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Generalised Sonic Porosity Transform for Upper Assam Oil FieldsLalaji Yadav, K. K. Prasad# & A. N. Bhattacharya

Well Logging Services, ONGC, Nazira, Assam#Well Logging Services, ONGC, Baroda, Gujrat

5th Conference & Exposition on Petroleum Geophysics, Hyderabad-2004, India PP 95-100

ABSTRACT : Several empirical and semi-empirical relationships between porosity and sonic transit time or velocity, have been

developed in the past for specific field conditions. Over the years Wyllie time average and Raymer–Hunt–Gardener equations

have been used for the determination of porosity from sonic transit time in Eastern Region of Oil & Natural Gas Corporation. For

unconsolidated sand reservoirs of Upper Assam, the transform needs modification using so called “Lack of compaction” correction,

an inconvenient, complicating factor. Another problem, associated with the transformation of sonic transit time into porosity

involves the selection of the proper matrix transit time, which is more difficult to ascertain in eastern region due to varying

complex lithology. An attempt has been made to develop such relationships through regression analysis of core derived porosity

data available over five hundred core measurements in the producing reservoirs of Upper Assam oil fields. It was not possible to

generate the relationships for every horizon field-wise due to paucity of data coupled with large scatter and having poor correlation

coefficient and high standard of errors. An effort has been made to pool all the available data together irrespective of horizons /

field to generate a generalized transform between porosity and sonic transit time. The equation has been generated through

regression analysis for the entire Upper Assam, ∅ = A + B∆T + C∆T2 + D∆T3, where A = - 0.922443, B = 0.02417986,

C = -0.0001697 and D = 4.8313 x 10-7. This empirical transform based on extensive field observations of transit time versus

porosity does not require the pre-knowledge of matrix transit time and compaction correction. This transform is essentially for

sandstones of Eastern Region and is presently valid up to 37% porosity.

INTRODUCTION

Sonic log has been overly favoured porosity log inEastern Region of Oil & Natural Gas Corporation Ltd. Severalempirical and semi-empirical relationships between porosityand sonic transit time or velocity, have been developed in thepast for specific field conditions (references). Wyllie timeaverage equation has been widely used for converting therecorded transit time to porosity, due to its simplicity. However,of late Raymer–Hunt–Gardener transform, which is based onfield observations, is also being used.

It has been generally experienced that the porosityvalues estimated from transit time values, using time averagerelationship are higher than core and cross plot porosities,viz., Neutron-Density as indicated in Fig. 1. A normalizationcorrection is generally imparted in the form of compactioncorrection assuming that the departure in porosity values fromthe expected values is only due to under compaction, which isnot totally valid. The magnitude of compaction correction isbased on shale transit time which exceeds 100 microsec/ft incase of under compaction. However, in the field of easternregion, the need of compaction correction is felt even thoughthe shale transit time recorded are in the neighborhood of 90microsec/ft. Moreover, the compaction correction is amultiplier only, which loses its utility at high transit times,

Figure 1: Comparison of neutron and density porosity with Corederived porosity

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Generalised Sonic Porosity Transform

Figure 2: Wyllie and Raymer Hunt transform with Core porosity –Upper Assam oil fields

where calculated porosity values are still more overestimated.Uncertainly on matrix transit time due to complex lithologyand uncertainty on shale correction in case of shaly sediments,have been other two disturbing factors.

In order to overcome these difficulties, the presentexercise of developing an improved transit time porositytransform for Upper Assam Valley was taken up by Yadav,etal 1986. It was not possible to generate the relationships forevery horizon field-wise due to paucity of data coupled withlarge scatter and having poor correlation coefficient and highstandard of errors. An effort has been made to pool all theavailable data together irrespective of horizons/ field togenerate a generalized transform between porosity and sonictransit time. Sonic porosity derived from the equationcorroborates with core measured porosity.

DATA ACQUISITION

The data has been taken from Lakwa, Lakhmani,Geleki, Rudrasagar and Other Areas. The compilation of coreporosity (conventional and sidewall) has been done from allthe available sources (see references). Corresponding transittime data was picked up from sonic logs. However, there areseveral wells where core porosity data was available, but nosonic log was recorded. Core data for such wells could not beused for obvious reasons.

In case of Lakwa-Lakhmani area, core data isavailable mostly from Tipam Sand 2, which is the mainproducing horizon in these fields. The core data in Gelekiand Rudrasagar fields is mainly confined to deeper horizonslike TS-3, TS-5 and Barails, where the porosity is lower thanthat of TS-2.

FITTING OF WYLLIE AND RAYMER HUNTTRANSFORMS TO THE EXISTING DATA

Both of these transforms are traced on the cross plotof transit time and core porosity for an easy reference(Fig. 2). The fluid and Matrix velocity values used for tracingthese transforms are 5300 ft/sec and 18000 ft/sec respectively.Both transforms appear to be over estimating the porosityvalues as compared to mean core porosity values.

DERIVATION OF THE GENERALISEDTRANSFORM

Wyllie time average equation :

∆tlog = φ∆tfluid + (1 - φ) ∆tmatrix — — (1)

or ∆t - (∆tf - ∆tma)φ = ∆tma — — (1a)

or φ = A + B ∆t — — (1b)

- ∆tma

1where A = —————— B = —————— ∆t

f - ∆t

ma ∆t

f - ∆t

ma

If we have n components of matrix, the equation 1,can be written as

∆t = φ ∆tf + V

1 ∆tm

1 V

2 ∆tm

2 + —— + Vn ∆t

mn — (2)

and φ + V1 + V

2 + ………….. + Vn = 1 — (3)

where V1, V

2 ……… V

n are volume fractions of n

components of matrix. From equation (2) and (3), it is clearthat each porosity value, are having different matrix transittimes, i.e. volumetric averages of the components which isagain a function of φ. To calculate φ by knowing ∆t and ∆t

f ,

∆tm1

∆tm2 …….. ∆t

mn, we further require (n – 1) independent

measurements to know V1, V

2 ……….. V

n. So A and B in

equation (1b) are no more constants and vary with φ. To

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account this variable portion, we have introduced anothervariable depth with a coefficient in equation 1.b as

φ = AD + B∆t + C — — — — (4)

where, D is depth.

To account different rates of variation with depth,we have also introduced logarithm of depth instead of depth.The equation (4) has been used for multivariable regressionanalysis.

Now, to look for a more generalized transformequation 1(b) can be written as :

1 1 1 —— = ————— = ——————— φ A + B∆

t A(1 + B/A ∆

t)

by simple Binomial expansion, we can write

1 1 B B2 B3

— = —— (1- — ∆t + — ∆

t2 - — ∆

t3 + —) — — (5)

φ A A A2 A3

The series in the bracket is of converging naturebecause B/A is always less than unity. So we can have aconverging series expression for 1/φ in terms of powers of ∆

t as

1/φ = a0 + a1∆t + a2∆

t 2 + a3∆

t 3 + - - - - — (6)

Similarly, a polynomial expression for φ can alsobe written because 1/φ is always a positive number and φ isalso a positive number less than 1, So

φ = a0 + a1∆t + a2∆

t 2 + a3∆

t 3 + - - - - — (7)

By rearrangement of equation 1b again we can havea polynomial expression for 1/∆

t as

1/∆t = a0 + a1φ

+ a2 φ 2 + a3

φ

3 + - - - - — (8)

Which is again of converging nature because φ isalways less than unity. These equations (7) and (8) form thebasis for polynomial curve fitting. It should be noted that a

0,

a1, a

2, a

3 are different in different equations 6, 7 and 8. Equation

(8) appears to be a generalized transform which reduces toWyllie and Raymer Hunt transforms, when we consider it upto second and third terms respectively.

METHODOLOGY

(a) Multivariable Linear Regression Fitting

Multivariable linear regression fitting of coreporosity with sonic transit time for different fields, takingdepth into consideration are presented in following table.

Generalised Sonic Porosity Transform

Apparent matrix transit time is decreasing as depthis increasing. For Lakwa and Lakhmani fields the standarderrors of estimate are high. If depths are kept constant as 2000,3000 and 4000 mts in these equations, these results bring outthe increase in porosity with depth for a constant transit time.The magnitude of porosity increase for each 1000 mts increasein depth is around 1.7%. This can be explained as the combinedeffect of lithology variation and depth of burial (Yadav, etal1986).

Due to high standard errors of estimate, depth wasreplaced by logarithm of depth and again multivariable linearregression fitting of core porosity with transit time and depthwas done and results are shown in following table.

Regression Analysis Results : phi = A*D+ B* Delt + C

Fields/Areas Regression Coefficients Correlation StandardA B C Coefficient Error

Lakwa and 0.190928 0.65295 - 41.0375 0.9193 4.3578Lakhmani

Geleki, 0.13840 0.740165 -45.6163 0.9799 2.9217Rudrasagar and other areas

Regression Analysis Results : phi = A*logD+ B* Delt + C

Fields/Areas Regression Coefficients Correlation StandardA B C Coefficient Error

Lakwa and 0.0174616 0.354616 - 8.68399 0.95 2.7491Lakhmani

Geleki, 5.117588 0.6035242 -48.2099 0.83 2.7350Rudrasagar and other areas

In this case standard errors of estimate andcorrelation coefficient have improved in Lakwa andLakhmani field. However, these are getting deteriorated inGeleki, Rudrasagar and Other areas.

(b) Least Square Polynomial Curve Fittings

Least Square Polynomial Curve Fitting of porositydata with velocity i.e. inverse sonic transit time is done andresults presented in the following table.

Regression Analysis Results1/DELT = A + B*Phi + C* phi **2 + D *phi**3

Fields/Areas Regression Coefficients Correlation StandardA B C D Coefficient Error

Lakwa 0.017966 -0.05531 0.127510 -0.08486 0.95 0.00056and Lakhmani

Geleki, 0.017267 -0.03506 0.047442 -0.02436 0.94 0.00063Rudrasagarand other areas

Upper 0.01733 -0.03103 0.017970 0.001021 0.96 0.00005Assam

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Generalised Sonic Porosity Transform

Figure 3 : Deduction of generalised transform through Regressionanalysis of core measurement

Figure 4: Extrapolation matrix and fluid transit times throughassam equation

Figure 5: Effect of under compaction on fluid transit time

RESULTS AND CONCLUSION

! It was not possible to generate the relationships for everyhorizon field-wise due to paucity of data coupled withlarge scatter and having poor correlation coefficient andhigh standard of errors. An effort has been made to poolall the available data together irrespective of horizons /field to generate a generalized transform betweenporosity and sonic transit time. The following equationshave been generated through regression analysis for theentire Upper Assam ( Fig. 3 ).

φ = A + B ∆t + C ∆

t 2 + D ∆

t 3

where A = - 0.922443, B = - 0.02417986,C = - 0.0001697 and D = 4.8313 x 10-7

! This transform does not require prior knowledge ofmatrix transit time and compaction correction. It isessentially for sandstones and is presently valid up totransit time of 120 microsec/ft i.e. 37% porosity.Behaviour of the equation at the boundadary conditionsi. e. 0 and 100 % porosity values are presented inFig. 4. Higher value of transit time at 100 % porosity isdue to uncompaction effects as illustrated on Neutron-Sonic transit time cross plot in Fig 5.

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Generalised Sonic Porosity Transform

Figure 6: Comparison of sonic porosity with core derived porosity

Figure 7 : Comparison of sonic porosity with core derived porosity Figure 8 : Comparison of Assam transform with other transforms

! Sonic porosity derived from the equation corroborateswith core measured porosity as indicated in Fig. 6 and7. Comparison of the transform with other transformsis presented in Fig 8.

! As this transform is derived from least square fitting ofdata, it represents only mean value of the porosityindependent of depth. High scatter of data above themean value represents data from deeper depths and lowerone represents data from shallower one in comparisonto mean value of 3000 m. So, this transform under andover estimates the porosity values by 1.7 % at depths of4000 m and 2000 m. respectively.

! It is hoped that proposed transform will be useful forthe estimation of porosity in Eastern Region. There isstill a scope for further refinement as more data comesin.

ACKNOWLEDGEMENTS

Authors are grateful to ONGC authorities for the kindpermission to present and publish this article. Technical

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Generalised Sonic Porosity Transform

discussions held with Sri. Aloke K. Bhanja and othercolleagues of well logging services are thankfullyacknowledged.

REFERENCES

Heysee, D.R., 1994 , Sonic porosity transforms andeffects of pore shape and clay distributionSociety of Professional Well Log Analysts; 2ndInternational Well Logging Symposium Transactions,Beijing Chapter; 149-15.

Koithara, J., Hashmy, K. H. and Mehra, Y.M., 1969, Laboratorymeasurements of logging parameters on Tipam sandstone’scores of Lakwa field ; ONGC unpublished report.

Nowhar, U.V.S. and Hansraj, H., 1980. Laboratory determinationof petrophysical parameters and their relationships forTS-5 sands of Geleki Field, IPE, ONGC, unpublishedreport.

Nowhar, U.V.S., Verma, R. P. and Hansraj, 1979. Laboratorydetermination of petrophysical parameters and theirrelationships for TS-1 and TS-2 sands of Lakhmani Field,IPE, ONGC, unpublished report.

Singh, M. and Hansraj, 1979. Laboratory determination ofpetrophysical parameters and their relationships forTS-1 sand of LT-22 ( Lakwa # 159 ), IPE, ONGC,unpublished report.

Verma, R. P. and Satyanarayana, M., 1982. Laboratory study fordetermination of petrophysical parameters and theirrelationships on core samples of Namti # 1, KDMIPE,ONGC, unpublished report.

Raymer, L. L., Hunt, E.R. and Gardner, J. S., 1980, An improvedsonic transit time-to-porosity transform, Society ofProfessional Well Log Analysts; 21st Annual LoggingSymposium Transactions; paper P, 1-13

Tao, G. and King, M. S., 1990, An improved acoustic velocities-to-porosity transform for sonic log interpretation, Societyof Professional Well Log Analysts; International WellLogging Symposium Transactions, Beijing Chapter; paperB, 1-19

Yadav, L., Prasad, K. K. and Bansal, C. R., 1986, Improved SonicTransit Time Porosity Transform for Upper Assam, ERBC,ONGC, unpublished report