Interpretation of Water Saturation Above the Transitional Zone in Chalk Reservoirs

Interpretation of Water Saturation Above the Transitional Zone in Chalk Reservoirs Jens K. Larsen* and Ida L. Fabricius, Technical U. of Denmark Summary The free water level (FWL) in chalk reservoirs in the North Sea may be hard to establish owing to strong influence from capillary forces and lack of pressure equilibrium across the reservoir. Even where wireline formation tester data on the FWL are available in one well, it is no straightforward task to predict the FWL in other parts of the field, where only conventional core analysis and logging data are available. It is thus difficult to predict the geometry of the hydrocarbon-bearing interval. This paper offers a simple model which, in a given well, allows us to predict the location of the FWL from conventional core analysis and logging data if wireline formation tester data are available from another well. Water saturation is averaged over the internal surface of the formation by applying Kozeny s equation, resulting in a pseudo water-film thickness (PWFT). The PWFT is larger than the equilibrium water-film thickness calculated from the augmented Young-Laplace equation because it includes water associated with grain contacts. The PWFT has a gradient with true vertical depth (TVD) that is related to the capillary pressure; this gradient is approximately the same for the wells investigated. Consequently, a unique relationship between the PWFT and the height above the FWL can be established, provided that the depth of the FWL is known from formation pressure data. The unique PWFT height above the FWL relationship can be used to establish the FWL in offset wells for which no reliable formation-pressure data exist. The estimation of the FWL in offset wells is inexpensive because it requires the use of log data and sufficient conventional coreanalysis data only. The model was tested on seven wells from the Gorm and Dan field in the North Sea and resulted in predictions of the FWL in the Dan field wells based on wireline formation tests in one Gorm field well. Introduction The interplay between capillary pressure and phase saturation in chalk is not well established; traditional normalization methods of capillary pressure curves are not able to model capillary pressure behavior within chalk reservoirs. 1 3 In North Sea chalk reservoirs, water saturations above the transitional zone vary frequently from values as low as 5% to those as high as 60% (Figs. 1a and 1b), depending on the capillary rock properties. Traditionally, the zone above the transitional zone is referred to as the irreducible zone because little or no water is produced. Experiments reveal that water saturations lower than those encountered in the irreducible zone can be obtained in the laboratory, provided that a sufficiently large difference in phase pressures is applied and sufficient experimental time is available. In fact, a water saturation close to zero can be obtained if a sufficiently high capillary pressure is applied and if the water phase remains continuous to provide an escape path for the water phase. In a field in equilibrium, the difference in phase pressures is caused by the difference in hydrostatic pressure (which itself is caused by the difference in fluid density). The capillary pressure, * Now with Maersk Oil Qatar. Copyright 2004 Society of Petroleum Engineers Original SPE manuscript received for review 24 April 2001. Revised manuscript received 10 February 2003. Paper 69685 peer approved 16 February 2004. defined as the difference in phase pressures for an oil/water system in equilibrium, can be calculated by Eq. 1, where h is the height above the FWL defined as the point at which the capillary pressure is zero. P c = p o p w = gh....(1) Consequently, the largest difference in phase pressures will occur immediately below the top of the reservoir and gradually decrease downward to the FWL, provided that capillary continuity is sustained. We assume that the hydrocarbon and water phases are continuous throughout the reservoir column and that no effective lower limit exists for the water saturation. Therefore, the terms irreducible zone and irreducible water saturation (S wi ) provide little meaning when interpreting the water saturation in chalk reservoirs. Throughout the paper, we will refer to the irreducible water saturation only while discussing or referring to traditional saturation models; otherwise, it should be assumed that no lower limit exists for the water saturation. Before a reservoir is filled with hydrocarbon, the formation is saturated with formation brine. Under the assumption that the formation is initially water-wet, oil will displace the formation brine in a drainage process. Given sufficient height for an oil column, water will be displaced from the center of all pores and will cover only the surface of the mineral grains of the formation. Considering a drainage process in which the water has been displaced from all pore centers, it is thus reasonable to expect a relationship between water saturation and the internal surface area of the formation. Wyllie and Rose 4 proposed a relationship between permeability, porosity, and irreducible water saturation and proved it valid for some sandstone reservoirs. Timur 5 suggested a generalized equation, k = A Y S,...(2) X wi where A, X, and Y are constants. Several authors have proposed similar relationships; 6 11 we refer to these kinds of models as water-film models because the volume of water is assumed to cover the surface of the rock in a thin water film. Wyllie and Rose s relationship, however, has not been tested on chalk. Moreover, there is no effective irreducible water saturation in chalk; consequently, we feel that establishing a relationship between the internal surface and the water saturation is a better approach. Kozeny s equation expresses a relationship between the specific surface of a porous medium and its permeability and porosity. 12 k = C 3 S,...(3) 2 where S is the specific surface with respect to total volume. The specific surface with respect to porosity is given by S p = S....(4) Mortensen et al. 13 have shown that Kozeny s equation (Eq. 3) is valid for chalk, which is reasonable to expect, considering the chalk s homogeneity. For porosities between 20% and 40%, the constant C is near 0.23 for chalk; 13 thus, there is a relationship between specific surface, porosity, and single-phase permeability that, when combined with a modified Wyllie and Rose equation (Eq. 2) in which S wi is substituted with S w, leads to a relationship between porosity, specific surface, and water saturation. 155

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. The Danian-Maastrichtian boundary (D-M) is marked by a thin solid line. (b) Water saturation of the Dan field wells MA-1, MD-1, and ME-1. Top chalk is marked by a thick solid line. The Danian-Maastrichtian boundary (D-M) is marked by a thin solid line. The GOC is marked by a thick dashed line. The inability of traditional capillary pressure models to capture water-saturation behavior has made determination of the FWL inaccurate. Moreover, studies have shown that the FWL can be tilted. 14 17 The dipping FWL of the Dan field has been proposed previously in two publications. Jacobsen et al. 15 mapped the FWL on the west flank of the Dan field by using data from a horizontal well and applying an equivalent radius model developed by Engstrøm. 2 Vejbæk and Kristensen 17 mapped the same area by using seismic inversion combined with a modified Leverett J-function as their saturation/capillary height relationship. We will test a water-film model similar to the one by Wyllie and Rose on seven wells from two different North Sea fields: three from the Dan field and four from the Gorm field (Fig. 2). We will apply log and core data from zones of Danian and Maastrichtian age, with a porosity range from 17% to 45%. We will discuss the interplay between capillary forces and the water-film model. Subsequently, we will discuss the effect of water located at grain contacts and the applications of averaging the water saturation with respect to the internal surface. This averaging leads to a unique relationship between the PWFT and the height above the FWL (HAFWL). We will use the PWFT and the formationpressure data from the Gorm field well N-22 to establish a PWFT/ HAFWL relationship; subsequently, we will use the relationship to estimate the height of the FWL in five of the six other wells. Microscopic Water-Film Model The concept of water-film models involves distributing a given volume of water on the surface of a porous, water-wet medium with a known internal surface area. As an initial step, we will assume that water covers the surface of the chalk grains in a thin 156

P c + z,...(7) where is the interfacial tension, is the total curvature of the interface, (z) is the disjoining pressure isotherm, and z is the thickness of a thin water film separating the hydrocarbon and mineral grain. The disjoining pressure isotherm depends on a number of parameters: capillary pressure, chalk composition, oil and gas composition, salinity and ph of formation water, pore geometry, and temperature. 18 21 On a field scale, only ph and salinity of formation water can be assumed constant; the remaining parameters vary significantly. The stability of a thin water film larger than a few nanometers is essentially a balance between capillary pressure, the curvature of the interface, the van der Waals forces (which always attract the oil/water and chalk/water interfaces), and the electrical forces of the oil/water and chalk/water interfaces, which can be either repulsive or attractive depending on the surface charge of the interface. 20 The surface charge depends on the ph, the composition of the chalk, and the CO 2 partial pressure of the brine. 18 For films less than a few molecules thick, repulsive structural forces dominate. Although the physical principles of calculating thin-film thicknesses are well established, we feel that on a field scale, more simple observations are needed. Therefore, we approach the problem by using a pseudo film thickness. The PWFT will be much larger than the real water-film thickness because the water located around the grain contacts will also contribute to the PWFT. Thus, the water-film model inherently assumes that the conditions are such that the water film is stable over the entire reservoir. Fig. 2 The Dan and Gorm fields in the Danish sector of the North Sea. layer with uniform thickness. Thus, if the specific surface, porosity, and water saturation are known, the thickness of the water layer can be calculated as h w = S w,...(5) S p where h w is the PWFT. Combining Eq. 5 with Kozeny s equations (Eqs. 3 and 4), the thickness of the water film can be expressed as h w = S w k C....(6) Notice that any water associated with grain contacts is averaged out over the surface of the grains. Thus, the water-film model basically transforms the water saturation into a PWFT. The thickness of a thin water film is controlled by intermolecular forces and can be described by the augmented Young- Laplace equation 18 Data We used log and core data from seven wells (three from the Dan field and four from the Gorm field), all of which have been cored. Although several other wells were considered, we selected these seven wells because of their abundance of core material. More than 2,071 measurements of porosity and permeability exist for the seven wells, approximately half from above the transitional zone. The wells contain data representing chalk of both the Danian and Maastrichtian ages. In the Gorm and Dan fields, the chalk of these two ages differs in that Danian chalk in general has a higher specific surface, S, than Maastrichtian chalk. 13 Røgen and Fabricius 22 studied chalk from hydrocarbon-bearing intervals and concluded that the difference in specific surface between the two formations is mainly a consequence of a higher content of silicates in 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 depth subsea (TVDSS), as reported by Megson. 23 In N-22, wireline RFT data were also used for estimation of FWL. Data Processing The basic logs involved in the calculation of the water saturation are the density log and the resistivity log. The porosity was calculated from ma b =....(8) ma S w w 1 S w h The water saturation was calculated by applying Archie s equation with cementation factor m2, Archie s constant a1, and the saturation exponent set to n2: S w n = a m R w R t,...(9) which leads to the following expression for water saturation: S w = 1 R w R t....(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 waterzone data of each well. The formation-water resistivities and fluid densities are given in Table 1. Porosity and permeability data from conventional core analysis were used to calculate the specific surface with respect to porosity, S p, from Eqs. 3 and 4 using C0.23. If the core porosity and log porosity deviated significantly [(i.e., more than 3 porosity units (p.u.)], the depth of the core was shifted to the nearest depth at which the log porosity and core porosity agreed. The deviation between core and log porosities is most likely caused by the uncertainty associated with core recovery and logging depth. The shifting was done to avoid data points at which the core porosity does not represent the porosity reflected on the log. Typically, this involved shifting the core depth between 1 and 3 ft. The maximum shift was 8 ft. Core data with permeability of 0.01 md or less were excluded from the data set because measurements of cores with such low permeability generally are not reliable. Permeabilities that were significantly larger than the surrounding permeability values also were excluded. Abnormally large permeability values are probably caused by fractures in the material; consequently, the measured permeability is not related to the specific surface of the matrix of such a sample. 157

Results The PWFT was calculated according to the water-film model in the chalk sections of the seven wells (Figs. 3a and 3b). In other words, we found the average thickness of all the water relative to the internal surface of the sample. Compared with the water-saturation plot, it is clear that large variations in water saturation above the transitional zone are smoothed out when transformed into a pseudo film thickness. The Danian zone of all seven wells contains larger variation in PWFT than the Maastrichtian zone. The pseudo film thickness above the transitional zone ranges from 6.0 nm (below the Danian Maastrichtian boundary of MA-1) to 30 to 50 nm (immediately above the transitional zone). However, the PWFT data exhibit significant scatter. The variation is typically 20% to 30% around the mean trend of the average PWFT in 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 the PWFT gradient with depth probably defines the crossover from the irreducible zone to the transitional zone, as defined in traditional capillary pressure models. The gradient above the transitional zone was established using a least-squares method. The average gradient was 3.5 nm per 100 ft, with correlation coefficients ranging between 0.3 and 0.7 (Table 2). In the transitional zone, the PWFT becomes more scattered, and the gradient of the PWFT is closer to 33 nm per 100 ft. In contrast, the water-saturation logs (Figs. 1a and 1b) exhibit a gradual change in water saturation. Consequently, the transitional zone and the irreducible zone (used in traditional capillary pressure models) cannot be identified from the S w logs. The change in gradient of the PWFT is clear for MA-1 in particular; it has a high oil column above the transitional zone and therefore covers a wide range in capillary pressure where the extent of the irreducible zone is large. The onset of the capillary transitional zone occurs at a PWFT of 30 to 50 nm for all wells except N-6, where the crossover is absent (Fig. 3a). Discussion Internal Surface and Grain Contacts. It is clear that the water saturation is related to the specific surface of the formation and that the large differences in water saturation above the transitional zone can be accounted for (to a first approximation) by the variation in specific surface. However, the local variations in film thickness indicate that the water-film model is too simplistic to describe the distribution of water on the chalk grains in detail. The pseudo film thickness is less well defined for the Danian zone compared to the Maastrichtian zone. In other words, the water saturation above the transitional zone is not controlled solely by the internal surface. Aggregates of chalk grains form irregular shapes and, consequently, the internal surface is not smooth; the water is present not only as a thin film but also as pendular rings residing around the grain contacts. To estimate the amount of water residing around grain contacts, we consider the water present as a thin film and as a pendular ring in a system of two spherical-shaped grains of dimensions similar to the size of natural chalk grains (Fig. 4). The amount of water present around a single grain contact depends on the capillary pressure, contact angle, and grain size. For a given capillary pressure, grain size, interfacial tension, and contact angle, the amount of water located around a single grain contact can be calculated analytically with moderate simplifications (see the Appendix). The total amount of water around the grain contacts is then the volume around a single grain contact multiplied by the number of grain contacts. Above the transitional zone, the volume of water can be divided into two components: the film phase and the grain-contact phase, which add up to the total volume of water (Eq. 11). To illustrate the contribution from the two components of water, the ratio between the surface-phase water and the grain-contact water is shown in Fig. 5 for a typical set of fluid parameters, assuming face-centered cubic (FCC) and body-centered cubic (BCC) packings of spheres as a function of capillary pressure. S w,total S w,contact + S w,surface....(11) Thus, the variation in pseudo film thickness within a limited depth range is a reflection of textural variations: intervals with relatively poor sorting will have a relatively high number of grain contacts for a given specific surface and, thus, a greater pseudo film thickness. Data points above the Danian-Maastrichtian boundary exhibit the largest local variations in water-film thickness, probably as a reflection of larger textural variability in a section locally rich in fine-grained silicates. Sensitivity of the Pseudo Film Thickness. By inserting Eq. 9 into Eq. 6, it is evident that the pseudo film thickness is sensitive to the accuracy of the porosity, the cementation factor, and the saturation exponent in Archie s equation. More specifically, a range of saturation exponents from 1.8 to 2.2 and cementation factors from 1.8 to 2.2 (common in chalks) 24 can change the pseudo film thickness from 27 to 52 nm for a sample of 25% porosity. This range in PWFT is purely associated with the use of Archie s equation for calculating the water saturation and is thus not related to the accuracy of the data. We have purposely kept our interpretation of the water saturation and the porosity deduction simple to remove unnecessary complexity from the water-film model. However, the accuracy of the estimation of the pseudo water-film model can be improved by a more rigorous interpretation of porosity and water saturation, including factors such as clay content, silica content, degree of cementation, and secondary porosity. In addition, the poor vertical resolution of the logs involved in the calculation of the water saturation also contributes significantly to the noise in the PWFT. However, the general trend in PWFT with depth can be recognized despite the low correlation coefficients. It is worth noting that for a given capillary pressure, the combination of Eqs. 6 and 9 predicts increasing irreducible water saturation for lower porosity, which is in accordance with the observations of Engstrøm 2 and in contrast to the Leverett J-function normalization of capillary pressure curves in which the irreducible water saturation is constant. 1,2 In MA-1, the gradient of the water-film thickness changes at a depth of 6,030 ft TVDSS, slightly above the GOC reported by Megson. 23 The pseudo water-thickness gradient in the gas zone of the Maastrichtian zone was established to be 15 nm/100 ft 158

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 transitional zone 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 PWFT in 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 is marked by a thick solid line. The Danian-Maastrichtian boundary (D-M) is marked by a thin solid line. GOC is marked by a thick dashed line. The transition from the irreducible zone to the transitional zone is marked by a thick solid line (T-Z). The gradient of the PWFT in the Maastrichtian oil zone is shown. The FWLs (FWL, V-K) proposed by Vejbæk and Kristensen 17 are shown. The gradient of the PWFT in the gas zone in MA-1 is shown. (Fig. 3b), which results from the greater difference in density between water and gas compared to water and oil. However, the gradient of the PWFT in the oil zone does not appear to be different for the Gorm field wells as compared to the Dan field wells, even though the difference in density is larger for the Gorm field. The lack of sensitivity to oil/water density difference is possibly caused by the smaller height of the irreducible zone and the smaller number of core data in the Gorm field. Consequently, a well-defined gradient is more difficult to establish in the Gorm field. Height Above the FWL. The existence of the gradient in pseudo film thickness is interesting because it shows that the water saturation of the irreducible zone is controlled partially by the lithology and partially by the capillary pressure. However, the variation of the PWFT gradient with depth is small. Consequently, a unique relationship between the PWFT and the height above the FWL can be established provided that the depth of the FWL is known (in other words, if the FWL is known, the PWFT can be predicted). However, FWL is not necessarily level but can be tilted. 14 17 It is therefore of more use to solve the inverse problem of establishing the FWL from the PWFT. We used RFT data to establish the FWL (7,269 ft TVDSS) of N-22 (Fig. 6). Thus, the pseudo film thickness of N-22 can now be related to the height above the FWL (Fig. 7) and compared to the PWFTs in the offset wells. To illustrate the principle, consider the 159

wells N-22 and MA-1. The PWFT points of N-22 can be plotted as height above the FWL of N-22; thus, this plot constitutes a unique relationship between the PWFT and the height above the FWL. Subsequently, the PWFT points of the MA-1 are plotted. Suppose that the FWL of MA-1 were equivalent to the FWL of N-22. In that case, the PWFTs of the zone above the transitional zone in MA-1 should fall on the PWFT/HAFWL relationship of N-22. However, it is seen (Fig. 7) that the PWFTs in MA-1 are larger as compared to the PWFT/HAFWL relationship established for N-22. Consequently, the FWL in MA-1 must be located higher than that in N-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 of N-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 of N-22. The zone above the transitional zone in N-6 is absent; consequently, 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 and Kristensen. 17 However, the relative depths of FWL between the wells are approximately the same as those reported by Vejbæk and Kristensen. 17 We feel, however, that the use of seismic inversion and a modified Leverett J-function introduces a significant source of error that easily can explain the difference between our estimation and that made by Vejbæk and Kristensen. 17 The estimation of the FWL by the pseudo film-thickness gradient can only be performed in wells in which a zone above the transitional zone is present. Whether such a zone exists should be evident from the gradient of the PWFTs. Consequently, the FWL can only be estimated in crestal wells that penetrate the zone above the transitional zone, and the method is thus not readily applicable to flanks of the reservoir containing only the transitional zone. However, it is worth noticing that the use of the PWFT/HAFWL relationship to establish the FWL is inexpensive because it only requires conventional core-analysis data, porosity and permeability, and log data, which are routinely measured in many exploration wells. The equivalent radius method and the Leverett J- function rely on capillary pressure measurements of cores, which are more expensive and time consuming and, consequently, fewer in number. Once the FWL has been established, the capillary pressure (Eq. 1) can be calculated, and the distribution of the water around the grain contacts and surface area (Eq. 11) in the simple spherical grain model can be established. The capillary pressure at the top of the transitional zone of MA-1 is 85 psi; it increases to 115 psi at the top of the oil column. These capillary pressures correspond to 45% of the water and 36% of the water present around the grain contacts at the top of the transitional zone and the top of the oil column, respectively, for the BCC model shown in Fig. 5 (r grain 0.5 micron, ow 0, and ow 40 mn/m). The BCC model assumes that the formation is composed of equal-diameter spheres, which is not the case. The calculations show that the water-film model is incapable of capturing all the features of the true water distribution in chalk. However, the water-film model still provides meaningful results because there is a correlation between specific surface area and graincontact density. Fig. 4 Schematic drawing of water located as a pendular ring around a grain contact. Fig. 5 Ratio between water present as pendular rings and water present as surface water as a function of capillary pressure for BCC and FCC closest packings of spheres with a grain radius of 0.5 microns. An oil/water contact angle of zero and an oil/water interfacial tension of 40 mn/m were used for the fluid properties. 160

Fig. 7 The principle of determining the FWL in an offset well from a well with known FWL. The FWL of N-22 was established by using RFT data. This plot thus constitutes the capillary pressure saturation relationship used in offset wells. Subsequently, the PWFT points of MA-1 are plotted. The PWFT gradient of N-22 is below the PWFT gradient of MA-1. Consequently, the FWL in MA-1 must be located higher than in N-22. Fig. 6 Determination of the FWL in N-22 from RFT data. The FWL was determined to be at 7,269 ft TVDSS. Change of Wettability With Height. Along with the thinning of the PWFT with increasing capillary pressure, it is expected that the electrical resistivity of the formation should increase. In wells MA-1, N-2X, and N-3X, very high resistivities are encountered in the upper part of the Maastrichtian zone. The resistivity reaches stable values larger than 100 ohm-m, which corresponds to resistivity indices larger than 100. Consequently, the electrical current is severely limited in its transport. The validity of the saturation exponent used in Archie s equation is therefore questionable. We suspect that part of the internal surface has become oil-wet because of the high capillary pressure in the top of the Maastrichtian zone. Large resistivities are never found in the Danian zone, probably because of the large number of grain contacts per volume, which ensures that bridges of water will carry the electrical current through the formation. A partial change of wettability is consistent with the model of Kovcsek et al. 25 The change of wettability is also in accordance with the observation of the increased Amott- Harvey wettability index with depth of the Dan field. 1 Thus, wettability change must be explained by a combination of capillary pressure and lithology. Conclusions 1. The amount of water above the transitional zone is controlled to a first approximation by the internal surface area of the formation. However, the grain-contact density also influences the water saturation. 2. The PWFT yields valuable information about the irreducible water saturation above the transitional zone and is able to smooth out large variations in irreducible water saturation. 3. There exists a pseudo water-film gradient with depth, which can be used as an alternative estimate of the FWL in water-wet reservoirs for which no reliable RFT data exist. The estimation of the FWL by calculation of the PWFT requires only conventional log- and core-analysis data. 4. The approach of using a pseudo water film can be refined by including the effects of water located as pendular rings around grain contacts, in addition to applying more specific values for the constants in Archie s equation for the different formations. The splitting of the water saturation into surface and contact water yields a more accurate interpretation of the water saturation above the transitional zone. Nomenclature a Archie s constant A constant C constant for Kozeny s equation C 1 center of grain (see Fig. 4) C 2 center of radius of curvature (see Fig. 4) g gravitational constant h w pseudo water-film thickness k permeability m cementation factor n saturation exponent Q position (see Fig. 4) p o oil pressure p w water pressure P interface grain intersection (see Fig. 4) P c capillary pressure r ow radius of curvature R, R grain radius of grain R t resistivity of the formation R w resistivity of the formation water 161

S w water saturation S w,contact water saturation of grain contacts S w,surface water saturation of grain surface S wi irreducible water saturation S specific surface with respect to total volume S p specific surface with respect to porosity X exponent for S wi in the Wyllie and Rose equation Y exponent for in the Wyllie and Rose equation z film thickness of real water film angle * angle curvature of the oil/water interface contact angle Fig. 8 PWFTs as a function of height above the FWL using the FWL of N-22 and the average PWFT gradient of 3.5 nm/100 ft to predict the depths of the FWL in offset wells with no RFT data. The FWLs of the different wells have been drawn in the graph. distance from C 1 to C 2 (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 C 2 (see Fig. 4) Acknowledgments Finn Engstrøm of Maersk Oil and Gas AS is acknowledged for stimulating discussions concerning the distribution of the water. Abdelhakim Chtioui made the first calculations of water-film thickness as part of his MS thesis at the Technical U. of Denmark. The Geological Survey of Denmark and Greenland is acknowledged for supplying data from the seven wells. References 1. Andersen, M.A.: Petroleum Research in North Sea Chalk, Joint Chalk Research Phase V (1995) RF-Rogaland Research, Stavanger. 2. Engstrøm, F.: A New Method to Normalize Capillary Pressure Curves, paper SCA 9535 presented at the 1995 International Symposium of the Soc. of Core Analysts, San Francisco, 12 14 September. 3. Leverett, M.C.: Capillary Behaviour of Porous Solids, Petroleum Technology (August 1941) 142, 152. 4. Wyllie, M.R.J. and Rose, W.D.: Some Theoretical Considerations Related to the Quantitative Evaluation of the Physical Characteristics of Reservoir Rock From Electrical Log Data, Trans., AIME (1950) 189, 105. 5. Timur, A.: An Investigation of Permeability, Porosity and Residual Water Saturation Relationship for Sandstone Reservoirs, The Log Analyst (July August 1968) 9, No. 4, 8. 6. Tixier, M.P.: Evaluation of Permeability From Electric-Log Resistivity Gradients, Oil and Gas J. (June 1949) 113. 7. Wyllie, M.R.J. and Gardner, G.H.F.: The Generalized Kozeny- Carman Equation. Its Application to Problems in Multiphase Flow in Porous Media, World Oil (March 1958) 146, No. 4, 121. 8. Mohaghegh, S., Balan, B., and Ameri, S.: Permeability Determination From Well Log Data, SPEFE (September 1997) 170. 9. Coates, G.R. and Dumanoir, J.L.: A New Approach to Improved Log-Derived Permeability, The Log Analyst (January February 1974) 15, No. 1, 17. 10. Coates, G.R and Denoo, S.: The Producibility Answer Product, The Technical Review, Schlumberger, Houston (June 1981) 29, No. 2, 55. 11. Ahmed, U., Crary, S.F., and Coates, G.R.: Permeability Estimation: The Various Sources and Their Interrelationships, JPT (May 1991) 578; Trans., AIME, 291. 12. Kozeny, J.: Über Kapillare Leitung des Wassers im Boden, Sitzungsberichte der Wiener Akademie der Wissenschaften (1927) 136, 271 306. 13. Mortensen, J., Engstrøm, F., and Lind, I.: The Relation Among Porosity, Permeability, and Specific Surface of Chalk From the Gorm Field, Danish North Sea, SPEREE (June 1998) 245. 162

14. Dennis, H. et al.: Hydrodynamic Activity and Tilted oil-water Contacts in the North Sea, presented at the 1998 NPF Conference, Haugesund, 29 30 September. 15. Jacobsen, N.L. et al.: Delineation of Hydrodynamic/Geodynamic Trapped Oil in Low Permeablity Chalk, paper SPE 56514 presented at the 1999 SPE Annual Technical Conference and Exhibition, Houston, 3 6 October. 16. Thomassen, J.B. and Jacobsen, N.L.: Dipping Fluid Contacts in the Kraka Field, Danish North Sea, paper SPE 28435 presented at the 1994 SPE Annual Technical Conference and Exhibition, New Orleans, 25 28 September. 17. Vejbæk, O.V. and Kristensen, L.: Downflank hydrocarbon potential identified using seismic inversion and geostatistics: Upper Maastrichtian reservoir unit, Dan Field, Danish Central Graben, Petroleum Geoscience (2000) 6, No. 1, 1. 18. Hirasaki, G.J.: Wettability: Fundamentals and Surface Forces, SPEFE (June 1991) 217; Trans., AIME, 291. 19. 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Kazatchenko, E. and Mousatov, A.: Primary and Secondary Porosity Estimation of Carbonate Formations Using Total Porosity and the Formation Factor, paper SPE 77787 presented at the 2002 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 29 September 2 October. 25. Kovscek, A.R., Wong, H., and Radke, C.J.: A pore-level Scenario for the Development of Mixed wettability in Oil Reservoirs, AIChE J. (1993) 39, No. 6, 1072. Appendix Calculation of Water Residing Around Grain Contacts. The objective of this appendix is to calculate the volume of a pendular ring of water around a grain contact. The volume of water depends on the curvature of the water/oil interface, the size of the two grains, and the contact angle between the grain surface and the water/oil interface. Fig. 4 shows a principal sketch of two grains, with contact point O located at the origin of a rectangular coordinate system. The z-axis points upward, and the x-axis points to the right. The two grains have radius R, and the center of the upper grain is located at point C 1. An oil/water interface with curvature r ow in one direction and center C 2 is shown. We observe from Fig. 4 that = C 1 C 2 = R 2 sin 2 + R cos + r ow 2 = R 2 2 + r ow + 2 R r ow cos...(a-1) R = OC 2 = tan * = r 2 ow + 2 r ow R cos....(a-2) We may thus express the angles and * as follows: sin = R sin....(a-3) cos = r ow + R cos....(a-4) sin * = R....(A-5) cos * =....(A-6) By using basic geometry, we obtain sin * = R r ow + R cos R sin....(a-7) The volume of water in a pendular ring is given as the difference between two integrals. V = 0 2 r ow z 2 2 dz R 2 R z 2 dz,...(a-8) 0 where r ow sin(* ) corresponds to the distance in the z- direction from the x-axis to the point at which the fluid interface intercepts the grain. The difference of the two integrals can be evaluated as V = 2 + r 2 ow r ow sin* Rr ow sin* 2 r 2 ow * 2 r ow 1 sin 2 *...(A-9) The capillary pressure is given approximately by P c 1 1...(A-10) r ow r ow. Calculation of the Amount of Water Present as Pendular Rings and Film Phase as a Function of Capillary Pressure. The volume of water present around the grain contacts is calculated as the number of grain contacts multiplied by the amount of water present around a single grain contact with a given capillary pressure, as shown previously. A BCC lattice and an FCC lattice have 8 and 12 contacts per unit lattice, respectively. Porosity equals 32% and 26%, respectively, for the unit lattices, and the specific surface areas with respect to porosity are 12 and 17 m 1. The FCC lattice is closest packed, which normally is not the case for chalk. However, the grain-size distribution probably makes a closer packing possible, which leads to a higher density of grain contacts. The amount of water present as surface water is calculated as the surface area times the thickness of the layer. We use a layer thickness of 6 nm, corresponding to 5% water saturation, which can be achieved in chalk samples during mercury injection. The ratio between the surface water and the volumes is taken as one volume divided by the other (Fig. 5). SI Metric Conversion Factors ft 3.048* E 01 m mn/m 1.0* E 03 N/m nm 1.0* E 09 m psi 6.894 757 E+00 kpa *Conversion factor is exact. Jens K. Larsen is currently a petrophysicist with Maersk Oil Qatar; he joined the company in 2001. His main interest lies within petrophysical evaluation and flow phenomena on pore and core scale. Larsen holds an MS degree in mechanical engineering and a PhD degree, both from the Technical U. of Denmark. Ida L. Fabricius is an associate professor at the Technical U. of Denmark, which she joined in 1985 after leaving Mærsk Oil and Gas AS. e-mail: ilf@er.dtu.dk. Her main interests are petrophysics, pore-fluid interaction, and acoustic properties. Fabricius holds an MS degree in geology from Copenhagen U. and a PhD degree from the Technical U. of Denmark. 163