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A Modif ied Levere t t J -Fun c t ion fo r the Dune and Ya tesCarbona te F ie lds : A Case S tu dy

Ali A. Ga rr ouch

Dep ar tm en t of Pet roleu m En gin eerin g, Ku wai t Un iv ers it y, P.O . Box 59 69 , S af at 13 06 0, Ku wai t

R eceiv ed J an u ar y 11 , 19 99 . R evi sed Man u scr ip t R eceiv ed May 5, 19 99

Effective medium theory (EMT) ha s been used to model capillary pressur e ( P c) as a function of water saturation ( S w) in porous media. The E MT model results show that both the m agnitudeand profile of the P c - S w profile are strongly affected by changing the pore-size distributionparameters and the rock pore geometry. Tortuosity, which is a function of these parameters, isused in developing a drainage capillary pressure model based on data from the Dune and Yatesfields. Capillary pressure is normalized with respect to an average pore radius that includestortuosity. To m inimize the uncertainty caused by th e contact angle term, only helium/watercapillary pressure curves were considered. The wetting-phase satura tion is normalized withrespect to an asymptotic irreducible wetting-phase saturation. For the Dune and Yates fields,the development of this type of dimensionless model may be useful as an input for reservoirsimulation studies.

In t roduc t ion

When two immiscible fluids are in contact in the poresof a hydrocarbon-bearing rock, a discontinuity in pres-sure exists across the interface separating them. Itsmagnitude depends on the interface curvature at thepoint. The difference in pressure between the wettingand nonwetting phase at the interface is called capillarypressure. Capillary pressure in porous media is givenby the Laplace equation

where nw is the interfacial tension between the wettingand nonwetting fluids and R 1 and R 2 are t he principalradii of curvature of the fluid interface. The capillarypressure which would develop if two immiscible reser-voir fluids existed in the same capillary would be

Here r i s the radius of the capi l lary and nw i s t hecontact angle . Since the nonwet t ing phase tends tooccupy the larger accessible pores first, capillary pres-

sure curves make an excellent indicator of the sequenceof pore filling by th e nonwetting ph ase du ring a drain-age cycle. It is therefore, a good indicator of the poresize distribution. 4

Although the capillary pressure magnitude in mosthydr ocar bon rocks is not la rge, kn owledge of th e effectsof capillary forces is extremely importa nt in u ndersta nd-ing fluid displacement in these rocks. 14 Indeed, thedistribution of various fluids in the reservoir rock isgreatly influenced by capillary forces during all recoveryphas es. On a microscopic scale, capillary forces a reimportant in determining the amount of t rapped orresidual oil in either laboratory or field displacement.This makes capillary pressure one of the most basicrock-fluid characteristics in multiphase flow, just asporosity and permeability ar e th e most ba sic propertiesin single-phas e flow. Capillary pr essure measu rement sin the laboratory are also useful for estimating a varietyof importan t petrophysical parameters. The measure-ments can be used to es t imate rock wet tabil ity byevaluating the USBM index, or the Amott ratio, theirreducible water saturation, depths of fluid contacts,height above the free water level, and transition zonethickness. 9

Leverett 13 proposed th e J -function for scaling drain-age capillary pressur e curves. This function incorporatesthe effects of interfacial tension but uses a simplerelation for the average pore radius ( k / )1/2 which doesnot account for the tortuous nature of reservoir rocks.

(1) Ao, S.; Xie, X. SPE - 21890, 1990 .(2) Bae, W. Th e In fluence of Macropore Heterogeneity on th e Petro-

ph ysical Propert ies of Car bonates . Dissertation, The University of Texasat Austin, 1992.

(3) Bebout, D. G.; Lucia, F. G.; Hocott, C. R.; Fogg, G. E.; VanderStoep, G. W. Report of Investigations No. 168. Bureau of EconomicGeology, The University of Texas at Austin, 1987.

(4) Collins, R. E . Flow of Fluids Through Porous Materials . Researchand Engineering Consultants Inc., Englewood, Colorado. 1990.

(5) Cornell, D.; Katz, D. L. Ind. Eng. Chem. 1953 , 45 , 2145 - 2152.(6) Fatt, I. Trans. AIME. 1956 , 207 , 141 - 181.(7) Focke, J. W.; Munn, D. SPE - 13735, 1985.(8) Galloway, W. E.; Ewing, T. E .; Garret t, C. M.; Tyler, N.; Bebout,

D. G. Atlas of M ajor Texas Oil R eservoirs . The University of Texas atAustin, Bureau of Economic Geology Special Publication, 1983.

(9) Garr ouch, A. A. In Si tu . 1996 , 20 , 1- 9.(10) Garr ouch, A. A.; Lababidi, H.; Gharbi, R. J . Phys. Chem . 1996 ,

100 , 16996 - 17003.(11) Heiba, A. A.; Sahimi, M.; Scriven, L. E.; Davis, H. T. SPE -

11015, 1982.(12) Larson, R. G.; Scriven, L. E.; Davis, H. T. Chem. Eng. Sci. 1981 ,

36 , 57 - 73.(13) Leverett, M. C. Trans. AIME 1941 , 142 , 152 - 169.(14) Longeron, D. G.; Argau d, M. J .; Bouvier, L. S PE- 19589, 1989 .

P c ) n w

(1

R 1+ 1

R 2)(1)

P c )2 n w cos nw

r (2)

1021 Energy & Fu els 1999, 13, 1021 - 1029

10.1021/ef990005l CCC: $18.00 1999 American Ch emical SocietyPublished on Web 07/15/1999

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This is given by

It is commonly accepted that the J -fun ction sat isfacto-ri ly correlates da ta from u nconsolidated san ds andsandstones data from the same formation. I t has beennoted tha t t he J -function is not satisfactory whencapillary pr essure da ta are scaled for r ocks tha t exhibita grea t dea l of het erogeneity such as in carbonat e rocks.Thomeer 19 introduced an improved nondimensionalmodel for relatively homogeneous and isotropic consoli-dated san dstone rocks. He described mercury capillarypressure with a hyperbolic model expressed by

Here S b is porosity, S b is porosity times hydrocarbonsatura tion, and F g is a dimensionless pore geometricalfactor. Swanson 18 related the coordinates of the pointon the capillary pressure curve given by the 45 tangentline (Figure 1) to th e pore geometr ical factor F g and thethreshold pressure P d

Here, S wA and P cA are the coordinates of the point onthe P c curve given by th e 45 line. Equ ations 4, 5, and

6 sh ow that a knowledge of porosity, pressure, an dsatu rat ion a t th is point suffices to determine the en tirecapillary pr essure curve. The missing link, h owever, isthe fact that these coordina tes ha ve not been related tothe pore size distribution characteristics of the mediumand its tortuosity.

Ao and Xie 1 formu lated an alytically a dimensionlessrelationship for capillary pressure that included param-eters such as sorting coefficient ( l ), cha racteristic pore

neck r adiu s coefficient ( ), and a parameter which isa function of the threshold pressure ( P d). It is given by

Although this model seems to incorporate characteris-t ics of the rock pore s ize d is t r ibut ion, the methodproposed for es tima ting th e coefficients l and is basedon i tera t ive techniques that use leas t -squares a p-proximations using capillary pressure experimentaldata. The derivation is also based on the assumption

that the pore size distribution is normal. This makesits use l imited to only relatively homogeneous, wellsorted sandstone rocks.

Though the current capillary pressure models canyield reasonable predictions for some cases, they lack generality and give little insight into the ph ysical causesaffecting rock capillary pressure. There is still a rudi-ment ary un dersta nding of the link between th e pore sizedistribution characteristics and the capillary behaviorof porous media . The purpose of t h is s tudy is t oillustrate the importance of this link by simulating rock capillari ty using the effective medium theory and todevelop a draina ge capillary pressu re m odel by includ-ing heterogeneity an d pore size distribution para meter s.This model is dimensionless and is called a modifiedLeverett J -function in this paper. Its development isbased on experimental data of carbonate rocks from theDune and Yates fields. Its use is, therefore, limited tothese two fields.

D e s c r i p ti o n o f t h e E x p e r im e n t a l D a t a , S e t u p ,a n d P r o c e d u r e s

Data for simultaneous capillary pressure and resistivityindex measurem ents obtained from 25 carbonate core sa mplesfrom the Dune and Yates f ie lds were used as a basis fordeveloping the nondimensional relationship between capillarypressure and water saturation. These fields, which have beenthe subject of extensive geological and petrophysical studies,are well documented by Bebout et al. , 3 and by Galloway eta l. 8 The Dune field is located on the east side of the CentralBasin Platform in the Permian Basin, northeastern Cranecount y in west Texas. The Yates field is located on t he southend of the Cent ral Basin Platform. The m ajor pa rt of the Yatesfield is in Pecos county with the eastern tip extending acrossthe Pecos River into Crockett county.

A petrographic data summar y of the core samples used inthis study is presented in Tables 1 and 2. The samples featurea high degree of variation in permeability ranging from 6 to611 md. The porosity varied from approximately 11% to 32%.For such porosity interval, the cementat ion exponent ( m )indicated a rat her sm ooth a nd moderate var iation between 1.4and 2.3. This is typical behavior for l ime and dolomitegrainstones with intergranular porosity, as well as dolomites

(15) Panda, M. N.; Lake L. W. AAPG Bull. 1995 , 79 , 431 - 443.(16) Pirson, S. J. Geologic Well Log A nalysis ; Gulf Publishing:

Houston, 1983 ; pp 136 - 138.(17) Sharma , M. M.; Garr ouch, A. A.; Dunlap, H . F. Log Anal. 1991 ,

32 , 511 - 526.(18) Swanson, B. F. J. Pet. Technol. 1981 , 33 , 2498 - 2504.(19) Thomeer, J . H . M. J. Pet. Technol. 1960 , 12 , 73 - 77.

Figure 1 . A schemat ic of Swansons represen tat ion of capil-lary pressure.

J (S w) )P c k

nw(3)

S bS b

) exp{- F glog(P cP d)} (4)

F g )[ln(1 - S wA)]

2

2.303(5)

P d ) P cA(1 - S wA) (6)

P c n w cos nw k ) 2 (

- 1

ln(1 - S w - S wr(1 - S wr )(1 + )))1/ l

(7)

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with int ercrystalline porosity. 7 The core samples from theDune field were collected from a producing zone at a depth of 3343 to 3532 ft a nd r epresent thr ee different facies. These arepellet grainstone, crinoid packstone/grainstone, a nd fusulinidwackestone. The latter facies consists of dolostone with an-hydrite and gypsum cement. The pore system is frequentlyvuggy and moldic, and its matrix has intercrystalline porosity.The former facies is a medium to dark brown dolostone withminor an hydrite n odules an d cements. It s porosity is t ypicallyinterpa rticle and moldic. The crinoid pa ckstone/gra instonefacies consists of a light-colored to medium gray and browndolostone with common gypsum cement and anhydrite nod-ules. Its porosity is typically vuggy, moldic, interparticle, andintercrystalline. Core sa mples from the Yates field are gener-ally characterized by a vuggy cavernous porosity. Dolomite isthe dominant diagenetic mineral in th ese rocks.

The cores were cleaned by toluene and xylene and then werecleaned for the second time by the Dean - Stark extractiontechnique using a mixture of 78% chloroform a nd 22% metha-nol (by volume). Extraction was conducted with fresh solventwhich was continu ously distilled an d condensed before redis-tribution to the extra ctors used. Draina ge capillary pressure -satur ation relationships were m easured u sing a special porousplate setup featur ing a number of experimental precautionstha t ensur e both pr ecision an d accura cy of the m easur ements.The porous plate used (Figure 2) consists of a closed cylinderwith a 2 mm thick ceramic porous disk which permits thewetting phase to drain from th e sam ple. The porous plate isimpermeable to the nonwetting phase a s long as t he th resholdpressure of the m embrane is not exceeded. The pressure onthe nonwetting fluid is increased stepwise to the thresholdpressure of t he porous plate , and the expelled volume ismeasured. Starting at a low pressure, fluid is displaced fromthe largest pores, and as the pressure is increased fluid isdisplaced from the smaller pores, progressively. At each step,

capillary equilibrium must be reached before a reading is madeof the produced volume of the wet t ing fluid. At higherpressures, the equilibrium outflow volume was estimated byplotting t he outflow measu rement versus 1/time. As time goesto infinity, 1/time becomes zero, and a n extrapolated value isused for the equilibrium volume displaced which is used tocalculate the equilibrium wat er sa tura tion. Measurementswere performed at a constant room temperature of 70 Fthroughout the experiment.

Cores used were 1 in. in diameter and 1.5 in. long. Thebottom of each core, which contacts the ceramic plate, isground with fine sand paper to make it as flat as possible.This helps in maintaining good capillary contact with theporous plate. Wet filter paper is placed between the sampleand the porous plate to ensure capi llary contact . Waterevaporation was minimized by placing mineral oil on top of the water column in the measuring pipets. Helium gas wasused as the nonwetting fluid. Helium was used because of itslow solubility in water. A synthet ic reservoir brine composedmainly of sodium chloride was used as the wetting phase. 2

To avoid polar ization and conta ct resista nce effects, the four -electrode t echnique was used t o measure rock resistivity atthe sam e time the outflow behavior was m onitored. Two copperO-ring voltage probes were placed in the middle of the coresample and spaced 0.5 in. apart. A brass mesh was used forthe t op curr ent electrode and t he m ain body of the core h olderis the bottom current electrode. The current passing throughthe rock was m easu red by pla cing a 1000-ohm (0.1% precision)resistance in series with the rock sample. The voltage differ-ence in t he middle part of the core sample was measur ed usinga voltmeter tha t h as 2 m egohms int ernal impedance. This high

impedance ensures accurat e voltage m easurements.9,17

To ensure one-dimensional displacement from the top of thecore to its bottom, the core sides were painted with Lucite,which consists of dissolved Plexiglass in chloroform. Lucite actslike glue and is favored over epoxy since it does not reversethe core sample wettability. A detailed description of t heexperimental procedures and a complete data set of capillarypressur e and resistivity measuremen ts for th ese core samplesis provided by Bae. 2 This data set has been used in this studyto develop a modified Leverett J -function. The method of ana lysis of these da ta as well as development of this J -functionare presented later in the text. The following section, however,illustrates the effects of pore size distribution parameters onthe P c - S w curves which is a centr al idea in th e developmentof the em pirical m odel.

Ta b le 1 . P e t r o g r a ph i c D a t a S u m m a r y o f C o r e S a m p l e sf rom the Dune F ie ld

depth (ft) k (md) (%) r j m m F

3343 a 67 13.7 2.38 4.67 1.87 41.33347 53 15.9 2.30 3.73 1.91 33.33348 156 16.5 2.01 5.49 1.77 24.53364 169 22.0 2.77 6.84 1.68 35.03365 143 23.0 2.03 4.48 1.96 17.83398 b 62 19.0 1.66 2.66 1.61 14.43419 12 18.0 1.67 1.21 1.60 15.6

3453c

50 21.6 1.88 2.54 1.83 16.43455 56 14.3 2.22 3.91 1.82 34.63460 67 18.4 1.71 2.89 1.63 15.83462 191 22.0 1.52 3.98 1.55 10.53470 6 11.5 3.35 2.15 2.12 97.73476 7 12.0 2.01 1.34 1.66 33.73501 12 14.7 1.87 1.50 1.65 23.83502 164 21.8 1.40 3.41 1.44 9.03504 127 16.8 1.70 4.14 1.59 17.23531 41 11.9 2.96 4.88 2.02 73.63532 14 17.5 1.69 1.36 1.60 16.4

a Beginning of pellet grainstone facies data. b Beginning of fusulinid wackestone facies data. c Beginning of crinoid pa ckstone/ grainstone facies data.

Ta b le 2 . P e t r o g r a ph i c D a t a S u m m a r y o f C o r e S a m p l e sf rom the Yates F ie ld

depth (ft) k (md) (%) r j m m F

1376.5 387 29.6 2.38 6.79 2.30 19.21377.5 305 29.0 3.27 9.42 2.29 36.91380.5 611 31.5 2.75 10.8 2.41 24.01389.3 296 23.9 2.09 6.54 1.87 18.31621.4 123 17.9 2.75 6.42 1.93 42.31639.8 55 19.6 1.86 2.76 1.93 17.61653.8 356 24.5 1.93 4.73 1.92 15.2

Figure 2 . A schematic view of the porous pla te cell.

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Model ing Capi l la ry Pressure Prof i les Us ing theEffec t ive Medium Theory

The effective medium theory (EMT) formulation fors imulat ing flu id t ranspor t in porous media i s wel ldocumented by Heiba et al. 11 and by Wang and Shar-ma .21 The following is a n ada ption of EMT for m odelingcapillar y pressu re of porous rocks. The EMT model wasdeveloped to quantify the individual effects of rock

stat istical par amet ers on th e capillary pressur e profile.The porous medium is represented by a network of randomly distributed pore throats (bonds) and porebodies (sites). Adjacent pore bodies are connected bypore throats. The local coordination number Z , whichis the nu mber of pore t hroats connected to a pore body,defines the connectivity. A larger value of Z impliesbetter conn ectivity. In th e limit, when Z goes to infinity,the model is identical to the bundle of capillary tubesmodel.

The elementary pore throat segment is assumed tobe a converging - diverging capillary tube characterizedby a throat radius ( r t) and a pore-body radius ( r b). Thepore thr oat is assumed to ha ve a sinusoidal shape a nd

the pore body can be approximated by a cube of size 2 r b

A schematic of the representation of a pore segmentis shown in F igures 3 a nd 4. The ra tio r b / r t is referredto as the aspect ra t io (a r). For a given a r value, r b isspecified by r t . As a r goes to 1, the pore reduces to acylindrical tube with constant radius. The relationbetween thr oat radius r t and pore length L is allowedto be of a form used by Fatt 6 in his network resistormodel:

where c and are arbi t rary constants . The actualporous m edium is r eplaced by a three-dimensionalnetwork ha ving pore th roats distr ibuted according to agiven probability function, and this network is in turnreplaced by a n effective network in which all porethr oats ha ve the sa me conductivity. For describing thestat istical var iation of pore sizes, a normalized Gaussian

throat-size distribution is used here:

This type of d is t r ibut ion would be expected i f thedepositiona l en vironmen t provided t he only source forgrain sorting. During the drainage process it is assumedth at oil and water occupy distinct flow chan nels (exceptfor the presence of nonflowing thin films).

Modeling of the draina ge process is based on a fluidfilling sequence. For example, in accordance with cap-illarity, the n onwetting phase flows in large pores a ndthe wetting phase flows in small pores. For the case of a st rongly water -wet rock, water forms th in films on thesurfaces of the pores drained by oil. As an example, letus consider primary drainage in water-wet rocks. Be-cause oil is the nonwetting phase, it enters the largestpores first. The fraction of pore segments allowed bycapil lar i ty to be occupied by oil a t a s tage dur ingdrainage is characterized by r d, and is given by

X d is the fraction of pore space allowed to be invadedby the nonwetting pha se during dr ainage. The fractionof pore segments accessible to the nonwetting phase(actually occupied by the nonwetting phase) is given by

Here X a( X d) is th e accessibility fun ction. For t his st udy,th e a ccessibility function of th ree-dimens iona l net workswas approximated by the accessibility function for aBethe tr ee with th e sam e percolation t hresh old. This isa good approximation and is useful to apply since X afor a Bethe tree can be calculated analytically. 12

The computational procedure for wetting-phase satu-rat ion a nd capillary pr essure is as follows: First we fix X d an d comput e the corresponding r d using eq 11. Then

(20) Toledo, P. G.; Scriven, L. E.; Davis, H. T . SPE Form. Eval. 1994 ,9 , 46 - 54 .

(21) Wang, Y.; Sharma , M. M. Presented in t he 29th SPWLA Annu alLogging Symposium, J une 5 - 8, 1988 , paper G.

Figure 3 . A schematic of a pore segment.

r ) r b - (r b - r t)sin x L

0 e x e L (8)

L ) cr t (9)

Figure 4 . A schematic of a two-dimensional representationof a pore segment.

f (r t) ) 2

1

[1 + erf ( 2)]exp[- 12(r t - )

2] (10)

X d ) r d

f (r )dr (11)

X nd ) X a( X d) (12)

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we comput e X n d using the following equa tion:

Here X * is solved for by setting

and b ) Z b - 1 where Z b is the equivalent Bethe treecoordination number. This is obtained by sett ing thenetwork percolation threshold equal to that of Bethetree. The water satu ration S w (drainage cycle an d wat er-wet rock) is given by

For a t hin film t hickness h , the volume of the t hin filmV f is given by

The volume of one pore segment is given by

The capillary pressure P c between the flowing fluids isthe key parameter for displacement. For strongly water-wet rock, the capillary pressur e is given by eq 2 with r replaced by r d. The primary drainage process ends whenthe displaced wat er loses its ability t o flow. When thishappens, water is left in the pores with throat radiuse r cd . In this analysis, r cd is determined by the percola-tion thr eshold X c ) 2/ Z , and r cd is solved for by setting

A relatively low value of coordination number ( Z varying between 5 and 15) and a relatively high valueof aspect ratio ( a r varying between 3 a nd 10) were us edto mimic qualitatively the behavior of porous media.Low coordination number values and high aspect ratiovalues are l ikely to represent a carbonate mediumheavily altered by diagenesis. 20 In general, carbonaterocks have an attra ctive porosity with either poorlyconnected segments or with pore bodies connectedmainly by small pore throats. 20 For varying the coef-ficient of skewness ( ), tr un cated log-normal pore sizedistributions were used such as

In th ese simulations r min ta kes th e value of 1.23 m andr ma x is equal to 49.1 m.

The model results assert the unequivocal relationshipbetween the pore size distribution characteristics and

the P c-

S w profile which appear s t o be sensitive to th emean ( ) , the standard deviation ( ) , the aspect ratio(a r) and the coordination number ( Z ). Figure 5 showsthe effects of aspect ratio variation which are morepronounced at low satura tions th an a t high satu rations.A change in both profile and magnitude takes place asa r is changed from 3 to 10. Figure 6 shows histogramsthat summarize the effects of , , , a n d Z on themagnitude of capillary pressure. These effects areassociated with a change in profile in some cases. Thehigher the value of is, the flatter the profile of P c -S w becomes and the smaller the value of P c becomes.Figure 6a suggests that the shape of P c - S w curveremains unchanged as is reduced from a value of 5 to

2.5 m; however, the capillary pressure increases invalue by app roxima tely 40% for a fixed sat ur at ion. Thisis true, since a rock with a large mean value indicatesthat all rock pore throats have larger values causingentry pressure for every pore t hroat to decrease. Theeffect of th e standar d deviation is, similar with anapproximat ely consta nt 10% increase in capillary pres-sure values a s t he st andard deviation is reduced from1.5 to 1 m. The effect of the skewness coefficient is,however, more pronounced at low S w values than at h ighS w values. As the coefficient of skewness ( ) varies fromzero (corresponding t o a bell-sha ped thr oat-size distri-bution) to a value of 0.5 (corresponding to a tru ncatedlog-normal distribution), t he capillary pressure in-

creased unevenly by as much a s 17% at high S w valuesto reach an increase of approximately 33% at low watersatu rat ion values. A rock with a small value of yieldsa flatter P c - S w profile. This is becaus e th e larger t hevalue of is , the more l ikely pore throats with smallsizes will be invaded by the nonwetting phase.

In summary, statist ical parameters for rock throatsize distribution have a significant effect on both themagn itude a nd p rofile of capillary pressu re curves. Thatbeing the case, capillary pressure is scaled with respectto tortuosity which, as will be shown in t he n ext section,turns out to be a direct function of , , and . Thefollowing section details t he scaling procedures an dintroduces th e m odified Leverett J -function.

X nd ) X d[1 - ( X * X d)2 b / b- 1] (13)

X *(1 - X *)3 ) X d(1 - X d)3 (14)

S w ) 1.0 -

X nd X d

r d

(V p - V f ) f (r )dr

0

V p f (r )dr (15)

V f )

2 hL [r b- 2

(r b-

r t)] (16)

V p ) [r b2 - 4 (r b - r t)r b +12

(r b - r t)2]Cr tR +

16 r b3

Z (17)

X c ) 0r cd f (r )dr (18)

f (r t) ) 2exp[- 12(ln (r t

+ r mi n ) )

2](r t + r mi n ) {erf [ln (r ma x) 2 ]- erf [

ln (r mi n )

2 ]}(19)

Figu re 5 . Comparing capillary pressure response for twowater-wet simulated porous media having different aspectratio values.

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S c a l in g P r o c e d u r e s

Rock pore geometry usually changes drastically upondiagenesis because of compaction and cementation.Compaction causes a reduction of the available inter-granular porosity while cements, depending on theirchemical an d crystallographic properties, fill th e int er-granular pores and increase the porous medium specificsurface ar ea a nd tortuosity. The end result is a majortra nsforma tion in both th e ma gnitude an d profile of thecapillary pressure - saturation curve of the rock. Theeffective tortuosity ( ) of a consolidated permeablemedium is deduced from P anda and Lake 15 as

Here, and k are the poros ity and permeabil ity,respectively; P b, P l, P f are the amounts of pore-bridging,

pore-lining, and pore-filling cements, respectively, ex-pressed as a fraction of total solid volume; a vb , a vl , a vf are the specific surface a reas of pore-bridging, lining,and filling cement, respectively; Dh p, C Dp , and representthe statistical parameters of the particle size distribu-tion, i.e., th e m ean grain size, its coefficient of varia tionwhich is the mean divided by t he st andard deviation,and the skewness coefficient of the pa rticle size distri-bution, respectively. The above equation gives a thor-ough rep resen ta tion of flow tortuosity of a consolidatedpermeable medium since it uses a significant numberof rock petrographic properties characterizing the throat-size distribution, porosity, and permea bility. In a n effortto separate the effects of th roat-size distribution, amodified Leverett J -function is proposed for drainagecapillary pressures. The capillary pressure is, therefore,normalized with respect to an average pore radius forthe rock that i s a funct ion of tor tuosity. A s implemat erial balance, using the capillary tu be model, leadsto the following expression for this mean pore radius:

Figure 6 . (a) Histogram showing the relative change in capillary pressure as varied from 5 to 2.5 m. The base case data isa r ) 3, h ) 0.001 m, Z ) 10, ) 2.5, ) 1.5, and ) 0.0. (b) Histogram showing the relative change in capillary pressure as varied from 1.5 to 1.0 m. The base case data is a r ) 3, h ) 0.001 m, Z ) 10, ) 2.5, ) 1.5, and ) 0.0. (c) Histogram

showing the relative change in capillary pressure as varied from 0 to 0.5. The base case data is a r ) 3, h ) 0.001 m, Z ) 10, ) 2.5, ) 1.5, and ) 0.0. (d) Histogram sh owing the r elative chan ge in capillary pr essure as Z varied from 15 to 5. The basecase data is a r ) 3, h ) 0.001 m, Z ) 10, ) 2.5, ) 1.5, and ) 0.0.

) Dh p2 3( C 3Dp + 3C

2Dp

+ 1)2 /

{2k(1 - )2[6(1 + C 2

Dp )(1 - 0)

(1 - )+ (a vbP b + a vlP l +

a vf P f ) Dh p( C 3 Dp

+ 3C 2Dp + 1)]2}(20)

r j ) 2 2k (21)

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The calculated mean pore radius for the core samplesused had an average of 4.3 m and a st anda rd deviationof 2.4 m (Tables 1 and 2). This is consistent withcarbonate rocks that have a positively skewed pore sizedistribution. The modified Leverett J -function is nowgiven by

Pirs ons m eth od 16 has been used in this analysis forestimating tortuosity values using rock resistivity mea-surements (Table 1). His relationship, which has beenwell substantiated by experimental evidence, is givenby 2 ) F , where F is th e rock forma tion factor definedas t he r atio of the fully brine-satu rat ed rock r esistivityto th e br ine r esistivity. According to Collins, 4 tortuositytakes a value of 1.5 for unconsolidated sands. For thecore samples u sed, tortuosity averaged 2.15 with asta nda rd deviation of 0.53. A ma ximum t ortuosity valueof 3.35 was obtained (Tables 1 and 2). There is still somecontroversy in the petroleum industry over adequatemodels that bes t represent reservoir rock tor tuos-ity. 5,16,22,23 Table 3 gives a summary of these models.For th is par ticular carbonate rock data set , Pirsonsmodel yielded the least data scatter.

Contact angle measurements, performed on mineralcrystals, usua lly do not reflect t he actual wettabili tyconditions of the rock. These measurements are verysensit ive to contaminants present in the nonwettingphase and to rock roughness. To remove the uncertaintycaused by th e conta ct-angle term in eq 22, only P c - S wdata generated using helium and water were used inthis st udy. For these cleaned core sam ples (Tables 1 and2), in the presence of helium, wat er a cts as t he wettingphase and the term cos nw is a pproximat ely one.

Drainage capillary pressure curves are transformedinto a plot of the modified Leverett J -function versus areduced wetting pha se sa tura tion S w

/ given by

All of the capillar y pressur e curves used in t his an alysisreached vertical asymptotic lines a t which th e wettingphase sa tura tion values rema ined consta nt even t houghcapillary pressu re values k ept on increasing. 2 The wet-t ing phase satu ration at the a symptotic line was ta kent o b e S wr . A Levenberg - Marquardt algorithm 10 isimplement ed to find a nonlinear model that adequat ely

describes the relationship between J and S w/

. T h emodel proposed for the modified J -fun ction is a s follows:

Here, R, , , , , are all posit ive constant s a nd aregiven in the Appendix. As shown in Figure 7, the model

appears t o mimic the Dune an d Yates field data reason-ably well . Originally these da ta appear to ha ve greatvariation in capillary p ressur e response which sh oweda lot of scat ter us ing the convent ional Leveret t J -function (Figure 8). A plot of modified J versus dimen-sionless wetting pha se satura tion on semilog pa per(Figure 9) illustr at es vividly th e conformity of th e modelto th e physics of capillar y beha vior in porous rocks. Themodel predicts increasing J values as the wetting phasesatura tion decreases. As the dimensionless wetting-phase saturation approaches zero, the J -function in-creases asymptotically to large values. At low J values,the m odel featu res a secondary plateau which indicates

(22) Winsauer, W. O.; Shearin, H. M.; Masson, P. H.; William, M. Bu ll. Am . As soc. Petr ol. Geol. 1952 , 36 , 253 - 277.

(23) Wyllie, M. R. J.; Spangler, M. B. AAPG Bull. 1952 , 36 , 359 -403.

J ) 2 2k P c n w cos nw (22)

S w/ )

S w - S wr1 - S wr

(23)

J ) R + exp[- + S w/ + S w/ + exp(S w/ ) -

sinh(S w/ )] (24)

Table 3 . Tor tuos i ty Models

authors yea r model

Wyllie and Spangler 23 1952 ) (F )2Winsauer et al . 22 1952 2 ) (F )1.2Cornell and Katz 5 1953 ) F Pirson 16 1983 2 ) (F )

Figu re 7 . The proposed J -function versus dimensionlesswetting phase saturation. A comparison between proposedmodel and experiment al data from th e Dune and Yates fields.

Figure 8 . Leverett J -function versus wetting phase satura-tion for the Dune and Yates fields.

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a bimodal throat-size distribution that is typicallyassociated with carbona te r ocks.

S u m m a r y

The s tudy presents an empir ica l nondimensionalmodel for capillary pressure based on data from theDune a nd Yates fields generally cha racterized by vuggy,moldic, interparticle, and intercrystalline porosities. Themodel account s for t he effects of inter facial ten sion an dpore size distribution. To minimize the uncertaintycaused by the contact angle, only water-wet sampleswere considered in th is stud y. The pore size distributionis accounted for by including rock tortuosity obtainedfrom electrical mea sur ement s. Tortuosity, a direct func-tion of the coefficients of variation and skewness of thepore size distribution, is included in scaling the modifiedLeverett J -function. The r esult is a norma lized capillarypressure that is independent of pore-size distributioneffects. The proposed formulation for the J -function isdependent on the wetting-phase saturation and involvesother petrographic parameters that can be obtainedeither experimentally or from field da ta. These consistof porosity, irredu cible water sat ura tion, permeability,and resistivity formation factor. This em pirical modelcan be u sed to provide a bas is for genera t ing onerepresentat ive capillary pressure curve for reservoirsimulation studies.

The dependence of capil lary pressure profi le onstatist ical parameters such as th e mean thr oat r adius,its st and ard deviation, an d t he coefficient of skewnessof the pore size distribution is illustr ated qualitat ivelyusing a n effective medium theory model. The EMTmodel i s free of spat ia l b ias and accounts for theexistence of thin films.

Acknowledgment . The author thanks Kuwait Uni-versity for its financial support of project EP-015.

Appendix : Cons tan ts for Eq 24

R ) 0.4 ) 74.8393

) 162.402 ) 0.005 ) 77.7771 ) 258.372

N o m e n c l a t u r e

a vb specific surface area of pore-bridging cementa vl specific surface area of pore-lining cementa vf specific surface area of pore-filling cementC pore-throat constantC Dp coefficient of var iat ion Dh p mean pore diameter f normalized pore size distr ibutionF resistivity forma tion factorF g dimensionless pore geometrical factorh thin film thickness J dimensionless capillary pressurek rock permea bility L pore lengthl sortin g coefficientm cementation exponentn saturation exponentP b amount of pore-bridging cement, a fraction of total

solid volumeP c capillary pressureP d threshold pressureP f amount of pore-filling cement, a fraction of total

solid volumeP l amount of pore-lining cement, a fraction of total

solid volumeP cA Swansons capillary pressur e pointr radius of capillary tuber j mean pore radiusr b pore body radiusr cd maximum radius that nonwetting phase can pen-

etrater d tube radius controlling drainage process R1 principal radius curvature of the fluid interface R2 principal radius curvature of the fluid interfacer min minimum throat radiusr ma x maximum throat radiusr t pore throat radiusSb porosity times h ydrocar bon satu ra tion in Thomeers

notationSb porosity in Thomeer s nota tionS w wetting phase satura tion (water sat urat ion)S wA Swansons wetting pha se satu ra tion pointS nw nonwetting phase satura tionS wr irreducible wetting phase saturationV p volume of a pore segmentV f volume of a t hin film X c percolation thr eshold X d fraction of pore space allowed to be invaded by the

nonwetting phase X nd fraction of pore segments accessible to the nonwet-

ting phase X a accessibility function Z coordination n umber Z b equivalent Bethe tree coordination number

Greek SymbolsR constant in the modified J -function constant in the modified J -function constant in the modified J -function rock porosity coefficient of skewn ess constant in the modified J -function constant in the modified J -function mean throat radius

Figure 9 . The proposed J -function model versus dimension-less wetting pha se satu ration.

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constant in the modified J -function throat size standard deviation b a function of coordina tion nu mber nw interfacial tension between wetting and nonwetting

fluids rock tortuosity

nw contact angle a pore-throat constant

a function of the threshold pressure characteristic pore-neck radius coefficient

EF990005L

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