GeoCh8

Well Control11 88Geological StatisticsC O N T E N T S

1. INTRODUCTION

2. MEASURES OF CENTRAL TENDENCY

3. MEASURES OF VARIABILITY

4. DISTRIBUTIONS

5. SAMPLE SUFFICIENCY

6. MEASURES OF SPATIAL CORRELATION

7. EXERCISES

12

LEARNING OBJECTIVES:

In this Chapter, we introduce some of the various statistical techniques used in theanalysis of reservoir data. These techniques play an important role in DevelopmentGeology in two areas:

Reservoir characterisation requires quantification of the geological description foruse in reservoir simulators.

Well data are few and far between and statistical techniques are used to populate theinterwell volumes.

The student will learn :

How to average permeability and porosity

How to explore the relationship between permeability measurements and geology

How to relate core and test permeability

How to measure permeability heterogeneity

Construct and evaluate measures of spatial correlation (variograms)

At the end of this Chapter the Petroleum Engineeer will be able to analyse core dataand prepare summary data for mapping or reservoir simulation or for general use inpetroleum engineering. The student should also be able to design a core samplingprogramme for porosity and permeability.

88Geological Statistics

Department of Petroleum Engineering, Heriot-Watt University 3

1. INTRODUCTION

This brief introduction to statistics is designed to give an overview for industryprofessionals who find themselves involved in reservoir characterisation. Reservoircharacterisation seeks to define quantitatively the input data needed to undertakepredictions of flow through permeable media (Lake, 1991). There is an obvious needfor hard numerical data and spatial information relating to the petrophysical proper-ties. This information is needed in numerical reservoir simulation.

Before commencing any discussion of statistics, some terminology needs to be clearlyunderstood. The properties of a reservoir unit for which the geologist or engineer isrequired to infer (or estimate or guess) values can be considered a population. Thispopulation may be the entire reservoir (e.g., the Brent Group reservoir in the NorthSea), a subdivision of the reservoir (e.g., the Etive, Rannoch Formations of the BrentGroup) or even a sedimentological entity within the reservoir (e.g., a bedform orlamina type). In each case, one can estimate the population parameter (e.g., mean,standard deviation, etc.) by statistical inference from a statistic computed from somedata. In this Chapter we will learn the definitions and how to calculate the simpleststatistical parameters.

The geologist usually estimates the population parameter (e.g., the mean) by anappropriate descriptive statistic (e.g., an average) from a sample. The sample (in thestatistical, rather than geological sense) can be a small set of measurements (e.g., coreplugs) taken from the reservoir or population of interest. The confidence with whichthe sample statistic can be taken as an estimate of the population parameter can alsobe quantified, expressing the uncertainty in the prediction of the property.

In the petroleum industry, the samples that are available are generally very small andnot necessarily representative. It is common to infer the parameters for an entirereservoir (order 108-1010 m3) from a few cores (10- 102 m3) from which limited samplesare taken (10-2-10-3 m3). The wells that are cored may be drilled in unrepresentativefield areas. In addition, only the exploration and appraisal wells (under non-optimumconditions of interval, recovery or mud chemistry) and the first few developmentwells are cored. Lack of representivity may arise from the fact that these wells aregenerally located in the crestal areas of the field which, possibly because of variablediagenesis within the hydrocarbon column, are often not representative of thereservoir population as a whole. Cores are rarely taken in the aquifer beneathhydrocarbon accumulations, but aquifer parameter estimation can be as important asparameters for the reservoir and the aquifer may have a different diagenetic history.

It is important to recognise that the estimates of the core population parameters (i.e.,average horizontal permeability or porosity) should be based on sufficient samples (innumber and size) taken from that core. If the core properties are poorly estimated, onecan expect the reservoir properties to be poorly modelled. Geologists and engineersshould aim to, at least, provide good estimates of core populations. The more variablea parameter is, the more samples are required to estimate it - permeability is commonlyvery variable and therefore most difficult to estimate.

14

A typical analysis of a core data set might include evaluation of:

summary numbers the variability how the summary numbers relate to the variability how good the numbers are how the variability of a property relates to location in the reservoir.

The next sections in this Chapter provide the tools to undertake such an evaluation.

2. MEASURES OF CENTRAL TENDENCYThe most commonly used descriptive statistics that are determined from a sample arethe measures of central tendency. By central tendency we mean the tendency of theobservations (measurements) in a sample to centre around a particular value ratherthan spread themselves across a range of values. When required to produce a set ofsummary numbers that describe our available set of variables, the average is the mosteasily determined.

Firstly, we need to discuss the types of variables that occur in petroleum engineering.These are illustrated in Figure 1.

Variables

CATEGORY QUANTITY

Nominal Ordinal

Ordered categories

Ranks

Discrete (counting)

Continuous (measuring)

Category variables refer to any variables that involve putting individuals intocategories. These can be: nominal (e.g., names such as make of bit, formation ) or, ordinal (e.g., a number to represent degrees of bit wear, or stratigraphic location).Ordinal data can be ranked (e.g., bits ranked by relative condition of wear, low= 1 tohigh = 7) or ordered (e.g., stratigraphic unit 1 is older than stratigraphic unit 2).

Quantity variables have a numerical quantity that can be either discrete (e.g., fossilspecimens, numbers of channels) or continuous (e.g., permeability, porosity, etc.). Itis the latter type of variables that we are most interested in.

Quantity variables can be either extensive or intensive. Extensive variables areadditive (e.g., volume). The volume-weighted average of the porosity for two samplesis the average porosity for the combined volume and a meaningful quantity.

Figure 1Types of variables (afterRowntree, 1981)



Permeability, on the other hand is an intensive variable, and is generally non-additive.The average of two plugs, e.g., a shaley core plug (1mD) and a clean sandstone(1000mD), combined, would depend on how they were combined. Intuitively, thecombined value could be either closer to 1000mD (in the case that flow is parallelthrough the two plugs) or closer to 1mD (flow through the two plugs in series). Theaverage permeability depends on the flow conditions and flow geometry. Inengineering the effective property is the property of an equivalent homogeneousmedium. Permeability averages are often used to estimate the effective permeability.

Whilst more than one measure of central tendency can be obtained for each type ofvariable, the numbers obtained may not always be meaningful. We now discuss thevarious measures and consider the relative merits of the measures in reservoircharacterisation. In this text, mean is the population parameter for central tendencyand average, the estimator of the population mean calculated by a sample statistic(such as the arithmetic average).

Arithmetic AverageThe arithmetic average of N data is obtained by adding the quantities and dividing bythe number of data in the sample. This is commonly expressed mathematically as:

kar =

11N

kii

N

=

where k represents permeability.

The arithmetic average is an estimator of the arithmetic mean is equally sensitive toall values. The arithmetic average (and the other averages) can be biassed. Bias is asystematic error in the estimator and may occur because of a number of reasons:

sands may be sampled more than other lithologies leading to high values for thearithmetic average, it may not be possible to plug very loose sands - often the best reservoir zones -leading to a pessimistic estimate of permeability, thin low permeability baffles (faults, thin siltstones) may not be sampled leadingto optimistic estimates of permeability.

Geometric averageThe geometric average is determined as the Nth root of the product of N data and isusually written as:

kgeom =

i=1

N ki

1/N

The geometric average of permeability can also be considered as the exponential ofthe arithmetic average of the natural log of permeability. This is easier to compute,as the product term in the above expression rapidily exceeds the capacity of mostcomputers. It can be written in this form as:

16

kgeom = exp

1

N i=1

N loge(ki)

The geometric average is indeterminate in the presence of zero data values and this cancause problems for a sandstone matrix containing shales. Very low values (0.001Mdor less) should be used instead of zeros.

Harmonic averageThe harmonic average for N permeability data is given by:

1khar = N -1N

i = 1

ki

Like the geometric average, the harmonic average is also indeterminate in thepresence of zero values and low values should be substituted.. All rocks (possiblyexcept salt) have some permeability , even if the measurement of it is below theresolution of the device in use.

The inverse of permeability (i.e., k-1) can be considered as resistance to flow. Theharmonic average is therefore the permeability that corresponds to the arithmeticaverage resistance to flow. It follows that the harmonic mean is sensitive to low valuesof permeability (i.e., large values of k-1). We have also seen that low permeability,fine grained, material commonly occurs in much thinner layers (e.g., micaceous orcarbonaceous laminae) than high permeability, coarse grained material (e.g., channelfill sandstones). This tends to result in a systematic (biassed) overestimate ofpermeability by the harmonic average (often used to estimate vertical permeability)due to undersampling of the low permeability layers.

MedianThe median of a distribution is equal to the value above and below which equalnumbers of samples lie (k50 see below). The median is most easily determined byranking the data and determining the middle value. The median may not be uniquefor a discrete distribution. The median is rather insensitive to the tails of thedistribution and, therefore, is often less sensitive to outliers (potential errors). Toillustrate this consider the following two samples of 5 permeability measurements:

A 120 133 210 220 350mDB 120 133 210 220 1350mD

In each case the median permeability is 210mD but, in sample A, the arithmeticaverage is 207mD and in sample B, 407md. In this example, the median could beconsidered a more robust estimator of population central tendency.



ModeThe mode of a distribution is the most commonly occuring value. A single mode isnot appropriate in be determined in bimodal (i.e., with two peak) or multi-modal(many peaked) distributions. The arithmetic average is not a meaningful number withcategory variables (e.g., average bit wear is 1 or 2; not 1.334 for example). In this case,the the mode is preferred.

Differences between measures of central tendencyFor a perfectly normal distribution all the above measures of central tendency willoverlie (Figure 2). Differences become increasingly marked as the distributionsbecome more skewed. Skewness is the term used to describe a distribution which isnot symmetrical. In this latter situation, one might ask which average is mostappropriate. Skewed distributions are commonplace in permeability data andestimating a single average measure may not be appropriate. The appropriate averagewill depend on what use is to be made of the estimator..

AverageMode

Median

SYMMETRICALDISTRIBUTION

Ar. Av.

ModeMedian

SKEWED DISTRIBUTION

Geom. Av.Har. Av.

Differences in the arithmetic (-kar), geometric (-kgeom), and harmonic (-khar) averages

are a function of the sample heterogeneity, and are commonly observed in permeabil-ity datasets. The differences are such that always:

khar kgeom kar

The differences can be exploited for permeability as each average is appropriate fordifferent flow conditions (refer to Archer and Wall, 1986, for derivation). Theseaverages are appropriate for the following conditions:

kar

bed parallel, single phase flow (i.e., horizontal flow in a horizontallylayered, bounded system)khar bed series, single phase flow (i.e., vertical flow in a horizontally layered,

bounded system)kgeom single phase flow in a random, 2-D field

The use of the various averages to estimate mean permeability is appropriate only forthe specific flow conditions described. Often the averages are used as (poor)

Figure 2Distibutions of measures ofcentral tendency

18

estimators under the wrong flow conditions (e.g., two phase flow, wrong dimensions,wrong boundary conditions, etc.), so extreme care is needed here to select theappropriate average. If the medium is homogeneous, the averages will be very similar.

Permeability measurements are often limited by measurement resolution. Frequentlyplug permeability data are listed, in core contractors reports, as 6000mD or no measurement possible", (NMP). The undetermined low values willhave an affect on the harmonic average (and also the population vertical permeability)and truncated high values on the arithmetic average (affecting the horizontal perme-ability estimates). Whilst there are no set procedures for handling this problem, hereare some suggestions: If the distribution is thought to be fairly symmetrical, the median will prove to bea better estimate of the population mean. If the distribution is significantly skewed it is most likely that the extreme missingvalues are the ones you really want! In this case, re-measure with a more accuratedevice - statistics cant help you.

It is always important to consider the ultimate use in engineering for any statistic. Thisinformation, together with knowledge of the geological structure and the degree ofvariability, should indicate the most appropriate average to use in reservoir characteri-sation. For averaging permeability: in a layered system at low dips (70o), us the arithmetic average for vertical flow andhorizontal flow along the beds. Use the harmonic average for horizontal permeabilityacross the beds. in a random system - use the geometric average for vertical and horizontalpermeability.

Instrument bias discussionThe Rannoch Formation in the North Sea is highly laminated and enables a test of theaveraging procedures. A series of experiments compares the average of a very smallscale measurement (taken with a probe permeameter - a device for measuringpermeability of a rock surface by injecting nitrogen through a very thin nozzle) withthe core plug measurement. The rock is laminated with the flow is parallel to thelamination. The procedure for layered systems used to average the small scalepermeability measurements and compare the averages with the measured plug results(Figure 3).

10001001 011

10

100

1000

Horiz. plug perm. (mD)

Prob

e ar

. av.

per

m. (m

D)

10001001 011

10

100

1000

Vertical plug perm. (mD)

Prob

e ha

r. av

. per

m. (m

D)

Figure 3Comparison of probepermeameter estimates ofhorizontal and verticalpermeability withmeasurements of core plugs



The probe permeameter cannot resolve the low permeability thin laminations. As aresult of this instrument bias, the harmonic average is a poor estimator of verticalpermeability. Estimating vertical permeability is often a problem in the industry,because the thin low permeability layers are under-represented in the sampling ordifficilt to measure. Care should be taken when using the harmonic average.

In contrast, the arithmetic average (because the high permeability layers tend to bethicker in these sediments) is a reasonable estimator of the horizontal permeability(Figure 3). This a fairly widespread problem in the measurement of the properties ofrocks - thin beds are under-represented because of volume and spacing considera-tions. Anisotropy in permeability is usually taken to be the difference betweenhorizontal and vertical permeability and is estimated by the ratio, k

v/kh . Sometimes

there is also anisotopy measured in horizontal permeability in orthogonal directions.Estimates of anisotropy in sediments depends on the measurement scale. For themeasurement scale , we consider the volume over which a measurement is made oran average is taken (Figure 4):

At the probe scale (a volume of 10-7m3), kv/kh ratios range between 0.7 and 1.

At the core plug scale (10-5m3) kv/kh ratios range from 0.2 to 0.4 with lamina-induced

anisotropy in the 2.5cm samples At the bed scale, it is difficult to take direct measurements of anisotropy but we can

use statistical estimators. Arithmetic averages of the horizontal plugs and harmonicaverages of the vertical plugs can be used to estimate bed-scale anisotropy (1m3).At this scale anisotropy can be 0.01-0.5, the former in more layered and the latterin the more massive reservoirs.

Gridblock anisotropies in the simulator will also vary with the scale of thegridblock. In most field simulations, the blocks are larger than the scale of bedding.In these cases, the anisotropy will be determined by the bedding and vary greatlyfrom that observed in core plugs. If a single gridblock is used to represent theRannoch Formation the anisotropy value will be as low as 0.007.

Knowledge of anisotropy is particularly critical in the performance prediction ofhorizontal wells.

10410210010 -210 -410 -6.001

.01

.1

1

Rannoch anisotropy

Sample volume (m3)

kv/k

h

ProbePlug

Plug averages

WB

Formation

SCSHCS

Probe average

Lamina Bed ParasequenceGrain

Figure 4Estimates of scaledependent anisotropy (k

v/kh

ratios) in the RannochFormation.WB, SCS andHCS refer to the rock typesof the Rannoch Formation.Formation referes to thewhole Rannoch interval.

110

Comparing well test and core plug average permeabilities.Well testing provides larger scale in-situ measurements of permeability. Largelybeyond the scope of this Chapter, well testing is defined as a perforated flowinginteval over which permeability is estimated from the analysis of pressure data whilstthe well is flowing ar once the flow has been stopped. Permeabilities derived fromthese measurements are often compared with core for ground truthing either sets ofdata.

There are a number of problems or issues to bear in mind when comparing well testand core plug average; Scale, flow geometry and boundary conditions of well test and core plug

permeabilities are all different. Plug data come from within the well, whereas well test data are dominated by a

region a few metres to a few 100metres from the wellbore. Well tests are in-situ measurements, whereas core plugs are laboratory measure

ments (often at different stress conditions and conducted with different fluids)However, even with these problems coupled with the uncertainty of the near wellregion, the reservoir engineer is often required to make such comparisons.

Well testing tends to be dominated (in clastic reservoirs) by the properties of beds.Lamina-scale features are usually undetectable or confused with wellbore phenom-ena. Larger scale features (e.g., formation boundaries) can be detected if the tests arelong enough.. Well test permeabilities are most closely associated with average bedpermeabilities. The appropriate average will therefore depend on the bed geometry(Figure 5) . A case study is discussed to illustrate the problem.

-kar

-kgeom

-khar1-5ft

5-10ft

10-50ft

Figure 5Alternative estimators forwell test permeabilitiesdepend on the geometry ofthe geology at the bed-scale.Well testing (in themiddle time region) isessentially a bedform scalemeasurement. Well testsmeasure effective scalepermeabilities at that scale



The accompanying sketch (Figure 5) shows how the averaging procedure might relateto bed scale and geometry in an imaginary reservoir. In many reservoirs, singleisolated high permeabilities can be filtered out as they represent lamina (i.e., verylocal) properties.

Two wells in the same braided stream reservoir have similar permeability distribu-tions (Fiure 6). The well test permeabilities are quite different in the two wells andcorrespond to the geometric average (Well A) or the (weighted) arithmetic average(Well B). Looking at the spatial distribution of the permeability (Figure 7) we see thereason why. Well A is characterised by isolated high permeability plug most likelycorresponding to thin, minor fluvial channels of limited extent. Well B on the otherhand, contains relatively thick, major channels of signifcant extent. The lowerpermeability matrix dominates the well test volume of investigation in well A(geometric average more appropriate for random discontinuous high permeabilityzones) and the channels, well B (arithmetic average for layer flow). This exampleshows the pitfalls of applying statistics without geology - one can always determinethe arithmetic average - but using it as the estimator of horizontal permeability issometimes dangerous! In well B it was appropriate because it is a layered system. Inwell B it would have overestimated the permeability as the layers are not continuous.

Figure 6Permeability distributionfrom two wells in the samebraided stream reservoirleft: Well (A) right: Well(B)

Figure 7Permeability distributionwith depth in the two wellsin Figure 6. left: Well (A)right: Well (B)

112

3. MEASURES OF VARIABILITY

In the previous section we reviewed measures of central tendency. The second classof descriptive statistics that can be used to describe a sample are measures ofvariability or dispersion. These are commonly used in other areas of data analysis buttend, traditionally, to be overlooked in petroleum engineering (particularly bygeologists). As we will see in this and subsequent sections, the measures of variabilityof permeability can:

define the level of heterogeneity determine the number of samples required indicate likely recovery process suggest likely flow performance

Because of these reasons, it is considered that measures of variability can be equally(if not more) useful than averages. Every estimate of central tendency (of permeabil-ity) should be accompanied by a measure of variability. The most commonly occuringmeasures of variability are reviewed in this section.

Standard deviationIn statistics, a deviation is a distance from the mean. The mean deviation is thus theaverage deviation for a sample. The variance is the average squared deviation. Thestandard deviation (SD) is given as the positive square root of variance:

N

i = 1

N

i = 1

SD

=

0.5

(ki

- k)2

N

SD

=

0.5

ki

N

2

k2

Standard deviation has the units of the measurement being considered (e.g., mD in thecase of peremeability). The lower the standard deviation the less the dispersion orspread of a distribution about the mean. 68% of all the observations in a normaldistribution lie within one standard deviation (SD) either side of the mean (2SD and 3SD include 95% and 99.7% of the observations, respectively). For small samplesizes the N in the denominator should be replaced by (N-1) to account for fewerdegrees of freedom caused by the less than optimum estimation of the mean.

Coefficient of variationThe standard deviation often tends to increase as the mean increases. An SD of 80mDis a high dispersion for a mean of 100mD, but a low dispersion if the mean is 1000mD.A more useful (in reservoir characterisation) absolute measure of dispersion is givenby the coefficient of variation (Cv), or normalised standard deviation:

Cv = SD /kar



For small samples (N < 10), the standard deviation needs to be multiplied by acorrection factor. The correction factor accounts for the fact that tha mean is not wellestimated with few samples, and therefore the SD is broader than if there were manysamples and is given by:

1+ 14(N-1)

The coefficient of variation is becoming more widely encountered in reservoirdescription, particularly in probe permeameter studies (Figure 8), and has been usedto define the level of heterogeneity:

0.0 < Cv < 0.5 Homogeneous0.5 < Cv < 1.0 Heterogeneous1.0 < Cv Very heterogeneous

Normal distributions have Cv < 0.5, for Cv > 0.5 the distributions become increasinglyskewed. Even under the latter conditions, the Cv appears to be a useful statistic.

43210Synthetic core plugs

Homogeneous core plugsAeolian wind ripple (1)

Aeolian grainflow (1) Shallow mar. low contrast lam.

Fluvial trough-cross beds (2)Fluvial trough-cross beds (5)

Mix'd aeol. wind rip/grainf.(1)Lrge scale x-bed dist chan (5)

Shallow marine SCSAeolian interdune (1)

Shallow marine Rannoch FmShallow mar. Lochaline Sst (3)

Shall. mar. high contrast lam.Shallow marine HCS

Heterolithic channel fillBeach/stacked tidal Etive Fm.

Dist/tidal channel Etive sstsFluv lateral accretion sst (5)

Sh. mar.rippled micaceous sstCrevasse splay sst (5)

S.North Sea Rotliegendes Fm (6)Carbonate (mix pore type) (4)

Cv

INCR

EASI

NG

SO

RTIN

G

Homogeneous

Heterogeneous

Very heterogeneous

Figure 8Coefficient of variation fora range of geologicalmaterials.(from Jensen etal, 1997)

114

Figure 9Graphical solution of theDykstra-Parsons coefficient(from Willhite, 1986)

Dykstra-Parsons coefficientA further measure of variability, developed by the oil industry, recognises thatpermeability is often log normally distributed. For permeability that is log-normallydistributed, the Dykstra-Parsons coefficient was defined in the 1950s as:

VDP = 1 - k

k0.5

k kk

0 5

0 5

.

.

where k is the permeability one standard deviation below the median permeability(k0.5) for a distribution of the logarithm (usually base 10) of permeability. Theseparameters are best determined by plotting a probability plot for log(k) and reading offthe 50 and 84th percentiles. This graphical procedure for the determination of VDP (forwhich probability paper is required) is illustrated in Figure 9. VDP is useful becauseof correlations with waterflood performance,enhanced oil recovery (EOR) potentialand its common occurence in the petroleum engineering literature. Note that estimatesof VDP for distributions that are not log normal can be unrepresentative.

Lorenz coefficientA third way of expressing heterogeneity in reservoirs is through the use of the LorenzCoefficient. The Lorenz Coefficient is defined as a specific area on a Lorenz Plot, auseful plot which involves the relationship between of core plug porosity () andpermeability(k) as a fraction of total reservoir porosity and permeability. The Lorenzplot is modelled on a mutch earlier plot (early 1900's) that was used to relate wealth(flow capacity) to population (storage capacity) - wealth is not evenly distributedacross people, neither is flow capacity in rocks!

To generate the Lorenz Plot, arrange the core data in decreasing order of k/ andcalculate the partial sums



Figure 10Lorenz plot showingdetermination of the LorenzCoefficient, L

c= 2A where A

is the shaded area

Fk h

k hjj jj

J

i ii

I= =

=

1

1

Ch

hjj jj

J

i ii

I= =

=

1

1

where 1 J I and there are I data points. Plot Fj (known as the flow capacity) versusCj (the storage capacity) on a linear plot (Fig. 10). The Lc can be calculated from theshaded area and varies from 0 - 1. Homogeneity is expressed on this plot by thediagonal (L

c= 0).

Fj

Cj0 1

1

0

Fraction of totalflow capacity

(permeability x thickness)

Fraction of totalstorage capacity

(porosity x thickness)

The plot can also be useful in well testing as it represents the balance between flowcapacity (or tranmissivity) and storage capacity (storativity) and can indicate thepossibility for double porosity behaviour. In a fractured reservoir, the transmissiveelements (the fractures) are often separate from the storage elements (the matrix) -these would have Lorenz Coefficients close to 1.

A modification to the Lorenz Plot is to plot the data with their natural ordering. TheModified Lorenz Plot ( Figure 11) can then give a good idea on the degree of layeringin the reservoir. The modified Lorenz Plot can be compared with the inflow logmeasured during production logging.

116

Figure 11Modified Lorenz Plotshowing stratigraphiclayering.

Figure 12Simple histograms

Fj

Cj0 1

1

0

Fraction of totalflow capacity

(permeability x thickness)

Fraction of totalstorage capacity

(porosity x thickness)

4. DISTRIBUTIONS

A distribution is a graphical representation of a set of frequencies (observed distribu-tion) or probabilities (theoretical distribution). Frequencies are presented on a barchart (histogram) in which the width of the bars are proportional to the class intervaland the height of the bars is proportional to the frequency it represents (Figure 12). Theclass interval is the interval between boundaries selected to subdivide the range intoa number of (usually equal) windows. Points falling at the boundaries aresystematically included in the class interval below or above. As a guideline, thenumber of class intervals should be between 7 and 25.

Freq

uenc

y

Variable

Probability is a measure of the relative frequency of occurrence of an event.Probability (P) is a number between 0 and 1. Probability 0 means impossibility, 1 iscertainty. Values can be derived from a theoretical distribution or by observation.For a discrete distribution, probability is defined as:



Figure 13Probability distributionfunctions underlying thesample histograms

Figure 14Cumulative distributionfunctions associated withthe above pdfs

number of required outcomes total number of possible outcomes

Thus the probability of picking a spade from a pack of cards is

1352 or 0.25

For a continuous variable, the probability is the relevant area under the graph of itsprobability density function (pdf). The total area under the graph is 1, i.e., a randomvariable will lie within the range of its pdf. The pdfs for the variables in the samplehistograms above can be derived as the sample size approaches infinity and the classinterval approaches zero (Figure 13).

Variable

Pdf

If there are sufficient observations in the sample, the sample histogram can be thoughtof as an estimate (or approximation) to the underlying variable pdf. For this reason,sample histograms are often referred to as pdfs (strictly, pdf is a populationparameter) (Figure 6).

The function that gives the cumulative probability or cumulative frequency (i.e., thefrequency with which a varible has a value less than or equal to a particular value) ofthe random variable is known as the cumulative distribution function (cdf) (Figure14).

Variable

Cdf

1

0

.5

k50 k50

118

Cdfs are the form of distributions that are commonly used as the input to Monte Carlosimulation. Random numbers between 0 and 1 are used to derive a number ofrealisations of the variable cdf (e.g., for porosity, volume, shale length, channel width,etc.). The pdf of the random variable will, with enough realisations, assume thesample pdf.

There are major benefits in identifying the form of the underlying pdf:

the pdf is the only statistical parameter that defines the extreme values and theprobability of their occurrence.

non-normal distributions can be transformed to normality if the underlying distribution is known. Parametric methods are appropriate and regression is enhancedfor normally distributed variables.

parametric (i.e., sensitive to the underlying distribution) statistical tests are morepowerful. Procedures where we dont know the form of the pdf are called non-parametric.

Distibutions that are not symmetrical are known as skewed. Consider the two pdfsin Figure 15, the one on the left is symmetrical whereas the one on the right ispositively skewed (i.e., tail - queue in French - to the positive side of the mode).

There are a set of power (p) transformations for 1 > p > -1 which will transformskewed distributions to normality. For kP where p = 1, the distribution is alreadynormal, for p = 0.5, root normal and for p = 0, log normal. These three distributionsare common for permeability within reservoir rocks.

There are other distributions more specific to small sample probabilities (e.g.,binomial, poisson) or truncated data sets (e.g., exponential, gamma) which will not becovered here.

5. SAMPLE SUFFICIENCY

The issue of sample sufficiency is not usually covered in basic statistical texts or evenconsidered in petroleum engineering. Core plugs, for example, are taken every foot,regardless - because thats the way it has always been done! In fairness to the corecontractors, geologists and engineers, this has, historically, been the practical (interms of cost, core preservation, etc.) sample limit. More recently, we have consideredthe appropriate number of measurements for our needs and the representatitivity ofthe sample population

With the development of probe permeameters, we are able to reconsider samplesufficiency and, because probe measurements are relatively cheap and non-destruc-tive, ensure that sufficient samples for our requirements are obtained.

A practical method for determining sample sufficiency comes from the central limittheorem which states that, if independent samples of size N are drawn from a parent



population with mean and standard deviation SD, then the distribution of theirmeans will be approximately normal (regardless of the population pdf) with mean and standard deviation, SD/N. From this, the probability that the sample mean (-k

s)

of N observations lies within a certain range of the population mean () can bedetermined for a given confidence interval. For a 95% confidence level (i.e., only amaximum of five times in 100 will the population mean lie outside that range) therange is given by t SE, where the standard error (SE) is given by SD/N. ( Thegreater the sample number, N, the more confident we can be about estimates of themean).

Standard error is the standard deviation (SD) of the sample mean, drawn from a parentpopulation, and is a measure of the difference between sample (-k

s) and population ()

means. Students t is a measure of the difference between estimated mean, for a singlesample, and the population mean, normalised by the SE. For normal distributions thet value varies with size of sample (N

s) and confidence level (95%), and these values

are well known (standard t tables in any basic statistics text). We can write the above,mathematically, as:

Prob (ks = t SDNs

) = 95%

We require a sample such that the average of the optimum sample size k0 withtolerance P%, satisfies the predetermined confidence interval, or:

Prob (k0 = P k0100 ) = 95%

When this condition is satisfied, Ns= N0 , and:

P k0100 = t

SDN0 .

Rearranging this gives an expression for the optimum number of specimens, N0:

N0 =

t SD 100

P k0

2

Now, for N > 30, t ~ 2 and with a 20% tolerance (i.e., the sample mean will be within20% of the parent mean, which we consider an acceptable limitation), the expressionreduces to:

N0 =

2 Cv 100

20 2

where Cv = SD /k0

N0 = ( )10 Cv 2

120

This rule of thumb is a very simple way of determining sample sufficiency. Althoughderived for the estimate of the arithmetic mean from uncorrelated samples by normaltheory, it has been found to be useful in designing sample programs in a range of coreand outcrop studies. Having determined the optimum number of samples (N0), thedomain length (D) will determine the optimum sample spacing (D0) as:

D0 = D / N0

An initial sample of 25 measurements, evenly spaced over the domain, which can bea lamina, bedform, formation, outcrop, etc. can be used to determine an initial estimateof variability. If the Cv, determined from this sample, is less than 0.5, sufficientsamples have been collected. If more are required, infilling the original with 1, 2 orn samples, will give 50, 75 or 25n samples. In this way, sufficient samples can becollected . When insufficient in very variable rock types (e.g., carbonates) thetolerance may have to be increased to reach achievable sample numbers). Note thatthe N-zero technique has really only been tested for permeability sufficiency.

The answer to the original question, posed in the opening paragraph of this section,can now be explained. Because formations contain facies of differing variability,some facies will be adequately sampled with 1ft core plugs, but some thin, highlyvariable and, possibly, significant facies can be under-sampled. This happens in theRannoch Formation (Middle Jurassic Brent Group, North Sea) where the criticalfacies at the Rannoch/Etive boundary in some wells is only 10ft thick with Cv = 1.Over 100 samples, therefore, are needed in such an interval and 10 core plugs areobviously insufficient (Figure 15).

When insufficient samples are known the sample tolerance(Ps) can be calculated:

Ps = 200 Cv

Ns

100010010-10058-10057-10056-10055-10054-10053-10052-10051-10050

Core Plugs

Arith av

Permeability (mD)

Cor

e dep

th (f

t)

PLUGS: N=9Cv = 0.74, Ar. Av. = 390mD

PROBE: N = 274,Cv = 0.99, Ar. av. = 172mD

Figure 15 Highly variable Rannoch(M. Jur.) interval showingthe concept of optimumsample density. N0 for thisinterval is 100 - satisfied bythe probe but inadequatelymeasured with core plugs.(From Jensen et al 1997)

Probe



6. MEASURES OF SPATIAL CORRELATION

We have encountered, in this geoscience course, correlation in the geological senseof drawing lines of equal stratigraphic significance between wireline logs. In thissection we consider spatial correlation (i.e., autocorrelation, correlation of a variablewith itself as a function of separation in space or time). In reservoir engineering, twoautocorrelation functions, the correlogram and the semivariogram (or variogram), arecommonly encountered (Figure 16) and are best defined by their equations below.The former tends to be used to measure the degree of similarity between neighbouringvalues (e.g, grid blocks in a numerical simulator) and the latter to examine thedifferences between neighbouring values (e.g., permeability in outcrop or corestudies). The latter is also used in a mapping procedure known as kriging which hasbeen adopted from the mining industry and has been used (with some success) in thepetroleum industry.

The correlogram function (r) is given by:

(h) = 1(N-h) (SD)2 [(k(x) -k)(k(x+h) -k)] where k(x) and k(x+h) are the permeabilities of any two points seperated by lag (thedistance between sample points, h) and N is the number of data points. As h tends tozero the correlation function tends to unity. A plot of the function against lag is thecorrelogram.

Lag distance

1

0 0

CORRELOGRAM SEMIVARIOGRAM

For comparison, the semivariogram function (g, referred hereafter and most com-monly as the variogram, the "semi-" is there because the function is symmetricalabout the origin and only positive lag distances are shown by convention) is given by:

(h)= 12N

[k(x) k(x h)]2 +

at a lag distance h. In this case, N is the number of pairs of data points. As h approacheszero the variogram (i.e., variance) approaches zero. Note that the variogram doesntrequire an estimate of the mean and is, therefore, more precise than the correlogram.

Figure 16Characteristic shapes ofautocorrelation funcions inthe presence of correlation

122

Figure 18Characteristic shapes ofautocorrelation functionsfor random (i.e.,uncorrelated) samples

Figure 17Variogram terminology43210

0

20

40

60

80

IDEAL SEMIVARIOGRAM

Lag (distance units)

Sem

ivar

iogr

am

(mea

s. un

its^2

)

SILL

RANG

E

432100

20

40

60

80

NUGGET

The variogram has some additional features (Figure 17). At some lag separation(known as the range) the variogram often approaches the variance of the data (the sill)and the statistical correlation between points at this and greater separation(s) is zero.If the variogram at the closest separation is away from the origin, a nugget is said toexist, often indicative of measurement inaccuracy. If the variogram at the closestseparation approaches the sill, the data are said to be uncorrelated (Figure 18, right).On a correlogram, uncorrelated data show the correlation function at or near zero fromthe shortest separation (Figure 18, left).

Lag distance

1

0 0

CORRELOGRAM SEMIVARIOGRAM

It is important to determine the correlation in a data set, as correlation effectivelyreduces the amount of "information" carried by each observation. This might resultin either additional samples being required. There is a paradox here, because we haveseen (section on sample sufficiency) that N0 samples (derived for uncorrelatedsamples) can give appropriate estimates of mean properties, even though permeabilitymeasurements can be seen to be correlated. Although the reason for this paradox isnot clear at the present time, it can be demonstrated that correlation in sedimentaryrocks exists at several scales. These scales are marked by significant decreases of thevariogram at some positive lag distance (holes).



Figure 19Probe (Minipermeameter)and core plug profiles fromthe Rannoch Fm North Sea

Figure 20Variogram for probe datashown in Figure 19

1000100101-10146

-10145

-10144

-10143

-10142

-10141

-10140

-10139

-10138

-10137

-10136

Core plug khMinipermeameter

Permeability (mD) Core depth (ft)

32100.0

0.5

1.0

1.5

Lag (m)

Sem

ivar

iogr

am m

agni

tude

The semivariogram can sometimes reveal average periodicities that are representedby a significant reduction in variance at some lag separation greater than the range. Anexample interval from the Rannoch formation shows well developed cyclicity (fig 20)and this is captured as a hole at approx. 4ft. (1.3m) in the accompanying variogram(Figure 20). This periodicity is related to the (hummocky cross-stratified) bedformthickness. There is evidence that the periodicity in the sediment can impact fluid flowand that the holes can therefore be used as a diagnostic engineering tool. This decreasein variance at certain separations reflects increased correlation and corresponds to thewavelength of a lamina or bedform.

124

It can also be seen in Figure 21 that each of these scales requires a tailor-madesampling plan, which may require more than N0 samples.

It has been oberved that many natural phenomena exhibit variograms that never attaina sill, suggesting correlation on a never-ending scale (Figure 22). Phenomena whichexhibit this phenomena are commonly known as fractals. Fractals exhibit variationat all scales, the closer you look the more you see. Variograms in a fractal mediumshould, therefore, exhibit the characteristic shape at all scales. Fractal behaviour overlimited length scales has been observed in rocks.

Lag distance

0

Figure 22Fractal behaviour in avariogram

Figure 21Multiple correlation scalesin sedimentary rocks, asshown by the variograms(from Jensen et al 1997)



Use of variograms in reservoir modellingThe main use of variograms in reservoir modelling is in the automated generation ofpermeability fields for simulation. A field in this sense is a grid of permeability values.With a few parameters (nugget, sill, range) and the variogram model can be used tomany equiprobable realisations from the same permeability distribution. These fieldswill all look different, but will have the same correlation structure as the inputvariogram. The input variograms can be generated from outcrop studies. These fieldsare called correlated random field (CRF). The CRF technique involves two steps:

(i) Randomly distributing permeabilities to all grid nodes.

(ii) Swapping the permeabilities around until their spatial correlation structureconforms with the given variogram structure.

The generation of many equiprobable permeability fields with millions of cells wouldbe very difficult manually! The are equiprobable because they are all equally likelyto represent reality. An example of correlated random fields for various isotropicfields is given in Figures 23.

Figure 23Correlated random fieldsgenerated from isotropicvariograms (Yuan andStrobl, 1991)

126

SUMMARY

Presented with some porosity and permeability data, the engineer should be able(using the material in this section) to carry out the following:

generate a suite of summary numbers - averages, mode, median, SD, CV - anddiscuss their engineering implications,

evaluate the variability in the data and decide the engineering and samplingimplications,

assess how the summary numbers relate to the varianbility,

determine how reliable the summary numbers are, and,

determine the correlation structure of the property in the reservoir.

The exercises at the end of this Chapter allow the student to try out these skills.

Figure 24Correlated random fieldsgenerated from anisotropicvariograms (Yuan andStrobl, 1991)



EXERCISES

1 Determine the arithmetic, geometric, harmonic averages, mode and median for theporosity and permeability in following set of core plugs, from an North Sea well. Tohelp determine the geometric and harmonic averages you may wish to determineln(perm) and the inverse of permeability first.

Depth

(m)

Porosity

(%)

Horiz. Perm - k.(mD)

3834.9 23.8 105

3835.2 24.8 140

3835.5 27.4 297

3835.8 26.4 236

3836.1 23.6 106

3836.4 24.6 140

3836.7 24.2 157

3837.0 24.8 144

3837.3 26.0 189

3837.6 24.8 111

3837.9 29.4 577

3838.2 27.6 318

3838.5 14.6 nmp

3838.8 22.5 52.4

3839.1 22.1 54

nmp = no measurement possible

128

2. If asked to provide an average of these data:

for reservoir simulation grid block, or, for comparison with a well test?

What would you need to consider? The variation of data with depth is given below.

3025201510-3840

-3839

-3838

-3837

-3836

-3835

-3834

Porosity (%)

Core

dep

th (m

)

6005004003002001000-3840

-3839

-3838

-3837

-3836

-3835

-3834

Horizontal Permeability (mD)

Core

dep

th (m

)

3. Plot histograms of permeability and porosity. Plot a Lorenz Plot. What can youdetermine from these plots?



4. From the plots of porosity and permeability vs depth given in Question 2,what canyou say about variability of porosity and permeability?

Calculate the coefficient of variation for both porosity and permeability.

For comparison, the Vdp for the plug permeability is given as 0.46

5. Do we have enough plugs to estimate the arithmetic mean permeability of thisinterval within 20%? To what tolerance can we estimate the arithmetic mean fromthese data?

6. Over this same 4m interval we also have probe permeameter measurements.These are plotted vs depth and displayed for comparison with the core plugs. Canyou see the permeability (geological) structure more clearly?

-3835

-3836

-3837

-3838

-38390 100 200 300 400 500 600

PROBE

PLUGS

Permeability (mD)

Dep

th (m

)

Given the following analysis of the probe data, can we expect an average within 20%of the arithmetic average? What is the true heterogeneity level of this interval andwhat do you think controls it? What are the implications for the grid blockpermeabilities?

130

Probe permeabilityNo. of meas. 320

Arith. av. (mD) 151Geom. av. (mD) 93Harm. av. (mD) 39

Cv 0.82N0Vdp 0.69

7.Determine the (semi)variogram function for the core plug permeability andcompare with that for the probe permeability. What is your interpretation of the twovariograms and of the differences between them?

2001801601401201008 06 04 02 000.0

0.5

1.0

1.5

2.0

(NB: You can plot your variogram on the above plot by dividing the function by thevariance. In this way you can compare variograms structure from various data sets.If a sill is present it should appear at the value one)

2001801601401201008 06 04 02 000.0

0.5

1.0

1.5

2.0

CORE 5 COARSE GRID3 8 3 5 . 0 - 3 8 3 9 . 0 m

Lag (cm)

Dim

. ex

per.

sem

i-var

.

(m

d^2/

Varia

nce)

VARIANCE15831(md^2)

dz = 1cm



SOLUTIONS

1.

Por PermN 15 14

Arith. Av 24.44 188Geo. Av 24.18 153Har. Av 23.86 126Mode 24.80 140

Median 24.80 142

2.To consider which average is the right one - for porosity it doesnt matter forpermeability there are differences which may be significant. A simulation grid blockmay be several 100m long so the possible layering of the interval would have to betaken into consideration (i.e., the scale of application).

If you look at the variation with depth plots, there is no strong evidence for layering,so one might choose the geometric average (appropriate for random geology). If youthink some of the data are outliers the mode or median might be a better estimate.

3. HistogramsThe histogram for porosity shows a more symmetrical distribution than that forpermeability.

12 1816 20 22 24 26 28 30

Porosity

Cou

nt

0

1

2

3

4

5

6

0 200100 300 400 500 600

Permeability

Cou

nt

0

1

2

3

4

5

6

7

8

Histograms

132

Un

ord

ere

dO

rde

red

Cu

mp

hi

Cu

mk

Ph

i (%

)k

(mD

)k/p

hi

Ph

i (%

)k

(mD

)k

/ph

iP

hi/to

tal

Ph

ik

/To

tal k

0.00

0.00

23.80

105

429.40

577

20

0.08

0.22

0.08

0.22

24.80

140

627.60

318

12

0.08

0.12

0.16

0.34

27.40

297

11

27.40

297

11

0.08

0.11

0.24

0.45

26.40

236

926.40

236

90.08

0.09

0.31

0.54

23.60

106

426.00

189

70.07

0.07

0.39

0.62

24.60

140

624.20

157

60.07

0.06

0.46

0.68

24.20

157

624.80

144

60.07

0.05

0.53

0.73

24.80

144

624.80

140

60.07

0.05

0.60

0.78

26.00

189

724.60

140

60.07

0.05

0.67

0.84

24.80

111

424.80

111

40.07

0.04

0.74

0.88

29.40

577

20

23.60

106

40.07

0.04

0.81

0.92

27.60

318

12

23.80

105

40.07

0.04

0.87

0.96

22.50

52

222.10

54

20.06

0.02

0.94

0.98

22.10

54

222.50

52

20.06

0.02

1.00

1.00

To

tal

Ph

iT

ota

lk

352

2626

Lorenz plot calculation



Un

ord

ere

d

Cu

mp

hi

Cu

mk

Ph

i (%

)k

(mD

)k

/ph

iP

hi/

tota

l P

hi

k/T

ota

l k

0.00

0.00

23.80

105

40.07

0.04

0.07

0.04

24.80

140

60.07

0.05

0.14

0.09

27.40

297

11

0.08

0.11

0.22

0.21

26.40

236

90.08

0.09

0.29

0.30

23.60

106

40.07

0.04

0.36

0.34

24.60

140

60.07

0.05

0.43

0.39

24.20

157

60.07

0.06

0.50

0.45

24.80

144

60.07

0.05

0.57

0.50

26.00

189

70.07

0.07

0.64

0.58

24.80

111

40.07

0.04

0.71

0.62

29.40

577

20

0.08

0.22

0.79

0.84

27.60

318

12

0.08

0.12

0.87

0.96

22.50

52

20.06

0.02

0.94

0.98

22.10

54

20.06

0.02

1.00

1.00

To

tal P

hi

To

talk

352

2626

Modified Lorenz plotcalculation

134

Lorenz Plot(A) ordered(B) unordered

1.00

1.00

0.90

0.80

0.80

0.70

0.60

0.60

0.50

0.40

0.40

0.30

0.20

0.200.00

0.00

0.10

Phih

kh

Lorenz Plot

1.00

1.00

0.90

0.80

0.80

0.70

0.60

0.60

0.50

0.40

0.40

0.30

0.20

0.200.00

0.00

0.10Ordered Plug Data

Ordered Plug Data

Phih

kh

Modified Lorenz Plot

A

B

The Lorenz Plot shows moderate heterogeneity, the unordered data (stratigraphicallyordered data) shows the occurrence of a high permeability zone.

4. They way the scales are drawn the plots of porosity and permeability appear to havethe same variability. This is because the scales relative to the mean are quite different



Por Permsd (STDEV; n-1) 3.35 138

Variance 11.23 18907Cv 0.14 0.73

The Cv is a normalised variability and shows the permeability variability to be muchgreater (heterogeneous) than the porosity (homogeneous). A Vdp of 0.46 forpermeability also suggests moderate heterogeneity.

5

Using N0 = (10Cv)2 the optimum number of samples (N0) = (7.3)2 53.

There are 14 plugs so the answer is "no".

The 14 plugs will estimate to

200 200 0 7314

CvNs

= .

i.e. 39%

6The probe permeameter picks out more distinct layering and low permeabilities in theregion where the core plugs showed no measurement possible (3838.5m). Thevariation can be related to the geological structure.

No = 67P = 9

Number of samples (320) is well in excess of the No (67) therefore the probe dataaverage will be within the 20% tolerance. In fact the average will be within 9%tolerance. This suggests the arithmetic average 151mD or geometric average 93mDmight be more appropriate. Note that the variability has increased with the probe data(this often, but not always, happens) and the differences between the averages hasbecome more pronounced than determined by the core plugs. The extra heterogeneityis due to the geological structure, some of which wasnt originally sampled by the plugs.

7Semivariogram calculation

Note the effect the missing data point (nmp) has on the numbers of pairs determined.The normalised gamma is determined by dividing the gamma by the variance of thepermeability (refer to Question 4).

136

Permeability Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 Lag 6

105140 1225

297 24649 36864236 3721 9216 17161106 16900 36481 1156 1140 1156 9216 24649 0 1225

157 289 2601 6241 19600 289 2704144 169 16 1444 8464 23409 16189 2025 1024 2401 6889 2209 11664111 6084 1089 2116 841 25 15625

577 217156 150544 187489 176400 190969 221841318 67081 42849 16641 30276 25921 31684

52.4 70543 275205 3434 18660 8391

54 2.56 69696 273529 3249 18225

Count 12 11 11 10 9 8Sum 340458 360443 604199 519434 265956 310150

Gamma 14186 16384 27464 25972 14775 19384Normalised

gamma0.75 0.87 1.45 1.37 0.78 1.03

Porosity Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 Lag 6

23.824.8 1.0

27.4 6.8 13.026.4 1.0 2.6 6.823.6 7.8 14.4 1.4 0.024.6 1.0 3.2 7.8 0.0 0.6

24.2 0.2 0.4 4.8 10.2 0.4 0.224.8 0.4 0.0 1.4 2.6 6.8 0.026.0 1.4 3.2 2.0 5.8 0.2 2.024.8 1.4 0.0 0.4 0.0 1.4 2.6

29.4 21.2 11.6 21.2 27.0 23.0 33.627.6 3.2 7.8 2.6 7.8 11.6 9.014.6 169.0 219.0 104.0 130.0 104.0 92.222.5 62.4 26.0 47.6 5.3 12.3 5.3

22.1 0.2 56.3 30.3 53.3 7.3 15.2

Count 14 13 12 11 10 9Sum 277.0 357.5 230.3 242.1 167.5 160.0

Gamma 9.9 13.8 9.6 11.0 8.4 8.9Normalisedgamma

0.88 1.22 0.85 0.98 0.75 0.79

These results are plotted below:



The permeability data show a very high nugget (0.65) suggesting that, whilst there issome correlation, the plug data may be interpreted as uncorrelated random.Note that the porosity semivariogram is not the same as the permeability variogramhowever the same interpretation might be made.

Comparison with the probe data shows the effect of the closely spaced data. Thenugget has dropped to 0.25 and there is a clear hole, indicating a repetitative structure(bedding). This suggests the interval is more layered, and that the arithmetic average ofthe probe data (151mD) might be more appropriate. Note that this is also close to thegeometric average of the plug data (153mD).

138

REFERENCES AND BIBLIOGRAPHY

Archer, J. S., and Wall, C.G., 1986, Petroleum Engineering: Principles and Practice,Graham and Trotman, Newcastle, 362p.

Jensen, J.L., Lake, L.W., Corbett, P.W.M., and Goggin, D.J., 1997, Statistics forPetroleum Engineers and Geoscientists, Prentice-Hall, NJ, 390p

Journal, A. G., and Huijbregts, C. J., 1978: Mining Geostatistics : Academic Press,London, 600p.

Lake, L. W., 1989, Reservoir Characterisation, vols 1 and 2, SPE Reprint No 27,SPE,

Linville, W., 1993, Reservoir Characterisation III, Penn Well Publishing Company1008p.

Willhite, G.P., 1986, Waterflooding, SPE Textbook Series Volume 3, RichardsonTx. 326p.

Porkess, R., 1988, Dictionary of Statistics: Collins, London, 267p.

Rowntree, D., 1981, Statistics Without Tears, Penguin, London, 199p.

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