interpretation of water saturation above the transitional...

Interpretation of Water Saturation Abovethe Transitional Zone in Chalk Reservoirs

Jens K. Larsen* and Ida L. Fabricius, Technical U. of Denmark

SummaryThe free water level (FWL) in chalk reservoirs in the North Seamay be hard to establish owing to strong influence from capillaryforces and lack of pressure equilibrium across the reservoir. Evenwhere wireline formation tester data on the FWL are available inone well, it is no straightforward task to predict the FWL in otherparts of the field, where only conventional core analysis and log-ging data are available. It is thus difficult to predict the geometryof the hydrocarbon-bearing interval.

This paper offers a simple model which, in a given well, allowsus to predict the location of the FWL from conventional coreanalysis and logging data if wireline formation tester data areavailable from another well.

Water saturation is averaged over the internal surface of theformation by applying Kozeny’s equation, resulting in a pseudowater-film thickness (PWFT). The PWFT is larger than the equi-librium water-film thickness calculated from the augmentedYoung-Laplace equation because it includes water associated withgrain contacts. The PWFT has a gradient with true vertical depth(TVD) that is related to the capillary pressure; this gradient isapproximately the same for the wells investigated. Consequently,a unique relationship between the PWFT and the height above theFWL can be established, provided that the depth of the FWL isknown from formation pressure data. The unique PWFT heightabove the FWL relationship can be used to establish the FWL inoffset wells for which no reliable formation-pressure data exist.The estimation of the FWL in offset wells is inexpensive becauseit requires the use of log data and sufficient conventional core-analysis data only.

The model was tested on seven wells from the Gorm and Danfield in the North Sea and resulted in predictions of the FWL in theDan field wells based on wireline formation tests in one Gormfield well.

IntroductionThe interplay between capillary pressure and phase saturation inchalk is not well established; traditional normalization methods ofcapillary pressure curves are not able to model capillary pressurebehavior within chalk reservoirs.1–3 In North Sea chalk reservoirs,water saturations above the transitional zone vary frequently fromvalues as low as 5% to those as high as 60% (Figs. 1a and 1b),depending on the capillary rock properties. Traditionally, the zoneabove the transitional zone is referred to as the irreducible zonebecause little or no water is produced. Experiments reveal thatwater saturations lower than those encountered in the irreduciblezone can be obtained in the laboratory, provided that a sufficientlylarge difference in phase pressures is applied and sufficient ex-perimental time is available. In fact, a water saturation close tozero can be obtained if a sufficiently high capillary pressure isapplied and if the water phase remains continuous to provide anescape path for the water phase.

In a field in equilibrium, the difference in phase pressures iscaused by the difference in hydrostatic pressure (which itself iscaused by the difference in fluid density). The capillary pressure,

defined as the difference in phase pressures for an oil/water systemin equilibrium, can be calculated by Eq. 1, where h is the heightabove the FWL defined as the point at which the capillary pressureis zero.

Pc = po − pw = ��gh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

Consequently, the largest difference in phase pressures willoccur immediately below the top of the reservoir and graduallydecrease downward to the FWL, provided that capillary continuityis sustained. We assume that the hydrocarbon and water phases arecontinuous throughout the reservoir column and that no effectivelower limit exists for the water saturation. Therefore, the terms“irreducible zone” and “irreducible water saturation” (Swi) providelittle meaning when interpreting the water saturation in chalk res-ervoirs. Throughout the paper, we will refer to the irreduciblewater saturation only while discussing or referring to traditionalsaturation models; otherwise, it should be assumed that no lowerlimit exists for the water saturation.

Before a reservoir is filled with hydrocarbon, the formation issaturated with formation brine. Under the assumption that the for-mation is initially water-wet, oil will displace the formation brinein a drainage process. Given sufficient height for an oil column,water will be displaced from the center of all pores and will coveronly the surface of the mineral grains of the formation. Consider-ing a drainage process in which the water has been displaced fromall pore centers, it is thus reasonable to expect a relationship be-tween water saturation and the internal surface area of the formation.

Wyllie and Rose4 proposed a relationship between perme-ability, porosity, and irreducible water saturation and proved itvalid for some sandstone reservoirs. Timur5 suggested a general-ized equation,

k = A ��Y

S wiX

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

where A, X, and Y are constants. Several authors have proposedsimilar relationships;6–11 we refer to these kinds of models aswater-film models because the volume of water is assumed tocover the surface of the rock in a thin water film. Wyllie andRose’s relationship, however, has not been tested on chalk. More-over, there is no effective irreducible water saturation in chalk;consequently, we feel that establishing a relationship between theinternal surface and the water saturation is a better approach.

Kozeny’s equation expresses a relationship between the spe-cific surface of a porous medium and its permeability and porosity.12

k = C ��3

S 2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

where S is the specific surface with respect to total volume. Thespecific surface with respect to porosity is given by

Sp =S

�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

Mortensen et al.13 have shown that Kozeny’s equation (Eq. 3)is valid for chalk, which is reasonable to expect, considering thechalk’s homogeneity. For porosities between 20% and 40%, theconstant C is near 0.23 for chalk;13 thus, there is a relationshipbetween specific surface, porosity, and single-phase permeabilitythat, when combined with a modified Wyllie and Rose equation(Eq. 2) in which Swi is substituted with Sw, leads to a relationshipbetween porosity, specific surface, and water saturation.

* Now with Maersk Oil Qatar.

Copyright © 2004 Society of Petroleum Engineers

Original SPE manuscript received for review 24 April 2001. Revised manuscript received 10February 2003. Paper 69685 peer approved 16 February 2004.

155April 2004 SPE Reservoir Evaluation & Engineering

The inability of traditional capillary pressure models to capturewater-saturation behavior has made determination of the FWLinaccurate. Moreover, studies have shown that the FWL can betilted.14–17 The dipping FWL of the Dan field has been proposedpreviously in two publications. Jacobsen et al.15 mapped the FWLon the west flank of the Dan field by using data from a horizontalwell and applying an equivalent radius model developed by Eng-strøm.2 Vejbæk and Kristensen17 mapped the same area by usingseismic inversion combined with a modified Leverett J-function astheir saturation/capillary height relationship.

We will test a water-film model similar to the one by Wyllieand Rose on seven wells from two different North Sea fields: threefrom the Dan field and four from the Gorm field (Fig. 2). We willapply log and core data from zones of Danian and Maastrichtianage, with a porosity range from 17% to 45%. We will discuss the

interplay between capillary forces and the water-film model. Sub-sequently, we will discuss the effect of water located at graincontacts and the applications of averaging the water saturationwith respect to the internal surface. This averaging leads to aunique relationship between the PWFT and the height above theFWL (HAFWL). We will use the PWFT and the formation-pressure data from the Gorm field well N-22 to establish a PWFT/HAFWL relationship; subsequently, we will use the relationship toestimate the height of the FWL in five of the six other wells.

Microscopic Water-Film ModelThe concept of water-film models involves distributing a givenvolume of water on the surface of a porous, water-wet mediumwith a known internal surface area. As an initial step, we willassume that water covers the surface of the chalk grains in a thin

Fig. 1—(a) Water saturation of the Gorm field wells N-2X, N-3X, N-6, and N-22. Top chalk is marked by a thick solid line. TheDanian-Maastrichtian boundary (D-M) is marked by a thin solid line. (b) Water saturation of the Dan field wells MA-1, MD-1, andME-1. Top chalk is marked by a thick solid line. The Danian-Maastrichtian boundary (D-M) is marked by a thin solid line. The GOCis marked by a thick dashed line.

156 April 2004 SPE Reservoir Evaluation & Engineering

layer with uniform thickness. Thus, if the specific surface, poros-ity, and water saturation are known, the thickness of the waterlayer can be calculated as

hw =Sw

Sp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

where hw is the PWFT.Combining Eq. 5 with Kozeny’s equations (Eqs. 3 and 4), the

thickness of the water film can be expressed as

hw = Sw �� k

C � �. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

Notice that any water associated with grain contacts is averagedout over the surface of the grains. Thus, the water-film modelbasically transforms the water saturation into a PWFT.

The thickness of a thin water film is controlled by intermo-lecular forces and can be described by the augmented Young-Laplace equation18

Pc�� + ��z�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

where � is the interfacial tension, � is the total curvature of theinterface, ∏ (z) is the disjoining pressure isotherm, and z is thethickness of a thin water film separating the hydrocarbon andmineral grain.

The disjoining pressure isotherm depends on a number of pa-rameters: capillary pressure, chalk composition, oil and gascomposition, salinity and pH of formation water, pore geometry,and temperature.18–21 On a field scale, only pH and salinity offormation water can be assumed constant; the remaining param-eters vary significantly.

The stability of a thin water film larger than a few nanometersis essentially a balance between capillary pressure, the curvature ofthe interface, the van der Waals forces (which always attract theoil/water and chalk/water interfaces), and the electrical forces ofthe oil/water and chalk/water interfaces, which can be either re-pulsive or attractive depending on the surface charge of the inter-face.20 The surface charge depends on the pH, the composition ofthe chalk, and the CO2 partial pressure of the brine.18 For films lessthan a few molecules thick, repulsive structural forces dominate.

Although the physical principles of calculating thin-film thick-nesses are well established, we feel that on a field scale, moresimple observations are needed. Therefore, we approach the prob-

lem by using a pseudo film thickness. The PWFT will be muchlarger than the real water-film thickness because the water locatedaround the grain contacts will also contribute to the PWFT. Thus,the water-film model inherently assumes that the conditions aresuch that the water film is stable over the entire reservoir.

DataWe used log and core data from seven wells (three from the Danfield and four from the Gorm field), all of which have been cored.Although several other wells were considered, we selected theseseven wells because of their abundance of core material. More than2,071 measurements of porosity and permeability exist for theseven wells, approximately half from above the transitional zone.The wells contain data representing chalk of both the Danian andMaastrichtian ages. In the Gorm and Dan fields, the chalk of thesetwo ages differs in that Danian chalk in general has a higherspecific surface, S, than Maastrichtian chalk.13 Røgen and Fabri-cius22 studied chalk from hydrocarbon-bearing intervals and con-cluded that the difference in specific surface between the twoformations is mainly a consequence of a higher content of silicatesin the Danian chalk. All the Gorm wells contain oil and water,while the Dan field wells also contain gas, with the gas/oil contact(GOC) ranging between 6,030 and 6,060 ft true vertical depthsubsea (TVDSS), as reported by Megson.23 In N-22, wireline RFTdata were also used for estimation of FWL.

Data ProcessingThe basic logs involved in the calculation of the water saturationare the density log and the resistivity log. The porosity was cal-culated from

� =�ma − �b

�ma − Sw�w − �1 − Sw��h. . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

The water saturation was calculated by applying Archie’s equationwith cementation factor m�2, Archie’s constant a�1, and thesaturation exponent set to n�2:

S wn =

a

�m�

Rw

Rt, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

which leads to the following expression for water saturation:

Sw =1

��Rw

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

Eq. 10 can be substituted into Eq. 8 and solved for porosity and,subsequently, water saturation for a known formation resistivity.

The formation-water resistivity was calculated from the water-zone data of each well. The formation-water resistivities and fluiddensities are given in Table 1.

Porosity and permeability data from conventional core analysiswere used to calculate the specific surface with respect to porosity,Sp, from Eqs. 3 and 4 using C�0.23. If the core porosity and logporosity deviated significantly [(i.e., more than 3 porosity units(p.u.)], the depth of the core was shifted to the nearest depth atwhich the log porosity and core porosity agreed. The deviationbetween core and log porosities is most likely caused by the un-certainty associated with core recovery and logging depth. Theshifting was done to avoid data points at which the core porositydoes not represent the porosity reflected on the log. Typically, thisinvolved shifting the core depth between 1 and 3 ft. The maximumshift was 8 ft. Core data with permeability of 0.01 md or less wereexcluded from the data set because measurements of cores withsuch low permeability generally are not reliable. Permeabilitiesthat were significantly larger than the surrounding permeabilityvalues also were excluded. Abnormally large permeability valuesare probably caused by fractures in the material; consequently, themeasured permeability is not related to the specific surface of thematrix of such a sample.

Fig. 2—The Dan and Gorm fields in the Danish sector of theNorth Sea.


ResultsThe PWFT was calculated according to the water-film model inthe chalk sections of the seven wells (Figs. 3a and 3b). In otherwords, we found the average thickness of all the water relative tothe internal surface of the sample.

Compared with the water-saturation plot, it is clear that largevariations in water saturation above the transitional zone aresmoothed out when transformed into a pseudo film thickness. TheDanian zone of all seven wells contains larger variation in PWFTthan the Maastrichtian zone.

The pseudo film thickness above the transitional zone rangesfrom 6.0 nm (below the Danian Maastrichtian boundary of MA-1)to 30 to 50 nm (immediately above the transitional zone). How-ever, the PWFT data exhibit significant scatter. The variation istypically 20% to 30% around the mean trend of the average PWFTin the Maastrichtian zone.

Although the data exhibit scatter in the pseudo film thickness,it is clear that a gradient with depth exists. A distinct change in thePWFT gradient with depth probably defines the crossover from theirreducible zone to the transitional zone, as defined in traditionalcapillary pressure models. The gradient above the transitional zonewas established using a least-squares method. The average gradi-ent was 3.5 nm per 100 ft, with correlation coefficients rangingbetween 0.3 and 0.7 (Table 2). In the transitional zone, the PWFTbecomes more scattered, and the gradient of the PWFT is closer to33 nm per 100 ft. In contrast, the water-saturation logs (Figs. 1aand 1b) exhibit a gradual change in water saturation. Conse-quently, the transitional zone and the irreducible zone (used intraditional capillary pressure models) cannot be identified from theSw logs. The change in gradient of the PWFT is clear for MA-1 inparticular; it has a high oil column above the transitional zone andtherefore covers a wide range in capillary pressure where the ex-tent of the “irreducible” zone is large. The onset of the capillarytransitional zone occurs at a PWFT of 30 to 50 nm for all wellsexcept N-6, where the crossover is absent (Fig. 3a).

DiscussionInternal Surface and Grain Contacts. It is clear that the watersaturation is related to the specific surface of the formation andthat the large differences in water saturation above the transitionalzone can be accounted for (to a first approximation) by the va-riation in specific surface. However, the local variations in filmthickness indicate that the water-film model is too simplistic todescribe the distribution of water on the chalk grains in detail. Thepseudo film thickness is less well defined for the Danian zonecompared to the Maastrichtian zone. In other words, the watersaturation above the transitional zone is not controlled solely bythe internal surface.

Aggregates of chalk grains form irregular shapes and, conse-quently, the internal surface is not smooth; the water is present notonly as a thin film but also as pendular rings residing around thegrain contacts. To estimate the amount of water residing aroundgrain contacts, we consider the water present as a thin film and asa pendular ring in a system of two spherical-shaped grains ofdimensions similar to the size of natural chalk grains (Fig. 4). Theamount of water present around a single grain contact depends on

the capillary pressure, contact angle, and grain size. For a givencapillary pressure, grain size, interfacial tension, and contact angle,the amount of water located around a single grain contact can becalculated analytically with moderate simplifications (see the Ap-pendix). The total amount of water around the grain contacts isthen the volume around a single grain contact multiplied by thenumber of grain contacts.

Above the transitional zone, the volume of water can be dividedinto two components: the film phase and the grain-contact phase,which add up to the total volume of water (Eq. 11). To illustratethe contribution from the two components of water, the ratio be-tween the surface-phase water and the grain-contact water isshown in Fig. 5 for a typical set of fluid parameters, assumingface-centered cubic (FCC) and body-centered cubic (BCC) pack-ings of spheres as a function of capillary pressure.

Sw,total�Sw, contact + Sw, surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

Thus, the variation in pseudo film thickness within a limited depthrange is a reflection of textural variations: intervals with relativelypoor sorting will have a relatively high number of grain contactsfor a given specific surface and, thus, a greater pseudo film thick-ness. Data points above the Danian-Maastrichtian boundary ex-hibit the largest local variations in water-film thickness, probablyas a reflection of larger textural variability in a section locally richin fine-grained silicates.

Sensitivity of the Pseudo Film Thickness. By inserting Eq. 9 intoEq. 6, it is evident that the pseudo film thickness is sensitive to theaccuracy of the porosity, the cementation factor, and the saturationexponent in Archie’s equation. More specifically, a range of satu-ration exponents from 1.8 to 2.2 and cementation factors from 1.8to 2.2 (common in chalks)24 can change the pseudo film thicknessfrom 27 to 52 nm for a sample of 25% porosity. This range inPWFT is purely associated with the use of Archie’s equation forcalculating the water saturation and is thus not related to the ac-curacy of the data. We have purposely kept our interpretation ofthe water saturation and the porosity deduction simple to removeunnecessary complexity from the water-film model. However, theaccuracy of the estimation of the pseudo water-film model can beimproved by a more rigorous interpretation of porosity and watersaturation, including factors such as clay content, silica content,degree of cementation, and secondary porosity. In addition, thepoor vertical resolution of the logs involved in the calculation ofthe water saturation also contributes significantly to the noise inthe PWFT. However, the general trend in PWFT with depth can berecognized despite the low correlation coefficients.

It is worth noting that for a given capillary pressure, the com-bination of Eqs. 6 and 9 predicts increasing irreducible water satu-ration for lower porosity, which is in accordance with the obser-vations of Engstrøm2 and in contrast to the Leverett J-functionnormalization of capillary pressure curves in which the irreduciblewater saturation is constant.1,2

In MA-1, the gradient of the water-film thickness changes at adepth of 6,030 ft TVDSS, slightly above the GOC reported byMegson.23 The pseudo water-thickness gradient in the gas zoneof the Maastrichtian zone was established to be 15 nm/100 ft


(Fig. 3b), which results from the greater difference in densitybetween water and gas compared to water and oil. However, thegradient of the PWFT in the oil zone does not appear to bedifferent for the Gorm field wells as compared to the Danfield wells, even though the difference in density is larger forthe Gorm field. The lack of sensitivity to oil/water density differ-ence is possibly caused by the smaller height of the irreduciblezone and the smaller number of core data in the Gorm field.Consequently, a well-defined gradient is more difficult to establishin the Gorm field.

Height Above the FWL. The existence of the gradient in pseudofilm thickness is interesting because it shows that the water satu-

ration of the irreducible zone is controlled partially by the lithol-ogy and partially by the capillary pressure. However, the variationof the PWFT gradient with depth is small. Consequently, a uniquerelationship between the PWFT and the height above the FWL canbe established provided that the depth of the FWL is known (inother words, if the FWL is known, the PWFT can be predicted).However, FWL is not necessarily level but can be tilted.14–17 It istherefore of more use to solve the inverse problem of establishingthe FWL from the PWFT.

We used RFT data to establish the FWL (7,269 ft TVDSS) ofN-22 (Fig. 6). Thus, the pseudo film thickness of N-22 can now berelated to the height above the FWL (Fig. 7) and compared to thePWFTs in the offset wells. To illustrate the principle, consider the

Fig. 3—(a) PWFT plotted against depth for the Gorm field wells N-2X, N-3X, N-6, and N-22. Top chalk is marked by a thick solid line.The Danian-Maastrichtian boundary (D-M) is marked by a thin solid line. The transition from the irreducible zone to the transitionalzone is marked by a thick solid line (T-Z). The FWL of N-22 based on RFT data is shown for the wells. The gradient of the PWFTin the Maastrichtian oil zone is shown. (b) PWFT plotted against depth for the Dan field wells MA-1, MD-1, and ME-1. Top chalk ismarked by a thick solid line. The Danian-Maastrichtian boundary (D-M) is marked by a thin solid line. GOC is marked by a thickdashed line. The transition from the irreducible zone to the transitional zone is marked by a thick solid line (T-Z). The gradient ofthe PWFT in the Maastrichtian oil zone is shown. The FWLs (FWL, V-K) proposed by Vejbæk and Kristensen17 are shown. Thegradient of the PWFT in the gas zone in MA-1 is shown.


wells N-22 and MA-1. The PWFT points of N-22 can be plotted asheight above the FWL of N-22; thus, this plot constitutes a uniquerelationship between the PWFT and the height above the FWL.Subsequently, the PWFT points of the MA-1 are plotted. Supposethat the FWL of MA-1 were equivalent to the FWL of N-22. In thatcase, the PWFTs of the zone above the transitional zone in MA-1should fall on the PWFT/HAFWL relationship of N-22. However,it is seen (Fig. 7) that the PWFTs in MA-1 are larger as comparedto the PWFT/HAFWL relationship established for N-22. Conse-quently, the FWL in MA-1 must be located higher than that inN-22. In a similar manner, the depths of FWL in N-2X, N-3X,MD-1, and ME-1 were predicted (Table 3) by using the FWL ofN-22 as a fixed point along with the 3.5-nm/100-ft gradient.Fig. 8 shows the heights of the PWFTs above the FWL of N-22.The depths of the FWL are also shown relative to the FWL ofN-22. The zone above the transitional zone in N-6 is absent; con-sequently, these data points are not included. The FWL in MA-1,MD-1, and ME-1 is located 390 ft, 140 ft, and 250 ft deeper(respectively) than the depths of the FWL reported by Vejbæk andKristensen.17 However, the relative depths of FWL between thewells are approximately the same as those reported by Vejbæk andKristensen.17 We feel, however, that the use of seismic inversionand a modified Leverett J-function introduces a significant sourceof error that easily can explain the difference between our estima-tion and that made by Vejbæk and Kristensen.17

The estimation of the FWL by the pseudo film-thickness gra-dient can only be performed in wells in which a zone above the

transitional zone is present. Whether such a zone exists should beevident from the gradient of the PWFTs. Consequently, the FWLcan only be estimated in crestal wells that penetrate the zone abovethe transitional zone, and the method is thus not readily applicableto flanks of the reservoir containing only the transitional zone.However, it is worth noticing that the use of the PWFT/HAFWLrelationship to establish the FWL is inexpensive because it onlyrequires conventional core-analysis data, porosity and permeabil-ity, and log data, which are routinely measured in many explora-tion wells. The equivalent radius method and the Leverett J-function rely on capillary pressure measurements of cores, whichare more expensive and time consuming and, consequently, fewerin number.

Once the FWL has been established, the capillary pressure(Eq. 1) can be calculated, and the distribution of the water aroundthe grain contacts and surface area (Eq. 11) in the simple spheri-cal grain model can be established. The capillary pressure at thetop of the transitional zone of MA-1 is 85 psi; it increases to115 psi at the top of the oil column. These capillary pres-sures correspond to 45% of the water and 36% of the water pres-ent around the grain contacts at the top of the transitional zoneand the top of the oil column, respectively, for the BCC modelshown in Fig. 5 (rgrain�0.5 micron, �ow�0, and �ow�40 mN/m).The BCC model assumes that the formation is composed ofequal-diameter spheres, which is not the case. The calcula-tions show that the water-film model is incapable of capturingall the features of the true water distribution in chalk. However, thewater-film model still provides meaningful results because thereis a correlation between specific surface area and grain-contact density.

Fig. 4—Schematic drawing of water located as a pendular ringaround a grain contact.

Fig. 5—Ratio between water present as pendular rings and wa-ter present as surface water as a function of capillary pressurefor BCC and FCC closest packings of spheres with a grain ra-dius of 0.5 microns. An oil/water contact angle of zero andan oil/water interfacial tension of 40 mN/m were used for thefluid properties.


Change of Wettability With Height. Along with the thinning ofthe PWFT with increasing capillary pressure, it is expected that theelectrical resistivity of the formation should increase. In wellsMA-1, N-2X, and N-3X, very high resistivities are encountered inthe upper part of the Maastrichtian zone. The resistivity reachesstable values larger than 100 ohm-m, which corresponds to resis-tivity indices larger than 100. Consequently, the electrical currentis severely limited in its transport. The validity of the saturationexponent used in Archie’s equation is therefore questionable. Wesuspect that part of the internal surface has become oil-wet becauseof the high capillary pressure in the top of the Maastrichtian zone.

Large resistivities are never found in the Danian zone, probablybecause of the large number of grain contacts per volume, whichensures that bridges of water will carry the electrical currentthrough the formation. A partial change of wettability is consistentwith the model of Kovcsek et al.25 The change of wettability isalso in accordance with the observation of the increased Amott-Harvey wettability index with depth of the Dan field.1 Thus, wet-tability change must be explained by a combination of capillarypressure and lithology.

Conclusions1. The amount of water above the transitional zone is controlled to

a first approximation by the internal surface area of the forma-tion. However, the grain-contact density also influences the wa-ter saturation.

2. The PWFT yields valuable information about the irreduciblewater saturation above the transitional zone and is able tosmooth out large variations in irreducible water saturation.

3. There exists a pseudo water-film gradient with depth, which canbe used as an alternative estimate of the FWL in water-wetreservoirs for which no reliable RFT data exist. The estimationof the FWL by calculation of the PWFT requires only conven-tional log- and core-analysis data.

4. The approach of using a pseudo water film can be refined byincluding the effects of water located as pendular rings aroundgrain contacts, in addition to applying more specific values forthe constants in Archie’s equation for the different formations.The splitting of the water saturation into surface and contactwater yields a more accurate interpretation of the water satura-tion above the transitional zone.

Nomenclaturea � Archie’s constantA � constantC � constant for Kozeny’s equation

C1 � center of grain (see Fig. 4)C2 � center of radius of curvature (see Fig. 4)

g � gravitational constanthw � pseudo water-film thickness

k � permeabilitym � cementation factorn � saturation exponentQ � position (see Fig. 4)po � oil pressurepw � water pressureP � interface grain intersection (see Fig. 4)

Pc � capillary pressurerow � radius of curvature

R, Rgrain � radius of grainRt � resistivity of the formation

Rw � resistivity of the formation water

Fig. 6—Determination of the FWL in N-22 from RFT data. TheFWL was determined to be at 7,269 ft TVDSS.

Fig. 7—The principle of determining the FWL in an offset wellfrom a well with known FWL. The FWL of N-22 was establishedby using RFT data. This plot thus constitutes the capillary pres-sure saturation relationship used in offset wells. Subsequently,the PWFT points of MA-1 are plotted. The PWFT gradient of N-22is below the PWFT gradient of MA-1. Consequently, the FWL inMA-1 must be located higher than in N-22.


Sw � water saturationSw,contact � water saturation of grain contactsSw,surface � water saturation of grain surface

Swi � irreducible water saturationS � specific surface with respect to total volume

Sp � specific surface with respect to porosityX � exponent for Swi in the Wyllie and Rose equationY � exponent for in the Wyllie and Rose equationz �film thickness of real water film � angle

* � angle� � curvature of the oil/water interface� � contact angle

� � distance from C1 to C2 (see Fig. 4)� � distance � 3.1415925∏ � disjoining pressure isotherm�b � bulk density�h �hydrocarbon density

�ma � matrix density�w � water density� � interfacial tension� � formation porosity� � distance from O to C2 (see Fig. 4)

AcknowledgmentsFinn Engstrøm of Maersk Oil and Gas AS is acknowledged forstimulating discussions concerning the distribution of the water.Abdelhakim Chtioui made the first calculations of water-filmthickness as part of his MS thesis at the Technical U. of Denmark.The Geological Survey of Denmark and Greenland is acknowl-edged for supplying data from the seven wells.

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Chalk Research Phase V (1995) RF-Rogaland Research, Stavanger.

2. Engstrøm, F.: “A New Method to Normalize Capillary PressureCurves,” paper SCA 9535 presented at the 1995 International Sympo-sium of the Soc. of Core Analysts, San Francisco, 12–14 September.

3. Leverett, M.C.: “Capillary Behaviour of Porous Solids,” PetroleumTechnology (August 1941) 142, 152.

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Fig. 8—PWFTs as a function of height above the FWL using theFWL of N-22 and the average PWFT gradient of 3.5 nm/100 ft topredict the depths of the FWL in offset wells with no RFT data.The FWLs of the different wells have been drawn in the graph.


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AppendixCalculation of Water Residing Around Grain Contacts. Theobjective of this appendix is to calculate the volume of a pendularring of water around a grain contact. The volume of water dependson the curvature of the water/oil interface, the size of the twograins, and the contact angle between the grain surface and thewater/oil interface.

Fig. 4 shows a principal sketch of two grains, with contact pointO located at the origin of a rectangular coordinate system. Thez-axis points upward, and the x-axis points to the right.

The two grains have radius R, and the center of the upper grainis located at point C1. An oil/water interface with curvature row inone direction and center C2 is shown.

We observe from Fig. 4 that

� = C1C2 = �R2 � sin2 � + �R � cos � + row�2

= �R2 + row2 + 2 � R � row � cos � . . . . . . . . . . . . . . . . . . . (A-1)

� = OC2 =R

tan *= �row

2 + 2 � row � R � cos �. . . . . . . . . (A-2)

We may thus express the angles and * as follows:

sin =R � sin �

�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-3)

cos =row + R � cos �

�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-4)

sin * =R

�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-5)

cos * =�

�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-6)

By using basic geometry, we obtain

sin �* − � = �R � �row + R � cos �� − R � sin� � ��

�� .

. . . . . . . . . . . . . . . . . . . . . . . . (A-7)The volume of water in a pendular ring is given as the differ-

ence between two integrals.

V = �0

�

�� − �row2 − z2�2

dz

− �0

�

�R2 − �R − z�2�dz , . . . . . . . . . . . . . . . . . . . . . . . . . (A-8)

where ��row·sin(*–) corresponds to the distance in the z-direction from the x-axis to the point at which the fluid interfaceintercepts the grain. The difference of the two integrals can beevaluated as

V = ��2 + row2 � � row sin�* − �

− R�row � sin�* − ��2� − � � � row2 �* − �

− 2 � � � � � � row�1 − sin2�* − � . . . . . . . . . . . . . . (A-9)The capillary pressure is given approximately by

Pc − � 1

row−

1

� − row�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-10)

Calculation of the Amount of Water Present as Pendular Ringsand Film Phase as a Function of Capillary Pressure. The vol-ume of water present around the grain contacts is calculated as thenumber of grain contacts multiplied by the amount of waterpresent around a single grain contact with a given capillary pres-sure, as shown previously. A BCC lattice and an FCC lattice have8 and 12 contacts per unit lattice, respectively. Porosity equals32% and 26%, respectively, for the unit lattices, and the speci-fic surface areas with respect to porosity are 12 and 17 �m–1.The FCC lattice is closest packed, which normally is not thecase for chalk. However, the grain-size distribution probablymakes a closer packing possible, which leads to a higher density ofgrain contacts.

The amount of water present as surface water is calculated asthe surface area times the thickness of the layer. We use a layerthickness of 6 nm, corresponding to 5% water saturation, whichcan be achieved in chalk samples during mercury injection. Theratio between the surface water and the volumes is taken as onevolume divided by the other (Fig. 5).

SI Metric Conversion Factorsft × 3.048* E–01 � m

mN/m × 1.0* E–03 � N/mnm × 1.0* E–09 � mpsi × 6.894 757 E+00 � kPa

*Conversion factor is exact.

Jens K. Larsen is currently a petrophysicist with Maersk Oil Qa-tar; he joined the company in 2001. His main interest lies withinpetrophysical evaluation and flow phenomena on pore andcore scale. Larsen holds an MS degree in mechanical engi-neering and a PhD degree, both from the Technical U. of Den-mark. Ida L. Fabricius is an associate professor at the TechnicalU. of Denmark, which she joined in 1985 after leaving Mærsk Oiland Gas AS. e-mail: [email protected]. Her main interests are petro-physics, pore-fluid interaction, and acoustic properties. Fabri-cius holds an MS degree in geology from Copenhagen U. anda PhD degree from the Technical U. of Denmark.


interpretation of water saturation above the transitional...

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