A predictive model for the wettability of chalk

Wettability is usually measured in special core analyses of limited plug samples according to typically costly and time-consuming procedures. For comparative purposes, wettability is considered an index. The two most frequently used wettability indices are the Amott–Harvey wettability index and the U.S. Bureau of Mines (USBM) index. The Amott–Harvey wettability index is linked to imbibition characteristics and the USBM index is associated with the area under capillary pressure curves. To provide a fast analytical method, a mathematical model for predicting the wettability of chalk is presented. The model is calibrated using experimental wettability data and subsequently applied to two wells in Danish chalk oil fields in the North Sea and to outcrop chalk samples. The model supplements traditional labor-intensive laboratory measurements and predicts water wettability variations with depth by modeling both depth and porosity dependencies; in addition, it provides estimates of the effects of the aging time and displacement temperature of chalk wettability measurements in the laboratory.


Introduction
Wettability is defined as the tendency of a fluid to adhere to the surface of a solid material or spread in the presence of other immiscible fluids [1,2]; it has a critical influence on multiphase flow properties both when one fluid displaces another and when fluids are at equilibrium in a static or flowing system [1,3]. Wettability remains an important reservoir property during the entire reservoir life cycle, from initial oil migration to field development and oil production [2,3]. The chalk fields in the Danish part of the North Sea are complex and challenging to develop due to the high-porosity (20-40%) and low-permeability (0.1-10 mD) nature of the reservoirs. Production is from the Upper Cretaceous and Danian low-permeability Tor and Ekofisk formations and the Lower Cretaceous very low-permeability reservoirs of the Tuxen and Sola Formations [4][5][6].
In chalk fields, water flooding is commonly applied to provide pressure support. The technique has been successfully applied in the Tor Formation to increase oil recovery, and it has technical potential for improved oil recovery in the Ekofisk Formation [7][8][9] and theoretically the Lower Cretaceous formations. The success of improved oil recovery in chalk fields by water flooding is highly dependent on the wetting conditions of the reservoir rocks [10,11].
Wettability is usually measured based on a limited sample set due to costly and time-consuming measurement procedures. For comparative purposes, wettability is generally considered an index [12]. The two most frequently used indices are the Amott-Harvey (AH) index and the U.S. Bureau of Mines (USBM) wettability index [13,14]. The AH index, I AH , is related to the imbibition characteristics of the sample (AH), and the USBM index is related to the area 2 Development of the wettability model Spiteri et al. [26] plotted I w and I AH indices versus the contact angle. The linear relationships between the Amott indices and static contact angle (degree) were extracted from the plots as follows: Considering its physical properties, chalk is a remarkably homogeneous sedimentary rock [24]. The liquid permeability (K l [m 2 ]) is related to a porosity-dependent factor (c), the specific surface area per unit grain volume (S s [m 2 m −3 ]) and the porosity (φ [fraction]) by the Kozeny [27]

equation as follows:
This equation is based on Poiseuille flow through a pipe, where the assumptions are that the fluid is incompressible and Newtonian; the flow is laminar through a pipe of constant circular cross-section that is significantly longer than its diameter; and there is no acceleration of the fluid in the pipe. Mortensen et al. [28] showed that the Kozeny equation is valid for chalk and proposed that c is calculated using the following equation: Equation 4 is derived analytically from a simple porosity circular tubes model and Poiseuille's law. The simplified correlation between c and porosity can be described by the following equation with a correlation coefficient (R 2 ) of 0.99: The effective pore throat radius (also called the mean hydraulic radius) [29] can be calculated as follows: Capillary pressure is influenced by the pore geometry, wettability and interfacial tension and demonstrates the interaction of rock and fluids [30]. Equation 7 expresses the relationship between the pore throat radius and capillary pressure [31]: where wo and P cwo are the interfacial tension between oil and water and the water-oil capillary pressure, respectively. By considering the effective pore throat radius in Eq. 7 and combining Eqs. 3, 6 and 7, the specific surface area per unit grain volume can be calculated from the following equation: Formulating Eq. 8 for two samples of the same fluid, i.e., a main sample and a reference sample (referred to as subscript 0), yields the following expression: wo is the same for both the main and the reference samples and therefore is not included in Eq. 9. The ratio of the water-oil capillary pressures is assumed to be proportional to the air-mercury capillary pressures.
where k is proportionality constant. The mercury injection capillary pressure (MICP) of chalk samples was measured by SKM Services Ltd. (Aberdeen, UK). Between the entry pore throat radius and the effective pore throat radius (calculated by Eq. 6), a correlation coefficient of 0.87 was obtained, indicating proportionality between the air-mercury capillary pressure at the effective pore throat radius and the air-mercury capillary entry pressure (Fig. 1). Therefore, we replace the ratio of the water-oil capillary pressures in Eq. 9 with k P ce P ce 0 and find the contact angle as follows: where P ce is air-mercury capillary entry pressure. A is a constant based on the properties of the reference sample and proportionality constant: Røgen and Fabricius [20] introduced an empirical correlation relating the specific surface area per unit grain volume, porosity and capillary entry pressure of chalk (in (7) P cwo = 2 wo cos wo R psi) (valid for chalk samples with calcite contents greater than 80%): By combining Eqs. 3, 5, 11 and 13, the contact angle can be calculated using the following formula: The term (20AK 0.5 l −0.5 ) is negligible compared to (10.11Ae 0.37 ).
Jakobsen et al. [5] developed linear relationships between the insoluble residual (IR) content and porosity for chalk formations in the Valdemar Field. The IR is primarily related to clay and silica. Extrapolating the linear relationships led to zero porosity at 79% and 96% IR for the Tuxen and Sola Formations, respectively. Strand et al. [22] reported a more water-wet condition in the presence of IR content due to the negative charge of silica and clay to stabilize the water film, coating the chalk surface and prohibiting the overlaying oil from reaching the surface. The fine pores and grain contacts are preferentially water-wet [32], and it is implicitly assumed that fine clay-rich rocks are characteristically water-wet [33].
If the IR content reaches high values, we expect nearzero porosity and, consequently, a completely wetted situation and a contact angle equal to zero. Therefore, the coefficient ( 10.11A ) in Eq. 14 must be one to obtain a zero contact angle at zero porosity. Therefore: Hence, the model is no longer dependent on the reference sample properties. The wettability of chalk can be predicted from Eq. 15 if the porosity is known, but it  should be noted that wettability is a state that is energywise favorable when chalk is in contact with a specific type of oil or water. The fluids, namely, oil and water, rearrange to decrease the energy state [3]. This relation is mathematically represented as follows [34]: Equation 16 is based on the theory of minimizing the energy in the solid/oil/water system, where so and sw are the oil-solid surface tension and water-solid surface tension, respectively. Because Eq. 15 is developed based on a reference oil and water system, it must be calibrated with a set of wettability measurements for specified fluids at specified measurement conditions. Therefore, an empirical correction factor, , is considered to specify the reference fluids and include the effects of different conditions, such as the aging time, temperature and depth, on wettability. This factor is introduced in Eq. 15 as follows: Equation 17 is valid for < 0 . For a positive , the contact angle is zero. The empirical correction factor, , is not unique and is used to overcome the deficiencies in considering various factors affecting wettability. is a dimensionless number that must be adjusted to improve the accuracy of the predictive wettability model. Figure 2 shows a sketch of the problem presented in this study. The outlined method can be combined with experimental wettability data obtained with calibration and porosity logging tools, such as neutron porosity and sonic devices, to evaluate the in situ wettability of chalk. In addition, the model can be used to define a new-format capillary number specific to chalk. This is a dimensionless number that describes the ratio of viscous to capillary forces acting across the interface of two immiscible fluids. Moore and Slobod [35] defined the capillary number as follows: where is the viscosity of the displacement phase and v is the velocity of the displacement phase. By combining Eqs. 17 and 18, the specified capillary number can be obtained as follows: The maximum oil recovery via the water flooding of chalk occurs under wettability conditions close to neutral-wet conditions [10]. According to Eq. 17 and the new expression of the capillary number (Eq. 19), increasing the porosity leads to a decrease in the capillary pressure and an increase in the capillary number. Therefore, increasing the porosity leads to increased oil recovery by water flooding.

Validating the model
Two Amott wettability index datasets were used to define and apply the developed method.

North Sea chalk reservoir samples
The first dataset includes 13 North Sea chalk reservoir samples from four major formations; Tor, Maastrichtian, Ekofisk and Hod, as listed in Table 1. The data were reported as part of the Joint Chalk Research Program by Christensen et al. [36]. Figure 3 exhibits a schematic diagram to illustrate the applied procedure for determining the Amott wettability index values. The samples were fresh and cleaned by flushing the plugs at 60 °C with 20-30 pore volumes of light synthetic oil (Isopar L). The plugs were then submerged in formation water, and spontaneous water imbibition was performed. As an equilibrium condition, no oil production by spontaneous imbibition within three days was

Fig. 2
A schematic general sketch of the problem observed. Forced water imbibition was then performed on the plugs in a centrifuge. Next, spontaneous imbibition in oil was performed. The plugs were again placed in a centrifuge, and a regular drainage capillary pressure curve was obtained in eight steps [36]. The measured I AH values were converted to contact angles by Eq. 1.
The α values (Table 1) were selected in such a way to result in linear behavior between α and depth, and the estimated contact angles from Eq. 17 matched the experimental data (the relative error is less than 10%). Very good agreement (R 2 = 0.95) between the modelestimated contact angle and measured Amott wettability index was observed, as listed in Table 1. The developed model correctly predicted water-wet conditions for samples 4, 5 and 6 and neutral wetting for the rest of the samples. Equation 20 expresses the relationship between depth and α as follows: where D is the TVDSS in meters.
Taking the derivative of Eqs. 17 and 20 with respect to depth yields the following equations: When the water wetness increases with depth, the derivative of the contact angle with respect to depth must be negative. Therefore, for 0 < < 90: By combining Eqs. 20 and 22 with inequality 23, the condition that leads to increased water wetness with depth can be expressed as follows:  . 3 Schematic diagram of the applied procedure to measure the Amott wettability index values [36] Using Eqs. 17 and 20, the contact angles were calculated for different depths and porosities, as shown in Fig. 4. Four main trends were detected. The first trend indicates (24) D < 0.0259 (47.666 − 0.0259D) that at constant porosity, water wetness decreases and the contact angle increases with depth. The second trend shows that at constant depth, water wetness decreases and the contact angle increases with porosity. These trends confirm and support each other.
The third trend indicates that water wetness decreases and the contact angle increases with depth and porosity. The fourth trend reveals that the water wetness increases and the contact angle decreases with increasing depth and decreasing porosity. The general trend in the chalk fields is a decline in porosity with depth [3]. Therefore, the fourth trend seems the most applicable to chalk reservoirs, but the third trend may also be applicable in some cases.

Rørdal outcrop chalk samples
The second Amott wettability index dataset includes 25 chalk samples from Rørdal Quarry in NW Denmark (Table 2).   where d and T d are expressed in days and ºC, respectively. Fitting analysis was performed using MS Excel.
The α values calculated from Eq. 25 for each aging time and displacement temperature are presented in Table 2. α decreases with the aging time and displacement temperature. Considering Eq. 17, a low α at constant porosity leads to weak water wetness. Therefore, increasing the aging time and displacement temperature decreases water wetness. Good agreement (R 2 = 0.80) between the model-estimated contact angle and measured Amott wettability index was observed, as listed in Table 2.
Taking the derivatives of Eqs. 17 and 25 with respect to the displacement temperature yields the following equations: Similarly, taking the derivatives of Eqs. 17 and 25 with respect to the aging time yields the following equations: Porosity does not change with variations in displacement temperature and aging time. Therefore, the derivatives of porosity with respect to displacement temperature and aging time in Eqs. 26 and 28 are equal to zero. Combining Eqs. 26 and 28 yields the following equation: Equation 30 shows that contact angle alteration due to a 6.1 ºC change in the displacement temperature is equivalent to its alteration due to a change of one day in the aging time.

Results and discussion
The chalk porosity dataset used to calculate the contact angles is derived from the M-10X and E-5X wells in the North Sea and the onshore Rørdal chalk quarry in Denmark. The M-10X well is in the Dan Field, and the E-5X well is in the Tyra SE Field in the Danish part of the North Sea. Figure 6 shows the porosity distributions. The average porosities of the Rørdal quarry, well E-5X and well M-10X samples are 44.3%, 33.2% and 28.3%, respectively. All samples are relatively pure chalk (CaCO 3 content > 80%), with the North Sea chalk being the purest (generally CaCO 3 content > 90%). The non-chalk components include clay and silica [20]. The North Sea wells are currently at maximum burial depth, but due to high pore pressures, the effective depth is less than that reflected by the TVDSS [37]. The Rørdal samples were never buried deeper than 1 km and are now exposed at the surface level following the Neogene uplift [38]. Figure 7 shows the predicted contact angles for the M-10X and E-5X wells versus depth using Eqs. 17 and 20. The predicted contact angles decrease with increasing depth and decreasing porosity in some cases (fourth trend). Nevertheless, increased contact angles with depth and porosity (third trend) can also be observed in Fig. 7. The average predicted contact angles of wells E-5X and M-10X are approximately 54 and 30 degrees, respectively. Well E-5X has a higher average porosity and is deeper than well M-10X and displays less water wetness.

Fig. 5
Schematic diagram of the applied procedure to determine the Amott water wettability index values [10] Some studies have reported that water wetness increases only slightly with depth [3], whereas others have shown a much stronger trend of increasing water wetness with depth [39][40][41]. According to Eqs. 17 and 20, depth and porosity have similar effects on wettability, and water wetness decreases. Porosity generally decreases with depth, and thus, we expect increased water wetness with depth. However, if porosity increases even slightly with depth, we expect that the water wetness will decrease. In the Danish North Sea, the chalk porosity does not decrease with depth due to the overpressured nature of the basin; thus, the usual depth trends are not expected [6]. The poor chalk reservoir properties are linked to the existence of clay layers and their adjacent pore-filling calcite, and the presence of large amounts of nano-quartz in the chalk. The effect on porosity is attributed to the contents of both clay minerals and nano-quartz [42]. Therefore, the water wetness tends to be higher for lower porosity due to the clay and quartz contents of chalk, which means that the surface is effectively less oil wetting [22].
Inequality 24 is valid for depths deeper than 1840 m. The derivative of porosity with respect to depth was calculated for wells M-10X and E-5X. Notably, 24% and 46% of D values satisfied inequality (24) for wells M-10X and E-5X, respectively. Thus, water wetness decreases with depth for 76% and 54% of the samples from wells M-10X and E-5X, respectively. Figures 8 and 9 show the predicted contact angles of outcrop chalk samples from Rørdal versus the aging time at displacement temperatures of 45 ºC and 90 ºC, respectively. The predicted contact angles increase with porosity and aging time. The contact angles become almost independent of the displacement temperature and reach values between 76º and 78º for 40% porosity when the aging time approaches 40 days. The essential aging time to re-establish reservoir wettability varies, depending on the brine, crude and reservoir rock. Generally, 40 days aging (1,000 h) at the reservoir temperature is well accepted to reach wetting equilibrium [43]. Figure 10 shows the cumulative distribution of the predicted contact angles of the Rørdal samples at aging times of 10, 20 and 40 days and displacement temperatures of 45 ºC and 90 ºC using Eqs. 17 and 25. The Rørdal samples exhibit relatively neutral wetting conditions, in line with the results of previous studies [21,22]. Samples at an aging time of 10 days and a displacement temperature of 45 ºC exhibit more water wetness. An aging time of 20 days and a temperature of 45 ºC and an aging time of 10 days and a temperature of 90 ºC show similar effects and neutral wetting. According to Eq. 30, the effect of a 45 ºC increase in the displacement temperature on the contact angle alteration is equivalent to the effect of a 7.4 day increase in the aging time.
The calculated contact angles are plotted versus the aging times in Fig. 11 at three different porosities and two displacement temperatures for the Rørdal samples. Figure 11 shows that water wetness decreases with aging time. By increasing the displacement temperature to 80 ºC at constant porosity, the water wetness decreases for all three porosities. In contrast to the mentioned trend, a general    observation for carbonate reservoirs is that the water wetness increases as the reservoir temperature increases due to reduction in the crude oil acid number at elevated temperatures [43,44]. Fluid-rock interactions rely on the components of all the involved phases. As the water imbibition tests for Rørdal outcrop chalk samples were performed with two different fluids, crude oil and n-decane, at two different displacement temperatures, 45 ºC and 90 ºC, the observed wettability trend with temperature is not comparable with the reported general effect of temperature on wettability of carbonate samples.
Water wetness decreases with aging time and displacement temperature because of enhanced adsorption of polar acidic and basic components of organic material that exist originally in the crude oil. The degree of water wetness reduction is determined by the interaction of the oil constituents, the mineral surface, and the brine chemistry [23,45]. By increasing the porosity at a constant displacement temperature, the water wetness decreases due to the low amount of negatively charged silica, which is not enough to stabilize the water film to inhibit oil covering the chalk surface.
There are several assumptions in the derivation of the proposed model. Specifically, the Amott index is converted to the contact angle, the ratio of the water-oil capillary pressures is considered proportional to the air-mercury capillary pressures, and an empirical correction factor, , is introduced, which might impose some limitations in terms of the applicability of the model.

Conclusions
Based on the developed wettability model and the application of the model to three large datasets, the following conclusions can be drawn.
1. An analytical-experimental wettability model is developed that includes porosity and parameters for predicting the wettability of chalk samples with calcite contents greater than 80%. 2. The contact angle increases with porosity and | | .
Additionally, α is dependent on the depth, aging time and displacement temperature. Increases in depth, aging time and displacement temperature lead to an increase in | |. 3. An analytical-experimental inequality is proposed to predict the water wetness trend with depth. 4. A new expression of the capillary number is introduced to explain the effects of porosity on oil recovery in chalk reservoirs. 5. The developed method can be combined with experimental wettability data and porosity logging tools to evaluate the wettability of the reservoir (despite limited applicability in some cases). Funding Open Access funding provided by NTNU Norwegian University of Science and Technology (incl St. Olavs Hospital -Trondheim University Hospital). This study was funded by the Centre for Oil and Gas-DTU/Danish Hydrocarbon Research and Technology Centre (DHRTC).

Fig. 11
Relationship between the aging time and calculated contact angle at porosities of 25%, 35% and 45% and displacement temperatures of 20 ºC and 80 ºC for the Rørdal samples Availability of data and material Not applicable.

Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of interest.
Code availability Not applicable.
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