertainty in pore volume is a function of the multivariate distribution of the three

contributing factors" C, net-to-gross ratio, and net porosity 5nference of this multivariate

distribution is difficult due to the poorly known dependencies such as the relationship between

porosity and surface interpretation ! particular model of this multivariate distribution can be built

assuming that the three factors are independent :e will adopt such a model The distributions of

uncertainty in the three controlling variables must be determined

The top and bottom surfaces will be stochastically modeled to $uantify the distribution of uncertainty

in the C This modeling is guided by well data and the best estimate of the surface from seismic

The uncertainty of the average net-to-gross ratio and the net porosity are determined by bootstrap

resampling from the best distribution that can be inferred from limited well data and supplementary

seismic and geologic data The gross roc vol9me is the reservoir volume above the oil*water contact

(.:') constrained by the top and bottom surfaces of reservoir ! gas-oil contact is needed for

reservoirs with gas 8igure 07 shows a cross section view of a reservoir

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8igure 07 ! cross-section of a hypothetical oil reservoir

The reservoir is constrained by a top and bottom surfaces (black curves) The .:' is represented by a

red hori6ontal line and the gas-oil contact G


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Lecture : +tructura" Mo%e"in, The Quiz

Question )

:e are not eplicitly trying to model surfaces that define geological structures :hat are we trying to

modelK :hyK

Question *

:hat kind of data is geophysics usefull for (in the contet of this lecture)K :hat is the place of

geostatistics, and how does it integrate with geophysical data to provide accurate modelsK

Question +

:hy model the surface closest to present-day surface firstK :hy not model each surface with the same

uncertaintyK :hat does this techni$ue preventK

Question

Dame two sources of velocity uncertainty, and suggest how these sources can impact the reservoir

model

Question -

mall scale faults are often ingnored in the reservoir model, whyK

solutions

Auly 31, 1777

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Question -

mall scale fratures are often ingnored in the reservoir model, whyK

5t is assumed that small scale fractures cam be approimately handled by effective flow properties

Auly 31, 1777

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Lecture : Ce"" Base% (acies Mo%e"in

&ethodology

e$uential 5ndicator &ethods

Truncated aussian &ethods

'leaning 'ell %ased eali6ations

#ecture ? Bui6

Introduction

eservoir simulation and decision making re$uires 39 distributions of petrophysical attribute like

porosity, permeability and saturation functions There is no direct need for lithofacies models

Devertheless lithofacies are considered important because petrophysical properties are often highly

correlated within the lithofacies types ;nowledge of the lithofacies types serves to constrain the range

of variability to the distribution of properties within the lithofacies #ithofacies are distinguished by

different grain si6e diagenetic alteration, or any other distinguishing feature 8or eample shale is

different than sandstone which is different than limestone which is in turn different that dolomite 5tshould be born in mind however that we are first and foremost concerned with making realistic

distributions for the purpose of decision making, not making pretty pictures

Met$odology

%efore even beginning to model lithofacies one should ask if it is a worthwhile venture &odeling

lithofacies may not always yield improved prediction of reservoir performance To make this decision

easier consider the following"

1 The lithofacies types must have significant control over the petrophysical properties

ignificant control is a contentious issue, however a useful guide would be to consider adifference in the mean, variance, and shape of at least 3O !s well the saturation function

should not overlap between lithofacies, and here there should also be a difference in the overall

average of 3O between lithofacies

2 9istinct lithofacies must be easily discerned in well log data and core data

3 the lithofacies must be at least as easy to model as the petrophysical properties implicity rules

here .verly elaborate models will be detrimental to the model

=ow many lithofacies should be modeledK The number of lithofacies to be modeled is a decision that

must be made at the time of modeling, however, there are some pieces of advice to be offered Two

@net@ and one @non-net@ (net means oil bearing) lithofacies often provide sufficient detail for most

reservoirs The limit of workability is three net and three non-net lithofacies %eyond this limit modelsbecome nearly unworkable

:ith the lithofacies distinguished select the modeling techni$ue"

1 cell based modeling (purely stochastic)

2 obect based modeling (partly stochastic and partly deterministic)

3 deterministic modeling (purely deterministic)

Dote that deterministic modeling is always preferred :hy leave things to chance when know the

responseK

'ell based modeling is by far the most preferred method for modeling lithofacies ome of the reasons

include"

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8igure ?+

Cleaning Cell Based Realisually the smaller proportions suffer

This is a product of the order relations carried out in the indicator kriging algorithm (correcting for

negative kriging weights)

The easiest way out of this dilemma is to post process the reali6ations to honor the target proportions

.ne useful method for cleaning is $uantile transformation Buantile transformation works well when

there is a natural nesting order to the lithofacies categories, such as a continuous variable transformed

to a categorical variable, however, artifacts can occur when dealing with more than two unorderedlithofacies

Ma7i.u. Posteriori #election

&aimum a posteriori selection (&!4) replaces the lithofacies type at each location uby the most

probable lithofacies type based on a local neighborhood The probability of each lithofacies type is

based on the following criteria"

1 'loseness of the data in the window to the location u

2 whether the data is a conditioning data

3 mismatch from the target proportion

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8igure ? The &!4 algorithm at work

8ormally, consider an indicator reali6ation,

where the proportions of each lithofacies type in the reali6ation are

(the probability of the indicator being state is between and 1) with the sum of all proportions being

e$ual to one, and the target proportions of each lithofacies bound by the same properties

Dow consider these steps to cleaning the reali6ation

and bringing the the proportions

closer to the target proportions

!t each of the=locations for al of the locations within the area of study, calculate the probability >(u)

for each indicator based on a weighted combination of surrounding indicator values"

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gLweight to ensure that the new lithofacies are closer to the target global proportionsG

specifically, increase the proportion if it is less than the target proportion, and decrease it if it is

too high"

where . are the target proportions and .()

are the reali6ation proportions

!s one would epect, the si6e of the window ?(uE) and the distance weighting have significant impact

on the @cleanliness@ of the result c(uE) has the effect of cleaning in favour of conditioning datagLdoes not impose the targ

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