network modelling of wettability and pore geometry effects on

Ž .Journal of Petroleum Science and Engineering 24 1999 255–267www.elsevier.nlrlocaterjpetscieng

Network modelling of wettability and pore geometry effects onelectrical resistivity and capillary pressure

H.N. Man, X.D. Jing )

Centre for Petroleum Studies, T.H. Huxley School, Imperial College of Science, Technology and Medicine, Prince Consort Road,London, SW7 2BP UK

Abstract

Recent research efforts have focused on using simple non-circular cross-sectional pore shapes to honour the physicsobserved at the pore scale. For example, there is evidence to suggest variations of wettability occur at this level. These porescan exhibit water-wet and oil-wet regions, depending on the physics of wetting films, and hence the porous medium maybeof mixed-wettability character. For low water saturations, electrical resistivity cannot be physically simulated at the porescale using cylindrical tubes, even though wetting film thickness’ and pore constrictions are taken into account.

A three-dimensional network model that investigates the petrophysical characteristics, electrical resistivity and capillarypressure, is presented. The influence of saturation history is also modelled. Key pore geometrical attributes such as pore

Ž .shape, aspect ratio, pore coordination number pore connectivity and pore size distribution are included in the model. Inaddition, pore constrictions are introduced which may result in phase trapping via snap-off within the tube itself.

Analysis of our developing network model starting from representing the pore shape as circular is presented. Using asimple non-circular cross-sectional pore shape we show bulk water retained in the crevices give rise to predictions that are inclose agreement with electrical resistivity and capillary pressure trends observed in experiments. Numerical results arepresented and compared with experimental data. q 1999 Elsevier Science B.V. All rights reserved.

Keywords: network model; electrical resistivity; capillary pressure; pore geometry; wettability

1. Introduction

The petrophysical parameters, such as electricalresistivity and capillary pressure, are very dependenton the complex irregular pore space of rocks. Oneapproach to simplify the real rock pore structure is touse an equivalent three-dimensional pore networksystem which captures the underlying physics andstill be able to predict the petrophysical parameters

) Corresponding author. Tel.: q44-171-594-7320; Fax: q44-171-594-7444.

Ž .E-mail address: [email protected] X.D. Jing .

in question. Network modelling should not be viewedas a complete substitute for experimental work butthe results are interpreted in order to improve theunderstanding of pore scale physics. For example,capillary pressure is a strong function of the narrow-est regions of the void space in porous media. Moresophisticated numerical models can implement pore

Žshape Yale, 1984; Jing, 1990; Kovscek et al., 1993;. ŽBlunt, 1997a,b and wettability alteration Kovscek

et al., 1993; McDougall and Sorbie, 1995; Dixit et.al., 1996; Blunt, 1997b . It has been realized that

these physical aspects also play an integral part inthe understanding of the petrophysical properties of

0920-4105r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0920-4105 99 00047-9

( )H.N. Man, X.D. JingrJournal of Petroleum Science and Engineering 24 1999 255–267256

rocks. Network modelling can help to resolve con-flicting experimental data or to provide data that isdifficult to obtain in the laboratory with a certainamount of confidence, for example, in order to up-scale.

Ž .McDougall and Sorbie 1995 and Dixit et al.Ž .1996 predicted capillary pressure and relative per-meability curves incorporating the effects of wetta-

Ž .bility alteration. Blunt 1997b , using a wettabilityŽ .alteration scenario proposed by Kovscek et al. 1993 ,

could designate an individual element having a varia-tion of wetting. His work drew on experimentalevidence that showed variations of wettability occur

Ž .at the pore scale Fassi-Fihri et al., 1991 . BluntŽ . Ž .1997b was able to reproduce Dixit et al.’s 1996results but because of the inclusion of pore scalenon-uniform wettability, he was able to deduce thatresidual oil saturation depended on the fraction ofelements exhibiting oil-wet regions.

Hydrocarbon saturation is an important parameterfor reserve estimates and prediction of future produc-tion performance. By using one of the more reliabletechniques, namely electrical logging, we can esti-mate the hydrocarbon saturation based on the Archietypes of equations, which relate the resistivity of arock to its porosity and water saturation. However,many studies have shown that the Archie equationsare not always obeyed.

ŽContrary to popular belief, several authors Di-ederix, 1982; Swanson, 1985; Worthington et al.,

.1989; Moss et al., 1999 have experimentally ob-served the so-called ‘‘non-Archie’’ behaviour forwater-wet rocks. This type of rock shows a point ofinflexion in the logarithmic resistivity index vs. satu-ration plots such that for the lower water saturationrange, the resistivity index curves toward the satura-

Ž . Ž .tion axis. Swanson 1980 and Longeron et al. 1986experimentally observed hysteresis for strongly wa-ter-wet sandstone samples during drainage and imbi-bition. They found resistivity indices were lowerduring imbibition than drainage. Other researchers

Žhave found that the reverse is true Dunlap et al.,.1949 . For strongly water-wet carbonate samples,

Ž .Longeron et al. 1986 observed virtually no hystere-Ž .sis in their resistivity curve. Wei and Lile 1991

focused on the combined effects of saturation historyand wettability. They have found for oil-wet sand-stones below a finite water saturation that the resis-

tivity indices between drainage and imbibition wereconsiderably different.

In terms of network modelling, Sharma et al.Ž .1991 looked at a range of factors that influencedthe shape of the resistivity index curves. They incor-porated wetting film thickness’ into their constricted

Ž .circular pore shape. Zhou et al. 1996 tackled theproblem using percolation concepts. Employing apore shape with crevices their model could simulatethe correct trends for different preferential-wet rocks.

To conclude, it has been recognized that pore sizedistribution, pore geometry, saturation history andwettability influences the resistivity of a rock. Thisindicates that the saturation exponent is very depen-dent on the distribution of fluids in the pore space.

2. Theoretical model

The current developing network model considersŽ . Ž .constricted pore tubes Sharma et al., 1991 Fig. 1 ,

so that a pore body can be imitated by controllingŽthe constriction factor pore body radius divided by

pore throat radius, which we denote as the narrowest. Ž .part of the pore tube Fig. 1 . To calculate capillary

pressure and electrical resistivity curves, the poretube is divided into sections of infinitesimal constantthickness by making cuts perpendicular to the x-axis.In cross-section, each section is star-like character-

Ž .ized by a radius Fig. 1 . Four closely packed uni-Žform rods form this cross-sectional shape Mason

.and Morrow, 1986 , which consists of crevices. It isŽ .referred to as a grain boundary pore GBP shape.

The pore tubes described above adhere to form aŽ .cubic lattice Fig. 2 . The lattice can have a maxi-

mum pore coordination number of six, i.e., six tubesare connected to each junction. The coordinationnumber is defined as the average number of poretubes adjoined to each junction.

Our network model into two-phase situations isŽ .based on Kovscek et al.’s 1993 work. At the pore

scale it has been recognized that crevices in the porespace may give rise to the mixed-wettability charac-ter. They proposed an idealized scenario where re-gions of the pore space contacted by oil, dependingon the physics of wetting films, may become oil-wet.Assuming the system is initially fully water satu-rated, the crevices contain water and remain water-wet.

( )H.N. Man, X.D. JingrJournal of Petroleum Science and Engineering 24 1999 255–267 257

Ž .Fig. 1. Different viewpoints from top clockwise: 3-D, circumferential and transverse of constricted pore tube. Pore constriction descriptionŽ .after Sharma et al. 1991 .

Ž .Fig. 2. Schematic picture showing part of the cubic lattice filled with oil left . Close-up of schematic picture showing pore constrictions andŽ .GBP shap right .


For water-wet regions using a circular pore shape,water films can be significant in calculating electri-

Ž .cal resistivity curves Sharma et al., 1991 . Using anapproximate solution involving hyperbolic trigono-

Ž .metric functions Hirasaki, 1988 , the contributionmade by water films was incorporated. For poreshapes that exhibit crevices, the contribution can beexcluded. This is because the bulk water retained inthe crevices dominates the volume.

In reality, wettability alteration at the reservoirscale occurs over geological periods of time. Thecritical capillary pressure, associated with wettability

Žreversal, depends on mineralogy Wolcott et al.,. Ž1993 , oil and brine properties, such as pH Dubey

.and Doe, 1993 , reservoir pressure and temperatureŽ .Hirasaki, 1988 . Experiments on core plugs showthat the impact of wettability alteration on hydrocar-

Ž .bon recovery is significant Salathiel, 1973 . Kamin-Ž .sky and Radke 1997 have showed mathematically

that wettability alteration may occur by diffusion ofsurface active components in the oil even throughthick water films. However, they do concede wetta-bility alteration is likely to follow water film rupture,otherwise diffusion theory predicts that all reservoirscontaining surface active polar compounds will be-come oil-wet. The actual mechanism of wettabilityreversal is still unclear.

Whether or not the pore shape contains crevices,common to all pore shapes the following wettability

Ž .alteration scenario proposed by Kovscek et al. 1993

was implemented in our network model. The systemwas initially considered to be completely water-wet.Thick stable water films reside next to the whole

Ž .solid surface Fig. 3a . If the critical capillary pres-sure, unique for each section has been exceeded, thethick water film then ruptures into a molecular thinfilm by which the portion of the pore wall wasseparated from the oil. As a result, these regionsallow polar surface active components in the oil to

Ž .adsorb andror deposit on the pore wall Fig. 3b .These regions are now oil-wet. In this paper, thisdual occupancy of both oil-wet and water-wet re-gions within an infinitesimally thin section of thepore tube is referred to as mixed-wet.

Ž .Similar to Kovscek et al. 1993 , if we assumeuniform mineralogy throughout the porous medium,the fluid system and the temperature is constant, thenthe curvature of the pore wall is the only dominatingfactor in determining the stability of the water film.Despite this, the idealized scenario proposed by

Ž .Kovscek et al. 1993 gives a theoretical understand-ing of wettability alteration.

2.1. Numerical solution

Considering the circumferential radius of curva-ture of the oilrwater interface near the corner of thecrevices, it is much less than the transverse radius of

Ž .curvature Fig. 1 . Therefore, the transverse radius ofcurvature in our calculations has been neglected

Ž . Ž .Fig. 3. a Water-wet GBP shape with thick water films coating the pore wall. b Mixed-wet GBP shape with molecular thin films, whichallow wettability alteration.


Ž .Chambers and Radke, 1991 . The electrical resis-tance of the pore tube is found by integrating thesections in electrical series.

The junction where each tube meets is assumed tobe a hypothetical mathematical point with no volumeand negligible resistance relative to the pore tubes.The volume is purely effective here because the poretubes take the volume into account. Simulations havebeen carried out for various network sizes of up to15=15=15 junctions. The lengths of all the poretubes are constant and are calculated to achieve thedesired porosity. The throat radii are randomly as-signed throughout our network either using variousuni-modal or multi-modal pore size distributions. Itis also possible to input more realistic pore sizedistributions deduced from, say, mercury porosime-try data.

To calculate the electrical resistivity of the model,an arbitrary potential difference is imposed acrossthe inlet and outlet faces. No-flow or periodic bound-ary conditions can be imposed in the direction per-pendicular to the potential gradient. The voltages ofthe junction where each tube meets are calculatedusing Ohm’s and Kirchoff’s laws. A successiveover-relaxation technique is employed to solve the

Ž .iterative algorithm Jing, 1990 .

2.2. Primary drainage

At the beginning of each simulation the model isfully saturated with water. The whole surface of themodel is designated as water-wet with a contactangle of zero. By increasing the capillary pressure,oil enters the network mimicking primary drainageinto a reservoir. The primary indicates the reservoiris fully saturated with water initially. For clarity,drainage will be consistently defined as oil displac-ing water and imbibition as water displacing oil,

Ž .irrespective which fluid preferentially wets coatsthe solid surface. The capillary pressure will bedefined as the oil pressure minus the water pressure.

However, oil may only enter a tube if one of theadjacent tubes has already been penetrated by oil. Anincrease in the capillary pressure pushes theoilrwater interface in discrete steps to the oppositeend of the pore tube by which it entered. Cross-sec-tionally, the oilrwater interface moves towards thecrevices. The fluids are assumed to equilibrate so

that the capillary pressure is constant across theoilrwater interface. This in turn determines themeniscus shape. Capillary equilibrium is acceptableas long as the capillary number is no greater than

y6 y7 Ž .10 to 10 Blunt, 1997a,b . The movement ofthe oilrwater meniscus as a function of water satura-tion was calculated based on Mayer and StoweŽ .1965 . Once capillary pressure equilibrium has beenreached, the electrical resistance of each constrictedpore tube is calculated by integration along the tubelength.

Infinitesimally thin sections of the pore tube whereoil has broken through and which have their criticalcapillary pressure exceeded, will exhibit a molecularwater film residing next to the pore wall. Hence, the

Žcontact angle is no longer zero Hirasaki, 1988;.Kovscek et al., 1993 . The cross-sectional area of

water and hence the volume of water retained is afunction of contact angle.

Not only does this affect volumetric and electricalresistivity calculations but using Kovscek et al.’sŽ .1993 wettability alteration scenario, these surfacesbecome oil-wet at the beginning of spontaneous im-

Ž .bibition Fig. 3b . To describe the meniscus move-ment as the capillary pressure is decreased the wet-tability variation of the pore space needs to bedistinguished. However, because our pore tube is ofvarying radii the idealized wettability pattern differsfrom an unconstricted pore tube. Once it is pene-trated by oil and the critical capillary pressure hasbeen exceeded the molecular water film spans thetube. Pore constrictions may prevent this from hap-pening.

2.3. Spontaneous imbibition

The drainage process in our simulations stops at amaximum capillary pressure. Portions of the porewall with a molecular thin water film were assumedto become oil-wet. The preceding arguments by

Ž .Kovscek et al. 1993 describe how oil-wet regionscould be created in initially water-wet systems. Thedisplacement of oil during spontaneous imbibitiontakes place only if there is a continuity of oil to the

Žoutlet provided by tubes penetrated by oil Lenor-.mand and Zarcone, 1984 . If oil enters a tube but

does not penetrate it, then oil may only escape by thejunction that it entered by. Since the contribution


from the pore nodes is considered implicitly by usingconstricted pore tubes and we assume strongly wet-ting conditions, this paper does not incorporate the

Žco-operative pore filling process Lenormand and.Zarcone, 1984; Blunt, 1997a; Øren et al., 1998 . The

co-operative pore filling process and its effect on thesaturation distribution and electrical conduction atcapillary pressure equilibrium is not quantitativelyclear.

In water-wet sections, as the capillary pressure isdecreased, water will spontaneously imbibe awayfrom the crevices along the same course as in pri-mary drainage up to a point. Oil may then form aninscribed circle within the pore shape. Further in-creases in water may result in oil losing contact withthe pore wall. When this happens an instability

Ž .occurs Ransohoff et al., 1987 . Oil then becomesŽ .disconnected snapped-off to achieve a stable con-

Ž .figuration Roof, 1970 . For small constriction fac-tors, the pore tube will almost spontaneously fill withwater. Other conditions are also necessary in order

Ž .for snap-off to prevail Chambers and Radke, 1991 .Water may also invade the pore tube by piston-like

Ž .advance Li and Wardlaw, 1986 . However, in ournetwork model we have implicitly assumed water-wetregions having a contact angle of zero and capillaryforces dominate the process. Therefore, snap-off is

Žthe dominant mechanism occurring here Dixit et al.,.1996; Blunt, 1997a .

ŽIn mixed-wet sections water-wet and oil-wet re-gions co-existing in an infinitesimally thin section of

.the pore tube there is no known trapping of oil.Surface active components adsorbed along the porewall remain. Therefore, water cannot imbibe alongthe same course as in water-wet sections.

3. Results and discussion

3.1. Effect of pore geometry

3.1.1. Circular pore shapeFig. 4 shows a typical simulation of capillary

pressure and resistivity index curve for a water-wetrock using circular pore geometry. The gradient is asexpected for water saturations greater than 20%, but

Fig. 4. Circular pore shape.


Fig. 5. GBP shape with no constrictions.

for water saturations less than this the resistivityindices are predicted to be too high. The reason forthis can be explained as follows. Although waterfilms can contribute to the conductivity of the rockŽ .Sharma et al., 1991 , the magnitude of the filmthickness is in the order of nanometres. A typicalpore radius is in the order of microns to tens ofmicrons. At a low saturation, most pore tubes are leftwith a thin water film residing next to their surface.These tubes with thin water films dominate thevoltage iterative equation and a large change ingradient is thus observed.

3.1.2. GBP shape with no constrictionsThe previous model, with circular pore shapes,

incorporated the contribution made to the volumeand electrical conductance by thin residual films. ForGBP shapes, the volumetric and electrical contribu-tion of thin residual films can be ignored withoutloss of generality because the bulk water retained inthe crevices dominates these properties.

ŽWhen no pore constrictions i.e., uniform pore.size along the tube length are implemented, the

most striking feature about the simulated resistivityindex curve is that the gradient is very close to unity

Ž .Fig. 5 . This is in agreement with theoretical expec-tation, because the electrical tortuosity factor is alsonearly one.

3.1.3. GBP shape with constrictionsIntroducing pore constrictions into the model gives

rise to predictions that are in close agreement withelectrical resistivity and capillary pressure trends ob-served in experiments.

A series of simulations have been performed toquantify the effects of pore constriction, coordinationnumber and pore size distribution on capillary pres-sure and resistivity index curves. Table 1 summa-

Table 1Input parameters used in the network model for Figs. 6–9

Figure 6 7 8 9

Porosity 0.25 0.25 0.25 0.25No. of normal distributions 1 1 1 2

aŽ .Mean mm 30 30 30 30, 10aŽ .Standard deviation mm 5 5 3 3, 1

Constriction factor 2 2 2 2Coordination number 6 5 6 6

aSecond normal distribution.


Fig. 6. Effect of higher constriction factor compared with Fig. 5.

rizes the model parameters used in simulations ofFigs. 6–9

The curvature of the electrical resistivity plotŽ .Fig. 6 can be explained as follows. Note that thereappears to be two regimes. The points of both curvesrefer to equally spaced capillary pressures. In thefirst regime, after the threshold capillary pressure isexceeded in order to invade the inlet face, a smallincrease in the capillary pressure results in a largedecrease in the water saturation. In the second regime,for the same increase in capillary pressure the watersaturation decreases by a relatively small amount.Therefore, relative to the first regime, the gradient islarger. It can be inferred in the first regime that oil ispenetrating the relatively large pore tubes and enter-ing part of some small pore tubes. In the secondregime, pore tubes penetrated by oil are dominatingthe electrical resistivity and a larger capillary pres-sure is needed in order to move the meniscus to-wards the crevices.

Several authors have experimentally observedŽsuch curvature Diederix, 1982; Swanson, 1985;

.Worthington et al., 1989; Moss et al., 1999 . Therealso does exist experimental data in which the curva-ture is not observed at all. This observation is gener-

ally true for data sets that have relatively high end-Ž .point water saturations. Diederix 1982 and Swan-

Ž .son 1985 explained these apparent conflictingtrends. They both attributed the curvature effect tothe micropore system in the porous media. Cores inwhich microporosity is not significant show no cur-vature. This is because the rough grain surface,which a non-micropore system lacks, retains water.Electrical conductance here is, therefore, morefavourable which explains the lower saturation expo-nent relative to high water saturations. The conclu-sions in the network modelling results, of a decreas-ing negative gradient, are very much analogous tothis. Note also that at low water saturations thegradient approaches unity, as expected theoretically.

It is known that excess conductivity associatedwith electrical double layers in shaly sands alsoresults in a curved resistivity index trend. However,double-layer excess conductivity is outside the scopeof this paper.

3.2. Effect of saturation history

Utilizing a contact angle of zero, our networkmodel predicts significant hysteresis in the capillary


Fig. 7. Effect of lower coordination number compared with Fig. 6.

Fig. 8. Effect of smaller standard deviation between pore radii compared with Fig. 6.


Fig. 9. Effect of bi-modal normal distribution.

Ž .pressure and resistivity index curves Figs. 6–9 . Fora given water saturation, the resistivity indices arelower in spontaneous imbibition than in primary

Ž .drainage Figs. 6–8 . This can be explained with thehelp of the corresponding capillary pressure curves.The large decrease in capillary pressure resulting in asmall decrease in water saturation signifies that thecapillary pressure has not decreased low enough forthe condition of snap-off to be met. The capillarypressure for the inscribed circle configuration, whichin most cases propagates snap-off, is lower than thecapillary entry pressure. The sudden increase in wa-ter saturation indicates that snap-off is occurring.

Ž .Note the different gradients in this cycle Figs. 6–9 .Ž .Longeron et al. 1986 also observed such a two-tier

gradient system. Hysteresis, between drainage andimbibition, is due to the different pore-level pro-cesses that occur during each cycle.

3.3. Effect of constriction factor

An increase in the constriction factor does show amarked decrease in the conductivity of the rockŽ .compare Figs. 5 and 6 . This can be explained by

Ž .the fact that for small constriction factors Fig. 5 ,for the same water saturation, the water phase is less

tortuous compared with large constriction factorsŽ .Fig. 6 . By setting a constriction factor of one, wederive the same capillary pressure and resistivitycurves as the model with no pore constrictions.

3.4. Effect of coordination number

The network can simulate random removal of theconducting branches. If some pore tubes are removedthen clearly the degree of interconnectedness be-tween them is poorer. Consequently, the coordina-tion number is lower and transport properties arereduced.

The results show, as expected, that when thecoordination number is decreased from 6 to 5, the

Žsaturation exponent increases compare Figs. 6 and.7 . Note also that for the lower coordination number

Ž .of 5 Fig. 7 , the resistivity indices are significantlyhigher than for the higher coordination number of 6Ž .Fig. 6 , i.e., the linear part of the curve is sustaineduntil a lower water saturation.

3.5. Effect of pore size distribution

When the standard deviation of the throat radii isŽ .decreased Fig. 8 , the resistivity curves displays


Ž .slight enhanced curvature compare Figs. 6 and 8 . Itappears Fig. 8 has a higher saturation exponent thanFig. 6. The reverse argument holds true for changing

Ž .the mean throat radii. In fact, Sharma et al. 1991has hypothesized that the ratio between the two poresize characteristics is the controlling parameter.

The investigation now extends to pore size distri-bution of the rock displaying two distinct normal

Ž .distributions Fig. 9 . Consistent with experimentalŽ .observations Worthington et al., 1989 our model

predicts ‘‘kinks’’ in both the electrical resistivity andŽ .capillary pressure curves Fig. 9 although the kink is

more pronounced in the resistivity curve. The kinksindicate that the oil at this stage is unable to invadethe smaller normal distribution. Once it does, thecurves revert back to the original pattern, because thewater saturation can again decrease.

3.6. Effect of wettability alteration

To study the effect of wettability alteration, re-gions of the pore space are assigned oil-wet duringprimary drainage by using a scenario proposed by

Ž .Kovscek et al. 1993 .At the beginning of each simulation the model is

fully saturated with water. The whole surface of the

model is designated as water-wet with a contactangle of zero. Regions with molecular films whichwe assumed to be created during primary drainageusing the wettability alteration scenario proposed by

Ž .Kovscek et al. 1993 , have a non-zero contact angleŽ .Hirasaki, 1988; Kovscek et al., 1993 . Rather thandetermining the contact angle for each molecular

Žfilm, which can be theoretically done Hirasaki, 1988;.Kovscek et al., 1993 , we have assumed a contact

angle of 208 for each molecular thin film. In subse-quent cycles these regions become oil-wet and wereassigned a contact angle of 1808.

Different sets of capillary pressure and resistivitycurves are observed. Fig. 10 shows capillary pressureand resistivity index curves with mixed-wet sections.As expected, the hysteresis in the capillary pressure

Ž .curves grows larger compare Figs. 6–10 .Comparing Figs. 6–10, it is interesting to note

that a different hysteresis trend has been found, inthe model where part of the open space has changedfrom water-wet to oil-wet. In Fig. 10, approximately32% of the total pore surface is computed to beoil-wet as a result of wettability alteration. Thepredicted resistivity indices are now higher than thatof primary drainage. This is due to mixed-wet sec-tions dominating the electrical resistance of the tube

Fig. 10. Effect of wettability alteration compared with Figs. 6–9.


because the resistance is found by integrating thesections of infinitesimally thin thickness in series.Note that snap-off is still possible, because 68% ofthe total pore surface remains water-wet. The electri-cal resistivity and capillary pressure trends agree

Ž .with our experimental data Moss et al., 1999 .

4. Conclusions

A three-dimensional pore network model has beendeveloped to simulate rocks with various wettingscenarios. This model includes the effects of poregeometry, pore connectivity, saturation history andwettability alteration. The model is capable in pre-dicting the generic behaviour of electrical resistivityand capillary pressure trends of non-shaly sands. Ourchosen representation of pore shape retains bulkwater in the crevices and leads to more realisticmodelling of pore scale wettability alteration.

We intend to predict electrical resistivity andcapillary pressure characteristics of a range of reser-voir samples and compare them with experimentaldata. This may require the implementation of contactangle effects and a further extension to secondarydrainage. Long-term aims include unifying the ef-fects of clay contents, clay distributions and theimpact of varying confining stress in this model.

Acknowledgements

We would like to thank Carlos Grattoni, AdamMoss, Paul Worthington, Jonathan Hastings and Car-olina Coll for their comments and stimulating discus-sions. We are also indebted to EPSRC for financialsupport.

References

Blunt, M.J., 1997a. Effects of heterogeneity and wetting onrelative permeability using pore level modeling. SPE J. 2,

Ž .70–87, March .Blunt, M.J., 1997b. Pore level modeling of the effects of wettabil-

Ž .ity. SPE J. 2, 494–510, December .Ž .Chambers, K.T., Radke, C.J., 1991. In: Morrow, N.R. Ed. ,

Interfacial Phenomena in Petroleum Recovery. Marcel Dekker,New York, pp. 191–255.

Diederix, K.M., 1982. Anomalous Relationships Between Resis-

tivity Index and Water Saturations in the Rotliegend Sand-Ž .stone The Netherlands . Transactions of the SPWLA 23rd

Annual Logging Symposium, Corpus Christi, TX, July 6–9,1982, Paper X.

Dixit, A.B., McDougall, S.R., Sorbie, K.S., 1996. A Pore-LevelInvestigation of Relative Permeability Hysteresis in Water-WetSystems. In: Symposium on Oilfield Chemistry SPE, Houston,TX. SPE 37233.

Dubey, S.T., Doe, P.H., 1993. Base number and wetting proper-Ž .ties of crude oils. SPE Reservoir Eng., 195, August .

Dunlap, H.F., Bilhartz, H.L., Shuler, E., Bailey, C.R., 1949. Therelation between electrical resistivity and brine saturation in

Ž .reservoir rocks. Trans. AIME 186, 259–264, October .Fassi-Fihri, O., Robin, M., Rosenberg, E., 1991. Wettability Stud-

ies at the Pore Level: A New Approach by the Use ofCryo-Scanning Electron Microscopy. In: 66th Annual Techni-cal Conference and Exhibition SPE, Dallas, TX. SPE 22596.

Hirasaki, G.J., 1988. Wettability: Fundamentals and SurfaceForces. SPE 17367. Symposium on Enhanced Oil RecoverySPErDOE, Tulsa, OK.

Jing, X.D., 1990. The Effect of Clay, Pressure and Temperatureon the Electrical and Hydraulic Properties of Real and Syn-thetic Rocks. PhD Thesis, Imperial College, London.

Kaminsky, R., Radke, C.J., 1997. Asphaltenes, water Films andŽ .wettability reversal. SPE J. 2, 485–493, December .

Kovscek, A.R., Wong, H., Radke, C.J., 1993. A pore-level sce-nario for the development of mixed-wettability in oil reser-voirs. AIChE J. 39, 1072.

Lenormand, R., Zarcone, C., 1984. Role of Roughness and EdgesDuring Imbibition in Square Capillaries. In: 59th AnnualTechnical Conference and Exhibition SPE, Houston, TX. SPE13264.

Li, Y., Wardlaw, N.C., 1986. Mechanisms of nonwetting phasetrapping during imbibition at slow rates. J. Colloid Interface

Ž . Ž .Sci. 109 2 , 473–486, February .Longeron, D.G., Argaud, M.J., Feraud, J.P., 1986. Effect of

Overburden Pressure, Nature and Microscopic Distribution ofthe Fluids on Electrical Properties of Rock Samples. In: 61stAnnual Technical Conference and Exhibition SPE, New Or-leans, LA. SPE 15383.

Mason, G., Morrow, N.R., 1986. Meniscus displacement curva-tures of a perfectly wetting liquid in capillary pore throats

Ž .formed by spheres. J. Colloid Interface Sci. 109 1 , 46–56,Ž .January .

Mayer, R.P., Stowe, R.A., 1965. Mercury porosimetry — break-through pressure for penetration between packed spheres. J.Colloid Interface Sci. 20, 893–911.

McDougall, S.R., Sorbie, K.S., 1995. The impact of wettability onwaterflooding: pore-scale simulation. SPE Reservoir Eng. 10,

Ž .208–213, August .Moss, A.K., Jing, X.D., Archer, J.S., 1999. Laboratory investiga-

tion of wettability and hysteresis effects on resistivity indexand capillary pressure characteristics. J. Pet. Sci. Eng. 24,

Ž .231–242, this volume .Øren, P., Bakke, S., Arntzen, O.J., 1998. Extending predictive

capabilities to network models. SPE J. 3, 324–336.Ransohoff, T.C., Gauglitz, P.A., Radke, C.J., 1987. Snap-off of


gas bubbles in smoothly constricted noncircular capillaries.Ž .AIChE J. 33, 753–765, May .

Roof, J.G., 1970. Snap-off of oil droplets in water-wet pores. SPEŽ .J., 85–90, March .

Salathiel, R.A., 1973. Oil recovery by surface film drainage inŽ .mixed-wettability rocks. JPT 10, 1216–1224, October .

Sharma, M.M., Garrouch, A., Dunlap, H.F., 1991. Effects ofwettability, pore geometry and stress on electrical conductionin fluid-saturated rocks. Log Anal. 32, 511–526,Ž .September–October .

Swanson, B.F., 1980. Rationalizing the influence of crude wettingon reservoir fluid flow with electrical resistivity behavior. JPT

Ž .32, 1459–1464, August .Swanson, B.F., 1985. Microporosity in Reservoir Rocks — Its

Measurement and Influence on Electrical Resistivity. Transac-tions of the SPWLA 26th Annual Logging Symposium, Dal-las, TX, June 17–20, 1985, Paper F.

Wei, J.Z., Lile, O.B., 1991. Influence of wettability on two- andfour-electrode resistivity measurements on Berea sandstone

Ž .plugs. SPE Form. Eval. 6, 470–476, December .Wolcott, J.M., Groves, F.R. Jr., Lee, H.-G., 1993. Investigation of

Crude-OilrMineral Interactions: Influence of Oil Chemistryon Wettability Alteration. In: Symposium on Oilfield Chem-istry SPE, New Orleans, LA. SPE 25194.

Worthington, P.F., Pallatt, N., Toussaint-Jackson, J.E., 1989. In-fluence of microporosity on the evaluation of hydrocarbon

Ž .saturation. SPE Form. Eval., 203–209, June .Yale, D.P., 1984. Network Modeling of Flow, Storage and Defor-

mation in Porous Rocks. PhD Thesis, Stanford University,Stanford, CA.

Zhou, D., Arbabi, S., Stenby, E.H., 1996. Effect of Wettability onthe Electrical Properties of Reservoir Rocks. SCA Conference,Paper Number 9624.

.

network modelling of wettability and pore geometry effects on

Documents