spwla 1995

SPWLA 36th Annual Logging Symposium, June 26-29,1995

An Integrated Approach to Saturation Height Analysis

Christopher Skelt, Scott Pickford Group Bob Harrison, Enterprise Oil

Introduction Abstract

One of the principal contributions made by petrophysicists to an understanding of hydrocarbon distribution within the reservoir is the saturation height function. Unfortunately, the shortage of specifically designed commercial software forces many petrophysicists to transform the input data into a domain where it is quasi-linear, so that least squares linear regression techniques can be used to derive coefficients in the equations. This can impose artificial weighting and undesirable constraints on the fitting process.

We present a simple, robust, non-linear formula- tion and optimization method, designed so that each term in the function can be related directly to a physical parameter such as irreducible water saturation, ratio of contact angle and surface tension between laboratory and reservoir conditions, threshold capillary entry pressure, and height dif- ference between free water level and oil water contact. The transformatioe applied to the function by altering each term is predictable, comprehensible, and independent of the other terms. This property allows petrophysicists to make optimal use of the log, capillary pressure and other special core data at their disposal, capitalising on the relative merits of each type of data.

When deriving a saturation height relationship petrophysicists need to be be aware of the varia- tion of field area with height above contact. The weighting option of the optimization process recog- nises this requirement, and fits the data best where each foot of vertical height represents the largest area1 extent of the field.

Finally, the implications of saturation-related phenomena connected with variations in rock properties observed in the wells on field-wide hydrocarbon distribution are discussed. The optimization method, backed by resistivity profile modelling, is used to distinguish artifacts arising from resolution incompatibilities from real rock property variations.

Traditionally, methods for predicting water saturation as a function of rock properties and height above contact have fallen into two groups, those based on capillary pressure curve averaging, and log-based methods2. The relative merits of each approach are well known. For example, capillary pressure based approaches allow the petrophysicist to relate saturation to observed pore throat distribution, for example by defining pore network parameters3, but require an accurate estimate of the ratio of the contact angle and interfacial ten- sions at laboratory and reservoir conditions. A number of log baaed formulE have appeared over the years, and many of these demonstrate consid- erable ingenuity in transforming the observed saturation, height and rock property data such that a curve fit can be achieved using least squares regression capabilities of popular spreadsheet packages. An unfortunate consequence of this is that coefficients in these equations cannot easily be related to petrophysical properties, and varying them can have unpredictable effects on computed saturation.

While development of the saturation height function is the responsibility of the petrophysicist, its application in determination of hydrocarbons in place is often the preserve of the computer mapping specialist. The petrophyisicist therefore may be unaware of two important factors, the extent of the reservoir above the highest point sampled by well data or capillary pressure data, and the varia- tion in field area with height above free water level. Taking these considerations in turn:-

l If the highest data point sampled in a well is below the top of the structure it is important that above the range of observed data the saturation height function behaves appro- priately.

l Especially in low relief structures a dispropor- tionately large percentage of reserves may lie in the lower part of the reservoir. An accurate evaluation therefore requires a particu-

NNN

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larly good fit where the bulk of hydrocarbons are present. This can be achieved by weighting the fit according to the field area, which t,he mapping specialist can easily supply.

Anecdota.1 evidence from the North Sea suggests t,ha.t oil companies spend roughly a.s much on cor- ing and core analysis as they do on logging and log analysis. Assuming that this represents a considered view on their part, it makes sense to attach importance to both core and log derived saturation data. This calls for a method of integrating the two data types, preferably in a way that allows the petrophysicist to make the most constructive use of their relative merits. Briefly we summarise these as:-

l Data Limitations A full discussion of t,he limitations of log and capillary pressure measurements is beyond the scope of this paper. Briefly however, petrophysicists are aware that commercial constraints sometimes prevent full pressure equilibrium from being reached in low permeablity plugs, and recognise that a core measurement charac- t,erizes only a tiny piece of rock, removed from the reservoir. The uncertainties in log based water saturation are related inter alia t.0 t,ool measurement accuracy, resolution lim- it.ations, and the validity of the chosen saturation equia.tion.

l Height/Pressure Origin Although hydrocarbon water cont,acts can be identified on open hole logs, the more fundamental free wa- t,er level cannot, other than by reference to formation pressures. This can be an important issue in fields with widely varying rock properties. It is not a problem with capillary pressure data where the origin is zero pressure, which is analogous to free wai,er level.

l Height/Pressure. Scaling Log based true vertical depth measurements provide an ob- vious reference for height input into the saturation height equation. Capillary pressures are converted using an estimate of the ratio of interfacial tension and contact angle at laboratory and reservoir conditions. This tends to be obtained from charts generated using typical fluids and cannot easily be verified for a particular field.

l Data Quantity Typically several orders of magnitude more data points are available from the log data set, although frequently

the transition zone is not well sampled if non- nett beds are present immediately above the contact. In contrast, the petrophysicist has fewer capillary pressure measurements at his disosal, although these have the advantage of yielding an essentially continuous profile for a particular rock type, and provide resolution t)hrough the transition zone.

Consequently, a method for integrating the two data types should make use of the consistent origin and continuous profile of the capillary pressure measurement, taking advantage of the height axis and possibly the larger data quantity of the wire- line log domain, while recognising the limitations in accuracy of both.

Functional Requirements

The function presented in this paper has been designed according to the following requirements, in the context of the considerations already outlined.

The function looks like a plausible saturation height profile and is well behaved over and beyond the height range of observed data.

It should be possible to weight the curve fit such that observed data is particularly well honoured where field area is largest, or the majority of hydrocarbons, or recoverable reserves, are present.

Although it is desirable to use both log and capillary pressure data, there should be no requirement for both data types to be present, and it should also be possible to use any other relevant data, such as Dean-Stark water saturations.

Changing terms in the equation has a predictable effect on the shape of the function, independent of other terms. Terms in the function should relate as closely as possible to physical phenomena or observations.

It should be possible to toggle between pressure and height domains by the application of scaling and displacement factors to optimize and confirm the fit.

The general form of the function relating hydrocarbon saturation sh to height above free-water level h is:-


For capillary pressure PC measurements the equiv- alent expression is:-

Sj, = 1 - S, = a exp ( >

,$ e

c

Depending on the context a, b, c and d are con- stants, or alternatively they may be simple functions of rock properties such as permeability. Fig- ure 1 shows the effect on the function of altering each coefficient.

l a is the asymptotic hydrocarbon saturation,

1 - sw,,,+

. b is a vertical scaling factor for the curve as a whole, which may be used to transform data between the laboratory pressure and height domains.

l c distorts the vertical axis to account for the saturations not following the simple form a exp -b/h. Both b and c are related to pore throat size distribution.

. d applies a vertical displacement to the entire curve, and can be used to locate the free water level.

Two important considerations in the process of fitting data to a generic algorithm need to be addressed.

l What criterion is used to characterize the best fit? Although least squares error minimization is popular, it is not necessarily the best method to use with data that is skewed, and often peppered with outliers. Having tried least squares minimization, we have chosen to fit by minimizing absolute differences which, although mathematially more diffi- cult, is less affected by the non-Gaussian distribution of a typical dataset.

l What, if any, weighting should be applied to the data? This depends on the purpose of the fit. The requirements for field-wide saturation mapping may be different from those of localising the hydrocarbon water contact. The program allows an individual weight to be attached to each data point.

In order to investigate these requirements a program has been written in Visual Numerics PV- Wave fourth generation programming and visual- ization environment to optimize any combination of

a, b, c and d, by minimizing the absolute differences between observed and calculated data. Various de- grees of freedom may be applied to the coefficients during the minimization process.

l No limits are set on the computed value of the coefficient.

l Upper and lower bounds are set on the computed value, constraining it within limits.

l The coefficient is set to a predetermined value.

Different criteria may be chosen for each coefficient. For example, a could be fixed, b and c allowed to vary over an infinite range, and d constrained with- ing limits of (say) *lO units.

The method should be considered a totally general approach which can be tailored according to the data available to solving a wide range of saturation height related problems. Rather than lay down specific rules, we present examples showing its application in different contexts.

Examples

Example 1 - Modelling Transition Zone

The example shown in figure 2 is a discovery well, drilled close to the top of the structure. In the ab- sence of geological control from other wells, hydrocarbons in place are to be mapped using constant rock properties throughout the accumulation.

The hydrocarbon water contact is clearly visible in a massive sand. Unfortunately, above about 95Oft, a comparison of core and log data suggests that computed water saturations in the rather thin sands are raised by the contribution to conductiv- ity by the adjacent shales. An evaluat)ion of hydrocarbons in place should therefore refer to air- brine capillary pressure data measurements, shown on figure 3. It has been established that these were made on rock that is representative of the field.

Rather than rely on empirical correlations to con- vert capillary pressure to height above free water level, the core a.nd log datasets are reconciled by allowing b and d in the general formula to vary.

First, an averaged a.ir brine capillary pressure curve is characterised, setting d to zero, by

-2.35 o.7g2 Sh = 1 -S, = 0.791exp 7

( > c

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Figure 3 shows that a close fit is achieved. If tra- dit,ional practice were to be followed, PC would be converted to height using the ratios

P Y cos ~,e,

c(m) - - Pc(lab) Y cos slab

P h=

c( ?-es)

dP water - Pod)

R.efereuce to literature5 suggests that interfacial tension at reservoir conditions is likely to be in the ra.nge 10 - 20dynes/cm, which implies a large un- certainty in transforming pressure to height, especially immediately above the contact, where satu- rat.ions change rapidly.

In terms of the general formula, it should be possible to rescale the capillary pressure axis into a height axis) and pick the free water level, by car- rying out the minimization process with log satu- rat,ions and vertical depths, fixing the values of a atltl c at the values found for the capillary pressure clat.a., a.ntl allowing the program to find the opti- 111~ m b and d.

In fact, this approach yielded invalid results. Sat- urat,ion build-up above t,he contact on figure 2 is a. relatively straight line, compared with the more curved capillary pressure characteristic. This anomaly can be explained by the lack of resolution of the 6FF40 induction tool. A more rigorous derivation of the saturation height function involves iteratively modelling (The Fastdip modelling software used in preparing this paper was supplied by Dr. R. Hardman of 6FF40 Inc.) a resistivity profile generated from the averaged capillary pressure curve. Each iteration comprised the following steps:-

I. A theoretical resistivity profile was created, startling at 1015ft with a constant value for R, of 0.6Qm for a distance of d feet, d being chosen as an estimated distance to the observed contact. Above d, resistivity was com- put,ed from wat.er air-brine water saturation using the Archie equation.

Rt = R,IS;

where..

0 792

2. A constant value of 0.9Rm was assigned to shale beds, identified using the gamma ray and core description.

This resistivity profile was convolved with the induction tool response and the resulting synthetic resistivity profile compared with the original log.

Modifications were made to b and d as neces- sary, and steps one to three repeated until an acceptable match, as shown on figure 4, was obtained.

The final values of the coefficients used for saturation mapping were

This places the free water level at 996.5ft , and implies a capillary pressure to height conversion factor of 0.733, which reference 6 suggests is within the large range of plausible conversion factors. Figure 4 shows that the modelled resistivity profile yields a synthetic induction curve that closely matches the original curve, and that the computed water saturations agree well with core derived water saturations in the thin sands towards the top of the interval.

Although the hydrocarbon water contact is visible in nett sand in this example, the same technique can of course be used to locate a down-dip contact, or one obscured by a shale bed.

Example 2 - Area Weighting

Figure 5 shows log data and computed results, in- dicating a hidden hydrocarbon water contact in the massive shale between 950 and 1000 ft. This region is characterised by low permeability, and hydrocarbon builds up slowly with height. Figures 6 and 7 show this in the context of the mapped area of the accumulation, with the heavy broken curve showing how the product of hydrocarbon saturation and area is distributed vertically. A function that accurately evaluates hydrocarbons in place needs to honour the observed data particuarly closely where the bulk of the hydrocarbons are present, achieved by weighting the data points according to their hydrocarbon saturation area product.

No special core analysis results are available for this well, although regional information suggests that values of about 0.93 and 0.82 are appropriate for a and c in the general equation. Imposing these values, and solving for the best value of b and d, not- ing that heights are reference to lOOOft, with data

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points weighted according to hydrocarbon saturation area product, yields the following values.

By comparison, solving with equal weights for the dat,a points yields the expression:

S,, =1-S~=0.93exp(~)og

A comparison of the two fitted curves on figure 8 is illuminating. Weighting the data points yields a contact nine feet deeper than the unweighted case. The unweighted fit matches the observed data bet- t,er over the top ca50ft of the interval, while the weighted fit better fits the deepest points. This exa.mple demonstrates the fragility of methods for extrapolat)ing to hydrocarbon water contacts, particularly as the vertical scaling factors, b in the general formula, differ by only about 2O%, a very small percentage error in an estimated conversion factor from pressure to height.

For comparison, the two computed hydrocarbon saturation area product curves are plotted on figure 9. As expected they differ particularly towards the base of the interval, and merge towards the top, further edorsing the decision to weight the data.

In pratical terms then, the effect of weighting in this case is to improve the fit to observed hydrocarbon saturations towards the base of the interval, resulting in a somewhat deeper estimated contact, and 5.2% more hydrocarbon in place.

Discussion

Our method shares some characteristics with ear- lier work314 but is distinguished by some important differences, which we believe render the important reconciliation between the capillary pressure and height domains easier. In fact, the method of reference 4 was developed in an environment where . capillary pressure data is usually not available. . whereas our technique arose from the problems of reconciling data from the two domains.

We acknowledge the pioneering work carried out by Thomeer in introducing a set of pore geometri- cal factors, relating saturation to capillary pressure. Using the notation of reference 6:-

sb = sbcs, exp -Fg 1% pc/pd

Our function shares the broad form, but differs in the behaviour of the exponent, the form of which has been developed to address difficulties encoun- tered by the practising petrophysicist. A discussion of the physical significance of each term beyond the introduction under Functional Requirements is outside the scope of this paper, which aims primarily to introduce practical methods to solve real problems, particularly in the area of making the most constructive use of all available log and core data.

We believe that iterative methods to solve non- NNN linear equations are under utilized in the petrophysical community. Indeed, reference 2 lists a number of relationships which demonstrate great ingenuity in their use of polynomials and logarithms to transform the data into a domain where it appears to be linear, before applying a linear regression method. Some advantages of our curve fitting philosophy are listed.

The effect of changing coefficients of the equation on the computed curve are comprehensible in their natural units of saturation and height or pressure than they would be if trans- formed.

A variety of criteria, such as minimization of least squares or absolute differences, can be compared, and the effect of outliers and non- Gaussian distribution assessed.

Weighting functions can easily be applied, to bias the fit to, for example, the height range where most rock, or the majority of the hydrocarbons are present.

These considerations have led to a technique that lends not only to accurately modelling saturation height behaviour, but perhaps more significantly, understanding and controlling the sources of error.

We have not presented a multi-well example, largely due to the difficulties of obtaining data re- lease for an entire field. Nonetheless, we can confirm that the technique has been used successfuly on multi-well datasets, and that in these circum- stances the implications of any weighting need to be clearly thought through.

Both examples presented here assume that saturation mapping is to be carried out with constant petrophysical parameters over the field. This is common practice when mapping a reservoir with little well control. If sufficient well dat,a is present to model variations in saturation behaviour with

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permeability, porosity, facies etc., the same general formula has been used, substituting simple math- ematical expressions for any or all or a, b and c. Derivation of these formulae can similarly be carried out using a combination of core and log data, and has been found to yield water saturation profiles which faithfully reproduce observed data.

Conclusions

A general function has been introduced to char- acterise saturation as a function of height above contact,. Several examples of its use are presented, although they represent only a small sample of the problems which have addressed by the method.

Although the terms in the formula may be related t,o physical characteristics of the rock and reservoir, this is not the primary purpose of the paper, which aims primarily to introduce a practical technique to the community of practising petrophysicists.

The function has be shown to be sufficiently pli- able to achieve a close fit to a dataset of saturation and height or capillary pressure. The use of area1 weighting allows the user to derive coefficients which model the observed data best over t.he depth range where the majority of hydrocarbons are present.

Glossary

Area of structure at a given elevation Acceleration due to gravity Height above reference Archie saturation exponent Capillary Pressure Wet formation resistivity True formation resistivity Hydrocarbon saturation Water saturation Surface tension Contact angle Density

Acknowledgements

us to the Esthetic pleasures of non-linear optimization.

References

1. Heseldin G.M., A Method of Averaging Capil- lary Pressure Curves, 1974, SPWLA 15th Annual Symposium. 2. Cuddy S., Allinson G. and Steele R., A Sim- ple Convincing Model for Calculating Water Satu- rations in Southern North Sea Gas Fields, 1993, Spwla 34th Annual Symposium, Paper H. 3. Thomeer J.H., Air Permeability as a Function of Three Pore Network Parameters, 1983, JPT Vol. 35 No. 4, pp 809-814. 4. Smith D., Predicting a Downdip Water Level Using Capillary Pressure Relationships, 1991, The Log Analyst Vol. 32 No. 5., pp 571-574. 5. Livingston H.K., Surface and Interfacial Ten- sion of Oil-Water Systems in Texas Oil Sands, 1938, AIME Tech Paper 1001. 6. Thomeer J.H., Introduction of a Pore Geo- metrical Factor Defined by the Capillary Pressure Curve, 1959, 34th Annual Fall Meeting of SPE, Dallas.

About the Authors

Christopher Skelt has an MA in Engineering from Emmanuel College Cambridge, and an MSc in Bioengineering from The University of Strath- Clyde, Glasgow. He entered the oil industry as a Field Engineer with Schlumberger in 1977. Since 1984 he has dedicated himself to petrophysics, with Schlumberger, Shell and, since 1991, with the Scott Pickford Group.

Bob Harrison has an MSc in Petroleum Engi- neering from Imperial College, London. He entered the oil industry in 1979. After a lengthy spell with British Gas Exploration and Production, he joined Enterprise Oil as a Staff Petroleum Engi- neerin 1993. He was Editor of the 1989 SPWLA North Sea R, catalogue, and the 1995 LPS Rus- sian Style Formation Evaluation Manual.

We t.hank our employers for providing the time and resources to write this paper, the contents of which represent our personal views. The oil companies which released data for the examples presented are gra.tefully acknowledged. Thanks are also due to Charles Wood, of Scott Pickford, for introducing

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Effect of Altering a -I

! I I

I I

i ! i !

I I I I *

\ I I

1 I I I

: plpcreasinga

I i I I I I I I

I U 1 u Hydrocarbon Saturation Hydrocarbon Saturation

Effect of Altering c

:I 1 II i ii

\I /

i ii

i /

I

I 0 Hydrocarbon Saturation

Effect of Altering b

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Effect of Altering d

+ Increasing d

\\ 1, \\ I), \\

\\

\\ A,. \\ .A---- ~ I ._ -------- ---._. ---. \. -\ , -._

... --A._. ---_ -------

1 0 Hydrocarbon Saturation

Figure 1 - Effect of Altering Parameters in S,,=a exp -(b/(h+d)) -7-


0 GR 150

960

980

0.2 ILD 20

1 SW-NE-IT o

1 CORE SW xxxx-ixx 0

:

. .

. .

.

fig2.ps 27.Mar-95 13:45:22

Figure 2 - Discovery Well Logs Core Data and CPI Results.

XXXX Cap Pressure Data Fitted Curve

/

:

i

0.0 0.2 0.4 0.6 0.8 1.0 Water Saturation

Figure 3 - Fit to Air/Brine Capillary Pressure Curve

960

980

fi&ps 27-Mar-95 12:21:25

Figure 4 - Resistivity and Saturation Profiles

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8.50

900

950

000

fig5.p~ 27-Mar-95 12:40:09

NNN

Figure 5 - Logs and CPI Results. -f&


800

G E.

c?

- Satn Area Product 850

\

- Structure Area

\ 900 \

c

\ 950

-

\ IOOOI

800

850

2 s 5 0 900 2 .o 5 >

950

1000

a

7 2 5 >

l-w 0 2 4 6 8

Structure Area 0.0 0.2 0.4 0.6 0.8 1 .O

Hydrocarbon Saturation

Figure 6 - Saturation vs. Height Figure 7 - Area and Hydrocarbon Area vs. Height

200 /-

u-l 8001

- Log Data

~ Weighted Fit

- Log Data

- Weighted Fit

- - -. Unweighted Fit

I

z M I 6 z

0.8 0.6 0.4 0.2 0.0 0.0 0.5 I.0 1.5 2.0 Hydrocarbon Saturation Hydrocarbon Area

Figure 8 - Results of Curve Fit Figure 9 - Hydrocarbon Area vs. Elevation

spwla 1995

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