Watfa1991.pdf

SPE Society of Petroleum Engineers

SPE 21415 Using Electrical Logs To Obtain the Saturation Exponent (n) in the Article Equation M. Watta, Schlumberger Middle East SA SPE Member

Copyright 1991, Society of Petroleum Engineers, Inc. This paper was prepared for presentation at the SPE Middle East Oil Show held in Bahrain, 16-19 November 1991. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A., Telex, 730989 SPEDAL.

ABSTRACT

INTROPUCTION

The saturation exponent (n) is an essential parameter that is required to solve the Archie Equation and obtain saturation values. This equation gives correlations between resistivity, porosity and saturations. When core measurements of n are unavailable, its value is assumed constant (usually 2). However, under certain conditions, this assumption of a constant value could lead to large errors in computed saturation values.

The saturation exponent (n) in the Archie Equation presents interpreters with some serious limitations in estimating accurate water saturation values. It has long been recognized that n is tied closely to rock wettability, and is thus an important factor in determining recovery and the optimum injection scheme for secondary recovery. The only practical means at present of obtaining n values is by using tedious, expensive, and time-consuming core measurements. This involves the slow injection of known mixtures of water and oil (Le. predetermined water saturations Sw) into the core sample, and the corresponding resistivity measurements. There has been doubts about the validity of the computed values of n from such core measurements, as these .measurements do not reproduce downhole conditions. Some core cleaning processes have been known to affect the true rock wettability.

This paper presents a technique whereby electrical logs are used to solve simultaneously for n and the water saturation values Sw and Sxo in the uninvaded and invaded zone respectively. This technique combines open hole log values which are sensitive to saturation, namely the deep reading resistivity Rt, the shallow resistivity Rxo , and the electromagnetic transit time tplt together with the through tUbing cased hole measurement of the formation capture cross-section (1:). Essentially four equations are used to solve for Sw, Sxo, n and the porosity exponent (m) at every log sampling point.

The main limitations of obtaining n from electrical logs is due to the fact that only one measurement, the deep resistivity Rt, is made in the virgin zone from open hole logs. This measurement is used to compute the water saturation (Sw) by assuming a value for n.

A data-base containing log data from 22 wells was constructed from the Upper Jurassic Arab Formation. This is a carbonate reservoir with anhydrite. The porosity distribution in sections of this formation is known to be heterogeneous, with the presence of vugs and fractures. The results of this study showed that the Arab Formation is water wet, and the computed values of n are consistent with the range of values obtained from core measurements. The n values were found to increase with increasing dolomitization, and were shown to be independent of both the pore type and of the porosity exponent m. The results presented demonstrates satisfactorily that this technique, under particular conditions, can give accurate computed values for n at downhole conditions.

Recent advances in through tubing saturation monitoring tools have made it possible to obtain very accurate values of water saturation (Sw), independently of the Archie Equation, and therefore, independently of the n and m exponents in this equation. This paper outlines a technique whereby cased hole measurements of the formation capture cross-section CI') can, under certain circumstances, be combined with other open hole logs, to compute in-situ values of n. * Mark of Schlumberger

References and illustrations at end of paper 679

2

USING ELECTRICAL LOGS TO OBTAIN THE SATURATION EXPONENT(n) IN THE ARTICLE EQUATION

A data-base comprising 22 wells was constructed. The open hole, cased hole, and well conditions for each well made it possible to s91ve the system of equations and obtain values for the saturation exponent n, as well as solving for other variables such as Sw and m. The data presented in this paper are for the Arab Formation, which is Jurassic. This formation is a mixture of dolomite, limestone and anhydrite with very little clay. Its pore structure is complex and heterogeneous, with isolated porosity and fractures. Moreover, core observations have showed that there is a large variation in particle size over these sections of the formation.

applies for values of either Sw or Sxo. The results of Fig.1 were derived from the following equation:

~= 2 Sw2

Effects of variations Computations.

a.

b.

How important are the parameters m and n in determining the values of water saturation in the invaded zone (references, 1,2,3)? The Archie Equation, for a clean formation can be defined as follows:

The same equation applies in the invaded zone, where the invaded zone parameters replace the uninvaded zone parameters:

. Sxo

n

m

on

Saturation

True value of m as compared to the value of used in the computations.

m

The true value of the porosity".

~

=

~1-( m/2)

(4)

Sw2

(2)

-~-

in

Using a similar procedure as in the preVious section, a value of m =2.0 is normally assumed. Charts are derived (Fig.-2) that show errors in the computed values of water saturations for variations in the true value of m from the assumed value of 2. and for a range of porosity values. Similar to Fig.-1, the saturation ratio presented on the horizontal axis in Fig.-2 applies for values of either Sw or Sxo' The results of Fig.-2 was derived from the following equation:

(1)

rt. m p

(3)

The effects variations in m on the computed values of water saturation Sw and Sxo in the uninvaded and invaded zones is governed by two parameters:

SENSITIVITY OF SATURATION COMPUTATIONS TO YARIATIONS IN THE VALUES OF nAND m

Rmf =-------'....

1.0

Where SW2 = Computed Sw values using a value of n=m=2 Swn = True Sw value for the correct n value, and assuming m = 2. Equation-3 can be derived easily from algebraic manipulations of the Archie Equation.

The results obtained from this study demonstrate that this approach and technique can yield representative values for n at normal downhole conditions. The results also showed that the formation is water wet. and the wettability is affected by the lithology. The study also demonstrated that there is no obvious correlation between the two exponents m and n and that n is not affected buy the pore type.

R xo

SPE21415

Where, Swm = true Sw value using correct m, and assuming n =2. Equation-4 can also be obtained from algebraic manipulations of the Archie Equation.

These two parameters, m and n will be evaluated separately in order to understand their individual effects on the computed values of saturation.

From Figs.-1 and 2 the following are apparent:

Saturation

a. The effect of n variations on saturation are most sensitive at low saturation values.

The effects of variations in n on the computed values of water saturation Sw and Sxo in the uninvaded and invaded zones is governed by two parameters:

b. The effects of m variations on saturation are most sensitive at low porosity values.

Effects of Variations Computations.

in

n

on

a - True value of n as compared to the value of in the computations.

c. The maximum error in computed water saturation values will occur when both the porosity and saturation are low. For example, in fig.- 2, if the true m value is 1.5. but a value of 2 is used, then the error in the computed value of saturation will be 75% when Sw=10 • as compared with a 20% error when Sw =70%.

n used

b - The true value of water saturation Sw or Sxo. Since a value of n =2 is normally assumed, charts are derived (Fig.-1) to show errors in the computed water saturation for variations in the true value of n and for the whole range of saturation variations from 0-100%. The saturation ratio presented on the horizontal axis in Fig.-1

680

SPE21415

DR. MOHAMED WATFA

.L = LW . 0

MEASUREMENTS AND ANALYSIS

. Sw + Lhc . 0 . (1-Sw) + Lma· (1-0) + Lei· Vcl

Traditionally, all saturation computations were made assuming constant values of m and n. This was dictated by practical limitations of the availability of continuous channels defining the variations of m and n at every sampling point. The limitations, and resulting inaccuracies caused by using constant values for m and n, were recognized from an early stage, and various techniques were tried to account for the variations of these parameters. The following is a summary of techniques that are in use to account for variations in m and n. :

(6)

Thus. when open hole measurements of Rt, Rt and tpl are available, and where subsequently measurements of .L are made after the filtrate effect has disappeared, and before changes in water saturation has taken place, then four equations become available (equations 1, 2 , 5 and 6) which respond to variations of water saturations in the invaded and uninvaded zones. Under such conditions, equations 1, 2, 5, and 6 can be solved simultaneously to obtain values for Sw. Sxo. m and n. The flow chart of Fig.3 shows the procedure used for this solution. The iterations for Sxo convergence shown on fig.-3 are necessary in order to obtain accurate lithology, porosity and clay volume values; the nuclear log measurements, which are used to determine porosity and lithology, could be affected extensively by light hydrocarbon. The porosity. lithology and clay volume inputs come from standard open hole log interpretations.

1 - Use of core data for m and n.: However, such data is usually incomplete and may not be representative of true downhole conditions. 2 - Empirical correlations between these parameters and other reservoir parameters:This was tried mainly for the parameter m where correlations between m and porosity and lithology were made (References 5, 6 and 7)

RESULTS and DISCUSSION

The original limitations of obtaining in-situ log evaluations of m and n was due to the absence of enough equations (or measurements) that responded appreciably to the volume of water. Up to recent times, only equations (1) and (2), representing resistivity measurements in the invaded and uninvaded zones, were available. To solve equations 1 and 2 for water saturations Sw and Sxo, values for m and n are needed; values of porosity (0) and clay volume (in shaley environment) are obtained from other log or core measurements. Likewise, the salinitv of the formation water in the uninvaded zone, and the filtrate salinity in the invaded zone, are obtained from cross-plots in 100% water bearing environments, or from sample analysis at the surface.

A data base containing data from 22 wells (reference 8) was constructed for the Arab Formation (Upper Jurassic). All the wells considered had original open hole Rt, Rxo and tpl logs, and sUbsequent through tubing .L log. The interpretation procedure outlined in fig-3 was made on all the data. Local reservoir knowledge, together with information from production data, were used to determine the intervals in each well where the value of Sw remained unchanged between the .L and Rt log's. Four examples are presented here of Sw. Sxo. m and n computations over sections of the Arab-B, C and 0 Formations. The examples are shown on Figures 4-7. Each figure Shows the following: • Raw data for .L and tpl on Track-1 (left). Raw data for Rt and Rxo in logarithmic Track-2. • The computed results for m and n on track-3. • Volumetric analysis ,of lithology, Water and hydrocarbon volumes in the invaded and uninvaded zones in Track-4 (right).

With the introduction of Electromagnetic Propagation Time (EPT*) Tool, a third equation was added to the two saturation equations. This equation can be defined as follows: tpl = t pw · 0 . Sxo + tphc. 0. (1-S xo ) + tpma (1-0) + tpcl . Vcl

3

(5)

Example-1, Fig-4: This is an example of the Arab-D formation. The variation in m is in the range 2-2.8; this is consistent with the observed value for m obtained from core analysis. The range of variations of n is in the range of 1.1-2. The variation in n appears also to be affected by dolomitization.

Originally, this new equation, together with equations (1) and (2) were used to solve for Sw, Sxo and m by assuming a constant value for n (reference 7). The introduction of the Dual-burst Thermal Delay Time (TDT-P* ) Tool gave a new dimension in measuring accurately the formation capture cross-section (E). Normally, E measurements are made few years after the original open hole logging of Rxo , Rt and tpl. If the E measurement is made over sections of the formation where the water saturation Sw is expected to be the same as that when the original open hole logging (Le. the same saturation as that seen by Rt), then a fourth equation for E can be introduced. This can be defined as follows:

Example-2, Fig-5: This is a second example of the Arab-D formation. The trends in m and n variations in this example are very similar to those observed in example-1. The value of n also shows an increase in value with the increase in dolomitization. Example-3, Fig-6: This is an example of the upper Arab-B and Arab-C formations. The variations in mare in the range of 2-4. This large variation in m, which was observed also from core analysis, reflects the complex 681

4

USING ELECTRICAL LOGS TO OBTAIN THE SATURATION EXPONENT(n) IN THE ARTICLE EQUATION

pore geometry structure of these upper Arab Formations. However, the variations in n remained in the same range of 1.1-2 which was seen for the lower Arab-D Formation.

The use of Rt, Rt and tpl in order to solve for Sw. Sxo and m, assuming a constant n, have been routinely applied in carbonate formations in the Middle East (reference 7). The results of the variable m obtained has on the whole being satisfactory when compared with core-derived data.

Example-4, Fig-7: This is another example of the upper Arab-B and C Formations. The trends here for m and n variations are similar to those of Example-3. No values for m or n are computed in the effectively zero porosity anhydrite beds.

The addition of the L log, to facilitate the computation of n, could introduce additional errors. Fig-10 shows the range of errors in the computed value of n as a function of the errors in the measured L (defined as AL). As shown in fig-10, there are two factors that define the errors of the computed n value originating from the L measurement:

The trends observed in the four examples were also apparent for the other wells evaluated in the data-base. There are few core analysis value for n in the Arab Formations and these are in the range 1.3-19. The observation of the variations of m and n in the Arab Formation of the wells considered can be summarized as follows: 1 - The variation of variations in m

n appears to

1 - Errors in the measured L (Le values of AL). 2 - Volume of water in the uninvaded zone (121 .Sw).

be independent of the

The range of irreducible volume of water in oil zones in the Arab formation are of the order of 3 pu. If we assume 121 .Sw= 3 pu, a value of AL=0.25 will introduce an error in n of the order of 10%. This error in n will increase to 20% if the value of AL=0.5. The chart of Fig-10 also shows that the errors in the computed value of n will decrease with the increase in the water volume. This suggests that the accuracy of the saturation computations using this technique will improve over the transition zone from water to oil. The errors presented on fig.-10 exclude errors in the other log measurements.

2 - The variation in n increases with dolomitization. Fig.-8 is a cross-plot of lithology versus the variation in n. This apparent increase in n with dolomitization was checked to ensure that it is not caused by the use of wrong parameters for dolomite; namely Ldol' the Capture cross-section of dolomite, and tpdol, the electromagnetic transit time of dolomite. However these two parameters need to be changed appreciably outside their recognized range in order to disguise the observed trend.

CONCLUSIONS

3 - The variation in n appears to be independent of the pore geometry structure and pore type. Le. vugs, fractures and grain size variations. This is in contrast with the variations in m which is highly affected by pore geometry variations.

A technique was presented that integrates open hole and cased hole logs to solve for the two exponents nand m in the Archie Equation. Values of nand m can be obtained for every log sampling point as a continuous curve. The results of a 22 well data-base study made on reservoirs in the Arab Formation showed the following:

4 - Fig-9 is a multi-well cross-plot of n vs. volume of water (121. Sw) made for the Arab-D Formation. This cross-plot shows a decreasing value of n with the decrease in volume water. This trend of n variations suggests that the Arab-D Formation in this reservoir is water wet. Core analysis have shown that a decrease in water volume will introduce large increases in n if the reservoir is oil wet (reference 9). ACCURACY OF THE COMPUTED

•

•

n VA LUES

The accuracy of the computed values of the following:

SPE21415

n is a function of

The Arab reservoirs evaluated here appear to be water wet. Wettability is affected by the lithology. In this case, the saturation exponent increases in value with increase in the dolomite volume. There is no definite correlation between the two parameters nand m in the results analyzed here. The variations of n appear to be independent of pore type and geometry.

The integration of the L equation with the other open hole logs, which made it possible to compute a variablen, did not affect appreciably the overall accuracy of the water saturation computations. For the reservoirs evaluated here, the errors introduced by using L are estimated to be within 15% of the true value of n.

1 - Accuracy of the four measurements Rt, Rxo , tpl and

L.

2 - Accuracy of the two assumptions made in order to use the L equation; namely, no changes in Sw between the Rt and L measurements, and zero filtrate effect.

682

SPE21415

DR. MOHAMED WATFA

NOMENCLATURE

ACKNOWLEDGEMENTS

m

The author would like to express his sincere thanks for the various oil companies in Abu Dhabi who cooperated by releasing, the log examples. The author would also like to thank Mahender Bhasin, senior technician in Schlumberger Dubai, for collecting the data and constructing the data-base.

n At

RxO Rw Rmf Sw Sxo SW2

Swn Swm

= Porosity exponent in the Archie Equation. = Saturation exponent in the Archie Equation. = Deep reading resistivity in the uninvaded zone (ohm-m). = Shallow reading resistivity in the invaded zone (ohm-m). = Formation water salinity (ohm-m). = Filtrate water salinity in the invaded zone (ohm-m). = Water saturation in the uninvaded zone. = Water saturation in the invaded zone. = Computed water saturation assuming IT1=n=2 = True water saturation in the uninvaded zone for correct value of n and assuming IT1= 2 = True water saturation in the uninvaded zone for correct value of m and assuming n =2

REFERENCES 1, 2, 3 - Watfa, M.:Seekjng the Saturation Solution, Middle East Well Evaluation Review, Number-3., 1987. 4 - Borai, AM.: A New Correlation fro Cementation Factor in Low -porosity Carbonates, SPE 5th Middle East Oil Show, Bahrain 1987, SPE-14401. 5 - Focke, J.W., and Munn, D.: Cementation Exponents (m) in Middle East Carbonate Reservoirs: SPE 4th Middle East Oil Show, Bahrain 1985, SPE-13735.

t pw , tphc, tpcl tpma = Electromagnetic transit time in water, hydrocarbon, clay and matrix respectively (n-seclm). = Electromagnetic transit time in dolomite (ntpdol seclm) = Measured electromagnetic transit time (nseclm) = Formation capture cross-section from the Dual-Burst TDT tools (cu). = Effective porosity (pu) Lw, Lhc, Lei, Lma = Capture cross-sections of water, hydrocarbon, clay and matrix respectively (cu). = Capture cross-section in dolomite (cu) = Errors in the measured E log.

6 - Neustaeder,R.: Log Evaluation of Deep Ellenburger Gas Zones, SPE Symposium, Monahans Texas, March 1968, SPE-2071. 7 - Amin, AT., Watfa, M, and Awad, M.A: Accyrate Estimation of Water Saturation in Complex Carbonate Reservoirs. SPE 5th Middle East Oil Show, Bahrain 1987, SPE-15714.

o

8 - Abed, AF., and Watfa, M.: A Dynamic Myltjwell Data Journal of Petroleum Engineering, November 1988.

~

9 - Anderson, W.G.: WeUability Literature Survey-Part-3: The effects of Wettability on the Electrical Properties of Poroys Medja, Jour. Pet. Tech., Volume-39, pp. 13711379.

683

5

$PE. 21 41

~

Charll oblalned 'or:

"..z

aad .llal Equalloa:

__ '"

=

3

~= 1.0

II

SW2.2.

~ I: "-'II g 0:: '::l

;

c:

.5

.."

E

u

~

:s

'"

ii

rn

2

-=u :;:s .. E ..." 8

Open hole evaluation to obtain:

o -+-----i-------,;-----;------j-1.0

1.5

2..0

2..5

3.0

Program ror Variable

m & n

Computation

True Value of saturation Exponent (n ) Fig-1 : Charts showing the effects of errors in the computed values of Water saturation as a function of the variation the value of the saturation exponent ( n) from a value of 2.0.

IIJ Fig-3: Flow chart showing the variable m and n computations. Sxo convergence is necessary for accurate initial porosity and lithlogy computations.

o

-f------t-----.. . ? . ------i-------i1.0

1.5

2..0

2..5

3.0

True Value of Porosity Exponent (m ) Fig-2: Charts showing the effects of errors in the computed values of Water saturation as a function of the variation the value of the porosity exponent ( m ) from 2.0.

684

ELECTRO-MAGNETIC TRANSIT TIME

n sec/m :~ ~

.'/!

,30

CU

ELECTRO·MAGNETIC TRANSIT TIME

Rl (ohm·mj.

..

CAPTURE CROSSSECTION (I: ) CU.

,

DEEP RESISTIVITY 0.1

10000

SHALLOW RESISTIVITY Rxo (ohm·m) _

I

0.1

n sec/m

_

I·················

10000

···· ..

DEEP . RESISTIVITY

:~

!~~

'30

cu

CAPTURE CROSS· SECTION (I: ) CU.

Rl (ohm·m) 0.1

~

10000

SHALLOW RESISTIVITY Rxo (ohm-mj 611_~:.1 .. _.... _.. }!I~~~

.

>

~

".. I I 1\ II, I I

Ql

011 UI

.... I

I

I':-! 'l!

I

'fIJFig-4: An Example of variable m and variable n computations over a section of the Arab-D Reservoirs (track-3). The variatJon in n appears to be affected by dolomitization.

Fig-S: A second example of variable m and variable n computations over a section of the Arab-D Reservoir. The variation in nand m follows the same trends as those observed in fig.-4.

~

rn

N

.... ..£:-

..-

.W\

ELECTRO-MAGNETIC TRANSIT TIME

VARIABLE

n sec/m .2.5

!~I

130 I

n sec/m

.....

.

CAPTURE CROSSSECTION (1: ) CU.

ELECTRO·MAGNETIC TRANSIT TIME

n

130

51 0.1 10000 1........ • .... • .. -_ .... ·-_·

CU

n

.. _.....5

:~ !~,._ CAPTURE CROSSSECTION (1: ) CU.

VARIABLEm

.,HALLOW RESISTIVITY Rxo (ohm-m)

VARIABLE

cu

:SHALLOW RESISTIVITY Rxe (ohm-m) 5~ . .O:~ .... __ .... _!~~o. o.

VARIABLEm

,--> I

I

~~

Ql

~

r

<-

\ I

Fig-6: An Example of variable m and variable n computations over a section of the Arab-B and C Reservoirs. The computed values of m vary over a wide range (1.6-4). This reflects the complex pore geometry in this section of the Arab Formations. The variations in n remains In the same range as that of the lower Arab-D Reservoirs.

"'

.

'U)

'"Q

rn

t')

Fig.-7: A second Example of variable m and variable n computations over a section of the Arab-B and C Reservoirs, The trends in m and n are consistent with those of fig.-6.

.... J:-

..~

SPE 2 1 41

t::

3

-

........ cQ): c:

+---------+---------+---------+---------+--------_. 1 1 1 1 1 1

0

c. 2 >< w c:

0 ;;

ns

a.. ~

ns

en

1

2

2

6

521 4 2 2 6 5 4 4 +---------+-- 4 343 3 534 2 2 11 2 3 1 4 4 4 464 2 2 2 3 6 343 22634 17 5 5 8 5 4 632 527 45 4 612 4 9 95223 2 + 2 2---- +---------49 8 9121411 734 2 2 1 1 5011 7 7 7 6 6 16 2 2 1---------+---------+---------+---------+--------_· 1 1 1 1 1 1 1 1 1

o

20

40

60

80

100

Vdol (volume of dolomite): 0/0 Fig. 8: A multi-well cross-plot showing the variation of n with the volume of dolomite in the Arab-D Reservoirs. There is a trend of increasing n value with increase in the fraction of dolomite.

7.5 -

I 1 I I

6.0

I

I

I

I I I

I I I

I

3.0

6 4 8 5 10 9 812 12141912 +--\510 313 6 5 I 31116·64573 I 5 1012 6 5 8 I 4 +- 4 I I I

0.0 -

I I I I

I I I

4.5

1.5

I

I I I

6 7 8 5

3 5 5 36334 3 8 5 3 6 9 12 4 9

6

I

I

I I I

I I I

I

I

1

I 1

I I I

I

I

I

I

1

I I I

I I I

1 I I

I I I

I I I

I I I

I 1.0

I

1.4

I 1.8

I 2.2

I 2.6

I 3.0

Fig.-9: A multi-well cross-plot of the computed n value versus the volume of water ~.Sw) in the uninvaded zone. This plot shows a trend of decreasing value of n with the decrease in the volume of water.

687

2

2.5

t1.L =

Errors in measured L log

1.5

1I Iilil il l i~l lil i l ljli l li l :il

en C» C»

.................:.:.:.:.:.:....

0.5

o Fig-10 : Estimations of the errors in the computed values of' n as a function of water volume in the uninvaded zone (0SW) and errors in the measured value of formation capture cross-section. Errors from other log measurements are assumed to be zero. I

'tJ)

-tI

rn

N

..... .s:::-

.....

ill