waxman smiths equation

Pore-Scale Anal y sis of the Waxman-Smits Shaly-Sand Con duc tiv ity Model1

Guodong Jin2, Carlos Torres-Verdín2, Sarath Devarajan2, Emman uel Toumelin2,3, and E. C. Thomas4

104 PETROPHYSICS April 2007

PETROPHYSICS, VOL. 48, NO. 2 (APRIL 2007); P. 104–120; 17 FIGURES, 3 TABLES

ABSTRACT

Waxman-Smits and dual-water mod els of elec tri calcon duc tiv ity of shaly sands account for the dual con duc -tive path ways formed by pore brine and clay min eralexchange cat ions, while Archie’s equa tions describe theelec tri cal con duc tiv ity behav ior of clay-free rocks. Theseempir i cal mod els are widely used in the inter pre ta tion ofresis tiv ity logs acquired in homo ge neous res er voir rocks.How ever, the mod els are not explicit in their pre dic tionsof elec tri cal con duc tiv ity with respect to rock struc ture,spa tial fluid dis tri bu tion in the pore space, wettability, orclay min eral dis tri bu tion. The objec tive of this paper is toquan tify the influ ence of exchange cat ions asso ci atedwith hydrated clay min er als, salin ity of pore bulk water,and water sat u ra tion on the elec tri cal con duc tiv ity ofshaly siliciclastic rocks. We accom plish this objec tive bycal cu lat ing excess con duc tiv i ties asso ci ated withhydrated clay min er als for explicit pore geom e tries ofbrine- and hydro car bon-sat u rated shaly gran u lar rocks. Inso doing we intro duce syn thetic pore-scale rep re sen ta -tions intended to repro duce exper i men tal obser va tions ofelec tri cal con duc tiv ity of shaly sands.

The syn thetic pore-scale mod els are con structed to rep -

re sent homo ge neous shaly sands that include the struc -tural effects of com pac tion, cemen ta tion, and dis tri bu tionof grain-coat ing clay min er als. Our pore-scale model isimple mented on the dig i tized rep re sen ta tive of rock sam -ples, which allows us to con sider arbi trary media bound -aries. Two-phase immis ci ble flu ids are geo met ri cally dis -trib uted in the pore space using the ordi nary per co la tionalgo rithm. We intro duce grain-coat ing hydrated clay min -er als and their cor re spond ing elec tric dou ble layer in thesyn thetic rock mod els with a grain “shell” of vari abledimen sions and diffusivity with respect to those ofpore-fill ing brine. Waxman-Smits for ma tion fac tors andresis tiv ity indi ces are cal cu lated directly with ran domwalk sim u la tions of late-time dif fu sion within the con duc -tive space formed by the pore water and clay min eralexchange cat ions in the dig i tized shaly-sand sam ples.Sim u la tions with this sim ple model cor rectly repro ducethe non lin ear behav ior of rock con duc tiv ity at low andhigh salin ity, and are con sis tent with reported lab o ra torymea sure ments.

Keywords: shaly sand, pore-scale model, con duc tiv -ity, clays

Manu script received by the Edi tor Novem ber 10, 2006; revised manu script received Jan u ary 26, 2007.1Orig i nally pre sented at the SPWLA 47th Annual Log ging Sym po sium, Veracruz, Mex ico, June 4–7, 2006, Paper M.2The Uni ver sity of Texas at Aus tin, Depart ment of Petro leum and Geosystems Engi neer ing, One Uni ver sity Sta tion, Mail Stop C0300,Aus tin, Texas, 78712; E-mail: [email protected], [email protected], [email protected] rently with Chev ron; 11111 S. Wilcrest Dr., Hous ton, TX, 77099; E-mail: toumelin@chev ron.com4Bayou Petrophysics; P. O. Box 1027, Tomball, TX, 77377; E-mail: ecthomas@char ter.net©2007 Soci ety of Petrophysicists and Well Log Ana lysts. All rights reserved.

INTRODUCTION

The most fun da men tal empir i cal rela tion ship for inter -pret ing elec tri cal con duc tiv ity mea sure ments in res er voirrocks was exper i men tally advanced by Archie (1942) as

Fa

Rw

m= =

s

s fo

, (1)

where the for ma tion resis tiv ity fac tor is denoted by FR, andthe con duc tiv i ties of the sat u rat ing water (or brine) and thewater-sat u rated rock are denoted by sw and so, respec tively.The for ma tion fac tor FR relates to the total inter con nectedrock poros ity f and the cemen ta tion (lithol ogy) expo nent m, with the lat ter depend ing mainly on the degree of con sol i da -tion of the rock. The coef fi cient a is a con stant that dependson tortuosity, shape of the pores, and their con nec tiv ity.

Archie’s law is appli ca ble only to water-wet clean or clay min eral-free sands, i.e. non-con duc tive rock matri ces(Klein and Sill, 1982). How ever, lab o ra tory mea sure ments(Waxman and Smits, 1968; Waxman and Thomas, 1974) for fully water-sat u rated shaly sands have shown that the rela -tion ship between so and sw is strongly non lin ear as reflected

by a con vex-upward increase in so as sw ®0. This non lin ear behav ior observed at low salin ity is attrib uted to the addi -tional con duc tiv ity result ing from the exchange cat ionsasso ci ated with hydrated-clay min er als (HCM) (Waxmanand Smits, 1968; Cla vier et al., 1977, 1984; Bussian, 1983;John son and Sen, 1988; Revil and Glover, 1997, 1998;Tabbagh et al., 2002). When clay min er als are pres ent, ionic sub sti tu tions within the lat tices result in an excess of neg a -tive charge near the HCM sur face. The counterions (or cat -ions) required to bal ance these charges form a thin “elec tri -cal dou ble layer” around the HCM sur face, where thehydrated cat ions are in rapid exchange with those in thepore bulk water (Cla vier et al., 1977, 1984). Elec tri cal con -duc tion within this layer can con trib ute sub stan tially to theelec tri cal con duc tiv ity of the whole porous sys tem at lowsalin ity or/and high tem per a ture.

Many empir i cal and the o ret i cal mod els were pro posed to describe the role of water and clay con tent on the elec tri calcon duc tiv ity of porous media (Waxman and Smits, 1968;Bussian, 1983; Cla vier et al., 1984; Sen and Kan, 1987;Lima and Sharma, 1990; Revil and Glover, 1998). To beuse ful for petrophysical inter pre ta tion, these mod els should be able to cap ture the non lin ear (con vex-upward) behav iorof the plot of so ver sus sw at low val ues of salin ity as well asthe non-zero value of so when sw = 0. Bussian (1983) pro -posed a gen eral model that related the elec tri cal prop er tiesof any het er o ge neous two-com po nent mix ture to the prop -er ties of the indi vid ual com po nents. Lima and Sharma(1990) con ceived of the sur face con duc tiv ity of clays as an

elec tri cally equiv a lent effec tive vol ume con duc tiv ity. Kimand Torquato (1990) and McCar thy (1990) cal cu lated theeffec tive con duc tiv ity of ran dom mix tures by reduc ingthem to a multi-com po nent con duc tiv ity prob lem. A com -mon fea ture of these mod els is that they assume a con duc -tive rock matrix. Using a dif fu sion cur rent model, Sen(1987) dem on strated that the nonlinearity of so ver sus sw

fol lows as a con se quence of the dis tri bu tion of the elec tricfield between the pore bulk water and clay-bound waterregions. Revil and Glover (1998) advanced an effec tivemedium the ory that accounts for the dif fer ent behav ior ofanions and cat ions in these two con duc tive regions.

The most widely used log-inter pre ta tion meth ods forshaly sands in the oil indus try are based on theWaxman-Smits and the dual-water mod els (Waxman andSmits, 1968; Cla vier et al., 1984). Both mod els assume thatexchange cat ions asso ci ated with clay min er als are theagents behind clay con duc tance. The pore space in shalysands is com posed of two zones (Cla vier et al., 1977, 1984;Klein and Sill, 1982). The first is occupied by water that isnot affected by sur face charges of clay min er als. The sec -ond is the elec tri cal dou ble-layer zone in near prox im ity tothe HCM sur face, where the ionic dis tri bu tion is con trolledby the unbal anced charge defects of clay min er als. The ini -tial cur va ture of the so – sw plot is attrib uted to an increaseof the HCM counterion mobil ity and to the com pres sion ofthe dou ble-layer thick ness with increas ing salin ity of theelec tro lyte.

An alter na tive inter pre ta tion of these elec tro chem i calcon cepts leads to the prop o si tion that dif fer ent val ues ofelec tri cal con duc tiv ity exist close to the grain sur face com -pared to the elec tri cal con duc tiv ity of the bulk pore fluid.Almon (1979) sug gests that the con duc tiv ity in the dou -ble-layer zone be an order of mag ni tude larger than the bulk pore water con duc tiv ity even for mod er ately saline waters.A decrease in salin ity is expected to empha size the rel a tivecon tri bu tion from the HCM counterions in the dou ble-layer zone, regard less of the model inter pre ta tion (Winsauer andMcCardell, 1953). Revil et al. (1998) and Leroy and Revil(2004) assign a spe cific sur face con duc tiv ity in lieu of aclay min eral con tin uum by inte grat ing the “anom a lous”sur face con duc tiv ity over the dou ble-layer thick ness. Ananal o gous approach has been used for mod el ing the con -duc tiv ity of the inter fa cial zone in cement mor tar (Garbocziet al., 1995).

How ever, the nature of the excess con duc tiv ity asso ci -ated with the pres ence of clay min er als requires fur therinves ti ga tion. There are prop er ties spe cific to the dou -ble-layer that war rant a care ful appraisal of param e ters cho -sen and results obtained. One such prop erty con cerns thechoice of the true thick ness d of an indi vid ual HCM cat iondou ble-layer. This is unlike the mor tar con duc tiv ity prob -

April 2007 PETROPHYSICS 105

Pore-Scale Anal y sis of the Waxman-Smits Shaly-Sand Con duc tiv ity Model

lem (Garboczi et al., 1995), where the width of the inter fa -cial zone is com pa ra ble to the grain radius. Clay min er alsdo stack up on top of one other to form lay ers as thick as 12mm. How ever, indi vid ual dou ble-layer thick nesses are stillin the sub-micron range. Thus, d is the thick ness of a rep re -sen ta tive vol ume con tin uum that would describe the cumu -la tive effect from sur face phe nom ena at each dou ble-layer.Like wise, there is no sim plis tic rela tion that asso ci ates sur -face con duc tance with vol u met ric excess HCM exchangecat ion con duc tiv ity. Param e ter cal i bra tion is cru cial owingto the freely vary ing nature of d.

The elec tri cal response of shaly sands is also deter minedby the dis tri bu tion, amount, shape and type of clay min er -als, the salin ity of pore water, and water sat u ra tion. It isessen tial to develop a clear under stand ing of the effect these fac tors have on the elec tri cal con duc tiv ity of shaly sands. In this paper we intro duce a new pore-scale method basedupon the three-dimen sional (3D) dig i tal rep re sen ta tion ofthe porous medium aimed at inves ti gat ing the effect of theelec tri cal con duc tiv ity con trast between bulk water andclay-bound water, and their rel a tive amount on the shalysand con duc tiv ity. First, we briefly describe theWaxman-Smits model, which forms the the o ret i cal basis ofour study. In order to illus trate our sim u la tion approach, wecon sider a clean, well-sorted, quartz-cemented, water-wetshaly-sand model. The geom e try of the model con sists ofelec tri cally insu lat ing spher i cal grains coated with a HCMcounterion dou ble-layer of thick ness d, con duc tiv ity scl,and the bulk pore elec tro lyte of con duc tiv ity sw. Elec tri calcon duc tiv ity of shaly sands is cal cu lated directly from ran -dom walk sim u la tions of late-time dif fu sion. Sub se quently,we describe the algo rithm of ran dom walk sim u la tion andour imple men ta tion on the dig i tized rock sam ples. Sim u la -tion results are pre sented and ana lyzed for two mod eledshaly sands with dif fer ent val ues of poros ity.

THE WAXMAN AND SMITS MODEL

Waxman and Smits (W-S, 1968) pro posed a phys i calmodel to describe the depend ence of shaly-sand con duc tiv -ity on clay con tent, expressed as cat ion exchange capac ityper unit pore vol ume. The W-S model has become the mostfully devel oped and widely accepted approach to the under -stand ing of the elec tri cal con duc tiv ity of shaly-sand for ma -tions. The main assump tions of the W-S model are(Bussian, 1983):1. The elec tri cal con duc tiv i ties asso ci ated with both bulk

water in the pore space and HCM exchange cat ions con -trib ute to the elec tri cal con duc tiv ity of shaly-sand for -ma tions in a man ner anal o gous to a par al lel cir cuit,

2. The elec tric cur rent trans ported by HCM exchange ionstrav els along the same tor tu ous path as the elec tri cal cur -

rent attrib uted to ions in the bulk water. This behav ior isdue to rapid exchange of HCM cat ions between the claymin eral sur face and bulk water. In other words, thetortuosity is the same for all ions,

3. The con duc tiv ity of HCM exchange cat ions, scl, isdepend ent upon the con duc tiv ity of bulk water, sw.

For fully water-sat u rated shaly sands, the W-S model isdefined as

s s so

R

w clF

= +1

*( ) , (2)

where the shaly-sand for ma tion resis tiv ity fac tor, FR* ,

relates to total inter con nected rock poros ity and is inde -pend ent of for ma tion-water salin ity. The value of FR

* can bedeter mined at high salin ity where sur face elec tri cal con duc -tion can be neglected. The con duc tiv ity of HCM exchangecat ions scl is expressed as

scl vBQ= , (3)

where Qv is the vol ume con cen tra tion of HCM exchangecat ions (equiv a lent/liter or meq/ml) and B rep re sents theaver age mobil ity of the HCM counterions near their sur -faces (mho cm2/meq), which at 25°C is given by

B w= - -0.046 0.6 0.013[ exp( / )] .1 s (4)

The W-S model cap tures the non lin ear behav ior of so ver sussw at low val ues of salin ity by allow ing the counterionmobil ity B to increase expo nen tially at low val ues of sw

until it attains a con stant and max i mum value at high val uesof salin ity. When sw = 0, the W-S model gives a finite valueof so, i.e., the rock is still con duc tive even if it is sat u ratedwith fresh water. While B accounts for the well under stoodelec tro chem i cal phe nom e non of ionic mobil ity, it is the onefree param e ter that is used in the empir i cal W-S regres sion.Regard less of the phenomenological accu racy of equa tion(4), it is worth while rec og niz ing that the prod uct BQv rep re -sents the exper i men tal excess con duc tiv ity due to the HCMexchange cat ions at var i ous val ues of salin ity and can beused to cal i brate a pro posed con duc tiv ity model in responseto type and amount of clay, and type of clay dis tri bu tion inthe pore space.

Another param e ter of inter est is the “excess con duc tiv -ity” asso ci ated with the HCM exchange cat ions. For betterunder stand ing, equa tion (2) is rewrit ten in its gen eral formas

ss

ow

RFX= +

*( ) ,1 (5)

where the dimensionless excess con duc tiv ity X, defined as


Jin et al.

XBQv

w

=s

, (6)

is a char ac ter is tic vari able used directly or oth er wiseincluded in most shaly-sand con duc tiv ity mod els(Worthington, 1985). Grain con duc tiv ity mod els (Bussian,1983; Lima and Sharma, 1990) use X in power laws involv -ing the cemen ta tion expo nent. Real is tic val ues of X at lowval ues of salin ity vary with bulk con duc tiv ity sw depend ingon Qv. At very high val ues of salin ity this ratio approacheszero. Fig ure 1 shows typ i cal val ues of X cal cu lated as afunc tion of sw and Qv.

MODEL OF SHALY SAND

We con struct a model of shaly sand with a dense ran dompack ing of mono-sized spheres. For sim plic ity, we use thereal pack ing con structed by Finney (1968), who mea suredthe spa tial coor di nates of 8000 balls com pris ing the cen tralpart of the pack ing, in which 25000 mono-sized spher i calball bear ings were packed densely. The size of ball bear ings is 200 mm. Our sim u la tion domain con sists of 1000 spheres, whose spa tial coor di nates were obtained from Finney’s data set (Fig ure 2). The ini tial poros ity is approx i mately 36.5%.In addi tion, we assume that grains are elec tri cally insu lat -ing.

Cemen ta tion is the pro cess of min eral nucle ation andpre cip i ta tion that binds the rock grains. Depend ing on thechem i cal and crys tal lo graphic prop er ties, cements pre cip i -tate on the grain sur faces and affect the spe cific sur face area and tortuosity of the rock. How ever, it is not an easy task topre dict the loca tion and chem i cal com po si tion of the var i -

ous cements inside the pore space of con sol i dated rocks(Bloch and Helmold, 1995). In this study, we assume thatcement, inde pend ent of its com po si tion, is homo ge neouslydepos ited on the grain sur faces within the sam ple accord ing to a pro posed cement over growth algo rithm (Rob erts andSchwartz, 1985; Bryant et al., 1993; Jin et al., 2003, 2006a). For sim plic ity, all grains in Fig ure 2 are allowed to growuni formly, i.e. with out mov ing their cen ters, to ren der lowval ues of poros ity. The increased sphere’s radius sim u latesthe growth of quartz cement (Schwartz and Kimminau,1987).

Dis persed clay min er als in sand stones are of three mor -pho log i cal types: (1) pore lin ing, (2) pore bridg ing, and (3)dis crete par ti cles (Wil son and Pittman, 1977; Neasham,1977). Pore-lin ing clays are attached to pore walls to form a

rel a tively con tin u ous and thin (£ 12 mm) clay min eral coat -ing (Neasham, 1977). In the model used here, we have cho -sen to describe the elec tri cal prop er ties of shaly sands witha “sim ple” grain coat ing model (John son et al., 1986; Limaand Sharma, 1990), in which a clay min eral shell with thick -ness d and cor re spond ing HCM cat ions is added to the elec -tri cally insu lat ing grains. Fig ure 3 shows the ide al ized grain coat ing model, where d is the thick ness of a rep re sen ta tivevol ume con tin uum intended to syn the size the cumu la tiveelec tri cal effect from sur face phe nom ena due to the cor re -



FIG. 1 Plots of dimensionless excess con duc tiv ity X as a func -tion of sw and Qv.

FIG. 2 Graph i cal illus tra tion of a sand pack based on 1000mono-size spher i cal grains. The dimen sions of the pack ing are2100 mm x 2100 mm x 2100 mm, with grain diam e ter equal to 220mm and the poros ity approx i mately equal to 36.5%.

spond ing dou ble layer. This thick ness is small com pared tothe dimen sions of both pores and grains.

Another inter pre ta tion of the clay shell thick ness d is that the grain sur face is elec tri cally charged where elec tri calneu tral ity requires an equal con cen tra tion of counterionsthat are con fined to the sur face within a dis tance d. Theseaddi tional HCM counterions con trib ute to the elec tri calcon duc tiv ity of the shaly sand beyond the con tri bu tion dueto bulk water. The water in the d zone is called theclay-bound water, sim i lar to the con cept of the dou ble layerdevel oped by Cla vier et al. (1977, 1984). As the salin ity ofthe pore water increases, d decreases and the sur face cat -ions are packed closer together near the sur face (John sonand Sen, 1988; Sposito, 2004).

To con struct a 3D model of the pore space, we discretizethe mac ro scopic cube with grains in the form of a 3D arrayof iden ti cal micro scopic cubes (voxels). Our con ven tion isthat a voxel is an ele ment of the pore space if its cen ter iswithin the pore space; oth er wise, it becomes an ele ment ofthe solid skel e ton. Fig ure 4 dis plays the cor re spond ingdiscretized pore space of the grain pack ing shown in Fig ure2, with voxel dimen sions of 250 x 250 x 250, and with theside length of each voxel equal to 6 mm. Poros ity can bedirectly eval u ated from the frac tion of void-space voxels inthe rock image, here approx i mately equal to 36.5%. Theporos ity cal cu lated this way depends on the spa tial res o lu -

tion of the discretization. How ever, such depend ence isneg li gi ble if the spa tial res o lu tion is suf fi ciently high. Forour study, we deter mined a small voxel size of 6 mm fromsen si tiv ity stud ies of cal cu lated per me abil ity (Jin et al.,2006b).

RANDOM WALK ALGORITHM ANDIMPLEMENTATION

Steady-state flow of elec tric cur rent through a porousmedium sat u rated with an elec tro lyte is gov erned by the dif -fer en tial equa tion (Revil and Glover, 1997)

Ñ× Ñ =[ ] ,s F 0 (7)

where F is the local elec tri cal poten tial. Addi tion ally, theno-flux bound ary con di tion is enforced at the solid-voidinter face when the solid phase is assumed to be insu lat ing,i.e.

n× Ñ =F 0 , (8)

where n is the unit nor mal vec tor out ward from the solid tothe pore space. The solu tion of this prob lem in terms of thecon duc tiv ity of the porous medium so is given by

j =- Ñs o F , (9)


Jin et al.

FIG. 3 Ide al ized grain-coat ing clay min eral mor phol ogy showsthe spa tial sep a ra tion between the HCM exchange cat ions (orclay-bound water) and bulk water regions. The size of the shellof HCM exchange cat ions rel a tive to the grain radius is exag ger -ated for better illus tra tion.

FIG. 4 Pore space vol ume of the grain pack shown in Fig ure 2.The num ber of voxels is 250 x 250 x 250. Voxel res o lu tion is 6mm and the poros ity is approx i mately 36.5%.

where j and ÑF are the mac ro scopic elec tri cal cur rentden sity and poten tial gra di ent vec tors, respec tively.

The dif fer en tial equa tion (7) can be numer i cally solved

via the finite-dif fer ence or finite-ele ment meth ods (Adler et

al., 1992; Martys and Garboczi, 1992). How ever, such cal -

cu la tions become computationally expen sive when the size

of the sim u lated sam ple increases or when dif fer ent con -

duct ing phases exist in the het er o ge neous sam ple with dif -

fer ent length scales. More over, they are dif fi cult to imple -

ment on syn thetic 3D grain packs because of both vari able

geo met ri cal sur faces of grain bound aries and complex

bound aries between immis ci ble flu ids.Alter na tively, the elec tri cal con duc tiv ity of het er o ge -

neous porous media can be cal cu lated by sim u lat ing the

Brownian motion of dif fu sive par ti cles in the porous media

where the trans port pro cess is gov erned by a dif fu sion equa -

tion together with the appro pri ate bound ary con di tions at

the multiphase inter face (Schwartz and Banavar, 1989; Kim

and Torquato, 1990; Ioannidis et al., 1997). The dif fu sion

equa tion is given by

Ñ× Ñ =[ ] ,Dt

YY¶

¶(10)

where Y is the local poten tial and D is the dif fu sion coef fi -cient. For late-time, i.e. steady-state dif fu sion, the char ac terof the above equa tion is iden ti cal to that describ ing thebehav ior of the static elec tric poten tial. An impor tantparam e ter included in the sim u la tion of the Brownianmotion of dif fu sive par ti cles is the time-depend ent dif fu sion coef fi cient (or diffusivity) D(t) in the porous media,

D tr t

t( )

( ),=

2

6(11)

where r t2 ( ) is the mean-square dis place ment of walk ers attime t (Tobochnik et al., 1990; Latour et al., 1993; Regier

and Schuchmann, 2005). In the long-time limit, i.e., t ® ¥,the dif fu sion coef fi cient reaches a con stant value D¥ for clay min eral-free rocks (Latour et al., 1995; Nakashima andWatanabe, 2002). This is expressed by the rela tion ship

D

Dw

¥=

12t

, (12)

where Dw is the dif fu sion coef fi cient of the bulk water, and tis the tortuosity of the pore space. Through the tortuosity,the dif fu sion coef fi cient is inti mately con nected to the for -ma tion fac tor FR (or rock con duc tiv ity so) (Latour et al.,1995; Toumelin, 2006), as

FD

DR

w

o

w= =

¥

s

s f

1. (13)

For sim plic ity of cal cu la tion, the bulk water con duc tiv ity sw

and dif fu sion coef fi cient Dw are assigned the value of unityin our sim u la tions.

Our imple men ta tion of ran dom walk sim u la tion is based on the voxel-based rock sam ple. There are two kinds ofvoxels used in this study: coarse and fine (Fig ure 5). Eachcoarse voxel can be occu pied by either the solid phase (A),oil (B), bulk water (C and D), or clay-bound water withsome oil or bulk water (E). We define the type of voxels E as those hav ing a com mon face, edge, or ver tex with voxelsoccu pied by the solid phase (A). The type of voxels D isdefined as those occu pied by bulk water and in contact withvoxels E. To explore the prop er ties of the HCM exchangecat ion region, voxels close to this region (type of voxels Dand E) are re-discretized into finer voxels. Fine voxels in Ecan be occu pied by clay-bound water, bulk water, or oil. Inthe sim u la tion, we assume that the rock is water wet.

In order to cal cu late a reli able ensem ble aver age of thedif fu sion coef fi cient, many walker tra jec to ries are gen er -



FIG. 5 Descrip tion of voxel types in the coarse and fine grids.There are five types of coarse voxels, each occu pied by eithersolid phase (A), oil (B), bulk water (C) and D), or clay-boundwater with either oil or bulk water (E). Each of coarse voxels Dand E is rediscretized into finer voxels. Each fine voxel in thecoarse voxels E may be occu pied by clay-bound water (darkcells), bulk water (medium cells), or oil (light cells). The rock isassumed to be water wet.

ated and eval u ated. In our model, the start ing points of walk -ers are ran domly spec i fied using a Monte Carlo algo rithm.Only those walk ers within the pore space occu pied by theelec tri cally con duc tive com po nents, i.e. bulk water andclay-bound water, are retained. The walker dis tri bu tion ispro por tional to the ratio of the vol ume of the clay-boundwater zone to that of the bulk water zone. Each walker is dis -placed for a suf fi ciently long time. Length steps of a ran domwalker are depend ent on the type of voxels or fluid in it,whereas the direc tion is dis trib uted equally and ran domly. Inthe coarse voxels C, a large step size is used, which is pro por -tional to the size of coarse voxels, while the step size invoxels D and E is pro por tional to the size of fine voxels. Aran dom step is pos si ble and the new loca tion of the walker isstored if the new loca tion is not occu pied by the non-con duc -tive phase (solid or oil). How ever, if the new posi tion iswithin solid or oil the step is not car ried out, i.e. the walker isreturned to the previous posi tion, and the clock is allowed toadvance by one time step (Schwartz and Banavar, 1989).

A ran dom walker can jump from the bulk water zone tothe clay-bound water zone with a given jump prob a bil itypb,cl, or stay at the orig i nal posi tion with prob a bil ity (1 –pb,cl). In either case, the sim u la tion time is incre mented byone unit. The reverse con di tion occurs for the case of walk -ers jump ing from the clay-bound water zone to the bulkwater zone. We enforce a peri odic bound ary con di tion inthe sim u la tions, that is, a walker that moves out of the porespace image at one side reap pears at the oppo site side.

We applied the ran dom walk algo rithm to cal cu late thefor ma tion fac tor of a sand stone com puted-tomog ra phy(CT) image to val i date this approach. There is no clay pres -ent in the rock core sam ple from which the CT image wasacquired. The sam ple has a mea sured poros ity of 20.3% and for ma tion fac tor of 17.5. The size of the CT image is

300´300´300 voxels with the side length of each voxelequal to 4.5 mm. The cal cu lated CT image poros ity isapprox i mately equal to 21.1%. We car ried out 10 inde pend -ent sim u la tions, each con sist ing of 1000 walk ers, on theimage and cal cu lated the effec tive diffusivity for each sim -u la tion. Fig ure 6 shows an exam ple of the vari a tion of theeffec tive diffusivity with sim u la tion time for one sim u la -tion, which describes an ensem ble aver age of 1000 ran domwalk ers. From equa tion (13), we cal cu lated the for ma tionfac tor for each sim u la tion. The mean value of the for ma tionfac tor from 10 inde pend ent sim u la tions is 18.5 with a stan -dard devi a tion of 0.21. Con sid er ing the dif fer ence in poros -ity and spa tial scales between the CT image and the coresam ple, our cal cu la tion is in good agree ment with theexper i men tal mea sure ment.

RESULTS AND DISCUSSION

We per formed the ran dom walk sim u la tions on twomod els of shaly sands with dif fer ent poros i ties, and inves ti -gated the effect of the dou ble layer thick ness d, diffusivitycon trast Dcl /Dw between clay-bound water and bulk water,and water sat u ra tion Sw on the elec tri cal con duc tiv ity of themod eled shaly sands.

Sin gle-phase sim u la tion

All grains in the Finney pack ing are uni formly grownwith out mov ing their cen ters to gen er ate two dif fer ent poros -i ties of 25.4% and 18.9%. A homo ge neous HCM exchangecat ion shell of thick ness d and diffusivity Dcl is added to each elec tri cally insu lat ing grain. The pore space is sat u rated withthe elec tro lyte, which is assigned a diffusivity equal to Dw.Note that dif fer ent val ues of elec tri cal con duc tiv ity existclose to the clay min eral sur faces, i.e. in the dou ble-layerzone, com pared to the elec tri cal con duc tiv ity of the bulk pore fluid, which could be an order of mag ni tude larger than thebulk pore water con duc tiv ity even for mod er ately salinewaters (Almon, 1979). We assign a spe cific con duc tiv ity tothe water in the dou ble-layer zone, and a dif fer ent value tothe bulk pore water, in which the elec tri cal con duc tiv ity isassumed to be pro por tional to the diffusivity of the cor re -spond ing water (Schwartz and Banavar, 1989).

Input param e ters in the sim u la tion are the bulk waterdiffusivity Dw, the clay-bound water diffusivity Dcl, theratio of the coarse voxel size to the ran dom step size rc for


Jin et al.

FIG. 6 Change of effec tive diffusivity with increased sim u la tiontime. The solid line describes an ensem ble aver age of 1000 ran -dom walk ers. Fluid bulk diffusivity is assumed equal to 1mm2/ms, and the cal cu lated effec tive diffusivity (asymp toticvalue) is equal to 0.256 mm2/ms (dash line), with the stan darddevi a tion equal to 1.86 x 10–4 mm2/ms.

the type of voxels C, the ratio of the finer voxel size to theran dom step size rf for the type of voxels D and E, the num -ber of walk ers N, the travel time of each walker tT, the thick -ness of clay-bound water d, and the prob a bil ity pb,cl (Table1). The cal cu lated val ues of the shaly-sand for ma tion fac tor FR

* and sat u ra tion of clay-bound water Sw,cl and bulk waterSw,b are shown for each con fig u ra tion of d (Table 2). Thevalue of FR

* is cal cu lated by assign ing Dcl = Dw dur ing thesim u la tion. It is dis cussed later that val ues of the diffusivitycon trast Dcl /Dw are close to 1 when the pore bulk water ishighly con duc tive. The value of FR

* from the W-S equa tioncor re sponds to the slope of the lin ear por tion of the so ver -sus sw plot that occurs at high val ues of sw.

If the bulk water con duc tiv ity sw and diffusivity Dw areassigned the value of unity in equa tion (13), then the rockcon duc tiv ity can be cal cu lated directly as so = fD¥. Equa -

tion (5) is used to com pare our sim u la tion results to the W-S model. The sim u lated val ues of dimensionless excess con -duc tiv ity X vary with the diffusivity con trast Dcl /Dw and thethick ness d for the mod eled shaly sands with poros ity of18.9% and 25.4%, respec tively (Fig ures 7 and 8). The HCM exchange cat ion layer can not be strictly treated as a con tin -uum of finite con duc tiv ity. The con duc tiv ity of HCMexchange cat ions decreases from a “sur face value” to thebulk value for both shaly-sand sam ples and is a func tion ofthe nor mal dis tance from the grain sur face (Schwartz et al.,1989). As d increases, the clay-bound water sat u ra tion inthe rock increases (Table 2), and its rel a tive con tri bu tion tothe over all rock con duc tiv ity increases, result ing in a higher excess con duc tiv ity X. With the poros ity increas ing, the rel -



TABLE 1 Parameters used in the ran dom walk sim u la tions.

Nota tion Val ues

Dw (mm2/ms) 1Dcl (mm2/ms) 1, 5 10, 25, 50, 100, 250, 500, 1000

rc 2 (coarse voxel size 6 mm)rf 2 (finer voxel size 0.6 mm)N 1000

tT (ms) 1.0´107

d (mm) 1.8, 3.0, 3.6, 4.8pb,cl (%) 50

TABLE 2 The cor re spond ing val ues of clay-bound watersat u ra tion Sw,cl, and bulk water sat u ra tion Sw,b for each con -fig u ra tion of d in the two mod eled shaly sands with dif fer entval ues of poros ity f and shaly-sand for ma tion fac tor, FR

*.

f(%) FR* Notation Val ues

d (mm) 1.8 3.0 3.6 4.818.9 16.6 Sw,cl (%) 18.1 29.0 33.7 42.0

Sw,b(%) 81.9 71.0 66.3 58.0

d (mm) 1.8 3.0 3.6 4.825.4 9.0 Sw,cl (%) 15.6 25.3 29.5 37.3

Sw,b (%) 84.4 74.7 70.5 63.7

FIG. 7 Sim u lated val ues of dimensionless excess con duc tiv ityX as a func tion of diffusivity con trast Dcl /Dw and thick ness d forthe shaly sand with poros ity equal to 18.9%.

FIG. 8 Sim u lated val ues of dimensionless excess con duc tiv ityX as a func tion of diffusivity con trast Dcl /Dw and thick ness d forthe shaly sand with poros ity equal to 25.4%.

a tive frac tion of the HCM exchange cat ion region in therock decreases, and hence one obtains lower val ues of X.

In the sim u la tions we use an unbi ased jump prob a bil ity(pb,cl = 50%) for walk ers step ping across the phase bound -aries, since HCM exchange cat ions on the min eral sur faceare in rapid exchange with those in the bulk water. Otherbiased prob a bil i ties could also be imple mented in our

model based on the rel a tive con duc tiv ity of the two phases,i.e., Dcl /Dw (Kim and Torquato, 1990) or the har monicmean of Dcl and Dw (McCar thy, 1990). How ever, a rig or ousjus ti fi ca tion for these two approaches is yet to be pro vided.

At low water con duc tiv ity the sim u lated rock con duc tiv -ity so is described as a func tion of the pore water con duc tiv -ity sw for the shaly sand with poros ity equal to 18.9% (Fig -ure 9). Val ues of Qv = 0.20 meq/ml and d = 3.6 mm are usedin this plot. The val ues of so are derived by rescaling thesim u la tion results with the cor re spond ing val ues of sw thatare obtained by choos ing the appro pri ate value of Dcl /Dw inFig ure 7 and by resolv ing the dimen sional excess con duc -tiv ity from Fig ure 1. Note that the sim u la tion curve ofshaly-sand con duc tiv ity is not lin ear at low val ues of sw. For clar ity, we cal cu lated the dif fer ences between the sim u la -tion results so, and the val ues obtained from the lin ear curve con nect ing the two end-points of the sim u la tion curve ofshaly sand, so,lin ear, and W-S cal cu la tion, so,W-S, from equa -tion (2) (Fig ure 10). As expected, so is an increas ing func -tion of sw, and exhib its con vex-up behav ior at low val ues ofsw (Waxman and Smits, 1968; Waxman-Thomas, 1974).The sim u la tion rock con duc tiv ity so is con sis tent with thecal cu la tions from Waxman-Smits model. It is also inter est -ing to note that the cho sen diffusivity con trasts are pro por -

tional to 1 / sw at high val ues of salin ity but pro por tional

to 1 / sw at low val ues of salin ity (Fig ure 11).Rea son able para met ric val ues can be sur mised from a

better under stand ing of the rela tion ship between diffusivity con trasts and pore fluid con duc tiv ity sw. Diffusivity con -trasts are strongly affected by dif fer ent val ues of d (3.6 mm


Jin et al.

FIG. 9 Plot of so ver sus sw for the shaly sand with poros ity equalto 18.9% at low val ues of salt con cen tra tion. Val ues of Qv = 0.20meq/ml and d = 3.6 mm were used to con struct the plot. Thedashed line (par al lel to that of clean sand) is used to empha sizethe nonlinearity of the rela tion ship between so and sw.

FIG. 10 Dif fer ences between the sim u la tion results so, val uesso,lin ear cal cu lated from the lin ear rela tion con nect ing the twoend-points of the sim u la tion curve of shaly sand in Fig ure 9, andWaxman-Smits cal cu la tions so,W-S from equa tion (2).

FIG. 11 Cho sen val ues of Dcl /Dw as a func tion of sw used tocal cu late the results shown in Figure 9. Val ues are selected toyield an excess con duc tiv ity con sis tent with W-S model pre dic -tions.

and 4.8 mm) in a syn thetic shaly sand with Qv = 0.20 meq/ml and f = 18.9% (Fig ure 12). It is clear that the desired valueof Dcl /Dw at dif fer ent val ues of salin ity is influ enced by thechoice of d. In the case of sw < 0.005 mho/cm, the value of d exerts a strong influ ence on the diffusivity con trast thatneeds to be cho sen. The ratio Dcl /Dw can be main tained rel -a tively con stant if d is allowed to increase with decreas ingval ues of salin ity. This is in keep ing with incre ments in dou -ble-layer thick ness that have been found to occur below acrit i cal value of salin ity (Cla vier et al., 1984; Sen, 1987;Bassiouni, 1994; Hill et al., 1979). Silva and Bassiouni(1988) cal cu lated the frac tional vol ume occu pied by thedou ble layer as a func tion of far-water con duc tiv ity. Adecrease of far-water con duc tiv ity below a value of 0.03mho/cm is accom pa nied by a dras tic increase in vol umeoccu pied by the dou ble layer. This fur ther jus ti fies theselec tion of higher val ues of d at low val ues of salin itywhile main tain ing invari ant diffusivity con trasts. In thecase of very shaly sands (Qv > 1 meq/ml) and under lowequilibrating brine con duc tiv ity (sw < 0.03 mho/cm) thedesired excess con duc tiv ity is real iz able only when d iscom pa ra ble to the bulk/pore dimen sions. The ratio Dcl /Dw

can then be as low as 1, which even tu ally reduces toWaxman and Smits’ assump tions of a uni formly enhancedpore elec tro lyte (Waxman and Smits, 1968).

The vari a tion of the appar ent for ma tion-poros ity expo -nent ma, defined as

Fa m

w

oa

= =1

f

s

s, (14)

is also of inter est. Excess con duc tiv ity gen er ated by theHCM counterions results in decreased val ues of ma at lowval ues of salin ity. As salin ity increases, ma reaches a con -stant value as the rel a tive con tri bu tion of HCM con duc tiv ity decreases. Fig ure 13 describes the vari a tion of ma with porefluid con duc tiv ity calculated for the shaly sand with f =18.9% and d = 3.6 mm. The vari a tion of ma closely resem -bles the trends in bilogarithmic plots of Fa/FR ver sus sw

reported by Worthington (1985).In addi tion to the rapid exchange with ions in the bulk

water, the HCM exchange cat ions on the clay min eral sur -face can move along the sur face when the shaly sand is sub -jected to an exter nal elec tri cal field (Revil and Glover,1998). This move ment results in finite rock con duc tiv ityeven if the shaly sand is sat u rated with fresh water (Dalla etal., 2004; Lima et al., 2005). Our sim u la tions con firm theexis tence of this finite con duc tiv ity. Fig ure 14 shows thevari a tion of the sim u lated rock con duc tiv ity so with thethick ness of clay-bound water d when the shaly sand is sat -u rated with fresh water (sw = 0 or neg li gi ble). One canobserve that so increases lin early as d increases, and that the con duc tiv ity becomes null when d = 0. This behav ior iscon sis tent with the prop erty that the con duc tiv ity of dryshaly sand is neg li gi ble (Bassiouni, 1994). The sim u latedrock con duc tiv ity can be rescaled back to the “true” value ifwe knew the con duc tiv ity of clay-bound water and itsdiffusivity. In the sim u la tion, we assume val ues of the con -duc tiv ity and diffusivity of clay-bound water equal to 1mho/cm and 1 mm2/ms, respec tively.



FIG. 12 Diffusivity con trasts cho sen to gen er ate W-S modelresults for the shaly sand with f = 18.9% and Qv = 0.20 meq/ml.

FIG. 13 Val ues of ma cal cu lated from sim u la tion results as afunc tion of sw and Qv for the shaly sand with f = 18.9% andd = 3.6 mm.

Two-phase sim u la tion

We inves ti gated the effect of water sat u ra tion on theelec tri cal con duc tiv ity of the mod eled shaly sands at lowand high val ues of water con duc tiv ity, sw. For the sim u la -tions, we main tain val ues of the clay shell thick ness con -stant at d = 3 mm, while the vol ume con cen tra tion of HCMexchange cat ions is set to Qv = 0.20 meq/ml for both sam -ples. At low sw, we choose the diffusivity con trast Dcl /Dw =5. From Fig ure 1, Fig ure 7 and Fig ure 8, we deter mine thatsw = 0.031 mho/cm for the shaly sand with poros ity of18.9% and 0.025 mho/cm for the sam ple with poros ity of25.4%. At high sw, we set Dcl /Dw = 1.

We assume that these two sam ples are water-wet sandand that a film of water is always main tained on the grainsur faces. Under such a con di tion, the wet ting phase (water)occu pies the cor ners of large pores and small pores, whilethe non-wet ting phase (oil) occu pies the cen tral parts of theinvaded pores. To assign the spa tial fluid dis tri bu tions inthe pore space, we use the tech nique of max i mal-inscribedspheres to obtain sev eral dis tri bu tions of water and oil foreach sam ple cor re spond ing to a dif fer ent value of cap il larypres sure (Silin et al., 2003, 2004). The ordi nary per co la tionalgo rithm is used for that pur pose. Fig ures 15(a) and (b) dis -play an exam ple of part of the pore space occu pied by waterand oil in the sam ple with poros ity of 18.9%, where watersat u ra tion is Sw = 0.668 includ ing clay-bound water and bulkwater. In the fol low ing sim u la tions, we only con sider thecases in which water is con nected through the pore space.

The resis tiv ity index IR is used to quan tify the effect ofpar tial sat u ra tion on the rock con duc tiv ity and is defined as


Jin et al.

FIG. 14 Plot of so ver sus d for the shaly sand with f = 18.9%under fresh water con di tions (i.e., con duc tiv ity of bulk watersw = 0). The fit ting dashed line is so,fit = 5.6 x 10–3d.

FIG. 15 Pore space occu pied by the wet ting phase, water (a),and non-wet ting fluid, oil (b), respec tively, in the mod eled shalysand with poros ity equal to 18.9%. Water sat u ra tion, includ ingclay-bound water and bulk water, is Sw = 0.668. The dimen sionof each image is 250 x 250 x 250 voxels, and the voxel res o lu -tion is 6 mm.

I bSRo

t

wn= » -

s

s, (15)

where so is the con duc tiv ity of the rock fully sat u rated withwater and st is the con duc tiv ity of the rock par tially sat u -rated with water (Archie, 1942). The value of the coef fi cient b is approx i mately equal to 1, and the sat u ra tion expo nent nis typ i cally about 2 for clean sands. How ever, the value of nvar ies with the vol ume of clay min er als pres ent in the rockand rep re sents an appar ent sat u ra tion expo nent when cal cu -lated using equa tion (15) for shaly sands. When the for ma -tion is par tially sat u rated, the HCM exchange cat ions aredensely packed within the clay min eral zone and the rel a tive con tri bu tion to over all rock con duc tiv ity from the HCMlayer is expected to increase. Based on exper i men tal evi -dence, Waxman and Smits (1968) pre dicted this excess con -tri bu tion to be inversely pro por tional to the value of watersat u ra tion. They for mu lated an empir i cal rela tion ship byinclud ing the effect of the HCM exchange cat ions throughthe func tion BQv/Sw as

I IBQ S

BQb SR R

w v w

w v

wn* *

/,

*

=+

+

é

ëê

ù

ûú» -

s

s(16)

where I R* and n* are the “clay-cor rected” resis tiv ity index

and sat u ra tion expo nent, respectively and the value of b* isclose to 1.0 (Diederix, 1982).

For each case of water sat u ra tion mod eled we com puted the bulk con duc tiv ity of the cor re spond ing part of porespace occu pied by water. Next, we cal cu lated the clay-cor -rected sat u ra tion expo nent n* from equa tion (16) for eachshaly-sand sam ple. In the ran dom walk sim u la tions, weassume that the elec tri cal con duc tiv ity of oil is zero. Fig -ure 16 shows the vari a tion of the com puted resis tiv ityindex I R

* with water sat u ra tion Sw for two sam ples at low sw. These two sam ples do not exhibit the straight-line rela tion -ship between I R

* and Sw pre dicted by Archie’s law on abilogarithmic plot. Instead, the rela tion ship is curved suchthat at low val ues of Sw the resis tiv ity index I R

* is rel a tively

low. We can rep re sent this curved rela tion ship as two lin earseg ments, with slopes n*L and n*H, respec tively, and across-over sat u ra tion Sw,cr. How ever, there is no imper a tiveto impose lin ear ity if some other rela tion ship is sug gestedby the data. Sim i lar anom a lous rela tion ships between I R

*

and Sw were reported by Diederix (1982) as due to sur facerough ness based on lab o ra tory mea sure ments of Rotliegend sand stone, which has a rough clay coat ing with Qv = 0.20meq/ml on the grain sur face. Other stud ies con firm the exis -tence of this rela tion ship (Tay lor and Barker, 2002; Dalla etal., 2004).

The val ues of coef fi cients b* and sat u ra tion expo nentsn* from equa tion (16) are cal cu lated for the two mod eledshaly-sand sam ples (Table 3). At low val ues of water sat u -ra tion, the con duc tiv ity enhance ment aris ing fromdensely-packed HCM exchange cat ions off sets the resis tiv -ity asso ci ated with lower val ues of Sw, thereby result ing in alower value of n* = n*L. At high val ues of water sat u ra tion,how ever, the rel a tive con tri bu tion from HCM exchangecat ions to the rock con duc tiv ity becomes smaller and ahigher value of n* = n*H is obtained. Com pared to Archie’sfor mula I SR w

* = -2, the coef fi cient b* is very close to 1(±0.09), val ues of sat u ra tion expo nent n*L and n*H are muchsmaller than the nor mally assumed value of n* = 2. Thisresult is in marked agree ment with Diederix’s exper i men talobser va tions on Rotliegend sand stone (Diederix, 1982).

We also note that the val ues of n*L and n*H depend on thesam ple poros ity. In our sim u la tions at low sw, we assumedthat both sam ples had the same char ac ter is tics in the HCM



FIG. 16 Com puted resis tiv ity index ver sus water sat u ra tion Sw

for two shaly sand sam ples for the case of low sw. Archie’s equa -tion used in the cal cu la tions is given by I SR w

* = -2. The fit tingcurves have the form I b SR w

n* **= - , where the val ues of b* and n*are listed in Table 3.

TABLE 3 Com puted val ues of coef fi cients b* and sat u ra tion expo nents n* in equa tion (16) for two mod eled shaly sandsam ples at low and high val ues of sw.

Low sw High sw

f (%) bL* n*L bH

* n*H b* n*

18.9 1.09 0.73 1.01 1.07 0.93 1.7225.4 1.05 1.07 1.01 1.33 0.95 1.93

exchange cat ion zone: d = 3 mm, Qv = 0.20 meq/ml, andDcl/Dw = 5. This implies that the rel a tive con tri bu tion on rock con duc tiv ity from HCM exchange cat ions is smaller in thesam ple with high poros ity than in the sam ple with low poros -ity. The sample with high poros ity will exhibit rel a tivelylarger val ues of n*L and n*H, as observed in our sim u la tions.

The cal cu lated val ues of cross-over crit i cal water sat u ra -tion Sw,cr are 0.80 for both sam ples. This cross-over could be larger and reach the value of 1.0 if the rock poros ity werelow enough or else if d were large enough. Such behav iorindi cates that only the HCM exchange cat ions con trib ute torock con duc tiv ity or that their con tri bu tion is rel a tivelylarge com pared to that due to pore bulk water. In the lat tercase, the two piecewise lin ear seg ments would become asin gle line with a slope equal to n*.

In con trast to the curved rela tion ship between log(I R* ) and

log(Sw) at low val ues of sw, our sim u la tions exhibit the nor -mal lin ear ity at high sw (Fig ure 17). One can observe that theeffect of water sat u ra tion on the expo nent n* is neg li gi ble.Although an increase in sw decreases the rel a tive con tri bu -tion from the HCM counterions to the rock con duc tiv ity,regard less of the model inter pre ta tion, this con tri bu tion stillmakes the value of n* smaller than 2. The value of n* islarger for the sam ple with high poros ity of 25.4% (n*=1.93)than for the sam ple with low poros ity of 18.9% (n*=1.72).Again, this result is due to the assump tion of equal prop er tiesin the HCM exchange cat ion zone for both sam ples.

Even though the anom a lous resis tiv ity index/water sat u -ra tion rela tion ships of very low n*-val ues obtained in our

sim u la tions were also observed in exper i men tal mea sure -ments (Diederix, 1982; Tay lor and Barker, 2002), we makethe fol low ing remarks stem ming from our sim u la tion results:1. The curved rela tion ship of log(I R

* ) ver sus log(Sw) in Fig -ure 16 was obtained at low sw, where the con duc tiv ityenhance ment aris ing from the HCM exchange cat ionsbecomes more prom i nent. We could not antic i pate thisbehav ior from Diederix’s exper i men tal results(Diederix, 1982).

2. The depend ence of the sat u ra tion expo nent n* on poros -ity could be more likely related to the rel a tive con tri bu -tion of HCM exchange cat ions on rock con duc tiv ity. For the rock with low poros ity, the rel a tive vol ume frac tionof the HCM exchange cat ion zone to the bulk (or free)water zone is large, which results in a sub stan tiallyhigher elec tri cal con duc tiv ity and a lower value of n*.

3. Water sat u ra tion may affect the elec tri cal behav ior ofshaly sands at low sw much more than at high sw. Thedepend ence of I R

* on water con duc tiv ity or salin ity wasalso observed in exper i men tal mea sure ments by Tay lorand Barker (2002). How ever, such a state ment can not be gen er al ized with out fur ther inves ti ga tion and ver i fi ca -tion by exper i ments.

CONCLUSIONS

We devel oped a pore-scale ran dom-walk model based on the 3D dig i tal rep re sen ta tion of a porous medium and usedit to sim u late pore-level phe nom ena that gov ern elec tri calcon duc tion in shaly sands. With this model, we repro ducedreal is tic val ues of excess con duc tiv ity asso ci ated with theHCM exchange cat ions at var i ous elec tro lyte con duc tiv i -ties. Our model can be extended to var i ous clay min eralmorphologies since we have made no assump tions regard -ing con tin u ous elec tri cal path ways for the HCM cat ions.Sev eral pre vi ous works on this sub ject did not con sider theeffec tive vol u met ric con tri bu tion to excess con duc tiv ityresult ing from sur face con duc tiv i ties asso ci ated with theHCM cat ions.

We have shown that the rela tion ship between inter fa cialclay min eral con duc tiv ity and effec tive excess con duc tiv ityof the HCM exchange cat ions needs to be cal i brated usingthe thick ness of the vol ume con tin uum, d. The require mentof increased val ues of d with a decrease in elec tro lyte salin ity is con sis tent with observed swell ing of the anion-free layer at low val ues of salin ity. By main tain ing con stant diffusivitycon trasts while allow ing only d to increase, we have sim u -lated the entire range of mea sured excess con duc tiv i tiesasso ci ated with mod er ately shaly sands. The fact that thethick ness of the clay min eral zone d needs to be com pa ra blein size to the pore dimen sions at low val ues of salin ity con -firms the W-S assump tions of a uni form pore elec tro lyte.


Jin et al.

FIG. 17 Com puted resis tiv ity index ver sus water sat u ra tion Sw

for two shaly-sand sam ples for the case of high sw. Archie’sequa tion used in the cal cu la tions is given by I SR w

* = -2. The fit tingcurves have the form I b SR w

n* **= - , where the val ues of b* and n*are listed in Table 3.

Clay in the rock matrix plays an essen tial role in elec tri -cal con duc tion, but only in the pres ence of an elec tro lyte.The con duc tiv ity of dry shaly sand is neg li gi ble. Our results con firmed the exis tence of a finite elec tri cal con duc tiv ity of shaly sand sat u rated with fresh water. It was found thatthere is no elec tri cal con duc tion when the thick ness of theclay min eral zone d is zero.

Sim u la tions con sid ered in this paper repro duced theanom a lous curved resis tiv ity index/water sat u ra tion rela -tion ships of very low clay-cor rected sat u ra tion expo nentn*, which are con sis tent with reported exper i men tal obser -va tions (Diederix, 1982; Tay lor and Barker, 2002). Thesim u lated val ues of n* are lower than the typ i cal value of 2.It was found that the rel a tive amount of the HCM exchangecat ion zone to the total con duc tive vol ume could dra mat i -cally affect the elec tri cal behav ior of shaly sands.

Our pore-scale model allows one to include arbi trarymedia bound aries and takes into account the effect of claymin er als on the elec tri cal prop er ties of the rock. The abil ityof our approach to cor rectly pre dict the so ver sus sw curvesand the sub se quent sen si tiv ity to model param e ters lends cre -dence to the phys i cal con sis tency of the method. Future work will involve study ing the effects of wettability, sat u ra tioncycles, and vary ing rock geom e try on the model param e ters.

NOMENCLATURE

a Coef fi cient in Archie’s equa tionb Coef fi cient in Archie’s equa tionb* Mod i fied coef fi cient in Archie’s equa tionbL

* Low value of b* coef fi cient in Archie’s equa tionbH

* High value of b* coef fi cient in Archie’s equa tion

B Mobility of HCM counterions, mho cm2/meq D Diffusivity, mm2/msDw Diffusivity of bulk water, mm2/msDcl Diffusivity of clay-bound water, mm2/msD¥ Asymptotic value of time-depend ent diffusivity, mm2/msFa Apparent for ma tion resis tiv ity fac torFR Rock for ma tion resis tiv ity fac torFR

* Shaly-sand for ma tion resis tiv ity fac torHCM Hydrated-clay min er alsIR Resis tiv ity indexI R

* “Clay-cor rected” resis tiv ity indexj Electrical cur rent den sitym Cementation expo nent in Archie’s equa tionma Apparent cemen ta tion expo nentn Unit nor mal vec torn Saturation expo nent in Archie’s equa tionn* “Clay-cor rected” sat u ra tion expo nentn*L Low value of n*

n*H High value of n*N Number of walk erspb,cl Jump prob a bil ity from the bulk water zone to the clay-bound water zoneQv Volume con cen tra tion of HCM exchange

cat ions, meq/mlr Displacement of a walker at time t, mm rc Ratio of the coarse voxel size to the ran dom

step sizerf Ratio of the fine voxel size to the ran dom step sizeSw Water sat u ra tionSw,b Saturation of bulk waterSw,cl Saturation of clay-bound waterSw,cr Cross-over crit i cal water sat u ra tiont Time, mstT Travel time of each walker, msX Dimensionless excess con duc tiv ity

Greek symbolsscl Conductivity of HCM exchange cat ions or clay-bound water, mho/cmso Con duc tiv ity of the water-sat u rated rock,

mho/cmso,lin ear Conductivity inter po lated from a lin ear

rela tion ship, mho/cmso,W-S Conductivity cal cu lated from Waxman-Smits’

model, mho/cmst Conductivity of the rock par tially sat u rated with

water, mho/cmsw Conductivity of the sat u rat ing water, mho/cmf Effective rock poros ityd Thick ness of the shell of HCM exchange

cat ions, mmF Local elec tri cal poten tialY Local dif fu sion poten tialt Tortuosity

ACKNOWLEDGMENTS

The work reported in this paper was funded by the Uni -ver sity of Texas at Aus tin’s Research Con sor tium on For -ma tion Eval u a tion, jointly spon sored by Aramco, BakerAtlas, BP, Brit ish Gas, ConocoPhillips, Chev ron, ENI E&P, ExxonMobil, Halliburton Energy Ser vices, Hydro, Mar a -thon Oil Cor po ra tion, Mex i can Insti tute for Petro leum,Occi den ta l Pet ro leum Cor po ra t ion , Pet robras ,Schlumberger, Shell Inter na tional E&P, Statoil, TOTAL,and Weatherford.

We are also thank ful to Pro fes sor Tad W. Patzek of theUni ver sity of Cal i for nia at Berke ley and Dr. Dmitry B. Silin of Law rence Berke ley National Lab o ra tory for their help in



par ti tion ing the images occu pied by wet ting and non-wet -

ting flu ids. A note of grat i tude goes to Dr. James J. Howard

and three anon y mous review ers of Petrophysics whose con -

struc tive edi to rial and tech ni cal sug ges tions improved the

orig i nal con fer ence paper.

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ABOUT THE AUTHORS

Guodong Jin received a PhD in Res er voir Engi neer ing andPetrophysics from the Uni ver sity of Cal i for nia at Berke ley in2006, and is cur rently a post-doc toral fel low in the Depart ment ofPetro leum Engi neer ing at the Uni ver sity of Texas at Aus tin. Dur -ing 1997-2000 he held the posi tion of Research Asso ci ate withInsti tute of Mechan ics, Chi nese Acad emy of Sci ences. Hisresearch inter ests include pore-scale mod el ing of petrophysicalresponse in res er voir rocks and shaly sands, rock frac ture and dam -age mech a nism, fluid flow in porous media.

Carlos Torres-Verdín received a PhD degree in Engi neer ingGeoscience from the Uni ver sity of Cal i for nia, Berke ley, in 1991.Dur ing 1991–1997 he held the posi tion of Research Sci en tist withSchlumberger-Doll Research. From 1997–1999, he was Res er voirSpe cial ist and Tech nol ogy Cham pion with YPF (Bue nos Aires,Argen tina). Since 1999, he has been with the Depart ment of Petro -leum and Geosystems Engi neer ing of The Uni ver sity of Texas atAus tin, where he cur rently holds the posi tion of Asso ci ate Pro fes -sor. He con ducts research on bore hole geo phys ics, for ma tion eval -u a tion, and inte grated res er voir char ac ter iza tion. Torres-Verdínhas served as Guest Edi tor for Radio Sci ence, and is cur rently amem ber of the Edi to rial Board of the Jour nal of Elec tro mag neticWaves and Appli ca tions, and an asso ci ate edi tor for Petrophysics(SPWLA) and the SPE Jour nal. He is co-recip i ent of the 2003 and2004 Best Paper Award by Petrophysics, and is recip i ent ofSPWLA’s 2006 Dis tin guished Tech ni cal Achieve ment Award.

Sarath Devarajan received his BS in Chem i cal Engi neer ingfrom the Indian Insti tute of Tech nol ogy, Madras and is cur rently aMas ter’s stu dent in the Depart ment of Petro leum Engi neer ing atthe Uni ver sity of Texas at Aus tin. His research inter ests includeres er voir char ac ter iza tion, elec tri cal con duc tion in shaly sands and sim u la tion of fluid flow through dis or dered porous media.

Emman uel Toumelin received a PhD degree in Petro leumEngi neer ing at the Uni ver sity of Texas at Aus tin and an engi neer -ing degree from the Ecole Centrale, Lille. From 2001 to 2005, hisresearch at UT focused on devel op ing pore-scale mod els of elec -tro mag netic and nuclear mag netic res o nance mea sure ments in sat -u rated rocks. He also held research intern ship posi tions with Baker Atlas, Schlumberger-Doll Research and ChevronTexaco between2001 and 2003. Emman uel joined Chev ron North Amer ica E&P in2006 as a petrophysicist for Midcontinent. He serves as a tech ni caledi tor for SPE Res er voir Eval u a tion and Engi neer ing and SPEJour nal.

E. C. Thomas received a PhD in Phys i cal Chem is try fromStan ford Uni ver sity in 1966, com pleted a post-doc toral year atPrince ton Uni ver sity, then joined Shell Devel op ment Co. and per -formed research in the elec tri cal behav ior of shaly sands and manyother areas of Petrophysical Engi neer ing. E. C. spent 32 years inthe Shell orga ni za tion in posi tions encom pass ing field work, oper -at ing divi sion engi neer, research super vi sor, tech ni cal train ing andtech ni cal over sight of Petrophysics. Upon retir ing, E.C. formedBayou Petrophysics to con tinue his work in Petrophysics. E.C. hasserved on numer ous SPE and SPWLA tech ni cal com mit tees andchaired sev eral SPE tech ni cal forums. He also served as an SPEDis tin guished Lec turer and pres ently serves as a tech ni calreviewer for SPE and an asso ci ate edi tor for SPWLA. In 2004 hewas awarded the SPWLA Gold Medal for tech ni cal achieve ment.


Jin et al.

waxman smiths equation

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