SPE-102249-PA

Relative Permeability CoupledSaturation-Height Models on the Basis of

Hydraulic (Flow) Units in a Gas FieldMaclean O. Amabeoku, SPE, David G. Kersey, SPE, Rami H. BinNasser, SPE, and Ali R. Belowi, SPE, Saudi Aramco

SummarySaturation/height functions on the basis of unique flow units havebeen developed as part of an integrated petrophysical analysis of agas field. Furthermore, coupling the saturation/height functionswith appropriate relative permeability models has effectivelyquantified hydrocarbon saturation, classified producibility of in-tervals, and defined critical water saturation. The results show thatlinking depositional and diagenetic rock fabric with flow units andthen linking the flow units with zones that have similar core cap-illary pressure and relative permeability relationships have en-hanced the utility of the saturation models. The saturation/heightfunctions provided more-accurate water saturation in the studyfield, and potentially they can overcome uncertainties associatedwith log interpretation by use of Archie or shaly-sand models.

The saturation/height models were developed from core capil-lary pressure (Pc) data to calculate water saturation vs. depth,which is independent of logs. The relative permeability modelswere obtained from special-core analysis (SCA). Consequently,the core-based saturation/height functions can be useful in thecalibration of log-based petrophysical models and with relativepermeability can also be used to estimate water/gas ratios (WGRs)and critical water saturation.

Capillary pressure and relative permeability curves from SCAstudies were distributed into corresponding flow units, on the basisof the calculated flow-zone indicators (FZIs). Saturation/heightfunctions were then developed for each unit and were used tocalculate water saturation in the study field. The most accurateflow-unit-based saturation model that evolved is a function only ofporosity and of height above the free-water level; it does not re-quire permeability in its application; and it performed better thanthe Leverett J-function in this field.

Coupled with hydraulic unit (HU)-based relative permeabilitycurves, the saturation models may provide more comprehensivepetrophysical interpretation in gas-bearing formations and mayhighlight potential differences in reservoir producibility.

IntroductionModels used to calculate water saturation from logs in this gasfieldcase study include the deterministic Archie equation (Archie1942), Waxman and Smits (1968), and an optimizing dual-water(DW) model presented by Eyvazzadeh et al. (2003).

Extensive laboratory measurements conducted by Amabeokuet al. (2005a) and Efnik et al. (2006) show variability of the satu-ration (n) and cementation exponents (m) vertically within the welland from well to well. This makes the use of single-valued (aver-age) parameters untenable.

The presence of illite, even though in small quantities, hasnecessitated the use of the DW model routinely to calculate po-rosity and water saturation in this field. Illite is filament-like, non-swelling clay that coats grain surfaces. It is thought that the DWmodel, which was optimized for this formation, provides more-accurate water saturation. The model uses as input all available

logs, mineral analyses, and electrical parameters, and it solves forclay-bound water and free fluids in the flushed and unflushedzones of the wellbore.

The relationship between capillary pressure (Pc) and watersaturation offers a means to estimate water saturation vs. depth,which is independent of wireline logs and provides the ability tocalibrate log-derived saturations. Saturation/height models, ifimplemented successfully, would also minimize, or eliminate, theuncertainties associated with electrical-parameter measurements.Some uncertainties that can impact the accuracy of electrical pa-rameters include electrode configuration, saturation and resistivityequilibration, and incomplete core cleaning. The experimental pro-tocols should also be designed to determine intrinsic saturation andcementation exponents (n* and m*, respectively) and not apparentproperties in shaly formations such as those discussed in this paper.

All that is required to develop these models are routine coreproperties and capillary pressure data.

Several saturation/height models have been proposed and usedin the petroleum industry with different degrees of difficulty andamount of data required. The Leverett J-function (Leverett 1944)is one of the most commonly used and is written as

JSw =0.2166Pc cos

k

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where Pc is the capillary pressure, is the contact angle, and isinterfacial tension between the fluid pair. The ratio (k/) implicitlyrecognizes the reservoir quality. A log-log plot of J(Sw) vs. Sw isnearly linear (Ibrahim et al. 1992; Cuddy 1993), as illustrated inFig. 1, and will yield a relationship of the form

Sw =a

JSwb, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

where a and b are regression coefficients. Substituting the func-tional form of J(Sw) in the height domain and solving explicitly forSw, Eq. 2 transforms to

Sw =a

0.2166Pcres cosres

kb . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

Cuddy (1993) developed the FOIL function by use of logdata from the Southern North Sea in net-reservoir intervals andaway from bed boundaries. Cuddy did not define the acronym. Healso demonstrated that FOIL can be applied equally to core cap-illary pressure data and does not require permeability. It providescorrelation between the bulk volume of water (BVW), which is theproduct of water saturation and porosity, and the height abovefree-water level. Cuddy (1993) presents the FOIL function as

BVW = Sw =A

HB, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

where H is height above free-water level and A and B are fittingparameters. The formulation was derived from the J-function. Aporosity/permeability relation was used to substitute for k in theJ-function. The detailed derivation is provided in Cuddy (1993).Cuddy showed examples in his paper to demonstrate that, for theNorth Sea gas fields he investigated, the BVW at a certain height

Copyright 2008 Society of Petroleum Engineers

This paper (SPE 102249) was accepted for presentation at the SPE Annual TechnicalConference and Exhibition, San Antonio, Texas, 2427 September, and revised for publi-cation. Original manuscript received for review 25 June 2006. Revised manuscript receivedfor review 16 May 2008. Paper peer approved 1 September 2008.

1013December 2008 SPE Reservoir Evaluation & Engineering

above the free-water level was virtually independent of the rockproperties and almost entirely dependent on the height.

The development is based on the observation that the productof porosity and saturation (Sw) will remain virtually constant inzones of irreducible water saturation and will increase toward thefree-water level. Although explicitly independent of permeability,Cuddy suggests separate functions may be used to refine the satu-ration/height relationships within a particular geological zone,lithofacies, or HU with fundamentally different pore geometries.After an appropriate function has been derived, water saturationcan then be calculated by use of

Sw =A

HB1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

Although the capillary pressure behavior is controlled by bothporosity and permeability, as recognized by the J-function, theimplementation of saturation-vs.-height functions, which directlyinclude permeability as a rock-quality discriminator, requires arobust permeability predictor. Permeability and porosity data re-quired to compute saturation with the J-function may be availablein cored wells. In such wells, a J-function can be developed andvalidated against the core data. Only a limited number of wells ina field are cored. Porosity can be obtained from logs, but perme-ability would have to be predicted and mapped in a 3D geocellularmodel in the field. This would be an arduous task. The choice ofthe FOIL function over the J-function would then be a matter ofexpediency. Cuddy suggested constructing separate functions fordifferent geologic units or lithofacies with distinct and coherent/k relationships.

Modified FOIL Function. Amabeoku et al. (2005a) first incor-porated HU zonation to overcome the initial limitations of theFOIL function (Cuddy 1993) and observed an improvement inwater-saturation calculation. Mitchell1 had suggested the FOILwater-saturation model could be modified to fit the core capillarypressure data by adjusting the porosity term. Further improvementin water saturation was obtained with the data from the gas field

under study when a porosity exponent was introduced. The result-ing empirical equation was presented by Amabeoku et al. (2005a)as the Modified FOIL function:

Sw =A

HB1C

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

A, B, and C are parameters that are obtained by robust nonlinearregression and other optimization schemes.

Geology of Study FieldPrudden2 described the study field as fault-bounded in the westand north and dipping gently away to the south and east. Thereservoir section consists of terrestrial clastics of Permian-Carboniferous age. The field is divided into two reservoir units: anapproximately 200-ft-thick upper Reservoir A and a lower Reser-voir B. The two reservoir units are separated by a siltstone unit ofvariable thickness (10 ft up to 115 ft). Stratigraphically, the rocksof Reservoir B continue below the known gas/water contact.

Reservoir A consists of an eolian sequence of fine- to coarse-grained, highly laminated, crossbedded dune sands and sheetsands. The dune and sheet sands are interspersed with interduneplayas of very fine silty sand with some paleosols near the top ofthe unit. The dunes prograde east (wind blowing west to east) as aseries of transverse dunes, which have a strong permeability an-isotropy. The original fabric is overprinted with a significantamount of quartz-overgrowth cements that have reduced porositysubstantially. Despite a composition of more than 95% quartz, theinternal architecture of Reservoir A is complex. The siltstone unitis also geologically complex and comprises a series of very finesilty sandstones and pure siltstones. This unit is a barrier betweenReservoirs A and B.

Reservoir B is dominantly sandstone of a fluvial origin, prob-ably braided-stream deposits related to glacial outwash. Theserocks are characteristically quartz-rich sandstones withextensive quartz overgrowths (>98% quartz). The reservoir sectionincludes very frequent, horizontal to low-angle stylolites withassociated tension gashes. Although the porosity is often slightlylower than that of Reservoir A, the lower clay content tends toimprove the permeability, and Reservoir B is regarded as the prin-cipal reservoir in this field. Although local reservoir architecture isalso quite complex, Reservoir B is a more continuous sequencethan the overlying Reservoir A. In general, the highest-porosity/-permeability units have the lowest amounts of quartz-overgrowthcements.

Hydraulic (Flow) Unit ZonationEbanks (1987) defined an HU as a volume of the total reservoirrock within which geologic and petrophysical properties that affectfluid flow are internally consistent and predictably different fromproperties of other rock volumes. It is a zone that is continuousover a defined volume of the reservoir and has similar averagerock properties that affect fluid flow and has similar bedding char-acteristics. Distribution of flow units is related to facies distribu-tion, but flow-unit boundaries do not necessarily coincide withfacies boundaries.

The HU-zonation scheme devised by Amaefule et al. (1993)was used to compute and distinguish the different flow units. Theequation is given as

logRQI = logz + logFZI, . . . . . . . . . . . . . . . . . . . . . . . . . (7)where

z =

1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

RQI = 0.0314k

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

1 From personal communication with P. Mitchell, 2004, PGLs capabilities with reference totight gas field study. 2 From personal communication with M. Prudden, 2005.

Fig. 1Illustration of development of equation to simplify rela-tionship between Sw and J-function.

1014 December 2008 SPE Reservoir Evaluation & Engineering

The FZI is calculated by

FZI =RQIz

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

The derivation of these simplified equations by Amaefule et al.(1993) originated from the generalized form of the Kozeny-Carmen (Kozeny 1927; Carmen 1937) relationship among poros-ity, permeability, surface area, tortuosity, and pore shape factor.Amaefules derivation shows that a log-log plot of RQI vs. z willyield a straight line with a unit slope. The intercept of this straightline at z1 is the FZI. Core samples that lie on the same straightline have similar pore-throat characteristics and therefore consti-tute a flow unit. Samples with different FZI will lie on otherparallel lines.

Fig. 2 shows the plot of reservoir-quality index vs. z calcu-lated from porosity and permeability data from some wells todetermine the HUs. Five units were identified, but the fifth HUwas not considered net-reservoir rock. Therefore, the samples inHU-5 were excluded from further analysis. Samples that were usedfor the determinations of capillary pressure and electrical param-eters are denoted by the red squares. Basic properties and com-puted hydraulic units of these samples are shown in Table 1.

Geologic Characteristics of SamplesConstituting the HUsThe clay content is generally low and ranges from 1% to a maxi-mum of 13% on a bulk basis. Initially, the clay fraction was sepa-rated and analyzed. The dominant clays from the analysis of theclay fraction are kaolinite, dickite, illite/mica, and chlorite.

HU-1. Fig. 3 shows representative thin-section (TS) photomicro-graph, scanning electron micrographs (SEMs), and pore-throat-size distribution of the rock that constitutes HU-1. This unit ischaracterized by medium-grained, moderately to poorly sortedquartz arenites. All the detrital grains are subrounded quartz withauthigenic clays comprising approximately 3% of the grain vol-

ume. Most of the clays are pore-filling and grain-replacing kaolin-ite. Lesser amounts of authigenic illite/mica and illite/smectite arealso present. Quartz overgrowths are the only cements. TS porosityis characterized by well-developed intergranular pores. Minoramounts of intraparticle and grain-moldic dissolution porosity arealso present. The pore throats can be described as macropores,with a large proportion of the pore throats in the 5- to 10-m range.

HU-2. This unit is a lower medium-grained, well-sorted quartzarenite. All the detrital grains are rounded to subrounded quartz,with authigenic clays comprising approximately 1% of the grainvolume. Most of the clays are pore-filling illite with lesser amountsof pore-filling kaolinite. Most of the grains are cemented by quartzovergrowths. Authigenic illite is also a common cement. TS po-rosity is characterized by isolated intergranular pores. The poresystem is reduced by quartz overgrowths and authigenic clays. TheTS and SEM are shown in Fig. 4. The dominant pore-throat size isin the 2- to 5-m range, followed by mesopores less than 2.5 m.

HU-3. HU-3 is characterized by laminated, lower medium- andfine-grained quartz arenites. As seen in Fig. 5, both medium andfine-grained laminae are poorly sorted. All the detrital grains aresubangular to subrounded quartz, with authigenic and detrital clayscomprising approximately 6% of the grain volume. TS porosity ischaracterized by isolated intergranular pores partially filled withquartz overgrowths and authigenic clay.

The detrital clays are commonly mica concentrated along bed-ding laminae. Most of the authigenic clays are pore-filling kaolin-ite. Smaller amounts of grain-coating illite are also present. Mostof the grains are cemented by quartz overgrowths. Authigenic illiteis also a common cement. The micropores are in the range of 0.025to 0.25 m.

HU-4. The rock in this unit is a fine-grained, well-sorted sublith-arenite. The general rock composition is detrital quartz (89%),

Fig. 2HU zonation for study-field core samples. Intercept of unit-slope lines at z=1 is FZI value.


rock fragments (6%), and authigenic and detrital clay (5%). Thedetrital grains are subangular to subrounded. Many of the rockfragments are deformed as a result of compaction. Clay mineralsinclude detrital chlorite associated with rock fragments and authi-genic pore-filling illite. Cements include quartz overgrowths,grain-rimming clay, and minor amounts of calcite and dolomite.The TS and SEM in Fig. 6 show most of the visible porosity isisolated intergranular pores and secondary porosity in rock frag-ments. The pores are dominantly micro with a cluster around0.025 micron.

Mineralogy. The presence of clay minerals can have implicationsin the measurements of electrical properties of the rock. Illite grain

coatings can trap water and provide continuous conductive phaseeven at lower water saturations.

SCA Data for Sw CalculationElectrical Parameters. An extensive SCA program was under-taken to acquire data that would be used in all renditions of thewater-saturation equations. Experiments were conducted to deter-mine apparent cementation and saturation exponents. Multiple sa-linity tests (Co, Cw) were also conducted to ascertain the effects ofclays on log analysis. These tests involved completely saturating acore sample with varying brine salinities. The conductivity (Co) ofthe fully brine-saturated sample is plotted against the conductivityof the saturating brine (Cw). From these experiments, the intrinsic

Fig. 3TS (left), SEM (middle), and pore-throat-size distribution (right) of HU-1 rock.


formation resistivity factor (F*), intrinsic cementation exponent(m*), intrinsic saturation exponent (n*), and volume concentrationof exchangeable cations (Qv) used for the Waxman-Smits shaly-sand water-saturation model can be estimated. Experimental de-tails and examples are given in Amabeoku et al. (2005a).

Reservoirs A and B have an average composite cementationexponent (m) of 1.89. The saturation exponent (n) for Reservoir Ais 1.79. For Reservoir B, the saturation exponent is 1.91.

Centrifuge Capillary Pressure. Ambient-condition high-speedcentrifuge air/brine capillary pressure data were obtained by use ofcore samples from three wells in the study field. The data are givenin Table 2, and the capillary pressure curves are shown in Fig. 7.

Strictly on the basis of the FZI values of the samples, thecapillary pressure data were assigned to their respective HUs.Samples X718.2, X713.8, and X742.8 belong in HU-1. SampleX727.3 was assigned to HU-4. The other samples were assigned toHU-2 and HU-3, as appropriate.

Saturation/Height Models by Use of CoreCapillary Pressure DataCentrifuge capillary pressure data were used in the developmentsof the saturation-height models that follow. The ambient-conditions air/brine centrifuge capillary data were converted toreservoir conditions by use of the following equation:

Pc-res = Pclab cosres coslab

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

The relationship between capillary pressure and the heightabove free-water level (H) is given by

Pc = 0.433*Hw g, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)where, in oilfield units, Pc is in psi; H is in feet; and w and gare both in g/cm3. 0.433 is a conversion factor from laboratory tofield units.

These definitions were then used to develop practical forms ofthe J-function, FOIL function, and the modified FOIL functioncited previously in Eqs. 1, 5, and 6, respectively, for each of thefour HUs.

Practical HU-Based Saturation FunctionsJ-Function. Eq. 12 was substituted for Pc in Eq. 3 for furtheranalysis. The J-function/saturation relationship was established byrobust nonlinear regression. The resulting model was transformedso that water saturation can be solved for in terms of the reservoirproperties and height. Eqs. 13 through 16 are the practical formsfor HUs 1 to 4, respectively

Sw1 =0.2249

0.0.094w gH cosres

k0.5150 . . . . . . . . . . . . . . . . . (13)

Sw2 =0.3739

0.094w gH cosres

k0.3784 . . . . . . . . . . . . . . . . . . (14)




Sw3 =0.2380

0.094w gH cosres

k0.4301 . . . . . . . . . . . . . . . . . . (15)

Sw4 =0.2038

0.094w gH cosres

k0.4082 . . . . . . . . . . . . . . . . . . (16)

Sw1, Sw2, Sw3, and Sw4 are the water-saturation calculationequations for HUs 1 through 4, respectively.

FOIL Function. The FOIL function is written as BVWSw.Having established the relationship with the height above the free-water level and the bulk volume of water, Sw can now be calculatedexplicitly for the different HUs by use of Eqs. 17 through 20:

Sw1 =0.2032H 0.5542

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

Sw2 =0.1727H0.3857

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

Sw3 =0.2132H0.4651

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)

Sw4 =0.1705H0.4082

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)

Modified FOIL Function. The multiple nonlinear regression al-gorithm was used to obtain the parameters for Modified FOIL. Thefinal forms of the function for the HUs are given in Eqs. 21through 24:

Sw1 =2.897H0.5485

1

0.2223. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)

Sw2 =0.5239H0.3765

1

0.5353. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (22)

Sw3 =2.9214H0.4604

1

0.1528. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23)

Sw4 =0.2476H0.4082

1

0.8486. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (24)

The water saturation can now be calculated with Eqs. 13 through24, after substituting the fluid properties provided in Table 3 or



those deemed to be appropriate for the field under study. Theregression parameters (A, B, C) shown in the equations above andthe fluid densities in Table 3 are specific to this field under studyand would vary for different data sets. The interfacial tensions andcontact angles are generally assumed values.

Comparison Between Saturation-ModelPerformance and Core DataWater saturations were calculated for each of the following mod-els: J-function, FOIL function, and Modified FOIL function, andthese were compared with the measured water saturations from thecore capillary pressure experiments. For illustrative purposes,Figs. 8 through 10 show plots of model-derived water saturationvs. core water saturation for HU-1. Rock properties of the sampleswere used in the calculations.

Among the three models, Modified FOIL performed the best,with a correlation coefficient of 0.985 between core water satura-tion and predicted water saturation; J-function followed, with cor-

relation coefficient of 0.968; and FOIL function had a correlationcoefficient of 0.952.

Applicability of the Saturation/Height Models forReservoir Water-Saturation CalculationsFrom HU classification, Reservoir B rock is of better quality thanthat of Reservoir A. Whereas the B reservoir is mostly HU-1, theA reservoir is predominantly HU-4.

A permeability-modeling approach developed in Amabeokuet al. (2005b) was used with sparse core data to develop a con-tinuous permeability trace in each well. This was done so that theperformance of the J-function could be evaluated. The laboratory-conditions air/brine centrifuge capillary pressure data were con-verted to reservoir conditions by use of the data in Table 2.

Routinely, water saturation in this field is calculated by use ofdeterministic Archie expressions and by an optimizing DW model.Of these two methods, DW is believed to be superior. Each of thesaturation models was run across Reservoirs A and B. Fig. 11compares the performance of each of the models to DW.

The lithology track (far left) shows that Reservoirs A and B arevery clean. Tracks 4 and 5 (from left) show the J-functions forHU-1 and HU-4, respectively. The J-functions are featureless inReservoir A and calculate water saturations that are higher than theDW water saturations. In Reservoir B, both models compare rea-sonably well with DW. In Track 6, the HU-1 FOIL model matchesDW quite well in Reservoir A and most of Reservoir B. It deviatesclose to the base of the reservoir. The HU-4 FOIL model shows a

Fig. 7Air/brine centrifuge capillary pressure curves.

Fig. 8Comparison between J-function predicted water saturation and core water saturation for HU-1.


good match in Reservoir A but fails in Reservoir B. Tracks 8 and9 show the performance of the Modified-FOIL functions for HU-1and HU-4, respectively. The Modified-FOIL HU-1 model exhibitsthe best match of all in Reservoir B. The water saturation inReservoir A is too low. That is not surprising: HU-1 was devel-oped for Reservoir B and was not expected or intended to beapplied to Reservoir A. Modified FOIL for HU-4 calculates goodwater saturation in Reservoir A. The water saturation in ReservoirB is higher and is considered to be wrong. Again, HU-4 modelswere not intended to be used in Reservoir B.

A catalog of all the wells evaluated shows the following com-mon observations:

1. The J-function models calculate water saturations reasonablywell in Reservoir B but unreliably in Reservoir A. Good perme-ability models are required, and this can limit their application.

2. FOIL for HU-1 performs very well in reservoir A but can bequestionable in Reservoir B. The FOIL HU-4 model works well inReservoir A but not in B.

3. The Modified-FOIL HU-1 model works the best in B. TheHU-4 model is also good in Reservoir A but not as consistentlygood as FOIL HU-1 in Reservoir A.

With the aforementioned observations, it became practical andnecessary to take advantage of the reservoir tops and develop ahybrid model that is fit for purpose in this field. The computer codewould use a simple logic, such as

If TopA, compute Sw with FOIL HU-1. If TopB, computeSw with Modified-FOIL HU-1.

Relative Permeability. Four sets of steady-state relative perme-ability data at reservoir conditions were obtained on single-core

Fig. 9Comparison between FOIL-function predicted water saturation and core water saturation for HU-1.

Fig. 10Comparison between Modified-FOIL-function predicted water saturation and core water saturation for HU-1.


plugs that are representative HU-1 and HU-4. Table 4 lists theproperties of the core plugs constituting the HUs.

The two sets of steady-state relative permeability data fromeach HU were normalized, averaged, and denormalized to obtainthe relative permeability curves used in these analyses. Techniquesfor normalizing relative permeability are available in Ahmed(2001). Figs. 12 and 13 illustrate averaged relative permeabilitycurves for HU-1 and HU-4, respectively.

Water/gas relative permeability ratios were also calculated andplotted. Fig. 14 is an overlay of HU-1 and HU-4 relative perme-ability ratios. The plots show that for any given water saturation,krw/krg is higher for HU-4. This indicates that HU-4 is more likelyto produce water in comparison with HU-1, if the water was abovethe irreducible saturation.

The relationship between the relative permeability ratio andwater saturation can be written as given by Ahmed (2001):

krwkrg

= aebSw, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25)

where a and b are coefficients that are obtained from regressionby use of the linear portion of the semilog plot of krw/krg vs.water saturation.

Relative permeabilities were also calculated from mercury-injection capillary pressure (MICP) and centrifuge capillary pres-sure data, by use of the Brooks-Corey (1966) method. This wasdone for comparison and to establish the feasibility of this ap-proach in areas where experimental relative permeability datamight not be readily available. The procedure is first to obtain from the capillary pressure data as follows:

Pc = PeS*w1

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (26)where Pe is the entry pressure and is the pore-distribution index.Sw* is the normalized wetting-phase saturation, and it is defined, forthe drainage case, as

S*w =Sw Swir1 Swir

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (27)

where Swir is the irreducible water saturation.Eq. 27 was used when the residual nonwetting-phase (gas)

saturation was low and could not be measured accurately. Other-wise, adaptation of the Corey model for the imbibition case wouldtake the form

Fig. 11Comparison of the HU saturation models in study field.


S*w =Sw Swi

1 Swi Snwr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (28)

where Swi and Snwr are the initial water saturation and the nonwet-ting phase saturation, respectively.

The wetting-phase and nonwetting-phase relative permeabili-ties (krw, krnw), respectively, are given by

krw = S*w2+3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (29)

krnw = 1 S*w21 S*w2+ . . . . . . . . . . . . . . . . . . . . . . . . . . . (30)

Fig. 15 is an overlay of relative permeability curves obtainedfrom capillary pressure data on the averaged steady-state data forHU-1. The abscissa uses the normalized water saturation, calcu-lated with Eq. 27, to reconcile the effect of the different irreduciblewater saturations obtained in the cores. In the steady-state calcu-lation of Sw*, the residual-gas saturation is subtracted from thedenominator, and the expression is that given in Eq. 28.

The relative permeability curves derived from MICP data andfrom centrifuge capillary pressure agree very well in nearly all theanalyses discussed in this field. This indicates that, for water-wet

and less-complex rocks, the less expensive and faster-to-measureMICP data can be used in place of the more expensive air/brinecapillary pressure data but should not be substituted for experi-mental relative permeability data. Moreover, pore-throat-size dis-tribution and structural information are also extracted from MICP data.

The difference between the steady-state gas relative permeabil-ity and those calculated with capillary pressure data is not signif-icant. The steady-state water relative permeability is generally less,as the normalized water saturation increases beyond 50%.

Fig. 16 confirms the similarity between the capillary pressurerelative permeability ratios. In the middle to upper ranges of watersaturation, all three ratios agree closely. The divergence betweenthe steady-state ratio and the capillary pressure ratios occurs at thelower water saturation. In spite of this, relative permeability fromcapillary pressure data provides very good approximation forsemiquantitative analysis. This is particularly relevant because it isthe linear part of the curve (middle section) that generally is usedto develop the relationship between the relative permeability ratioand water saturation given in Eq. 25.Coupling Relative Permeability WithSaturation FunctionsSteady-state relative permeability data were used in these analyses.The data derived from capillary pressure could equally be used.The functional forms of the relative permeability ratio relationshipare given by the following:

for HU-1,krwkrg

= 3.2 106e19.48Sw1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (31)

for HU-4,krwkrg

= 6.98 106e22.06Sw4, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (32)where Sw1 and Sw4 are the water-saturation values calculated withthe saturation/height models for HU-1 and HU-4, respectively.WGRWater producibility in this analysis is presented as the volume ofwater that would be produced with 1 scf of gas. This is the WGR,with units of STB/scf, and it is given as

Fig. 12Averaged steady-state relative permeability of HU-1 samples.


WGR =BgBw

gw

krwkrg

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (33)

where Bg and Bw are gas and water formation factors, respectively,and g and w are gas and water viscosities, respectively. Substi-

tuting the relative permeability ratio for HU-1, for example,WGR becomes:

WGR1 =BgBw

gw

3.2 106e19.48Sw1. . . . . . . . . . . . . . . . . . . . . (34)

Similarly, for HU-4, the ratio would be calculated as:

WGR4 =BgBw

gw

6.98 106e22.06Sw4. . . . . . . . . . . . . . . . . . . . (35)

With the applicable values of Bg, Bw, g, and w, as shown inTable 5, the WGR can be calculated for saturation at each depth.

Assessment of WGR and Verification WithTest DataHaving established the utility of the HU-based saturation func-tions, we shall now incorporate relative permeabilities to calculateWGR. These calculations are based on the single hybrid saturationmodel from the preceding section. Remarks on water producibilityare based on the saturation/height model. The DW water saturationis plotted alongside for comparison purposes.

In Fig. 17, the mirror image of the gamma ray (GR) log inTrack 2 (from left) shows the thick porous sands in Reservoirs Aand B in Well 5X. The density-neutron overlay in Track 3 andthe water saturation in Track 6 show these sands to be gas-filled.Track 7 shows the WGR in STB per million scf of producedgas. This analysis indicates that there will be no water productionfrom completions in the reservoir intervals. Spikes in WGR occurat very low porosity and at the base of Reservoir A. The conven-tional analysis also indicates that the water is bound and will notbe produced.

Well 4X in Fig. 18 has cased-hole test data obtained acrossReservoir A and part of Reservoir B. There was no water produc-tion. The test showed very high gas rate. Relatively small quantityof condensate was also produced. As in Well 5X, the apparentlyhigh water saturation is a computational artifact because of the lowporosity in the denominator (Eqs. 21 through 24). Consequently,WGR spikes in Track 7 can be regarded as noise. Moreover, thedensity-neutron overlay near the top of Reservoir A shows thespiky intervals not to be gas sands. These are shaly or silty. In the

Fig. 13Averaged steady-state relative permeability of HU-4 samples.

Fig. 14Relative permeability ratios of HU-1 and HU-4 samples.


reservoir sands, the saturation and WGR are well behaved and areconsistent with the test data.

Fig. 19 (Well 1X) has more-extensive test data. DST #1 acrossthe top half of Reservoir A produced all gas and no water. CH Test#1 across the top 60 ft of Reservoir B produced all gas. Another

CH test (CH Test #2), across all of Reservoir B and approximately5 ft below the free-water level, produced gas.

Discussion of ResultsThrough the HUs concept, saturation/height relationships havebeen developed to honor differences in lithofacies or pore geom-etries. Generally, the Modified-FOIL model is the most direct andthe easiest to use. The respective HU-based Modified-FOIL mod-els also performed the best in the wells analyzed. It was observedthe HU-1 FOIL model developed for reservoir B worked very wellin Reservoir A, which has predominantly HU-4-type rock. Withthese observations, it became practical and necessary to take ad-vantage of the reservoir tops and develop a hybrid model that is fitfor purpose in this field. The computer code was modified in a waythat allows the Modified-FOIL HU-1 model to be used for satu-ration calculations in Reservoir B, while the HU-1 FOIL model isused to calculate saturation in Reservoir A, all in one pass withoutintervention by the user. Both the FOIL and Modified-FOIL mod-els do not require permeability in their applications. But applica-tion of the J function is constrained by the requirement that goodpermeability models be available.

Water/gas relative permeability curves were developed for eachHU. These were then used successfully to calculate the potential ofwater mobility in each reservoir. A very noteworthy observation is

Fig. 15Overlay of relative permeabilities calculated from capillary pressure data by the Brooks-Corey method with steady-staterelative permeability data.

Fig. 16Comparisons of relative permeability ratios calculatedfrom capillary pressure data by the Brooks-Corey method withsteady-state relative permeability ratio.


that relative permeability calculated from capillary pressure dataagreed closely with steady-state relative permeability. This obser-vation has important time and cost implications. Steady-state rela-tive permeability measurements are time-consuming, whereas cen-trifuge capillary pressure data can be acquired in a relatively shorttime at a significantly lower cost. Capillary pressure from this gasfield can now be used confidently to develop not only for satura-tion functions but also for relative permeability in this field.

Comparison between calculated WGR and well-test data con-firm the robustness of the relative permeability-coupled saturation

models. Generally, WGR is less than 0.1 STB/million scf, exceptwhere the porosity is low (less than the cutoff value) and near thegas/water contact. At no point did the models contradict the testresults. These models can serve as guides to augment well tests andcan also be used to make well-completion decisions.

ConclusionsSaturation/height models from core capillary pressure (Pc) data arean effective way to determine fluid saturation. This study devel-oped saturation/height models by

Fig. 17Calculation of WGR and impact on water mobility in a well that was not tested.


1) Linking depositional and diagenetic rock fabric to HUs.2) Linking the HUs to zones with similar capillary pressure relationships.3) Determining saturation/height functions for each HU with the

modified-FOIL function.Improved fluid-saturation profiles derived from Modified FOIL

can be used to validate log models and parameters. Because thismethod does not require permeability in field application, it can beused in uncored wells.

HU-based relative permeability models have been developedfor each of the reservoirs in this gas field. Capillary pressure datahave presented a cost-effective way to develop both saturation andrelative permeability models in this gas field.

Coupled with HU-based relative permeability curves, thesaturation models will provide more-comprehensive petro-

physical interpretation in gas-bearing formations and will predictwater producibility. The range of WGR indicates no water wouldbe produced in the tested intervals. This is validated by thetest results.

Nomenclaturea fitting parameter in relative-permeability-ratio/

water-saturation relationshipA fitting parameter in saturation functionsb fitting parameter in relative-permeability-ratio/

water-saturation relationshipB fitting parameter in saturation functions

Bg gas formation volume factor, RB/Mscf

Fig. 18Water-mobility prediction vs. well-test results.


Bw water formation volume factor, RB/STBC fitting parameter in saturation functions

Co conductivity of 100%-brine-saturated sand,mhom1

Ct total conductivity of partially brine-saturated sand,mhom1

Cw conductivity of brine solution, mhom1F apparent formation-resistivity factor (1/m)

F* intrinsic formation-resistivity factor (1/m*)H height above free-water level, ft

J(Sw) Leverett J-functionk permeability, md

krg relative permeability to gaskrnw relative permeability to nonwetting phasekrw relative permeability to water

m cementation exponent for clean sandm* intrinsic cementation exponent

n saturation exponent for clean sandn* intrinsic saturation exponentPc capillary pressure, psi

Pc-lab capillary pressure in laboratory fluid pair, psiPc-res equivalent capillary pressure in reservoir fluid pair,

psiPe capillary entry pressure, psiQv volume concentration of clay exchange cations,

meq/m?Snwr irreducible gas saturation, fraction

Sw water saturation, fractionS*w normalized water saturationSwi irreducible water saturation, fraction

Swir, irreducible water saturation, fraction contact angle, degrees

lab contact angle at laboratory conditions, degreesres contact angle at reservoir conditions, degrees

Fig. 19Validation of mobility calculations with isolated cased-hole (CH) test results.


pore-distribution indexg gas viscosity, cpw water viscosity, cpg gas density, g/cm3w water density, g/cm3 interfacial tension, dyne/cm

cos product of interfacial tension and contact angle,dyne/cm

coslab product of interfacial tension and contact angle atlaboratory conditions, dyne/cm

cosres product of interfacial tension and contact angle atreservoir conditions, dyne/cm

porosity, fractionz pore-volume/grain-volume ratio

AcknowledgmentsThe authors thank the management of Saudi Aramco for theirsupport and for permission to publish this paper.

ReferencesAhmed, T. 2001. Reservoir Engineering Handbook, second edition, 298

301, 304309. Oxford, UK: Elsevier.Amabeoku, M.O., Kersey, D.G., BinNasser, R.H., Al-Waheed, H.H., and

Al-Belowi, A.R. 2005a. Incorporating Hydraulic Units Concepts inSaturation-Height Modeling in a Gas Field. Paper SPE 93763 presentedat the SPE Asia Pacific Oil & Gas Conference and Exhibition, Jakarta,57 April. DOI: 10.2118/93763-MS.

Amabeoku, M.O., Lin, C., Al-Khalifa, A., Cole, J., Dahan, M., Jarlow, J.,and Ajufo, A. 2005b. Use of Fuzzy-Logic Permeability Models toFacilitate 3D Geocellular Modeling and Reservoir Simulation: Impacton Business. Paper IPTC 10152 presented at the International Petro-leum Technology Conference, Doha, 2123 November. DOI: 10.2523/10152-MS.

Amaefule, J.O., Altunbay, M., Tiab, D., Kersey, D.G., and Keelan, D.K.1993. Enhanced Reservoir Description: Using Core and Log Data toIdentify Hydraulic (Flow) Units and Predict Permeability in UncoredIntervals/Wells. Paper SPE 26436 presented at the SPE Annual Tech-nical Conference and Exhibition, Houston, 36 October. DOI: 10.2118/26436-MS.

Archie, G.E. 1942. The Electrical Resistivity Log as an Aid in DeterminingSome Reservoir Characteristics. Trans., AIME, 146: 5462.

Brooks, R.H. and Corey, A.T. 1966. Properties of porous media affectingfluid flow. ASCE J. of Irrigation and Drainage 101: 8592.

Carmen, P.C. 1937. Fluid Flow Through Granular Beds. Trans., Institute ofChemical Engineers London, 15: 150166.

Cuddy, S. 1993. The FOIL FunctionA Simple, Convincing ModelFor Calculating Water Saturations In Southern North Sea Gas Fields.Paper H1-17 presented at the SPWLA Annual Logging Symposium,Calgary.

Ebanks, W.J. Jr. 1987. Flow unit conceptIntegrated approach to reser-voir description for engineering projects. AAPG Bulletin 71 (5): 551552.

Efnik, M.S., Dernaika, M., and Kalam, M.Z. 2006. Evaluation of watersaturation from laboratory to logs. Paper SCA2006-56 presented at theInternational Symposium of the Society of Core Analysts, Trondheim,Norway, 1216 September.

Eyvazzadeh, R.Y., Cheshire, S.G., Nasser, R.H., and Kersey, D.G. 2003.Optimizing Petrophysics: The Ghawar Field, Saudi Arabia. Paper SPE81477 presented at the Middle East Oil Show, Bahrain, 912 June.DOI: 10.2118/81477-MS.

Ibrahim, A., Bassiouni, Z., and Desbrandes, R. 1992. Determination ofrelative permeability curves in tight gas sands using log data. PaperSS presented at the SPWLA Annual Logging Symposium, 1417June.

Kozeny, J. 1927. ber Kapillare Leitung des Wassers im Boden. Sitzungs-berichte der Wiener Akademie des Wissenschaften, 136: 271306.

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Waxman, M.H. and Smits, L.J.M. 1968. Electrical Conductivities in Oil-Bearing Shaly Sands. SPEJ 8 (2): 107122; Trans., AIME, 243. SPE-1863-PA. DOI: 10.2118/1863-PA.

SI Metric Conversion Factorsbbl 1.589 873 E01 m3cp 1.0* E03 Pas

dyne 1.0* E02 mNft 3.048* E01 m

ft2 9.290 304* E02 m2psi 6.894 757 E+00 kPa

*Conversion factor is exact.

Maclean Amabeoku is Coordinator of Special Studies in thereservoir description and simulation department at SaudiAramco. E-mail: [email protected]. In thatcapacity, he formulates and oversees petrophysical studies,with the sole objective of improving parameters and modelsused in formation evaluation and minimizing uncertainties.Amabeoku has done extensive work on core-log data com-parison and integration, permeability prediction using innova-tive soft computing technologies, saturation-height modeling,and relative permeability normalization. He holds a BS degreefrom the University of Kansas, and MS and PhD degrees fromthe University of Southern California, all in petroleum engineer-ing. David Kersey is a senior petroleum engineering consultantin the reservoir description and simulation department, SaudiAramco. E-mail: [email protected]. He currentlyoversees strategic planning, training, and evaluation of newtechnology. Kerseys previous position with Saudi Aramco wasSupervisor, Petrophysical and Special Studies Unit, ReservoirDescription Division. He did his undergraduate work at the Uni-versity of Southern California and his graduate work at HarvardUniversity. Rami BinNasser is Supervisor of Special Studies in thereservoir description and simulation department at SaudiAramco. E-mail: [email protected]. In that capac-ity, he sets future direction for petrophysics by evaluating newlogging and data processing technologies to improve forma-tion evaluation. BinNasser has had several assignments aspetrophysicist in the Gas & Exploration Unit, production engi-neer and reservoir management engineer. He holds a BS de-gree in mechanical engineering from King Fahd University ofPetroleum and Minerals in Dhahran, Saudi Arabia. Ali Al-Belowiis Supervisor of Udhailiyah area petrophysics in the reservoirdescription and simulation department at Saudi Aramco. E-mail: [email protected]. In this position, he ensures thatproduction and openhole logs are analyzed promptly, utilizingthe most complete and accurate petrophysical models. Al-Belowi has extensive experience in petrophysical evaluationsof exploration and development fields. He holds a BS degree inpetroleum engineering from King Saud University in Riyadh,Saudi Arabia.


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