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Water saturation determination
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Water saturation (Sw) determination is the most challenging of petrophysical calculations and is used to quantify its more important complement, the hydrocarbon saturation (1 – Sw). Complexities arise because there are a number of independent approaches that can be used to calculate Sw. The complication is that often, if not typically, these different approaches lead to somewhat different Sw values that may equate to considerable differences in the original oil in place (OOIP) or original gas in place (OGIP) volumes. The challenge to the technical team is to resolve and to understand the differences among the Sw values obtained using the different procedures, and to arrive at the best calculation of Sw and its distribution throughout the reservoir vertically and areally. In OOIP and OGIP calculations, it is important to remember the relative importance of porosity and Sw. A 10% pore volume (PV) change in Sw has the same impact as a 2% bulk volume (BV) change in porosity (in a 20% BV porosity reservoir).
Techniques for calculating water saturation
Sw in wellbores can be determined by the following primary methods:
- Sw calculations from resistivity well logs by application of a model relating Sw to porosity, connate-water resistivity, and various rock electrical properties.
- Sw calculations from laboratory capillary pressure/saturation (Pc/Sw) measurements by application of a model relating Sw to various rock and fluid properties and height above the free-water level.
- Sw calculations using oil-based mud (OBM)-core-plug Dean-Stark water-volume determinations.
- Combinations of these methods.
This listing is the chronological order in which data are likely to become available, not in a ranked order based on the accuracy of the various methods. The choice of which Sw-calculation approach to use is often controlled by the availability of the various types of data. If no OBM cores have been cut, then this technique cannot be used unless funds are spent to acquire such data from one or more newly drilled wells. This is not a high incremental cost when OBM use is planned for other purposes. Resistivity logs are run in all wells, so these data are available for making standard-log-analysis Sw calculations. A key consideration when making calibrated Sw calculations is the availability of special-core-analysis (SCAL) data on core samples from the particular reservoir; that is, the number of laboratory electrical-property and Pc/Sw core-plug measurements that have been made.
The technique chosen to calculate Sw is often a hybrid that combines the use of two of these basic data sources. For example, the OBM-core Sw data can be used in combination with the resistivity logs to expand the data set used to include all wells and the whole of the hydrocarbon column. Alternatively, the OBM-core Sw data can be used in combination with the Pc/Sw data. In this way, the OBM-core Sw data define the S w values for the majority of the reservoir, whereas the Pc/Sw data define the Sw values in the interval just above the fluid contact and perhaps in areas of the field where Pc data are available but OBM-core data are not.
Data availability and data quality
This section discussed the input-data availability and data quality issues for each Sw technique. These considerations often control the initial choice of methodology to calculate Sw and need to be addressed at the start of the project to determine whether it is practically possible to fill gaps in the database in order to use a more accurate Sw-calculation approach. This discussion assumes that accurate porosity values are available from the routine-core-analysis database and that porosity is calculated point by point from the well logs. The discussion focuses on particular aspects that affect the choice of Sw methodology. Many of the database considerations are discussed on the petrophyscial database page.
Wells generally have one variety or another of laterolog or induction resistivity log because they are broadly useful and because government regulations typically demand that they be recorded. This generally provides point-by-point data from the top of the hydrocarbon column down through any aquifer intervals that are present. However, in many fields, the early wells are spread thinly over the reservoir area, but the later development wells are drilled only in areas chosen to maximize rate and recovery while minimizing costs. This means that, often, few wells are drilled downdip where the hydrocarbon column thins because of an underlying aquifer, or in the potentially thin updip limits of the reservoir. In such areas, there may, therefore, be few resistivity logs.
Laterologs are preferred to induction logs when the drilling mud has moderate to high salinity. This limitation of induction tools arises because of the excessive conductivity signal from the borehole and the mud-filtrate-invaded zone. Deep laterolog tools read too high when measuring immediately beneath anhydrite and salt,  and alternative resistivity curves should be selected. When formation resistivity, Rt, is very high, previous generations of induction tools had limited accuracy, but current tools are much improved. Although the deep induction measurement is a running average over many vertical feet, modern tools include systems to deconvolve the raw log and provide a final log with a good vertical resolution.
Deep invasion of water-based mud (WBM) filtrate affects all resistivity logs, and, in the extreme, the available resistivity log may be used only qualitatively. At the opposite extreme, when oil-based mud (OBM) filtrate invades a hydrocarbon reservoir, the invading OBM filtrate generally displaces only the reservoir oil and gas, leaving the Sw unchanged. Here, invasion of OBM does not usually change the deep-formation or the invaded-zone resistivity. For moderate invasion depths, the logging company charts are sometimes used to correct the deep-reading log to provide a better estimate of Rt.
Pad-mounted shallow-reading microresistivity logs measure Rxo, the resistivity of the mud-filtrate-invaded zone. When used together with the deeper-reading tools, these logs provide valuable information about the mobility of the reservoir fluids, including the presence of tar. In WBM wells, they also provide an estimate of the residual-hydrocarbon saturation, Sorw.
Connate-brine resistivity data
An accurate value of connate-brine resistivity, Rw, or its values and distribution throughout the reservoir, are required for accurate Sw calculations using resistivity logs. Temperature estimates are also required.
A first check on the Rw of the aquifer is to back-calculate the apparent Rw with the Archie equation using the invasion-corrected resistivity logs and the best estimates of a and m parameters. Because Sw is typically 100% PV in the aquifer interval, the n value is not relevant here.
The spontaneous-potential (SP) log provides a second method to calculate Rw in wells drilled with WBM. Information on the mud-filtrate composition and temperature is used with the SP deflection to calculate Rw. The moderately accurate calculation process is valid in the aquifer but is also valid in the hydrocarbon column if high resistivity does not suppress the SP response. When OBM-core salinity measurements are not available, the SP log provides the only evidence of possible Rw variations in the hydrocarbon column.
A third estimate of aquifer-water composition and Rw is often taken from samples recovered during flow tests of the aquifer interval; however, the Rw of the oil and/or gas column is not always the same as that of the aquifer interval.  Aquifer-interval flow tests must be validated and checked for contamination from mud-filtrate invasion.
For the oil or gas column, the determination of the Rw value or values is far more of a challenge because the reservoir water will not flow. The typical, but not necessarily correct, first assumption is that the hydrocarbon-column Rw is the same as that of the underlying aquifer. If wells have been cored with OBM, core plugs from the hydrocarbon- and water-bearing intervals can be analyzed for both their water volume and their salt content, particularly the chloride ion that in almost all cases dominates the anion side of the salinity determination. 
Fig. 1 shows variation of chloride concentration with depth for a reservoir in Ecuador.  The chloride value can generally be used to quantify the reservoir-water salinity, from which the Rw at reservoir conditions can be calculated using standard water-resistivity vs. chloride charts or algorithms. For reservoirs in which there is a considerable CO2 content (3+ mol%), the ion distribution at surface conditions will differ from that at reservoir temperature and pressure. Equilibrium ion-distribution calculations need to be made when adjusting the surface-salinity measurement to reservoir conditions.
Fig. 1 – Formation-water salinity variation within the hydrocarbon column (Villano oil field, Ecuador). Hollin is the name of the oil-reservoir formation, and WOC is the water/oil contact. Chloride concentration in the reservoir brine varies from approximately 2,000 to 35,000 ppm. Lowest invasion refers to limited OBM-filtrate invasion into the cores.
Formation temperature affects the Sw estimates because, for constant formation-water composition, Rw varies with temperature.  Maximum downhole temperature is measured with most log runs and drillstem-tool (DST) tests, and these are widely used to estimate a temperature vs. depth profile. It can be argued that the temperature required for resistivity-derived Sw estimates is the prevailing temperature in the rock volume seen by the tool at the time of logging. At this time, the relevant rock is likely to be cooler than the original formation temperature. The error induced by the usual maximum temperature simplification is not large, and the cooling issue is generally ignored.
Electrical-property SCAL data
The third aspect of making these Sw calculations is the choice of the model for the "electrical network" within the rock. These models relate Sw to several formation variables including the bulk-formation resistivity and the formation-water resistivity. A number of models have been published, for example :
- Waxman-Smits-Thomas (WST)
- Dual-water (DW)
Laboratory measurements of two or more types of electrical properties are taken. All of these models assume a homogeneous rock sample.
First, a set of cleaned core plugs with a range of porosities are fully saturated with brine of known resistivity, and the bulk resistivity of each core plug is measured. For this simplest model, the slope of a line fitted to a log-log plot of the data set gives the cementation exponent, m, and the intercept is the cementation constant,  a (see Fig. 2, where a = 1 and m = 1.77). These parameters are used to predict point-by-point F from porosity; leading to predictions of R0 and Sw.
where F = formation factor, Rw = brine-water resistivity, and R0 = rock resistivity with zero oil and gas saturation (100% PV Sw). The plotted logarithmic data (log10F and log10ϕ) are fitted with a linear model of the form,
where ϕ = porosity, a = cementation constant, and m = cementation exponent. Therefore, m = − change in log 10 F /change in log 10 ϕ (the slope of the line-fit) and a = F at 100% BV porosity (the line-fit intercept).
Fig. 2 – Laboratory-measured formation resistivity factor (F) vs. porosity for core plugs from Dulang field, Malaysia.
This model was developed by Archie,  who proposed a = 1.0 and m = 1.8 to 2.0 for his data set. Subsequent work by Exxon researchers for several sandstone rocks recommended a = 0.61 and m = 2.15 (the Humble formula).  Carbonates also have been studied and yielded a recommendation to use m = 1.87 + 0.019/ϕ below 9% BV (the Shell formula).  However, carbonate pore and fracture networks vary greatly, and m values from 1.0 to 3.0 may be required. Clearly, m is not a constant, but varies with rock type.
When plotting these formation-factor data, it is typically assumed that the rock samples have similar pore geometry, but with differing levels of porosity and diagenesis. Reservoir-specific exponent values are likely to provide more-accurate Sw results than worldwide correlations. However, before reservoir-specific values are determined, descriptive and experimental data need to be studied to determine whether they need to be subdivided into various groupings that relate to distinct differences in lithological properties like:
- Grain size
- Clay-mineral content
In partially brine-saturated rocks, a related experimental study involves measuring electrical properties as a function of water saturation. In these experiments, the resistivity index (IR), the ratio of the desaturated-rock resistivity to the 100% PV brine-saturated rock resistivity (Rt/R0), is measured as a function of brine saturation. For example, in a porous-plate apparatus, Sw is changed by increasing gas pressure, and therefore capillary pressure, at the gas/water interface in the pores. Brine flows from the base of the plug via a porous-plate. From the measurements on each core plug, a log-log plot of IR vs. Sw is made (see Fig. 3, where n = 1.64). The slope of the line (almost always forced through IR = 1.0 at Sw = 100% PV) is the Archie saturation exponent n (see Eqs 3 and 4). On the basis of experimental data, Archie recommended that n = 2.0, and this value is still widely used when no experimental data are available. Although cementation exponents can be determined from log analysis, saturation exponents cannot and, therefore, require external information from core data.
where n = saturation exponent, the slope from the origin of a line-fit of several data points; IR = resistivity index; and Sw = fractional brine water saturation.
Fig. 3 – Laboratory-measured formation resistivity index (IR) vs. brine saturation for core plugs from Dulang, Malaysia.
A straight line-fit is usually used, but curved line-fits can be considered where necessary. Curvature is often the result of the clay-mineral content but may also result from an inhomogeneous water distribution at the pore scale (e.g., when microporous rock grains are present). When significant amounts of clay minerals are present in the rocks, other models are required to extend the Archie relationships. The WST model, discussed next, is based on laboratory SCAL measurements including cation-exchange capacity (CEC).
Waxman-Smits-Thomas exponents and cation-exchange capacity
WST cementation and saturation exponents (m* and n*) are required to apply the WST shaly-sand-model equation discussed below. The quantity of cation-exchange sites per gram of rock sample (CEC) may be measured in the laboratory by several methods and, after converting to CEC per unit PV, is used as the model parameter QV. The most reliable measurement of QV involves carrying out bulk-rock resistivity, R0, tests at several brine resistivities and, therefore, is time consuming. The rock conductivity values (1/R0) are plotted vs. brine conductivity (1/Rw) to identify the excess conductivity resulting from the shales and clay minerals. The slope of the fitted line is the reciprocal of F*, the WST formation factor. The excess conductivity is modeled as being equal to BQv/F*, and B is presumed in this model to be always positive. The parameter B is the equivalent counter-ion conductance,  which is a function of temperature and the free-water resistivity. Qv is estimated from the values of F* and B. Core resistivities are also measured when Sw is less than 100% PV and both the WST exponents m* and n* are derived (see Fig. 4 and Fig. 5, respectively). It should be noted that m* > m and n* > n, except in "clean" sands.
Fig. 4 – Laboratory-measured formation resistivity factor (F) vs. porosity for core plugs from the Shannon sand, Wyoming, U.S.A. (after Keelan and McGinley). F and m are Archie parameters, and F* and m* are Waxman-Smits-Thomas parameters.
Fig. 5 – Laboratory-measured formation resistivity index (IR) vs. brine saturation for core plugs from the Shannon sand, Wyoming, U.S.A. (after Keelan and McGinley). n is the Archie saturation exponent, and n* the Waxman-Smits-Thomas saturation exponent.
Other CEC methods require the breaking up, disaggregation, and consequent partial loss of the real geometry of the rock’s electrical network. These simpler methods, such as the ammonia method, use analytical-chemistry methods to measure CEC. After measuring porosity and grain density, this practical laboratory unit is converted to the required Qv parameter.  These simpler CEC measurements are often made on sidewall cores and are used together with exponent values measured on cores from neighboring wells.
Numerous other shaly-sand models have been developed, and, unlike WST, many are calculated from effective porosity. These types of models are generally applied using Archie exponents. When using SCAL electrical-property data, there must be consistency between the electrical-network model used to derive the laboratory parameters and the model used in the final Sw calculations from the porosity and the resistivity logs (e.g., if the laboratory provides standard Archie n values, these are not appropriate input to the WST equation).
Capillary pressure SCAL data
Pc data are a different type of SCAL data that can be taken experimentally in several ways. All Pc saturation tests respond to the pore-size distribution of the rock and the interfacial properties of the various solid/fluid systems. These data are obtained by desaturating core plugs, either using a centrifuge or a porous-plate apparatus. Initially, cleaned and dry plugs are saturated with either water or oil. The liquid is then displaced by air or nitrogen. Because air is very nonwetting compared with either water or oil, using these fluid pairs (air/water or air/oil) means that, as the Pc increases, the air will first occupy the largest pores. As the Pc and air saturation increase, the air will occupy smaller and smaller pores. The core plug begins the experiment saturated with the wetting phase, so the desaturation process provides data for the drainage Pc curve. After completing the drainage process, the core plug can be spun under the liquid in a centrifuge experiment, the liquid saturation will increase, and the imbibition Pc curve will be generated. Usually, only drainage Pc/Sw data are taken, and for most reservoir situations, these are the relevant data because they correspond to the original oil (or gas) trap-filling process.
Mercury-injection capillary pressure (MICP) data are taken on cleaned and dried irregular core pieces. The core pieces are evacuated to a low vacuum, and mercury is injected with increasing pressure, up to 20,000 psi and sometimes higher. Corrections for clay-mineral adsorbed-water removed during drying can be made with the Hill-Shirley-Klein method.  The MICP experiment has the advantage of being run rapidly but is not a true wetting/nonwetting system. The sample cannot be used for subsequent SCAL tests because some mercury is retained within the core pieces at the end of the testing sequence. MICP data are widely used to measure pore-size distribution, but, when considering whether they should be used for accurate Sw calculations, MICP should be compared with air/water or air/oil Pc/Sw data.
The Pc/Sw data are usually compared first on a Leverett "J-function" basis.  The Pc data are converted to the J-function basis by multiplying each Pc value by the square root of its permeability divided by porosity and then dividing by the fluid-pair IFT multiplied by the contact angle (see Eq. 5). J-function values differ depending on whether they are calculated in oilfield or metric units. The J-function approach assumes similar pore-size distribution in all cores tested. In this way, the various Pc/Sw data tend to converge when the underlying assumptions are met; however, there may still be enough scatter to suggest that the data need to be divided into two or more groupings (see Fig. 6).
From a J-function vs. Sw plot, the technical team can determine whether enough data have been obtained, whether new data need to be gathered to fill in portions of the data ranges, and whether the data indicate that subgroupings are appropriate and needed. Also, this plot indicates whether there are significant outliers that should be excluded or examined in more detail. A drawback to this averaging method is the introduction into Sw determination of four measured parameters and their associated errors (i.e., porosity, permeability, IFT, and contact angle).
Fig. 6 – Example plots of centrifuge air/brine capillary pressure (Pc/Sw) data and its conversion to a J-function basis; data from an Asian gas field. The air/brine Pc value of 100 psi is equivalent to a height above free-water level of 200 to 350 ft, depending on the reservoir fluid’s properties and the temperature and pressure conditions.
Capillary pressure data may also be averaged by various models.  The relationship of Sw with permeability, and then porosity, is examined and is followed by examination of the height dependency.
OBM core water saturation data
The last type of Sw data discussed here is that obtained from routine core analysis of core plugs cut from OBM cores, either preserved as whole cores or else with core plugs cut at the wellsite and preserved individually. These data are taken foot-by-foot and are direct measurements of reservoir Sw values.  Many fields may never have had any wells cored with OBM; others may have only one or two OBM-cored wells. Even a single OBM coring of the full reservoir interval offers significant data that may impact the technical team’s methodology for making the Sw calculations. It is better to have at least two wells fully cored with OBM from different areas of the reservoir.
To evaluate the OBM-core Sw data, they should be plotted as Sw vs. log10 (permeability) or vs. porosity to identify outliers and trends in the data. Particularly, the low-porosity/low-permeability data range should be examined for potential measurement problems. Sometimes, the raw laboratory measurements of water volume and PV data need to be reviewed for problem points and recalculations made where appropriate. Finally, the data should be divided into various possible interval groupings so that any needed zonation can be identified.
If reservoir connate water has flowed out of the core plug at any stage before the laboratory measurement, the OBM-core Sw data are clearly not representative of the in-situ reservoir Sw. This certainly occurs in water-bearing formations and can also occur in the lowest intervals of the oil/water or gas/water transition zones. These lowest intervals, which may be a few feet to approximately 30-ft thick, are precisely the same intervals in which a water cut is expected with the initial oil production. The mobile-water intervals can be identified in OBM wells where the shallow-reading induction-log resistivity is higher than the deep-reading induction-log resistivity. This pattern indicates higher oil saturations in the invaded zone compared with the original oil saturations. Where mobile water is observed, the OBM-core Sw measurements do not represent in-situ Sw and are too low.
Application of each water saturation technique
Methodologies for quantifying Sw at the wellbore are discussed here. The main features of each approach are described; however, in some cases, there are variations that are not addressed. For each technique, its strengths and weaknesses are discussed.
Resistivity well logs and relational model
The most common technique for calculating Sw is the use of resistivity logs with a model (empirical or theoretical) that relates Sw to Rt, Rw, and porosity. As mentioned previously, a large number of Rt/Sw models have been published. The models are applied at every data point in the reservoir where deep resistivity, porosity, and shale-volume estimates, if required, are available. The evaluation of all other necessary parameters (constant or variable Rw values, a, m, n, QV, Vsh, R0 = F • Rw, etc.) has been discussed previously. Several commercial software packages are available that perform these S w calculations for a variety of log models.
Clean sand (Archie) model
This model is used for field studies in the many sandstone and carbonate reservoirs in which the clay-mineral content is low. This decision is strengthened after SCAL data have demonstrated that the simplest solution is satisfactory. When a significant fraction of smectite (montmorillonite) is present and where finely laminated sand and shale sequences occur, one of the shaly-sand models is very likely to be required. Low-resistivity pay is an issue in several oil-producing areas, such as the US Gulf Coast, Egypt, and Indonesia, and hydrocarbon reserves can be missed and left undiscovered as a result of the resistivity suppression by clay minerals and shales.
In the clean-sand model, the formation water is the only electrically conductive medium. In shaly rocks, Rt is suppressed and Archie Sw calculations are too high. As clay-mineral-rich rocks were studied and experimentally tested, more-complicated electrical models were developed to account for the effects of the geometries of conductive clay minerals and shale on rock resistivity. The primary goal of the shaly-sand models is to determine a working relationship between Sw using parameters similar to the Archie model, but also incorporating the quantity and specific electrical properties of the clay-mineral/shale. All of the shaly-sand models reduce to the Archie equation when the shale component is zero. For simplicity, in all of the shaly-sand models, the cementation constant, a, is taken to be 1.0 but, if required, can be easily associated again with the Rw term.
Laminated sand/shale model
A parallel resistor model might be used for laminated sands, with multiple thin parallel layers of 100% shale interbedded with clean-sand layers. Thin, in this context, means that there are several beds within the vertical resolution of the resistivity-logging tool.
The value of ϕsd may be assumed from neighboring thick sands, and all of the parameters, except the Sw of the sand, Swsd, can be estimated.
Poupon-Leveaux (Indonesia) model
The Indonesia model was developed by field observation in Indonesia rather than by laboratory experimental measurement support.  It remains useful because it is based on readily available standard log-analysis parameters and gives reasonably reliable results. The formula was empirically modeled with field data in water-bearing shaly sands, but the detailed functionality for hydrocarbon-bearing sands is unsupported, except by common sense and long-standing use. Sw results from the formula are comparatively easy to calculate and, because it is not a quadratic equation, it gives results that are always greater than zero. Several of the other quadratic and iterative-solution models can calculate unreasonable negative Sw results.
The Indonesia model,  and other similar models, are often used when field-specific SCAL rock electrical-properties data are unavailable but are also sometimes used where the SCAL exponents do not measure the full range of shale volumes. Although it was initially modeled on the basis of Indonesian data, the Indonesia model can be applied everywhere. The inputs are the effective porosity, ϕe, shale volume and resistivity (Vsh and Rsh), and water and deep resistivities (Rw and Rt). The Sw output is usually taken to be the water saturation of the effective porosity, but it has been recently suggested that the output is likely to estimate Swt. Many other log-based shaly-sand models have been proposed but, for brevity, are not discussed here.
Waxman-Smits-Thomas and dual-water models
Swt, the water saturation of the total porosity, is calculated at each reservoir data point by iterative solution of the complex multiparameter Waxman-Smits-Thomas (WST) and dual-water (DW) equations (Eqs. 7 and 8). For brevity, the details of the solution methods are not presented here. The WST and DW models are total-porosity/Sw system models.
where Swt = water saturation of the total porosity as shown schematically in Fig. 1, B = specific cation conductance in (1/ohm•m)/(meq/mL), and QV = CEC in meq/mL of total PV. The exponents m* and n* apply to the total PV.
The DW model is also based on the WST data. It uses clay-bound-water conductivity instead of WST’s BQv factor (see Eqs. 7 and 8) and an alternative shale-volume descriptor, Swb, the saturation of physically bound water in the total PV (see Fig. 1).  When Vsh is zero, Swb is zero; and when Vsh is 100% BV, Swb and Swt are also 100% PV.
where Rwb = resistivity of clay-bound water in the shales, and Rwf = resistivity of free formation water in the shale-free water zones. Because of the different model assumptions, DW exponents mo and no must always be smaller than the WST exponents and may be values similar to "clean" sand exponents. Where the WST and DW models have been properly applied, the hydrocarbon pore volume (VHCP) results should be equal. All Swt calculations from the WST and DW methods must be checked to ensure that they are greater than Swb. After this check, they are used with ϕt to obtain the VHCP. For the DW model, when the outputs require conversion to effective porosity, ϕe, and effective water saturation, Swe, the properties are converted with Eqs. 9 and 10, respectively.
Strengths and weaknesses of resistivity log-based calculations
The greatest strength of Sw calculations from the Rt logs is that these calculations can be made at each net-pay depth with valid data for all wells within the log database. The calculations can account for any subsets of input parameters related to the individual zones.
The weaknesses of the Rt-based Sw calculations are that one has to select a model to describe the relationship of Sw to Rt, Rw, and a variety of other input parameters. Any model is an approximation to the real nature of the reservoir pore system and, typically, has limitations such as how the clay-mineral conductivity is modeled. Log-analysis estimates of Vsh are rather uncertain, so sands that are substantially free of clay minerals can easily, and incorrectly, be assigned significant clay volumes. In these circumstances, complex shaly-sand models may have been applied when it is more appropriate to model the sand as clean sand. Effective porosity is also impacted by the uncertain Vsh estimates. The Rw is often assumed to be constant within the hydrocarbon column, and usually there is little data regarding Rw other than from aquifer samples. In several cases in which the Rw distribution has been studied in depth, it was found to vary in systematic ways within the hydrocarbon column and not necessarily be the same as in the underlying aquifer. 
CEC can be measured in the laboratory, but in the reservoir it must be estimated by correlations with porosity or Vsh. For the laboratory CEC measurements, there are fundamental uncertainties such as the degree to which the clay-mineral geometry is altered by the disaggregation of the core. The total surface area and CEC may be enhanced by comminution (i.e., grinding to grain-size particles). 
The other input parameters for the Sw/Rt models are either based on "worldwide experience" (such as default exponent parameters in commercial software packages) or developed from SCAL rock-electrical-property measurements on a relatively small number of core plugs from the reservoir interval. Hence, there are relatively few data determining the parameters that are used for the log point-by-point Sw calculations. It has to be assumed that the manner in which the water saturation is distributed in the core plugs during these laboratory experiments is like that of the real reservoir. Because water is present during the laboratory measurements, clay minerals are rehydrated at the time of the tests.
Laboratory capillary-pressure/saturation measurements
A second Sw method that is totally independent of the resistivity logs uses laboratory-measured Pc/Sw data. The underlying concept of the use of capillary pressure data is that the reservoir has come to capillary equilibrium over geologic time (the millions of years since hydrocarbons have entered and filled the reservoir trap). This equilibrium is reproduced in laboratory experiments using the centrifuge, porous-plate, and mercury-injection capillary pressure (MICP) methods. The Pc/Sw data are measured on a selected set of reservoir core plugs representing a range of porosity and permeability values (and possibly also lithologies).
Centrifuge experiments are typically made on 1-in. core plugs over a period of several days in the intense gravitational field (up to 1000 G) of the centrifuge and are assumed to be equivalent to what occurs in a hydrocarbon reservoir over millions of years in a 1-G gravitational field and over lengths of 10 to hundreds of feet. These assumptions are broadly accepted as being reasonable, provided that the samples are not damaged during testing in the centrifuge. The reported Pc/Sw values are not the raw laboratory data. In the laboratory, the average saturation is determined at each centrifuge speed, and those raw data are input to a mathematical model to convert them to a tabulation of endface saturations and Pc values.
Porous-plate Pc tests are made on core plugs at several different gas pressures and are generally carried out at the same time as the resistivity experiments. After reaching equilibrium with no further brine flow at each pressure, the Sw is constant along each plug and is calculated from its weight loss.
MICP tests are made on dried core pieces and the volume of injected mercury, the nonwetting phase, is converted to an Sw value. This is considered to be total Sw if, at high enough pressures, mercury enters both the microporosity and dry clay-mineral porosity. Conversely, for centrifuge or porous-plate tests, where brine is present as the wetting phase, clay minerals probably hydrate, and their physically bound water is unlikely to be displaced during the test. Brine-related Pc/Sw measurements may give total or effective Sw, depending on the specific porosity measurement method used (i.e., whether the porosity occupied by the clay-mineral physically-bound water is included or excluded from the porosity calculation). Effective Sw values are always lower than total Sw values and should be very low at high capillary pressures if there is little nonclay-mineral-related microporosity. 
The conversion of the laboratory Pc/Sw data to reservoir conditions requires knowledge of the IFT and contact angle of the fluid pair used in the laboratory and properties of the brine and hydrocarbon fluids at reservoir conditions. These are needed to calculate the density of each phase and to estimate the interfacial-tension (IFT) between the fluid pair at reservoir conditions. The Pc values (in psi) are converted to vertical height above the hydrocarbon/water contact, Hhwc (in feet), with the following formula:
where fluid densities (ρ) are in g/cm3, and the subscripts are r = reservoir, s = surface, h = hydrocarbon, and w = water. Table 1 lists some typical values for IFT, σ, and contact angle, θ, used in 11.  and provides approximate ranges for the factors for converting Pc-laboratory data to height above a reservoir free-water level. Height-Pc conversion factors are similar for many oil and gas reservoirs; the footnotes in Table 1 describe the values that were assumed to calculate these ranges. More details of correlations for brine/hydrocarbon IFT as a function of oil or gas gravity have been published.  The reservoir-condition contact angle, θ, is usually taken as 0 for gas reservoirs and 0 or 30° for oil reservoirs because, generally, data are not available at reservoir conditions.
The suite of Pc/Sw data is typically converted to a mathematical relationship between Sw as the dependent variable and the independent variables—porosity, permeability, and Howc or Hgwc. Because permeability is usually determined as a function of porosity, it is often not included as an independent variable. Two of the mathematical forms that have been used are
where A, B, C, D, and E are curve-fit constants. In Eq. 13, B permits the removal of singularities at zero height.
In developing the coefficients for these relationships, any zonation of the reservoir intervals needs to be applied, and then separate sets of coefficients developed for each zone. The zonation can be based on geological interpretation of the reservoir depositional and diagenetic history and/or variation in the Pc/Sw curves for different parts of the reservoir interval.
The depth of the reservoir’s oil/water contact (OWC) or gas/water contact (GWC) must be known in order to make Sw calculations using the Pc/Sw methodology. The calculations of S w are made only above this depth. In reality, the Howc or Hgwc is referenced to the free water level (FWL) (i.e., the depth at which Pc =0 and which is deeper than the observed OWC or GWC). For a gas reservoir consisting of good-quality rocks, the difference between the FWL and the GWC is typically 1 ft or less. However, for an oil reservoir containing a heavier oil, this difference can be 10 feet or more, and, given four-way closure on an anticlinal structure, the impact on the OOIP volume between using the FWL vs. the observed OWC as the Howc = 0 depth can equal a few percent of OOIP.
Once the various sets of coefficients have been developed and the Pc to Howc (or Hgwc) conversion made, an Sw value can be calculated at each data point within the log database that has a valid porosity value and is above the OWC or GWC. Hence, there will be the same number of, or more, Sw values available from this Sw methodology as when using the Rt logs.
Strengths and weaknesses of capillary pressure-based calculations
The strength of Sw calculations from Pc/Sw data is that, after making a correlation with porosity and height, a unique Sw value is available for all wells at all net-pay depths with valid porosity values in the log database. This also applies to the whole hydrocarbon column anywhere in the reservoir once the wellbore porosity values have been propagated into the full geocellular model grid. These calculations can account for any zonation and subsets of input parameters related to the individual zones.
A potential weakness in the Pc approach to Sw calculations is whether the laboratory measurements have been allowed sufficient time to reach equilibrium. If not, the Sw values, particularly at high P c values, will be too high. Another potential weakness is the accuracy of the IFT value used in converting from surface to reservoir conditions; fortunately, these values vary over a limited range for most hydrocarbon/brine pairs. A third potential weakness is the definition of the FWL depth compared with the observed OWC or GWC. A fourth potential weakness is whether enough data have been taken to be representative, both vertically and areally, of the zones in the reservoir. 
The fifth potential weakness concerns the complexity of the reservoir’s hydrocarbon-filling and structural history. In simple oil-reservoir situations and most gas-reservoir situations, this is not an issue. However, for oil reservoirs with tar mats and heavy-oil zones there is a complication because of the varying oil density near the OWC, including the possibility that the tar mat has a hydrocarbon density very close to that of the connate brine. Another aspect may be whether all or portions of the hydrocarbon column are on the imbibition cycle where imbibition Pc/Sw data are needed for the Sw calculations, not the typical drainage Pc/Sw data. 
OBM-core-plug Dean-Stark water-volume determinations
The third method for determining the Sw in a reservoir’s hydrocarbon column is to cut OBM cores and perform Dean-Stark water-volume determinations on the routine core plugs. Foot-by-foot Sw values can be calculated from these water volumes and the associated core-plug PVs. OBM cores are typically cut only in a few wells in a particular field. These Sw data can be applied to other, uncored wells in the reservoir if strong correlations between these values and porosity and/or permeability are identified. These data are not valid in the oil/water or gas/water transition zone or in the aquifer, intervals in which the connate brine is mobile. OBM-core Sw values may be found to be either higher or lower than those from the other two methods described previously.
Strengths and weaknesses of OBM-core values
The strength of Sw values from routine-core-analysis Dean-Stark Sw data is that these data are the most direct measure of reservoir connate Sw values above the oil/water or gas/water transition zone. Relative to the two methods discussed previously and the variations of these methods, the OBM-core Sw approach is a direct Sw determination and the other methods are indirect Sw-calculation approaches that require many more assumptions and inferences.
The weaknesses of the OBM Sw method are that it does not apply to the lowest parts of the oil/water or gas/oil transition zone where the brine phase has mobility and that, generally, the amount of OBM core Sw data is limited because the operator cuts cores with OBM only in a limited number of wells because of the expense. The first of these weaknesses can be overcome if the OBM Sw data are used in combination with either the resistivity logs or with Pc/Sw data.
Another consideration is that the whole project, from the mud formulation to the core-handling and -preservation procedures on through the routine-core-analysis measurements, needs to be monitored and reviewed in detail to ensure all steps were executed properly. This demands that considerable time and effort be spent by the technical team to ensure success; however, to some extent the same comment applies to the Pc/Sw and resistivity-log/Sw calculation approaches discussed previously.
Integration of water saturation data from different methods
Depending on the data availability in a particular reservoir situation, a combination of the various Sw approaches may prove superior to the use of a single type of data. The first step in going to a combination approach is to review the reservoir’s database to identify any significant gap in vertical, or areal, coverage. The most obvious gap often occurs near the fluid contact, because there is little reason to drill wells in downdip locations, particularly during a reservoir’s development phase. Three examples of combination approaches are described below.
Resistivity-log and capillary pressure data
Resistivity-log-derived Sw results may not be available throughout the hydrocarbon column of a reservoir. To fill gaps and average the point-by-point data set, it is common practice to plot Sw as a function of height, to omit nonpay points, and to identify various porosity ranges by coding the data points. Resistivity-log-derived Sw data frequently shows V- or U-shaped patterns on these plots because of the shoulder/bed effects near nonpay sections (shales). The most accurate Sw values in such patterns are usually at the lowest Sw values where the thin-bed correction is minimized. In a manner similar to that described in the previous Pc/Sw section, height/saturation curves are often fitted to these resistivity-log-derived Sw data to enable reservoir hydrocarbons-in-place volumes to be calculated. The function forms are similar to or are the same as those described above for Pc/Sw. 
Routine OBM core with capillary pressure data
Because there is a need to define the Sw characteristics of the oil/water or gas/water transition zone and because the OBM-core Sw data can be incorrect and too low in this interval, one approach is to use Pc/Sw data in combination with the routine OBM-core Sw data. This can be done by first correlating the OBM-core Sw data to porosity and assuming that this relationship is valid above the oil/water or gas/water transition zone. The functional form of this first relationship might be
The second step is to create a tabular data set in which the Sw/porosity correlation is used to calculate an array of Sw values for large Howc or Hgwc values and a range of porosity values. For this part of the data set, Sw is assumed to be independent of the Howc or Hgwc values. The Pc/Sw data converted to reservoir conditions is used to provide data points for low Howc or Hgwc values and various porosity values. Statistical calculations are applied to the whole of this data set. The functional form of this second relationship might be
With this functional form, the boundary conditions of the first step are automatically met in the second step.
Routine OBM-core with resistivity-log data
To address the lack of valid OBM-core Sw data in the oil/water or gas/water transition zone discussed previously, it is also possible to combine OBM-core Sw data with the resistivity-log data to develop an overall Sw methodology. This approach assumes that a number of wells have been drilled through the OWC or GWC so that there are log resistivity values through the oil/water or gas/water transition zone. In this approach, the OBM-core Sw data are used to back-calculate the saturation-exponent, n, values over each zone so that the core-based VHCP value equals that calculated from the resistivity logs (see Eq. 2). Then the core-based saturation-exponent, n, values are applied to the noncored well’s resistivity logs to calculate Sw point-by-point throughout the reservoir interval in all wells.  This approach assumes that the Rw, a, and m values have been determined from other experimental and fluid-sample data so that R0 can be calculated.
where R0 is the bulk resistivity at Sw = 100% PV and is calculated with Eqs. 4 and 5. Rt is the deep-reading resistivity-log reading, and Swc is the OBM-core Sw above the mobile-water transition zone. The resulting back-calculated n values at the core-plug depths are averaged for the zone. In some instances, n may be found to have an areal variation within a zone that should be taken into account in subsequent calculations.
Adjustments to water saturation data from different methods
We have described three methodologies water saturation determination. They are basically independent methods; hence, they can be used together to determine the accuracy of the Sw calculations throughout the hydrocarbon column. Because the methods are based on very different technical approaches and assumptions, if the different methods give essentially the same Sw answer, then it is highly likely that this is the correct Sw.
However, the challenge comes when, as is often the case, the different methods result in different Sw values and distributions. The OBM-core Sw values might be either higher or lower than those from the other two methods. The common misunderstanding that OBM-core Sw is likely to be too low is unsubstantiated. In a very large reservoir, it could go both ways depending on where one is in the reservoir.  If the values are quite different, two aspects of the calculations need to be reviewed in depth. First, the quality of the input laboratory data needs to be checked and how it was converted from raw data into the input values to the Sw calculations needs to be reviewed. Second, the assumptions and models used for the Sw calculations need to be checked. For example, with the Pc/Sw data, the assumed oil/water density difference may be considerably in error, or the shaly-sand Sw model may be inappropriate for the particular reservoir. As well as the Sw averages, the zone-average VHCP values from the various methods should be compared, which includes porosity in the comparison calculations.
Core, total, and effective systems compatibility
The Archie Rt-based Sw equation models "clean" sands. Various other shaly-sand models use either the effective or the total-porosity systems. It is well known that these basic models, if applied properly to the same formation, must produce the same final VHCP from their different calculation procedures (see Fig. 1 and Eqs. 6 through 8).  ϕt is greater than or equal to ϕe; however, at the same time, Swt is greater than or equal to Swe and, when used together, the appropriate combinations must give the same VHCP result. For the total-porosity system, VHCP = ϕt × (1-Swt), whereas for the effective-porosity system, VHCP = ϕe × (1-Swe).
The VHCP can also be estimated from a combination of core porosity and Dean-Stark Sw measured on preserved OBM cores. The several systems—core, total, and effective—must all give the same fundamental results, and the most accurate of them (the OBM-core method) can be used to calibrate and test the less accurate methods. When properly adjusted and applied (e.g., by improving the Vsh estimates or IFT values), all three methods give the same final VHCP. If they do not agree, the likely sources of uncertainty and error must be examined.
It is clearly inconsistent and incorrect to mix the systems by, for example, reporting an effective porosity with a total Sw, a total porosity with an effective Sw, or a standard-core porosity with an effective Sw. System compatibility must also be maintained by correct use of the SCAL measurements and log-analysis formulae, when these are used to calibrate the resistivity logs and Pc/Sw methods. The differences should be resolved as much as possible. To the extent that they are not, the differences can be considered to be a measure of the uncertainty in the Sw calculations.
It is the uncertainty of the hydrocarbon saturation (1 − Sw) that is economically important, not the absolute uncertainty in Sw. When uncertainties in Sw are evaluated, their importance in terms of So and Sg should be accounted for. The uncertainties of the several Sw-evaluation methods vary widely.
OBM-core water saturation data
The water volume extracted from a single core plug may have a random and known systematic uncertainty of ± 0.05 cm3, where each uncertainty refers to one SD. The PV of a typical 1-in. core plug is 4.0 cm3 if the porosity is 20% BV. The water-volume uncertainty alone equates to an Sw uncertainty of ± 1% PV (0.05/4.0). The uncertainties in porosity have a further effect on this calculation.  An OBM-core Sw of 20% PV, therefore, has a combined 1-SD range from approximately 18 to 22% PV. At lower porosities and higher Sw values, the water-volume uncertainty may be ± 0.1 cm3, leading to an Sw uncertainty of ± 3% PV, when the porosity is 15% BV. As porosity decreases, the uncertainty grows. Before the measurements are made, any water in the toluene and the Dean-Stark apparatus must be removed, or the Sw values will be overstated. The extraction time required to recover the water adsorbed on the clay minerals adds to the uncertainty.
The uncertainty of the average core Sw will be improved when plugs are selected at one or two per foot with equal spacing and without regard for the rock quality. However, as discussed previously, plug samples are not always selected at random, so care must be exercised, especially regarding the Sw values predicted at depths where core is not available. From a broader perspective, it must also be remembered that 1-in. core plugs only sample approximately 2% of the full-core volume. Because of these many factors, the authors estimate that uncertainties similar to those given concerning porosity also propagate to the zone-average OBM-core Sw values. Measurements in which larger core plugs are analyzed will reduce several of the uncertainties.
Resistivity-log-derived water saturation values
The log readings, typical SCAL-derived Archie exponents, and all of the other associated parameters are uncertain. For example, the resistivity-log uncertainty may be ± 50% when R t is 500 ohm•m. The most important uncertainty contributors at low Sw values are likely to be Rt and n. Sw uncertainty in this circumstance is estimated at ± 5% PV (i.e., if Sw is calculated as 10% PV, the 1-SD range is 5 to 15% PV).  At lower porosity values and higher water saturations, similar methods led to uncertainty estimates of ± 9% PV. Given that further uncertainty in the final calculated Sw may arise from shaly-sand effects and many other sources, the authors believe that the ranges given apply equally to the overall systematic uncertainty of the Sw zone-average values. These estimates are all 1 SD; therefore, in 32% of cases, zone-average uncertainties are considered likely to be greater than the ranges given.
Capillary pressure-derived water saturation values
The uncertainty estimates are the sum of several factors. Most of these factors have their greatest impact on the Sw calculations in the first 100 to 200 ft of the hydrocarbon column above the fluid contact. Therefore, because the transition zone is considerably longer in many oil reservoirs than in a gas reservoir, their impacts will be greater in most oil reservoirs. Above 200 ft, the Sw values are usually only changing slowly; hence, the primary consideration above the transition zone is whether the laboratory measurements are taken at equilibrium conditions.
The first factor in the uncertainty analysis is the fundamental assumption as to whether the drainage or imbibition Pc/Sw data should be used. In most cases, the drainage curves should be used, but, in a few situations, the reservoir may be on the imbibition cycle. In these situations, the improper choice of using the drainage Pc curve can lead to a +5 to 20% PV Sw error in the first 100 to 200 ft above the OWC. 
The second factor concerns the laboratory Pc/Sw measurements. If the measurements are not taken to equilibrium, then the Sw values at a particular Pc value will be too high. This can be +1 to 10% PV effect for the large Howc or Hgwc range. The other key aspects for reported centrifuge laboratory results are how the raw laboratory measurements of water volumes were determined and how these data have been converted to the reported endface saturations. The water-volume measurements have the same-size potential error as discussed for the OBM Dean-Stark Sw measurements (± 1 to 3% PV). Differences in the laboratory calculation procedures can result in further variations of ± 1 to 3% PV in reported Pc/Sw results when using the same raw laboratory data. For porous-plate tests and others, the repeated handling of poorly cemented or uncemented core plugs can cause grain loss, which, after the final calculations, translates into small errors in Sw.
The third factor is how the suite of raw laboratory data for a particular reservoir interval are curve-fitted and presented in the final laboratory report as tabulations of Pc/Sw values for each core plug. Uncertainty in the application arises from how these reported values are averaged for use in the Sw calculations over the full range of reservoir porosity and permeability values. This uncertainty includes how the data are weighted and whether some potential outlier data from one or two core plugs distorts the averaged Pc/Sw curves. These uncertainties primarily affect the first 100 to 200 ft above the Howc or Hgwc so that their impact depends on how thick the hydrocarbon column is and its distribution as a function of Howc or Hgwc.
The final factor is the conversion of the averaged Pc/Sw curves (or equation) from surface to reservoir conditions, all of which affect the conversion of Pc values to Howc or Hgwc values. This includes a number of subfactors, each with its own uncertainty level: IFT at surface and reservoir conditions, fluid-pair density difference at reservoir conditions, contact angles, and depth of the actual in-situ FWL compared with the OWC or GWC. The contact angles at surface and reservoir conditions are generally taken to be the same because no data are available to proceed otherwise. For these other factors, the uncertainty is considerably greater for an oil reservoir than for a gas reservoir; because the IFT values can be low and compared with those for a gas reservoir, the density differences are significantly less particularly if there is a vertical oil-gravity variation that results in a heavy-oil interval just above the OWC. All of these factors affect the Howc or Hgwc values; therefore, their impact on the S w calculations is predominantly in the first 100 to 200 ft above the fluid contact.
In summary, the use of Pc/Sw data can result in Sw uncertainty of ± 5 to 15% PV in the oil/water or gas/water transition zone. Above that transition zone, the uncertainty is related to whether the laboratory data were taken at equilibrium conditions and how the various Pc/Sw curves have been averaged together. In this range, the uncertainty is likely to be 3 to 10% PV.
|a||=||Archie cementation constant|
|a*||=||Waxman-Smits cementation constant|
|A||=||Coefficient in various equations of this chapter|
|B||=||Specific cation conductance, [(1/ohm•m) / (meq/mL)]|
|C||=||Coefficient in various equations|
|D||=||Coefficient in various equations|
|E||=||Coefficient in various equations|
|F||=||Archie formation factor|
|F*||=||Waxman-Smits-Thomas formation factor|
|FHCP||=||hydrocarbon pore feet, L, ft [m]|
|Hgwc||=||height above the gas/water contact, L, ft [m]|
|Hhwc||=||height above the hydrocarbon/water contact, L, ft [m]|
|Howc||=||height above the oil/water contact, L, ft [m]|
|k||=||permeability, L2, md [μm2]|
|m||=||Archie cementation exponent|
|m*||=||Waxman-Smits-Thomas cementation exponent|
|mo||=||dual-water cementation exponent|
|n||=||Archie saturation exponent|
|n*||=||Waxman-Smits-Thomas saturation exponent|
|no||=||dual-water saturation exponent|
|Pc||=||capillary pressure, m/Lt2, psi|
|Pce||=||entry capillary pressure, m/Lt2, psi|
|Qv||=||cation-exchange capacity of total PV, meq/mL|
|R0||=||rock resistivity with 100% PV water saturation, ohm•m|
|Rsd||=||clean-sand resistivity, ohm•m|
|Rsh||=||shale resistivity, ohm•m|
|Rt||=||true resistivity of uninvaded, deep formation, ohm•m|
|Rw||=||connate-brine resistivity, ohm•m|
|Rwb||=||clay-bound water resistivity, ohm•m|
|Rwf||=||free-formation-water resistivity, ohm•m|
|Rxo||=||shallow-reading invaded-zone microresistivity, ohm•m|
|Sg||=||gas saturation, %PV|
|So||=||oil saturation, %PV|
|Sorw||=||residual-oil saturation to water displacement, %PV|
|Sw||=||water saturation, %PV|
|Swb||=||saturation of clay-bound water in the total porosity, %PV|
|Swc||=||connate water saturation, %PV|
|Swc||=||core water saturation, %PV|
|Swe||=||water saturation of the effective porosity, %PV|
|Swsd||=||sand water saturation, %PV|
|Swt||=||water saturation of the total porosity, %PV|
|Vcl||=||clay content, %BV|
|VHCP||=||hydrocarbon pore volume, L3, ft3 [m3]|
|Vsh||=||shale content, %BV|
|θ||=||contact angle, degrees|
|ρb||=||formation bulk density, m/L3, g/cm3|
|ρfl||=||fluid density, m/L3, g/cm3|
|ρh||=||hydrocarbon density, m/L3, g/cm3|
|ρma||=||matrix or grain density, m/L3, g/cm3|
|ρw||=||water density, m/L3, g/cm3|
|σ||=||interfacial tension, m/t2, dynes/cm|
|ϕc||=||core porosity, %BV|
|ϕcl||=||clay porosity, %BV|
|ϕe||=||effective porosity, %BV|
|ϕsd||=||sand porosity, %BV|
|ϕsh||=||shale porosity, %BV|
|ϕt||=||total porosity, %BV|
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