PETROPHYSICS, VOL. 50, NO. 2 (APRIL 2009);; PAGE 102-­115;; 14 FIGURES
Connectivity Theory – A New Approach to Modeling Non-­Archie Rocks1
Bernard Montaron2
ABSTRACT
So-­called Archie rocks are characterized by a resistivity index (RI) versus water saturation that plots as a straight line on log-­log scale, the line’s slope EHLQJHTXDOWRín where n is the saturation exponent. All rocks for which the RI has curvature qualify as non-­Archie rocks. This paper presents a relationship (equation 6) that links the true formation resistivity Rt to water saturation Sw and formation porosity ij. This differs from Archie’s equation in its simplest form by the introduction of a small term called “water FRQQHFWLYLW\FRUUHFWLRQLQGH[´:&,Ȥw to correct for water connectivity effects. This term helps to stabilize INTRODUCTION
:KHQ*($UFKLHLQWURGXFHGKLVHTXDWLRQD
new era for the oil and gas industry began, where for WKH ¿UVW WLPH LW EHFDPH SRVVLEOH WR PDNH TXDQWLWDWLYH
hydrocarbon reserve estimates using resistivity and porosity logs. In its simplest form
Rw
,
( S wM ) where Rt is the resistivity of the uninvaded, virgin reservoir rock, Rw is the water (or brine) resistivity, and ijis the total porosity of the rock. Archie’s equation works remarkably well in “clean” water-­wet sandstone formations.
In more general form
R
Rt  n w m .
S wM
Rt 
the single exponent in the model called the conductivity exponent µ – equivalent to Archie’s equation when n = m. $PHWKRGWRGHULYHDQDO\WLFDOH[SUHVVLRQVIRUȤw for various formations is introduced. Effective medium theory is used ZLWKDPRGL¿HG&5,0PL[LQJODZWRPRGHOWKH:&,DVD
IXQFWLRQRIFRQGXFWLYHVWUXFWXUHVDQGÀXLGSDUDPHWHUVLQ
the rock. The model agrees with predictions of Waxman-­
Smits and Dual-­Water models for shaly sands (concave down curves) as well as experimental data obtained on strongly oil-­wet rocks (concave up curves). Connectivity theory also provides insight as to why so many rocks satisfy Archie’s model.
Archie’s equation requires the determination of two H[SRQHQWV WKH VRFDOOHG FHPHQWDWLRQ H[SRQHQW m and saturation exponent n. The value of water saturation Sw LV TXLWH VHQVLWLYH WR WKHVH H[SRQHQWV )RU H[DPSOH IRU D
relatively small-­sized oil reservoir using m = n LQVWHDG
RI DV LQ HTXDWLRQ HDVLO\ UHWXUQV VHYHUDO DGGLWLRQDO
billions of dollars in reserves. These two exponents are known to take different values for various reservoir rocks. This is especially true for carbonates, for which rock typing and pore geometry characterization are essential for petrophysical modeling. Indeed, in carbonates n can range IURPOHVVWKDQWRJUHDWHUWKDQDQGm can exceed 4 in some vuggy rocks. To make things worse, various rock types tend to be distributed in some carbonate formations with a high level of heterogeneity. Determining a suitable average value to use for n and m in such heterogeneous formations, in order to obtain accurate estimates for hydrocarbons in place, is quite a challenge.
0DQXVFULSWUHFHLYHGE\WKH(GLWRU1RYHPEHUUHYLVHGPDQXVFULSWUHFHLYHG)HEUXDU\
2ULJLQDOO\SUHVHQWHGDWWKH63:/$WK$QQXDO/RJJLQJ6\PSRVLXPKHOGLQ(GLQEXUJK6FRWODQG0D\
6FKOXPEHUJHU&RQYHQWLRQ7RZHU/HYHO32%R['XEDL8$((PDLOEPRQWDURQ#GXEDLRLO¿HOGVOEFRP
‹6RFLHW\RI3HWURSK\VLFVDQG:HOO/RJ$QDO\VWV$OO5LJKWV5HVHUYHG
102
PETROPHYSICS
April 2009
Connectivity Theory – A New Approach to Modeling Non-­Archie Rocks
Besides understanding the stability of Archie’s
exponents, or the lack thereof, it is also important to understand when Archie’s equation “works” for a given rock, and when it doesn’t work so well. One obvious IHDWXUHRIHTXDWLRQLVWKDWWKHUHVLVWLYLW\LQGH[SORWWHG
versus water saturation in log-­log scale must be a straight OLQHZLWKVORSHín/LQHVZLWKVORSHV±±DQG±DUH
VKRZQLQ)LJXUHLQEODFN7KHUHIRUHDVVRRQDVDVOLJKW
curvature is observed on the resistivity index function, Archie’s equation no longer applies. Such rocks are called “non-­Archie” rocks.
The blue curve is characteristic of rocks that tend to remain more conductive than “normal” when the water saturation is lowered. That is the case, for example, with shaly sands. The red curve is observed on strongly oil-­
ZHWURFNV$VLOOXVWUDWHGLQ)LJXUHLWLVQRWSRVVLEOHWR
GH¿QH D XQLTXH VDWXUDWLRQ H[SRQHQW RQ VXFK FXUYHV ,Q
this example the slope picked locally at different points DORQJWKHFXUYHWDNHVYDOXHVUDQJLQJIURP±WR±LH
from n WRn ZLWKn calculated using
DV SRVVLEOH LH WR SXW WKHP EDFN WR YLUJLQ UHVHUYRLU
conditions. The curve obtained when measuring
resistivity versus water saturation while slowly injecting crude oil into a water saturated core is called the “drainage curve”. The curve obtained while injecting water into an oil saturated core is the “imbibition curve”. These two curves can be quite different for some rocks, for example oil-­wet rocks. The model developed in this paper is intended to apply to virgin reservoir rocks and to drainage curves obtained from SCAL when such curves are considered representative of the reservoir.
PERCOLATION THEORY AND EFFECTIVE MEDIUM THEORY (EMT)
A tremendous amount of work was accomplished by SK\VLFLVWV LQ WKH V V DQG V WR XQGHUVWDQG
universal scaling laws for random percolation. See, for H[DPSOHWKHWH[WERRNV6WDXIIHUDQG$KDURQ\DQG
/DJHVDQG/HVQH7KHUHLVDQLQWHUHVWLQJSDUDOOHO
CONNECTIVITY THEORY – A NEW APPROACH TO MODELING
between percolation on random resistor networks and S w d (NON-­ARCHIE
d ln( RI )
RI )
transport properties of natural porous rocks. This link was ROCKS
n .
highlighted in a large number of publications in the last d ln( S w )
RI d ( S w )
)LJXUHV2QO\ \HDUV)LJXUHDLOOXVWUDWHVWKHVHVLPLODULWLHV
The objective of this article is to propose a theory to A porous rock can be modeled using a simple three-­
explain why Archie’s equation seems to work in many GLPHQVLRQDO'FXELFJULGZKHUHDSURSRUWLRQp of cells cases, and why it doesn’t for some rocks. The objective is EOXH DUH VHOHFWHG DW UDQGRP WR EH ¿OOHG ZLWK EULQH RI
also to propose a model that is applicable to non-­Archie conductivity ıw. The other cells (white) represent the non-­
rocks, and that reduces to Archie’s equation for the most conductive solid. The conductivity of the cube shown probable values of its parameters. LQ)LJXUHDPHDVXUHGEHWZHHQWZRRSSRVLWHPHWDOL]HG
The characterization of n and m is usually done on cores sides, is a function of the conductive paths linking the by performing special core analysis (SCAL). Cores two sides. The cube conducts electricity only if the
brought back to the lab from downhole are submitted to various treatments in order to “restore” them as much 1000
n = 10
n = 2
Resistivity Index R.I. 100
n = 4.2
n = 3
10
n = 2.7
n = 1
1
0.1
Water Saturation Sw
1
Fig.
Fig. 1 Resistivity index vs. Sw in log-­log scale. Archie’s Fig.
2a.2a 1XPHULFDO H[SHULPHQW IRU UDQGRP VLWH
SHUFRODWLRQRQDVLPSOHFXELFJULGîî2QO\EOXH
HTXDWLRQFRUUHVSRQGVWRVWUDLJKWOLQHVLQEODFNVORSHV±
Fig. 1.
cells are conductive and are here in proportion p ±±DQGGRHVQRWDSSO\WRFXUYHVEOXHDQGUHG
April 2009
PETROPHYSICS
103
Bernard Montaron
proportion p exceeds a percolation threshold pc which in WKHFDVHRIDVLPSOHFXELFJULGLVHTXDOWR7ZR
conductive cubes are connected only if they share a square side. The maximum coordination number of this FXELFQHWZRUNLVHTXDOWRWKHQXPEHURIVLGHVRIWKHFXEH
i.e., 6. If cubes are allowed to connect through corners, edges and sides the coordination number can be increased XSWRDQGWKHSHUFRODWLRQWKUHVKROGEHFRPHVVOLJKWO\
OHVVWKDQ
Of course the numerical modeling of rocks can be made PXFK PRUH UHDOLVWLF WKDQ WKH FXEH PRGHO LQ )LJXUH D
:HJHQHUDWHGWKHURFNPRGHOVKRZQLQ)LJXUHEXVLQJD
UDQGRPSURFHVVZLWKVSHFL¿FFRQVWUDLQWV,QWKLVFDVHWKH
URFNLVPDGHZDWHUZHWWKHFXELFFHOOVWKDWDUHLQFRQWDFW
with the surface of pores contain water.
*UDLQVDUHJURZQUDQGRPO\VWDUWLQJIURPD'GLVWULEXWLRQ
of solid nuclei. The growth process stops when the desired SRURVLW\ YDOXH LV UHDFKHG KHUH SHUFHQW7KH /DSODFH
partial differential equation can be solved on this model XVLQJDFRPSXWHU8QIRUWXQDWHO\WKLVDSSURDFKGRHVQRW
meet the objective of deriving analytical formulas for the conductivity of porous rocks.
5HWXUQLQJWRWKHVLPSOHFXEHPRGHORI)LJXUHDQHDUWKH
percolation threshold the conductivity is known to follow a power law with a critical conductivity exponent, usually denoted tYHU\FORVHWRLH
Fig. 2a.
t
§ p pc ·
V Vw ¨
¸.
© pc ¹
E\ %RULV .R]ORY DQG 0LFKHO /DJHV DW (63&, LQ 3DULV
SHUV FRPP t “ 7KH\ KDYH
DOVR FRQ¿UPHG WKDW WKH FULWLFDO UHJLRQ LQ ZKLFK WKH
universal behavior is valid is, however, very small.
In his fundamental paper “Percolation and Conduction” 6FRWW .LUNSDWULFN XVHG HIIHFWLYH PHGLXP WKHRU\
(07 DSSOLHG WR FRQGXFWLYLW\ RI PL[WXUHV WR ¿QG
expressions of the conductivity of a random resistor QHWZRUN RQ D VLPSOH ' FXELF JULG IRU ERQG DQG VLWH
percolation. Reformulated for the site percolation example JLYHQ LQ )LJXUH D WKH .LUNSDWULFN (07 FRQGXFWLYLW\
model takes the form
V |Vw p
pa
,
a
(4)
where aLVHTXDOWR
This model is valid for pFORVHWREXWLWDSSHDUVIURP
0RQWH&DUORFRPSXWHUVLPXODWLRQVWKDWLWLVDQH[FHOOHQW
approximation for p ! ,W LV LQWHUHVWLQJ WR QRWH WKDW
close agreement can be obtained between equation (4) P
and the power law
§ p b ·
V Vw ¨
¸ ,
© b ¹
by adjusting the exponent µ and the threshold b.
7KLV DJUHHPHQW LV VKRZQ LQ )LJXUH 7KH UHG FXUYH
ODEHOHGµ&RQQHFWLYLW\¶SORWVHTXDWLRQZLWKµ and b 7KHDJUHHPHQWLVEHWWHUWKDQSHUFHQWRI
1.0
According to numerical simulations performed recently Kirkpatrick
Connectivity
0.9
0.8
Conductivity
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
Fig. 2b.
1
F
ig. 33. An excellent match is observed between Fig.
Fig. 2b $PRUHHODERUDWHFXEHPRGHOZLWKîî
cubic cells. This plane section shows water in blue, oil in black and rock in gray.
104
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
p
.LUNSDWULFN(07FRQGXFWLYLW\PRGHOIRUVLWHSHUFRODWLRQ
on a cubic grid and a power law with an exponent equal WR
PETROPHYSICS
April 2009
Connectivity Theory – A New Approach to Modeling Non-­Archie Rocks
full scale for all values of pDERYH7KH¿WLVHYHQEHWWHU
LQWKHLQWHUYDO>@ZLWKµ DQGb ZLWK
DQHUURUOHVVWKDQSHUFHQWRIIXOOVFDOH
This example of the random cube model is interesting because fundamental physics have established the conductivity relationship for both ends of the pUDQJHLH
at the percolation threshold pc DQG FORVH WR 7KH WZR
HTXDWLRQVDUHGLIIHUHQWEXWLWKDSSHQVWKDWHTXDWLRQFDQ
EH¿WWHGWRERWKFDVHV2IFRXUVHWKHUHLVDELJGLIIHUHQFH
between the random cube model and real porous rocks. 'HVSLWHWKDWZHZLOOVKRZKHUHWKDWHTXDWLRQFDQEH
used as one key element of a consistent approach to derive analytical models for the conductivity of rocks.
(TXDWLRQVVLPLODUWRHTXDWLRQKDYHEHHQSURSRVHGE\
several authors to model the conductivity of natural URFNVIRUH[DPSOH=KRXHWDO0RQWDURQ
+XQWDQG.HQQHG\%HUJJLYHVDQ
H[FHOOHQW VXUYH\ RI YDULRXV (07V DSSOLHG WR UHVHUYRLU
rocks.
THE CONNECTIVITY EQUATION
,QHTXDWLRQWKHSURSRUWLRQp of conductive sites is HTXDO WR WKH SUREDELOLW\ RI ¿QGLQJ ZDWHU ± WKH RQO\
conductive phase in the medium – at a random point in the cube volume. Therefore, in natural porous reservoir rocks p is equal to the bulk volume fraction of water ij6w where ij is the total porosity of the rock, and Sw is the ZDWHUVDWXUDWLRQLHWKHIUDFWLRQRIWKHSRURVLW\RFFXSLHG
by water.
)RUUHDVRQVWKDWZLOOEHFRPHFOHDULQWKHUHVWRIWKHSDSHU
the parameter b is renamed Ȥw and called the “water connectivity correction index” or “water connectivity LQGH[´ :&, :H FDOO HTXDWLRQ ZULWWHQ LQ D IRUP
VLPLODUWRHTXDWLRQWKHFRQQHFWLYLW\HTXDWLRQLH
Rw '
where Rw ( S wM F w ) P
Rt
$VVKRZQLQ)LJXUHWKH:&,FRQWUROVWKHFXUYDWXUHRI
the resistivity index (RI) curve. The RI curve is concave down for negative values of the WCI and concave up RWKHUZLVH%\GH¿QLWLRQWKHUHVLVWLYLW\LQGH[LVWKHUDWLR
of Rt to the resistivity Ro of the rock fully water saturated. This can be expressed by
P
§ Sc ·
RI  ¨
¸ ,
© S w Sc ¹
by introducing the critical saturation Sc GH¿QHGE\
Sc  F w / M .
1000
V w S wM F w P where V w V w F w P D
April 2009
Xw = -­0.03
Xw = -­0.01
Xw = 0
Xw = +0.01
Xw = +0.03
(6)
The exponent µ is called the conductivity exponent. When modeling reservoir rocks with the connectivity equation WKHW\SLFDOUDQJHREVHUYHGIRUWKHH[SRQHQWLVWR
i.e., a range much reduced compared to the range observed for Archie exponents n and m. This stability has positive practical consequences. The parameter Ȥw (WCI) takes VPDOOYDOXHVW\SLFDOO\UDQJLQJIURPí”Ȥw”
Indeed the WCI is a corrective term that adjusts the water YROXPH IUDFWLRQ LQ HTXDWLRQV DQG WR DFFRXQW IRU
positive or negative water connectivity effects in the rock. Several examples of these effects are presented in the remainder of this article.
With such values for the conductivity exponent and the When Sc is positive (concave up curve) the resistivity takes very high values when the water saturation is reduced toward Sc )RU H[DPSOH LQ )LJXUH WKH FULWLFDO
saturation Sc FRUUHVSRQGVWRWKHVROLGUHGFXUYHZLWK
Ȥw 6XFKDQH[WUHPHEHKDYLRUFDQEHREVHUYHGLQ
strongly oil-­wet rocks with very high oil saturations (at OHDVWSHUFHQW
:KHQWKH:&,DSSURDFKHVWKHFRQQHFWLYLW\HTXDWLRQ
reduces to Archie’s equation with n = m = µ. This is 100
Resistivity Index R.I. V
Rw F w P WCI the resistivity Rw¶ LV DOPRVW DOZD\V ZLWKLQ ¿YH
percent of the true water resistivity Rw. The uncertainty on RwLVXVXDOO\ODUJHUWKDQWKLV¿JXUHDQGLWLVDFFHSWDEOH
for some calculations to simplify equation (6) by replacing Rw' by Rw.
(TXDWLRQ D LV WKH VDPH DV HTXDWLRQ EXW ZLWK
conductivities instead of resistivities. In some of the developments made later in this paper we will use the approximation
E
V | V w ( S wM F w ) P .
P = 2.0
I = 25%
10
Sc = 0.12
1
0.1
Water Saturation Sw
1
Fig. F4ig.Resistivity Index curves obtained from the 4.
connectivity equation with exponent µ 7KHSDUDPHWHU
Ȥw controls the curvature.
PETROPHYSICS
105
Bernard Montaron
an important feature of this model. The large amount of experimental evidence showing the applicability of Archie’s equation for many natural rocks makes it a requirement for any good petrophysical conductivity model to contain Archie’s equation as the normal case.
The Case of “Clean” Water-­Wet Sandstone Rocks
,Q WKH VLWH SHUFRODWLRQ PRGHO )LJXUH D ZDWHU LV
distributed at random in space. The distribution of water in naturally porous rocks is actually much more constrained HJDVVKRZQLQ)LJXUHE
)LUVWZDWHULVLQWKHSRUHVSDFHDQGWKHSRURVLW\KDVD
VSHFL¿FJHRPHWULFDOVWUXFWXUHZLWKVSDWLDOFRUUHODWLRQV)RU
example the porosity of sandstone results primarily of the stacking of sand grains having a certain size distribution resulting from erosion due to transport. The pore space has subsequently been submitted to the effect of the overburden pressure and cementation. A huge difference with the site percolation model is the fact that the porosity in sandstone rocks is almost entirely connected. Second, wettability is a self-­organizing mechanism that KDVDGUDPDWLFHIIHFWRQWKHFRQQHFWLYLW\RIÀXLGSKDVHV
LQVLGH WKH SRUH VSDFH )RU H[DPSOH ³FOHDQ´ VDQGVWRQH
rocks tend to remain strongly water-­wet even when exposed to crude oil at reservoir temperature and pressure conditions. The surface of pores is always covered by D FRQWLQXRXV ¿OP RI ZDWHU KRZHYHU VPDOO WKH ZDWHU
saturation may be. Reducing the water saturation results in DORZHUDYHUDJHWKLFNQHVVRIWKLV¿OPEXWWKHFRQQHFWLYLW\
of the water phase is essentially unchanged. Anderson SXEOLVKHGDYHU\GHWDLOHGWZRSDUWVXUYH\RIWKH
literature on wettability.
)RUDJLYHQURFNVDPSOHWKHVHWZRSURSHUWLHV±SRURVLW\
entirely connected and water-­wet pore surface – imply that the conductivity remains positive for any non-­zero value of the water volume fraction however small Sw may be. Therefore, in that case Ȥw must be negative or equal WR LQ HTXDWLRQ 7KH YDOXH Ȥw OHDGV WR$UFKLH¶V
equation that applies well to most “clean” sandstone UHVHUYRLUURFNV1RWHWKDWWKHWHUP³FOHDQ´KHUHUHIHUVWR
clay-­free rocks. The effect of clay will be discussed later in the article.
So even though there is a link between the connectivity equation and percolation physics, the WCI should not be interpreted in general as a percolation threshold. Ȥw or ScFDQEHLQWHUSUHWHGDVVXFKRQO\IRUYHU\VSHFL¿FFDVHV
such as – for example – strongly oil-­wet rocks provided there is no other mechanism in the rock enhancing the connectivity of the water phase. Other authors have made VLPLODUFRPPHQWV6HHIRUH[DPSOH=KRXHWDO
DQG .HQQHG\ ZKR FDOOV WKH :&, D ³SVHXGR´
percolation threshold (PPT).
106
THE MODIFIED CRIM MIXING LAW
0L[LQJODZVKDYHEHHQLQWURGXFHGLQHIIHFWLYHPHGLXP
theories to calculate a physical parameter of a mixture as a function of the volume fractions of the components
of the mixture and the values of the parameter for each component. 7KH &RPSOH[ 5HIUDFWLYH ,QGH[ 0HDVXUHPHQW &5,0
PL[LQJ ODZ ZDV LQWURGXFHG E\ %LUFKDN HW DO WR
model the complex conductivity of mixtures at high IUHTXHQF\ 6HH DOVR %HUU\PDQ $SSOLHG WR WKH
complex conductivity of a rock made of a mixture of k components with respective volume fractions x, …, xk &5,0WDNHVWKHIRUP
V  xV  xV  ...  xkV k ,
with
x  x   xk  This mixing law was shown to give excellent predictions when compared to dielectric measurements on rocks – 6HOH]QHY HW DO +RZHYHU LWV DSSOLFDWLRQ UDQJH
JRHVDOOWKHZD\WR]HURIUHTXHQF\'&)RUH[DPSOHDW
'&FRQVLGHUDSRURXVURFNPDGHRIDPL[WXUHRIVROLG
JUDLQVZDWHUDQGRLO
Then
V V  V w,
V x M x  S wM ,
.
x S w M
V  V w ( S wM )
$SSO\LQJHTXDWLRQZHJHWZKLFK
LV$UFKLH¶V HTXDWLRQ LQ LWV VLPSOHVW IRUP HTXDWLRQ 7KHWUHDWPHQWDERYHLVDQDwYHDSSOLFDWLRQRIWKH&5,0
mixing law which includes the assumption that water has the same connectivity throughout the rock and at all scales. The requirement for a good model to include Archie’s law as a limiting case leads naturally to consider D PRGL¿HG YHUVLRQ RI WKH &5,0 PL[LQJ ODZ ZKHUH WKH
H[SRQHQWLVUHSODFHGE\µ for a certain value of µ
V P  xV u  ...  xkV k u .
A much more general form that could be considered is
f (V )  x f (V )  ...  xk f (V k ),
for some continuous function, f (ı). This equation is HQWLUHO\V\PPHWULFDOLHDOOFRPSRQHQWVDUHWUHDWHGRQ
an equal footing – a property that is highly desirable when modeling random mixtures. To make physical sense it should be possible to change units for conductivity and HTXDWLRQVKRXOGVWLOOEHYDOLG,QRWKHUZRUGVIRUDQ\
constant c one must have
f (cV )  x f (cV )  ...  xk f (cV k ).
PETROPHYSICS
April 2009
Connectivity Theory – A New Approach to Modeling Non-­Archie Rocks
)XQFWLRQV VDWLVI\LQJ HTXDWLRQ IRU DOO c are called
‘homogeneous’ or ‘scale independent’. We have proved ± 0RQWDURQ ± WKDW DOO PDWKHPDWLFDO VROXWLRQV
RI WKLV IXQFWLRQDO HTXDWLRQ OHDG WR HTXDWLRQ LH
WR VDWLVI\ HTXDWLRQ IRU DOO c the function f must be of the form f (ı) = ı µ for some real number µ. This UHVXOW JLYHV HTXDWLRQ D YHU\ VSHFLDO VWDWXV LQ WKH
world of mixing laws because it is the only mixing law that is both symmetrical in its variables and KRPRJHQHRXV1RWHWKDWWKLVLQFOXGHVWKHOLPLWLQJFDVH
µ ĺ ’ ZKLFK FRUUHVSRQGV WR WKH JHRPHWULFDO PHDQ
lim  xV P
P of
 ...  xkV k P   V x ...V k xk .
P
The mixing law based on the geometrical mean is obtained with the function f (ı) = ln (ı) and is also called the Lichtenecker mixing law. The mixing law in equation ZDV ¿UVW LQWURGXFHG E\ /LFKWHQHFNHU DQG 5RWKHU
WRFDOFXODWHWKHFRPSOH[SHUPLWWLYLW\RIPL[WXUHV
VDQGUHVHUYRLUURFNPRGHOXVHGKHUHLVVKRZQLQ)LJXUH
6. The clay-­bound-­water has a much higher conductivity compared to the ‘free’ brine contained in the pores.
Therefore, clay must be considered as a separate phase in the model. To simplify only four phases are assumed to be present in the medium. (In principle, different types of clays, with different conductivities associated to each W\SHPLJKWEHGLVWLQJXLVKHG8VLQJ&ODYLHUHWDO
notation the bulk volume fraction of clay water is Scwij where ij is the total porosity, the conductivity of clay water is ıcw , the bulk volume fraction of ‘free’ water is (Sw-­ Scw)ij where Sw is the total water saturation, and ıw is the conductivity of ‘free’ water (brine). The other two phases – solid quartz grains and oil – have ]HURFRQGXFWLYLW\(TXDWLRQWKXVWDNHVWKHIRUP
V P
V
with
APPLICATION TO SHALY SANDS
1RZ DSSO\ HTXDWLRQ IRU WKH PRGHOLQJ RI VKDO\
sands. I use the experimental data published by Clavier HW DO LQ WKHLU SDSHU RQ WKH 'XDO :DWHU PRGHO
Clavier’s paper refers to the original work of Waxman DQG 6PLWV DQG :D[PDQ DQG 7KRPDV 6KRZQLQ)LJXUHLVDWKLQVHFWLRQRIDW\SLFDOH[DPSOH
of shaly sand. A schematic representation of the shaly u
ScwMV cw
( S w Scw )MV w u , Fw
V w ( S wM F w ) P ,
§ § V · P ·
ScwM ¨ ¨ cw ¸ ¸ ¨© Vw ¹
¸
©
¹
(TXDWLRQLVDQDSSUR[LPDWLRQRIHTXDWLRQEZLWK
WKH:&,SURYLGHGE\HTXDWLRQ6LQFHWKHFRQGXFWLYLW\
of clay water is higher than the conductivity of brine, Ȥw WDNHV QHJDWLYH YDOXHV WKHUHIRUH WKH 5, FXUYH LV
FRQFDYHGRZQ7KHFRUUHVSRQGLQJ'XDO:DWHUPRGHOLV
c
c
a
a
c
a
Grain
Clay
Water
Oil
Fig. 5 6(0SLFWXUHRIDFUHWDFHRXVVDQGVWRQHLQ7H[DV Fig. 6 This shaly sand model assumes that the medium Abundant pore lining authigenic chlorite (a) and patchy Fig. LVDPL[WXUHRISKDVHVJUDLQVFOD\ZDWHUDQGRLO7KH
6
SRUH ¿OOLQJ FKORULWH F FOD\ DUH YLVLEOH 7KH GDUN FRORU conductivity of the clay bound water is different from the between quartz grains is epoxy.
conductivity of free water.
April 2009
PETROPHYSICS
107
Bernard Montaron
V
§ S §V
··
V w S M ¨¨ cw ¨ cw ¸ ¸¸ Sw © V w
¹¹
©
n
w
m
$VSHU&ODYLHUHWDOWKHFOD\ZDWHUVDWXUDWLRQLV
equal to the product of the counterion concentration Qv and not by vQ GH¿QHG DV WKH FOD\ ZDWHU YROXPH
DVVRFLDWHGZLWKXQLWRIFOD\FRXQWHULRQVPHT&ODYLHU
HW DO SRLQW RXW WKDW LQ RUGHU WR REWDLQ FRUUHFW
values for the conductivity the parameter Qv used in WKHLUPRGHOLVUHTXLUHGWREHSHUFHQWKLJKHUWKDQWKH
true “chemical” counterion concentration measured in the lab. This “electrical” Qv – as they call it – also takes slightly different values for the Dual Water and
the Waxman-­Smits models. Regardless of which case is FRQVLGHUHGLWLVD¿WWLQJSDUDPHWHU
8VLQJWKHGDWDLQ&ODYLHUHWDOn m ij percent, ıw PKRPıcw PKRPDWƒ &WKH¿W
VKRZQLQ)LJXUHLVREWDLQHGE\DGMXVWLQJWKHFRQGXFWLYLW\
exponent to µ DQGWKHFRXQWHULRQFRQFHQWUDWLRQE\
SHUFHQW FRPSDUHG WR WKH 'XDO :DWHU PRGHO :LWK
these values the water connectivity index is equal to Ȥw ± DQG WKH FULWLFDO VDWXUDWLRQ Sc LV ± 7KH
PD[LPXPUHODWLYHHUURUEHWZHHQWKHWZRFXUYHVLV“
percent.
,W LVVKRZQKHUHKRZWKHPRGL¿HG&5,0PL[LQJODZ
FDQ EH XVHG WR FRPELQH IRXU GLIIHUHQW SKDVHV WR
model the conductivity of a shaly sand reservoir rock. The application of the mixing law leads directly to the connectivity equation and provides an analytical H[SUHVVLRQ RI WKH :&, DV D IXQFWLRQ RI IRUPDWLRQ
properties. The agreement between this model and Waxman-­Smits and Dual-­Water models is remarkable. 100.00
Rt -­ Ohm-­m
Dual Water
Connectivity Eq.
However, all three models use values of the counterion concentration that are not equal to measured chemical counterion concentrations. This is an indication that these models are incomplete. It is however, possible to develop a more sophisticated connectivity model that allows use of the true counterion concentration.
OIL-­WET CARBONATE ROCKS
A detailed approach for modeling oil-­wet rocks is
GHVFULEHG LQ 0RQWDURQ 6WDUW E\ FRQVLGHULQJ WKH
PRGHOVKRZQLQ)LJXUH7KHVSDFHFDQEHGLYLGHGXS
in ‘oil-­wet cells’ and ‘water-­wet cells’. Oil-­wet cells have SHUFHQW RI WKH VROLG VXUIDFH FRYHUHG E\ RLO :DWHU
ZHW FHOOV KDYH SHUFHQW RI WKH VROLG VXUIDFH FRYHUHG
by water. Presented here is a different way to partition the pore space, as compared to 0RQWDURQ 7KH RLOZHW FHOOV DUH VKRZQ LQ )LJXUH ZLWK UHG
boundaries. They are entirely contained in the porosity. The cells are polyedra constructed with respect to the center of gravity of surrounding grains. The water-­wet cells are the rest of the porosity with black boundaries.
The pore volume fraction of oil-­wet cells is xo and the SRUH YROXPH IUDFWLRQ RI ZDWHUZHW FHOOV LV í xo . $FFRUGLQJ WR WKHLU GH¿QLWLRQ ZDWHUZHW FHOOV DQG
RLOZHW FHOOV KDYH SRURVLW\ EXW WKH\ PD\ KDYH
GLIIHUHQW DYHUDJH ÀXLG VDWXUDWLRQV 6XFK D ³FOHDQ´
medium made only of water-­wet cells – shown in )LJXUH ± IROORZV WKH FRQQHFWLYLW\ HTXDWLRQ ZLWK D
:&, HTXDO WR ]HUR LH LW IROORZV $UFKLH¶V HTXDWLRQ
$PHGLXPPDGHRQO\RIRLOZHWFHOOV)LJXUHFDQQRW
conduct electricity unless the amount of water in the V\VWHP LV VXI¿FLHQW IRU LVRODWHG ZDWHU EOREV WR EHFRPH
connected. This medium has a clear percolation behavior with a percolation threshold for the water volume fraction Brine
10.00
Oil
Scw = 10% Vcw/Vw = 3 I = 20%
DW: n=1.77 m=1.85
Grains
CE: P=1.81 S’cw/Scw =1.055
1.00
0.10
Water Saturation Sw
1.00
Fig.
7 Comparison between the Dual Water model for Fig.
7
VKDO\VDQGVDQGWKHFRQQHFWLYLW\HTXDWLRQ7KH¿WLVGRQH
E\ DGMXVWLQJ FRXQWHULRQ FRQFHQWUDWLRQ SHUFHQW
108
Pore Surface Covered by:
Water
Oil
Fig. 8
Fig. 8 A simple model for oil-­wet porous rocks. The volume is considered a mixture of “water-­wet cells” (black boundaries) and “oil-­wet cells”(red boundaries).
PETROPHYSICS
April 2009
Connectivity Theory – A New Approach to Modeling Non-­Archie Rocks
equal to
F w  F w  Sc M .
Ȥw is the maximum possible value for the WCI and Sc is the maximum possible value for the critical saturation. Sw must be higher than Sc for the medium to EHFRQGXFWLYH)URPLQLWLDOUHVXOWVSUHVHQWHGLQ0RQWDURQ
LWVHHPVWKDWScLVRIWKHRUGHURISHUFHQWWR
percent. Oil-­wet natural rocks, even strongly oil-­wet rocks, DUH IDU IURP KDYLQJ SHUFHQW RI WKHLU SRUH VXUIDFH
covered by oil. In most cases the surface fraction covered by oil is actually very small – typically a few percent. The reason is that micro pores are generally fully water saturated and water-­wet. Only macro pores and a fraction of the meso pores can be oil-­wet. However, the vast majority of the pore surface area is in micro pores, while the majority of the pore volume is in macro and meso pores. This is a direct consequence of the pore size distribution. The medium described above is a mixture of three phases (solid grains, oil-­wet cells (ow), water-­wet FHOOVZZZLWKWKHIROORZLQJSDUDPHWHUV
V sol V ow | V w ( S wow Sc ) P ,
V ww V w S www P xsol M xow xoM ,
xww
7KLV UHVXOW LV FRQVLVWHQW ZLWK 0RQWDURQ ZLWK D
different space partitioning method. It can be shown that xo remains identical in both methods. $FFRUGLQJWRHTXDWLRQDPHGLXPZLWKSHUFHQW
SRURVLW\DQGDQRLOZHWEXONYROXPHIUDFWLRQRISHUFHQW
DV GH¿QHG LQ )LJXUH ZRXOG KDYH D :&, RI WKH RUGHU
RIDQGDFULWLFDOVDWXUDWLRQScHTXDOWRSHUFHQW
Such a large positive value of the WCI would create a pronounced concave up curvature for the RI curve. That W\SH RI H[WUHPH EHKDYLRU FDQ EH FUHDWHG DUWL¿FLDOO\ RQ
carbonate cores.
$IDPRXVGDWDVHWIURP6ZHHQH\DQG-HQQLQJVLV
VKRZQ LQ )LJXUH LQ FRPSDULVRQ WR WKH FRQQHFWLYLW\
HTXDWLRQ7KH5,FXUYHVVKRZQDUHEDVHGRQHTXDWLRQ
with various values of the critical saturation Sc. The black straight line corresponds to Archie’s equation (Sc DQG ¿WV ZHOO WKH ZDWHUZHW GDWD SRLQWV 7KH GRWWHG DQG
solid black curves correspond to positive values of Sc XS WR 6ZHHQH\ DQG -HQQLQJV XVHG URFNV IURP WKH
same carbonate formation for these experiments. The black dots were obtained on a rock rendered completely water-­wet using a chemical treatment. The blue and red dots correspond to the same rock type made strongly oil-­
wet. The blue and red dots match nicely the connectivity xo M The total water saturation is equal to
Sw
xo S wow xo S www $SSO\LQJ HTXDWLRQ WR WKH FRQGXFWLYLWLHV DQG
volume fractions of the three phases leads to the connectivity equation with a WCI equal to
F w  xo Sc M .
Fw
Fw
F w ! Fig. 9 7KHZDWHUZHWPHGLXPIROORZV$UFKLH¶VODZ
$ SHUFHQW RLOZHW PHGLXP H[KLELWV
w a large positive percolation threshold Ȥwo. In these PHGLDSRURVLW\LVRQO\SRUHERXQGDULHVDUHSUHVHQW
Fig.
9.Ȥ
i.e., April 2009
Fig. 10 'DWD IURP 6ZHHQH\ DQG -HQQLQJV compared to the connectivity equation. The same exponent µ FDQEHXVHGWRPRGHODURFNPDGHHLWKHU
water-­wet or strongly oil-­wet.
PETROPHYSICS
109
Bernard Montaron
equation with Sc FORVH WR EOXH GRWV DQG UHG
dots). It is remarkable here that in all cases the same conductivity exponent µ LV XVHG 0RGHOLQJ WKH
black dots with Archie’s equation requires n IRUWKH
black dots, n IRUWKHUHGGRWVDQGn IRUWKH
blue dots with a poor match in this last case due to the curvature that cannot be represented by Archie’s equation.
Archie Exponents Revisited
One way the saturation exponent n can be related to the parameters of the connectivity equation is by equating HTXDWLRQWRWKH5,GHULYHGIURP$UFKLH¶VHTXDWLRQ
OQ S w Sc OQ Sc nP
.
D
ln S w
8VLQJ WKH H[DPSOH VKRZQ LQ )LJXUH IRU Sc HTXDWLRQDJLYHVn DWSw DQGn DWSw just above the critical saturation. Another method is to use the logarithm derivative
Sw
d ln( RI )
n P
.
E
d ln S w
S w Sc
(TXDWLRQEJLYHVWKHORFDOVORSHn at the point (RI, Sw (TXDWLRQ D JLYHV WKH VORSH RI WKH OLQH MRLQLQJ
WKH SRLQW WR WKH SRLQW RI, Sw 1RWH WKDW IRU ERWK
expressions the limit value of n at Sw LV µ Sc). )RU VPDOO YDOXHV RI _Sc_ HTXDWLRQ E ZLWK Sw gives a good approximation of the best matching n for $UFKLH¶VODZ6LPLODUO\RQHFDQ¿QGWKHH[SUHVVLRQIRUm 0RQWDURQ
OQ Sc OQ M Sc mP
.
ln M
of the pore structure of carbonate rocks and of the FRQQHFWLYLW\RIZDWHULQWKHVHV\VWHPV6ZDQVRQ
ZDVRQHRIWKH¿UVWWRXQGHUOLQHWKHLQÀXHQFHRIPLFUR
porosity on the conductivity of rocks. Petricola et al. have proposed a mixing model for micritic carbonate rocks based on a partition of the pore size GLVWULEXWLRQ LQ JURXSV PLFUR PHVR DQG PDFUR SRUHV
This type of partitioning has been widely adopted by the industry for carbonate petrophysical modeling. Considered here is a model for micritic carbonates based RQ WKLV SDUWLWLRQ 6HQ HW DO KDYH VKRZQ WKDW WKH
FRQGXFWLYLW\H[SRQHQWIRUVSKHUHSDFNLQJVLVHTXDOWR
This is also the value known for micrite crystal packings IRXQGLQPLFULWLFFDUERQDWHURFNVDVVKRZQLQ)LJXUH
The size of these crystals is limited to a few microns. The sub-­micron pores are so small that very high capillary
pressure is required for oil to penetrate them, so high in fact that micritic grains generally remain fully water saturated and perfectly water-­wet, at least in formation layers located not too far above the oil-­water contact. The low value of the conductivity exponent for micritic grains LVGXHWRWKHKLJKFRQQHFWLYLW\RIWKHZDWHU¿OPVLQWKHVH
micro-­pores.
$UHDVRQDEOHPRGHOIRUPLFURSRUHVLVxm = bulk volume IUDFWLRQ RI PLFULWLF JUDLQV ijm = porosity inside micritic
)RUH[DPSOHDVVXPLQJSHUFHQWSRURVLW\µ DQG
Sc WKHQ m 7KH KLJK VWDELOLW\ RI WKH
conductivity exponent and the fact that it does not depend on Sw makes the connectivity equation model simpler to use than Archie’s equation.
MIXED-­WET MICRITIC CARBONATES The simple model presented above for oil-­wet rocks leads systematically to a concave up curvature for the RI curve. The connectivity of water is always reduced by the presence of oil-­wet surfaces that FUHDWH ³FXWV´ LQ WKH FRQWLQXLW\ RI WKH ZDWHU ¿OP
However, some mixed-­wet carbonate rocks do not have RI plots with concave up curvature. In fact some mixed-­
wet carbonate rocks nicely follow Archie’s law (no curvature) and some can even present a concave down FXUYDWXUH VLPLODU WR VKDO\ VDQGVWRQHV 7KH H[SODQDWLRQ
of this apparent paradox requires a detailed analysis 110
Fig.
1111.
Thin section of a micritic limestone from the Fig.
Thamama formation (Abu-­Dhabi). The oil patches were added to the picture. Grains are made of tight packings of fully water saturated micrite crystals.
PETROPHYSICS
April 2009
Connectivity Theory – A New Approach to Modeling Non-­Archie Rocks
JUDLQVµm FRQGXFWLYLW\H[SRQHQWRIPLFULWLFJUDLQV
Swm IXOO\ ZDWHU VDWXUDWHG DQG Ȥwm $UFKLH¶V
equation). The conductivity of micritic grains is V m  V wMm .
7KHRLOLVFRQWDLQHGLQPHVRDQGPDFURSRUHVLHSRUHV
other than pores located in micritic grains. With this VLPSOL¿HG PRGHO WKH PHVRPDFUR SRURVLW\ ijM must be HTXDO WR RU ORZHU WKDQ ± xm . The remaining volume IUDFWLRQ ± xm – ijM corresponds to solid grains (e.g., dolomite crystals or large calcite crystals) that may be present. The total porosity is ij = xmijmijM. 0HVRPDFURSRUHVKDYHZDWHUZHWDQGRLOZHWVXUIDFHV
7KHVDPHGH¿QLWLRQVDVLQ)LJXUHDQG)LJXUHDUHXVHG
WRGH¿QHWKHPHVRPDFURSRUHYROXPHIUDFWLRQxo of oil-­
ZHWFHOOV)LQDOO\WKLVPRGHOLVDPL[WXUHRIIRXUSKDVHV
PLFULWLFJUDLQVRLOZHWFHOOVZDWHUZHWFHOOV
and (4) solid grains, i.e., grains with no micro-­porosity. Its parameters are
x  xm ,
x  xoM M ,
x xo M M x4 xm M M V  V wMm P ,
V | V w ( S wow Sc ) P ,
V  V w ( S www ) P ,
V4  m
ZKHUHWKHDSSUR[LPDWLRQJLYHQE\HTXDWLRQELVXVHG
for the conductivity, ı, of oil-­wet cells. Also assume that the exponent µLQWKHPL[LQJODZJLYHQE\HTXDWLRQ
is the exponent of the meso-­macro pore network. Water saturations in oil-­wet and water-­wet cells are Swow and Swww. The total water volume fraction is equal to
S wM
xoM M S wow xo M M S www xmMm 8VLQJWKHH[SUHVVLRQIRUSwijDERYHLQHTXDWLRQOHDGV
WRWKHFRQQHFWLYLW\HTXDWLRQEZLWK
F w | xm (Mm Mmum / m ) xo Sc M M .
(TXDWLRQFDQEHXVHGLQHTXDWLRQDWRJHWWKH¿QDO
expression for the conductivity of the medium.
Numerical Example
With ijm DQGWKHEXONYROXPHIUDFWLRQRIPLFULWLF
grains xm WKHWRWDOPLFURSRURVLW\LVxmijm Assuming a total porosity of the rock sample ij the meso-­macro porosity is ijM DQGWKHYROXPH
IUDFWLRQIRUVROLGJUDLQVLV8VLQJµ µm Sc DQGxo WKHQȤw LHGHVSLWHWKHIDFW
WKDWSHUFHQWRIWKHPHVRPDFURSRUHYROXPHLVRLOZHW
the presence of a large amount of micrite compensates exactly the effect of oil-­wetness and Archie’s equation April 2009
applies beautifully. Of course the value µ ZDV
chosen here on purpose to get Ȥw LQRUGHUWRPDNHWKH
point.
8VLQJRw RKPPDQGWKHQXPHULFDOGDWDDERYHWKH
resistivity of this formation is Rt RKPP7KHZDWHU
contained in the micro-­pores is non-­movable, and the oil
VDWXUDWLRQLQWKHPHVRPDFURSRUHYROXPHLVSHUFHQW
and even higher in macro pores. This is a typical example of low resistivity pay (LRP) carbonate formation such as described by 3HWULFROD
Discussion: Archie or Not Archie? The model presented above for mixed-­wet micritic FDUERQDWHV LV D VLPSOH PRGHO )RU H[DPSOH LW GRHV
QRW DFFRXQW IRU YXJV WKDW PD\ EH SUHVHQW LQ VXI¿FLHQW
TXDQWLWLHVWRFKDQJHVLJQL¿FDQWO\WKHSHWURSK\VLFDOPRGHO
Vugs could have been accounted for by adding another term in the WCI resulting from the addition of a “vug component” in the mixing model. +RZHYHU HTXDWLRQ LV D TXDQWLWDWLYH PRGHO WKDW
greatly helps understand water connectivity effects in these rocks. The presence of oil-­wet surfaces in meso-­
macro pores reduces the connectivity of water and the FRUUHVSRQGLQJ:&,WHUPLVHTXDOWRLQHTXDWLRQ
6XFKDODUJH:&,VKRXOGJHQHUDWHDYHU\SURQRXQFHG
concave up curvature if grains had no micro-­porosity. But the presence of micro-­porosity in micritic grains and the fact that the conductivity exponent of micrite crystal packings is
only µm JHQHUDWHVD:&,WHUPWKDWLVQHJDWLYH,QWKH
numerical example chosen above the two terms compensate exactly and Archie’s equation ‘works’ with n = m :LWKGLIIHUHQWQXPEHUVDOOSRVVLEOHFXUYHVVKRZQLQ)LJXUH
FDQEHREWDLQHG&RQFDYHXSRUGRZQDQGVWUDLJKWOLQHV
The conductivity of micritic carbonates can behave like shaly sandstones. They can also behave like strongly oil-­wet carbonates if, for example, the micrite volume IUDFWLRQLVQRWVXI¿FLHQWWRFRPSHQVDWHWKHHIIHFWRIRLO
wetness. And they can also follow Archie’s equation in all situations where Ȥw LV FORVH WR HJ LQ WKH UDQJH
í”Ȥw”
The fact that Archie exponents n and m, as well as the conductivity exponent µ DUH JHQHUDOO\ FORVH WR LV
FRUUHODWHG ZLWK WKH &5,0 PL[LQJ ODZ H[SRQHQW This also has to do with the critical conductivity exponent IRU'VLWHSHUFRODWLRQ±HTXDOWR±DQGWKH.LUNSDWULFN
(07PRGHO±DSRO\QRPLDORIGHJUHH
Application to a Middle East Carbonate Reservoir
When SCAL data is unavailable, or unconvincing, a good VWDUWLVWRDSSO\HTXDWLRQRUHTXDWLRQZLWKµ WR DQG WR XVH ORJ GDWD WR GHWHUPLQH WKH :&, 7KLV
PETROPHYSICS
111
Bernard Montaron
SURFHGXUH ZDV XVHG RQ D GDWD VHW IURP D 0LGGOH (DVW
carbonate reservoir.
The water saturation, the porosity and the resistivity of the formation must be known in order to calculate Ȥw from HTXDWLRQ 8QIRUWXQDWHO\ WKH PHWKRGV DYDLODEOH WR
estimate water saturation independently of deep resistivity UHO\RQLQVWUXPHQWUHVSRQVHVLQÀXHQFHGE\GULOOLQJLQGXFHG
changes in formation properties. The deepest response is from the neutron capture sigma tool with about a 6-­inch
depth of investigation. At “wireline time” the near-­
wellbore formation logged has probably been completely LQYDGHGZLWKPXG¿OWUDWH-LD%XFNOH\DQG0RUURZ concluded that surface-­active components used in water based muds adsorb on clays and are removed from the ¿OWUDWHE\IRUPDWLRQRIWKHPXGFDNH&KDQJHVLQZHWWDELOLW\
GXHWRPXG¿OWUDWHLQYDVLRQDUHWKHUHIRUHH[SHFWHGWREH
minor in carbonate rocks drilled with water based muds. On the other hand, wettability alteration would be severe for oil-­based muds due to oil-­wetting surfactants used in WKHVH ÀXLGV )RUWXQDWHO\ PRVW FDUERQDWH URFNV DW OHDVW
LQ WKH 0LGGOH (DVW DUH GULOOHG ZLWK ZDWHU EDVHG PXGV
The water saturation was provided by an EPT tool. This high frequency tool measures the dielectric constant of the formation that is a direct function of the water saturation. This very shallow measurement was combined with the resistivity provided by an Rxo tool. The WCI was calculated using the equation
F w | S xoM ( Rmf / Rxo ) P ,
0.09
0.08
Cw ( P = 1.90 )
0.07
0.06
0.05
0.04
Fw 0.03
0.02
0.01
0.00
-­1.0
Oil-Wet
-­0.5
0.0
0.5
Normalized Amott-­Harvey Index
1.0
Water-Wet
Fig.
Fig.12
12.1RUPDOL]HG $PRWW+DUYH\ ZHWWDELOLW\ LQGH[
PHDVXUHPHQW RQ FRUHV FRPSDUHG WR WKH :&, GHULYHG
IURPZLUHOLQHORJVLQD0LGGOH(DVWFDUERQDWHUHVHUYRLU
A good correlation is observed.
112
where Rmf LV WKH UHVLVWLYLW\ RI WKH PXG ¿OWUDWH NQRZQ
accurately, and where µ The Ȥw and ScORJVDUHVKRZQLQ)LJXUH,QRUGHUWR
highlight oil-­wet zones the intervals with negative WCI or Sc are truncated. Only positive spikes are shown. The depth is increasing from left to right. The light blue curves DUH WKH IW VOLGLQJ DYHUDJHV IRU HDFK SDUDPHWHU DQG
show a normal trend, with high values towards the top of the reservoir and low values at the bottom of the reservoir showing a water-­wet zone close to the oil-­water contact.
The logs show a layered reservoir structure with strongly positive WCI zones alternating with mixed-­wet zones and some layers that appear to be water-­wet with Ȥw close to zero at the top of the zone, in the middle and at the bottom. A complete suite of logs was run in this well and DQ(/$1LQWHUSUHWDWLRQPDGH
)LJXUH VKRZV WKH DJUHHPHQW EHWZHHQ WKH (/$1
and the Sc ORJV 7DU ]RQHV ZHUH LGHQWL¿HG LQ (/$1
E\ DVVHVVLQJ QRQPRYDEOH K\GURFDUERQV XVLQJ 105
measurements independent of the ones used for the determination of Ȥw and Sc.
Tar zones are known to be strongly oil-­wet. The tar mat DW [ IW LGHQWL¿HG ZLWK (/$1 FRUUHODWHV SHUIHFWO\
with the WCI positive spike shown by the Sc log. The WLJKW OLPHVWRQH EHG DW [ IW FRQWDLQV RQO\ ZDWHU DQG
correlates with a water-­wet marker on the Sc and Ȥw logs. All the apparently water-­wet zones at the bottom of the UHVHUYRLU DUH LQGHHG FRQ¿UPHG WR FRQWDLQ PRVWO\ ZDWHU
E\ (/$1 7KH DSSDUHQWO\ ZDWHUZHW OD\HUV DW WKH WRS
of the reservoir actually have negative values of Ȥw
DSSUR[LPDWHO\±7KLVUHTXLUHVDQH[SODQDWLRQ
A comparison of the Sc log with an ECS log (nuclear VSHFWURVFRS\WRROIRUPLQHUDORJ\LGHQWL¿FDWLRQLVVKRZQ
LQ)LJXUH$OOWKHOD\HUVZLWKVOLJKWO\QHJDWLYHȤw agree perfectly with anhydrite layers. This is easy to understand IURPHTXDWLRQ7KHSRURVLW\RIDQK\GULWHLV]HURRmf ZDVDSSUR[LPDWHO\RKPP$OWKRXJKWKHUHVLVWLYLW\
of anhydrite is expected to be extremely high – i.e. mega-­
ohm-­m – Rxo being a very shallow measurement measured RQO\RKPPGXHWRWKHLQÀXHQFHRIWKHVDOW\PXGLQ
the borehole. We should have found Ȥw LQIURQWRIHDFK
anhydrite layer, but instead we found the value provided E\HTXDWLRQLH±½ ±7KLVVKRZV
that a zero value for Ȥw or Sc can also correspond to a very tight zone with zero porosity. In practice this could be taken care of by applying a cut-­off to ignore anhydrite layers.
)LJXUHVKRZVDOOWKHRWKHUWDUOD\HUVLGHQWL¿HGZLWK
(/$17KH\DOOPDWFKSRVLWLYH:&,DQGScVSLNHV)LYH
cores from this well were tested for wettability in the lab XVLQJWKH$PRWW+DUYH\PHWKRGVHH$QGHUVRQKLV
second citation). A good linear correlation was obtained between Ȥw and the normalized Amott-­Harvey index PETROPHYSICS
April 2009
Connectivity Theory – A New Approach to Modeling Non-­Archie Rocks
Measurements: Sxo from EPT, Rxo, Phi_T (Rhob + ECS) Rmf = 0.024 ohm-­m
P = 2
0.15
Critical Water Fraction Cw
Xw Log (Water Connectivity Index) 0.10
0.05
0.00
9200.0
9300.0
9400.0
9500.0
9600.0
9700.0
10000.0
10100.0
10200.0
10300.0
10400.0
10500.0
;;;
; 9800.0
; 9900.0
;;; ;;
;;;;
1
Depth (ft) 0.60
Critical Saturation Sc
Sc Log
0.50
0.40
0.30
0.20
0.10
0.000
1
9200.0
9300.0
9400.0
9500.0
9600.0
9700.0
10000.0
10100.0
10200.0
10300.0
10400.0
10500.0
;;;
;;
; 9800.0
; 9900.0
;;;;
;;; 100-­ft Sliding Average
Depth (ft)
Fig. 13.
Fig. 13 )LUVWȤ
and ScORJV/LJKWEOXHFXUYHVDUHIWVOLGLQJDYHUDJHVRIȤw and Sc7KHWUHQGLVQRUPDO2LOZHWDW
w
the top of the reservoir and water-­wet at the bottom. The highest ScVSLNHVWRSDWDERXW2QO\WKHSRVLWLYHYDOXHVRI
Ȥw and Sc are shown to highlight oil-­wet zones
with simple water-­based muds.
1
0
1
Anhydrite
Tar
Critical Saturation Sc
0.60
0.50
0.40
0.30
0.20
0.10
0.000
x200 x300 x400 x500 x600 1
9200.0
9300.0
9400.0
9500.0
9600.0
x700 x800 x900 x000 x100 x200 x300 x400 x500
9700.0
9800.0
9900.0
10000.0
10100.0
10200.0
10300.0
10400.0
10500.0
Depth (ft)
Fig. 14.
Fig. 14 7KHWDUPDWDW[IWLGHQWL¿HGZLWK(/$1FRUUHODWHVSHUIHFWO\ZLWKWKH:&,VSLNHVKRZQE\WKHS
log. The c WLJKWOLPHVWRQHEHGDW[IWFRQWDLQVRQO\ZDWHUDQGFRUUHODWHVZLWKDZDWHUZHWPDUNHURQWKHȤw and Sc logs. All the WDUEHGVLGHQWL¿HGE\(/$1FRUUHODWHZLWKRLOZHWVSLNHVRQWKHȤw and Sc logs. Anhydrite beds detected by the ECS tool correspond to slightly negative values of Ȥw (and Sc ) due to borehole effects on Rxo measurements.
April 2009
PETROPHYSICS
113
Bernard Montaron
using µ )LJXUH7KLVFRUUHODWLRQLVZHOOLQOLQH
ZLWK WKH PRGHO RU WKH PRUH HODERUDWH PRGHO expressing the WCI as a linear function of the oil-­wet volume fraction in the rock.
CONCLUSION
The connectivity equation provides a nice and practical DOWHUQDWLYHWR$UFKLH¶VHTXDWLRQ,WLVDVLPSOHHTXDWLRQ
Like Archie’s equation it has only two petrophysical parameters – the conductivity exponent and the water connectivity index. It reduces to Archie’s equation in the limit ȤmĺDQGLWDOORZVWKHPRGHOLQJRIQRQ$UFKLH
rocks such as strongly oil-­wet rocks and shaly sands. I have shown that the connectivity equation is consistent ZLWKWKHPRGL¿HG&5,0PL[LQJODZ
7KH FRQQHFWLYLW\ HTXDWLRQ DQG WKH PRGL¿HG &5,0
mixing law are the two main analytical tools used in the theory of connectivity. A porous formation is described
DV D PL[WXUH RI YDULRXV VROLG DQG ÀXLG SKDVHV HDFK
phase being characterized by its volume fraction in the medium and its electrical conductivity. This approach accounts for variations of water connectivity among the different phases. The method was successfully aplied to develop analytical models for the conductivity equation of rock types as different as shaly sands, oil-­wet rocks,
and mixed-­wet micritic carbonates. All these rocks were shown to follow the connectivity equation. The conductivity exponent in this model seems much more stable than Archie exponents n and m. The variability in the connectivity theory is transferred to the water connectivity index (WCI). However, this parameter can be expressed analytically as a function of rock properties such as wettability, clay content or the pore structure of rocks.
114
REFERENCES
$QGHUVRQ :* :HWWDELOLW\ /LWHUDWXUH 6XUYH\ ±
3DUW 5RFN2LO%ULQH ,QWHUDFWLRQV DQG WKH (IIHFWV RI
&RUH+DQGOLQJRQ:HWWDELOLW\63(
$QGHUVRQ :* :HWWDELOLW\ /LWHUDWXUH 6XUYH\ ±
3DUW:HWWDELOLW\0HDVXUHPHQW63(
$UFKLH*((OHFWULFDO5HVLVWLYLW\/RJDVDQ$LGLQ
'HWHUPLQLQJ 6RPH 5HVHUYRLU &KDUDFWHULVWLFV $,0(
7UDQVS
%HUJ &5 $Q (IIHFWLYH 0HGLXP $OJRULWKP IRU
Calculating Water Saturations at Any Salinity or )UHTXHQF\*HRSK\VLFV9RO1RSS((
%HUU\PDQ -* 0L[WXUH 7KHRULHV IRU 5RFN
Properties, Rock Physics and Phase Relations – A KDQGERRNRI3K\VLFDO&RQVWDQWVSS±
%LUFKDN - *DUGQHU / +LSS - DQG 9LFWRU - High dielectric constant microwave probes for sensing VRLOPRLVWXUH,(((3URFHHGLQJVYROSS
&ODYLHU & &RDWHV * DQG 'XPDQRLU - Theoretical and Experimental Bases for the Dual-­Water 0RGHOIRU,QWHUSUHWDWLRQRI6KDO\6DQGV63(-RXUQDO
SS
+XQW$*3HUFRODWLRQ7KHRU\IRU)ORZLQ3RURXV
0HGLD6SULQJHU
-LD'%XFNOH\-6DQG0RUURZ15$OWHUDWLRQ
RI:HWWDELOLW\E\'ULOOLQJ0XG)LOWUDWHV3UHVHQWHGDW
WKH6HSW6&$6\PSRVLXP1RUZD\
.HQQHG\ ' 7KH 3RURVLW\:DWHU 6DWXUDWLRQ
&RQGXFWLYLW\5HODWLRQVKLS$Q$OWHUQDWLYHWR$UFKLH¶V
0RGHO3HWURSK\VLFV9RO1RSS
.LUNSDWULFN63HUFRODWLRQDQG&RQGXFWLRQ5HYLHZV
RI0RGHUQ3K\VLFV9RO1RSS
/DJHV0DQG/HVQH$Invariances d’Echelle – Des changements d’états à la turbulence, Belin Editions.
/LFKWHQHFNHU . DQG 5RWKHU . 'LH +HUOHLWXQJ
GHV ORJDULWKPLVFKHQ 0LVFKXQJVJHVHW] HV DXV
allgemeinen Prinzipien der stationären Strömung, 3K\VLNDOLVFKH=HLWVFKULIWYSS
0RQWDURQ%3HUFRODWLRQ7KHRU\&DUERQDWH5RFN
Petrophysics, and Wettability Logging. Schlumberger 5HVHUYRLU6\PSRVLXP&DPEULGJH8.
0RQWDURQ%$4XDQWLWDWLYH0RGHOIRUWKH(IIHFW
of Wettability on the Conductivity of Porous Rocks. 63(
0RQWDURQ%7KH0DWKHPDWLFVRI0L[LQJ/DZV
,QWHUQDO QRWH WR WKH $SSOLHG 0DWKHPDWLFV 6,*
(Schlumberger Eureka).
3HWULFROD 0 7DNH]DNL + DQG $VDNXUD 6 6DWXUDWLRQ(YDOXDWLRQLQ0LFULWLF5HVHUYRLUV5DLVLQJWR
WKH&KDOOHQJH63(
6HOH]QHY1%R\G$DQG+DEDVK\7'LHOHFWULF
0L[LQJ/DZVIRU)XOO\DQG3DUWLDOO\6DWXUDWHG&DUERQDWH
PETROPHYSICS
April 2009
Connectivity Theory – A New Approach to Modeling Non-­Archie Rocks
5RFNV63:/$WK$QQXDO/RJJLQJ6\PSRVLXP
6HQ 31 6FDOD & DQG &RKHQ 0+ $ 6HOI
6LPLODU0RGHOIRU6HGLPHQWDU\5RFNVZLWK$SSOLFDWLRQ
WR WKH 'LHOHFWULF &RQVWDQW RI )XVHG *ODVV %HDGV
*HRSK\VLFV9RO1RSS
6WDXIIHU ' DQG $KDURQ\ $ ,QWURGXFWLRQ WR
Percolation Theory. CRC Press. Revised Second Edition.
6ZDQVRQ%)0LFURSRURVLW\LQ5HVHUYRLU5RFNV±
,WV0HDVXUHPHQWDQG,QÀXHQFHRQ(OHFWULFDO5HVLVWLYLW\
63:/$th Annual Logging Symposium.
6ZHHQH\ 6$ DQG -HQQLQJV +< 7KH (OHFWULFDO
Resistivity of Preferentially Water-­Wet and Preferentially Oil-­Wet Carbonate Rock, Producers 0RQWKO\1RSS
:D[PDQ 0+ DQG 6PLWV /-0 (OHFWULFDO
Conductivity in Oil-­Bearing Shaly Sands, SPE-­J, June SS7UDQV$,0(
:D[PDQ 0+ DQG 7KRPDV (& (OHFWULFDO
Conductivities in Shaly Sands – I. The Relation Between +\GURFDUERQ6DWXUDWLRQDQG5HVLVWLYLW\,QGH[,,7KH
7HPSHUDWXUH &RHI¿FLHQW RI (OHFWULFDO &RQGXFWLYLW\
63(-37)HESS
=KRX'$UEDEL6DQG6WHQE\(+$3HUFRODWLRQ
Study of Wettability Effect on the Electrical Properties RI 5HVHUYRLU 5RFNV 7UDQVSRUW LQ 3RURXV 0HGLD SS
April 2009
ABOUT THE AUTHOR
Dr. Bernard MontaronMRLQHG6FKOXPEHUJHULQDQG
worked in various assignments in R&D and marketing LQ(XURSHWKH86$DQGWKH0LGGOH(DVW+HLVFXUUHQWO\
the Schlumberger theme director for carbonates and QDWXUDOO\ IUDFWXUHG UHVHUYRLUV EDVHG LQ 'XEDL 8$(
%HUQDUG0RQWDURQKROGVDGHJUHHLQSK\VLFVIURPµ(FROH
Supérieure de Physique et de Chimie Industrielles de Paris’ (ESPCI), and a PhD in mathematics from Paris VI 8QLYHUVLW\)UDQFH
PETROPHYSICS
115