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Compute Saturation SWE & SWT Generalities The calculation of water saturation is one of the most troublesome aspects of Shaly Sand log analysis. Many equations have been developed over the years based on known physical principles or on empirically derived relationships. SW can be determined if one or more measurements can be made that are strongly influenced by the fluid properties. Resistivity measurements are, by far, the most commonly used measurement to determine SW. In the very earliest days of well logging, it was recognized that the presence of hydrocarbons was indicated by anomalously high resistivity in porous intervals. The presence of clay can suppress resistivity and sometime mask the hydrocarbon effect. We will review here some of the methods used to calculate SW from resistivity data. There are two general groups of Water saturation equation commonly used today and both have strong suites of followers. Equations by Group 1- Vshale or resistivity model equations (Uses PHIE) Laminar Simandoux Modified Simandoux Poupon Leveaux

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Compute Saturation SWE & SWTGeneralitiesThe calculation of water saturation is one of the most troublesome aspects of Shaly Sand log analysis. Many equations have been developed over the years based on known physical principles or on empirically derived relationships. SW can be determined if one or more measurements can be made that are strongly influenced by the fluid properties. Resistivity measurements are, by far, the most commonly used measurement to determine SW. In the very earliest days of well logging, it was recognized that the presence of hydrocarbons was indicated by anomalously high resistivity in porous intervals. The presence of clay can suppress resistivity and sometime mask the hydrocarbon effect. We will review here some of the methods used to calculate SW from resistivity data. There are two general groups of Water saturation equation commonly used today and both have strong suites of followers. Equations by Group1- Vshale or resistivity model equations (Uses PHIE) Laminar Simandoux Modified Simandoux Poupon Leveaux Fertl & Hammack. Indonesia Etc. 2- Cation Exchange or conductivity model equations (Uses PHIT) Waxman-Smits Modified Waxman Smits Juhasz Dual Water Charlebois etc. What equation to use:?It is generally accepted that conductivity models using PHIT will give results that are more consistent than resistivity type equations however conductivity models have inherent problems to the users for the following reasons: PHIT cannot be measured without calibration with core analysis since the dry clay matrix points does not exist in nature and is therefor not seen by the logs. Different laboratory methods to calculate PHIT in clays will give different PHIT for the same core. It can take months before sufficient core results are available to determine the actual PHIT. Oil companies do not always take cores in the shales and the PHIT from low volume clays in a sand do not necessarily have the same properties as in the adjacent shales. The determination of QV and the establishment of a correlation between logs and QV is not always evident and can induce errors.Recommendations:1. In a development field, and provided sufficient core results are available in the zone of interest only, we recommend the use of a conductivity saturation equation to determine SWT and SWE2. Resistivity model equations should preferably be used in all wildcat wells or where insufficient core results can be used to determine PHIT accurately.Input Logs: PHIE PHIT VCLAYInput log should all be environmentally corrected before proceeding here below. Output curves: SWE SWTZone parameters: a m n a1 m1 n1 RW RCLAY B Earth electrical conductanceElectrical currents can flow in various ways.1. Metallic conductance is probably understood by most. It requires that a voltage potential exist across a conductor (light bulb) and that the conductor be comprised of elements which have free electrons that can be passed from one atom to another.2. Ionic conductance is not so well understood. Current flow though a conductive ionic medium (e.g. salt water) requires the actual movement of charged ions (Such as Na+ and Cl-) through the medium (water). The SP measurement is derived from this phenomenon. However, when an alternating current is induced in the medium, the direction of the potential is constantly reversing. The Cations and Anions are still in motion but their behaviour is different.When a field reaches a peak in one direction, the displacement of the charged ions is set to a maximum. As the field collapses and then builds in the reverse direction, the ion displacement is in the opposite direction. At higher frequencies, the distance travelled by an ion between + and -peaks becomes extremely small. Thus, it can be seen that the conductance in the formation is due to the ability of charged ions to migrate through the water within the pore spaces.The presence of hydrocarbon impedes the movement of charged ions.Many electrical tools measure conductivity but, traditionally, resistivity is recorded on the log. Thus most equations have been written in terms of resistivity instead of conductivity.Resistivity is the reciprocal of conductivity. R = 1/CR = resistivity in Ohm2/mC = Conductivity in mhosResistivity tools measure resistivity as a series circuitConductivity tools measure conductivity as parallel circuits.The equation for total conductance in a parallel circuit isCT = C1 + C2 + Cn EQ 1The equation for total conductance in a series circuit isCT = 1 / (1/C1 + 1/C2 + 1/Cn) EQ 2For total resistance in a parallel circuit we getRT = 1 / (1/R1 + 1/R2 + 1/Rn) EQ 3For total resistance in a series circuit we use:RT = R1 + R2 + Rn EQ 4Various Shaly Sand models differ in the manner in which the two current paths are defined. Generally, the formation water in the pore space is treated as one current path and the clay network is treated as the second current path.BasicsArchie characterized the conductivity of porous media having a non-conductive matrix as a function of the porosity and of the conductivity of the saturating fluid.Archies equation is in the form:F = a/PHI^m EQ 5The conductivity of a clean sand saturated with a conductive brine can be defined as:Co = CW/F EQ 6WhereCo: Conductivity of a 100% water saturated rockCW Conductivity of the saturating brineF Formation resistivity factorPHI: Porositya and m are constantsWhere electrical current can be conducted by a second path, the above equation becomes invalid. This condition exists where the matrix is conductive of where a second conductive path exists in the pore network. Clay provides such a conductive path. Where clays are present, the conductivity can be expressed asCo = K * CW/F + X EQ 7WhereK Interactive clay function (Unity in many models)X The additional conductivity due to the presence of clayAnalysis of Eq. 7 will show that, for any given value of X, the magnitude of the shale effect decreases as the formation water conductivity CW increases.Thus, where formation waters are very saline, the contribution of the shale term becomes relatively small.Conversely, where formation water conductivity is low (formation water is fresh) the contribution of the shale term is relatively large.It is then obvious that accurate characterization of the shale term becomes increasingly important as formation waters become fresher. It is in the brackish and fresh water applications that most shaly sand analysis models encounter difficulties.Nearly every shaly sand log analysis model is in basic agreement with EQ 7, however they differ in the manner in which X and F are derived.We address here only some of the most commonly used equations.Elements of the water saturation modelsThere are two different approaches to compute SW in Shaly Sand models1-V-Shale or resistivity Models These are the most commonly used today as they can use recorded logs input only.2-Cation Exchange or Conductivity Models These models can give better results as they can be matched closely to laboratory measurements. It is not as popular as the cost of doing the laboratory tests and the lack of core data often precludes the use of these models.The V-Shale Models work reasonably well in sands where the water is very saline.The V-Shale Models fail to account for significant non linear changes in formation conductivity as a function of conductivity where fluid conductivity is less than a certain critical value. However this non-linearity can be overcome using advanced techniques such as the dual clay resistivity option.The -Cation Exchange Models are based on the Waxman-Smits or derivatives of it.The Waxman-Smits and Modified Waxman-Smits equations have much in common and the question of the value of one relative to the other has sparked lengthy and sometimes bitter debates.However, both equations account for non-linear changes in conductivity in the fresh to brackish water reservoirs. Since they also account for the linear conductivity changes in the saline reservoirs, they are more acceptable as general Shaly Sand models.The Waxman-Smits model was the first technique to demonstrate broad application based upon core and core data from a variety of reservoir conditions.The Modified Waxman-Smits model was later proposed as a practical application of the lessons learned from the Waxman-Smits model.The Waxman Smits model can give better results but must be calibrated against core data which is not always available. This problem will be discussed in more details further down this presentation.ARCHIEThe very earliest research established that for a formation with constant porosity and water salinity, an increase in resistivity indicated the presence of hydrocarbons. Archie quantified this relationship as: SW^n = F * RW /RT EQ 8aorSW^n = F * CT/CW EQ 8bWhere:F Formation resistivity FactorRW Formation water resistivityCT Conductivity of the formationRT Resistivity of the formationSW Formation water saturation (fraction)n Saturation exponentEQ 8 can be rewritten as:CT = K * CW * SW^n / F EQ. 9Formation Factor FThe key to this relationship is the Formation resistivity factor which is defined in equation 10F = Ro/RW Eq 10 or F = CW / Co EQ 10bWhere:Ro Resistivity of the shale free formation 100% saturated with water of resistivity RWCo Conductivity of the shale free formation 100% saturated with water of conductivity CWElectrical current will flow through any conductor if there is a difference in voltage across this conductor. Salt water is a good conductor while Quartz is a good insulator.Thus, when a potential is applied across a clean-sand (no Clay) with salt-water filling the pore space, all of the current will flow through the water in the pores.High porosity sands with large pore throats will flow currents easily (low resistance) but sands with low porosity will restrict or block off the flow of current (high resistance to currents).This restriction of the flow path is referred as the tortuosity of sand. More tortuous flow paths will result in higher resistivity and higher formation resistivity factor.Water SaturationIn a clean water-wet sand, (SW =1.0) and EQ 8 and 10 becomeRo/RW * RW/RT = 1.0 = Ro/RT or CW/Co * CT/CW = 1.0 = CT/Co EQ 11For clean sands where Ro = RT and Co = CT, EQ 8 can be rewritten as:SW^n = Ro/RT = CT/Co EQ 12EQ 12 states that the ratio of the resistivity of a wet formation Ro to its resistivity measured while only partially saturated RT can be related to SW by an exponent n.The ratio RT/Ro is referred to as the resistivity indexSince there is a large number of logging tools which measure RT it should be obvious that, knowing n we can calculate SW if we can determine Co or Ro.Determining Co or Ro from FWe can determine Co or Ro if we know RW or CW and can determine the formation resistivity factor F from another down-hole measurement. Fortunately there is a known relationship between F and PHI F = a/PHI^m Eq 13 Where PHI Porosity m Cementation exponent a Porosity intercept With F derived from porosity, we can solve Co or Ro in terms of F and CW or RW by rewriting EQ 10 Ro = F*RW or Co CW/F Eq 14a, b Determining F from porosityEQ 13 shows that F can be determined from the porosity if the values of a and m are known.For most beginners, the values of a and m can be confusing and need to be carefully explained.The values of a and m are best derived by cross plotting the log of porosity versus the log of the formation factor.The value of F is obtained from log or core analysis. A core sample is saturated with a brine of known RW. The resistivity of the water saturated core is measures and the resistivity of the water saturated sample Ro is divided by the resistivity of the saturated brine RW. The resulting ratio plot F is then plotted against the porosity as measured on the sample.

Figure 1Figure 1 illustrates the cross plotting of F versus Porosity from core. The value of m is the slope of the best fit line through the data and a is the intercept where porosity is set to 100%. The value of m is referred to as the cementation factor or the tortuosity factor because it increases with the complexity of the current path through the pore network.

Figure 2Figure 2 illustrates the effect of variation in PHI, m and RW on Ro and FSaturation exponent nHaving determined F then Ro and having measured RT and PHI we are now ready to determine SW. This last step requires knowledge of the exponent n.The value of n is empirically derived and can be estimated either from well logs or core studies.In practice, n is often used as a fudge factor to satisfy the whims of management if reserves are a little low.The exponent n is used to take into account such variables as the electrical properties of the water/oil interface, the wettability of the matrix and increased current paths.If the matrix is partially oil wet so that the thin veneer of formation water is interrupted, (see Figure 3) then n can be significantly altered. Figure 4 shows the effect or varying n on the final SW.

Figure 3

Figure 4The saturation exponent n can be estimated from core analysis.Once the core porosity has been measured, it is saturated with a brine of known resistivity. The resistivity of the fully brine saturated core sample is then measured.Some of the fluid is then removed and the resistivity again measured.The amount of fluid removed is carefully measured so that the partial saturation can be calculated.After several repetitions the measured data are plotted as in figure 4.The slope of the line is the saturation exponent n.Several samples representing a wide range of porosities and facies type for a particular reservoir should be so measured in order to obtain a statistically significant value of n.A separate n for each facies should be expected for most shaly sand reservoirs.Effect of clay and salinity on the Archie equationThe Archie equation works relatively well as long as salinity is high and there exists only one current path through the formation.However, the presence of trace amounts of clay minerals provides a second current path and requires alteration of EQ 8.The Archie equation will be in error in clean sands if the formation water salinity is extremely low. This is due to surface conductance effects.Reservoirs having extremely fresh water cannot properly be evaluated using conventional resistivity logs. However a growing number of dielectric constant and electromagnetic propagation tools offer means of evaluating very fresh water reservoirs.Evolution of shaly sand modelsNumerous approaches have been taken over the years to predict Ro for shaly sands.We will not go into details here but Table 1 and Table 2 list some of the earlier efforts to develop reliable SW equations.For more background information we refer you to:SPWLA Shaly Sand reprint Section III by Patchett and HerrickThe evolution of Shaly Sand Concepts in Reservoir Evaluation by Paul F Worthington (1985)Cation exchange modelsThe publication in 1967 of the Waxman-Smits equation opened a new era in formation evaluation.This equation removed some of the errors associated with RCLAY (from adjacent shales) techniques.This model requires extensive coring and core analysis and thus is not universally accepted as an economic mean of calculating SW.Clavier et Al attempted to incorporate the concept of the Waxman-Smits approach into a general model that requires minimal core data. Ref: The Theoretical and Experimental Basis for the Dual Water Model for the interpretation of Shaly Sands SPE Journal, 1984. Waxman SmitsHill and Milburn (1950) established the relationship of cation exchange capacity CEC and the suppression of resistivity on water bearing shaly sands.Waxman and Smits (1967) refined the relationship propose by Hill and Milburn and presented a general equation for SW which required CEC data for core analysis.Waxman and Thomas (1974) expanded the model with explanations of the influence of partial saturation by hydrocarbons and of the influence of temperature on the resistivity of shaly sands.Koerperich (1975) focused on the application of the Waxman-Smits model to solve for SW in reservoirs containing fresh and brackish water.Juhasz (1979) ties the various Waxman-Smits concepts together with modern logging measurements to illustrate the use of the Waxman-Smits model to derive SW directly from logs.Waxman-Smits Overview, summary:A parallel conductance path exists in shaly sandsSome of the electrical currents will flow though the cations associated with the clay and some will pass through the salt solution in the pore system.Knowledge of the boundary between the clay associated water and the pore water is not necessarily in this model because the equation treats the two waters as a mixture.This is possible because the formation resistivity factor used in the model reflects the net conductivity from the two current paths.The conductivity through the cation system is expressed by the terms B and QV. B is temperature and salinity dependent.QV quantifies the cation concentration per volume of formation water.The ion concentration is usually expressed in equivalents per litre.The combined term B*QV expresses the conductivity due to the presence of compensating cations in the water filling pores.The conductivity of a wet shaly sand can be expressed as:Co = CW/F1 + B*QV /F1 Eq 15Where F1is the formation factor for the Waxman Smits modelBecause the equation treats the two conductance paths as a mixture having a common F, SW cannot be solved in terms of the resistivity index.Instead, SW must be included in the body of the equation to balance the mixture relationship. Thus the Waxman-Smits equation can be written in the familiar form:CT = CW * SW^n1 /F1 + B * QV*Sw^(n1-1)/F1 Eq 16Conductivity of wet shaly sands

Figure 5Figure 5 is a plot of the relationship of the conductivity of a water saturated rock as a function of the conductivity of the saturating fluid.The solid line shows a typical shale response.The slope of the line is the formation resistivity factor FAt higher conductivities the relationship is linear.At lower fluid resistivity (Fresh water) the relationship is non-linear and any slight increase in fluid conductivity results in a sharp increase in rock conductivity.Water saturation less than 100%The Waxman-Smits general equation for saturation can be expressed as: SW^n1 = F1*RW / (RT * (1.0 + RW * B * QV / SW)) Eq 17a or SW^n1 = F1*CT / (CW * (1.0 + B * QV /(CW * SW))) Eq 17b The term QVQV is expressed as cation concentration per unit volume of fluid in the pore space. (Equivalents per litre or meq/m)QV is derived from laboratory measurements of the cation exchange capacity of the rock (CEC)CEC is expressed as milli-equivalents per gram of sample.To determine QV, CEC must be corrected for density and porosity by the relationship:QV = CEC * RHOMA * (1.0-PHIT) / PHIT EQ 18WherePHIT = Total Porosity which includes clay associated waterRHOMA = Matrix densityEQ 18 should be applied with caution as the matrix density used in the relationship must include the proportional clay and clean matrix densities. This is best achieved by obtaining matrix density values from pieces of the same core samples on which the CEC measurements are made.Where no CEC is available, Lavers has suggested the following relationship:QV = Aq * PHIT^Bq EQ 19WherePHIT Total Porosity which includes clay associated waterAq Constant: Default = 0.0029Bq Exponent: Default = -3.0590This is the value of QV used by PETROLOG when an external QV is not supplied as an input curve. This relationship was used in the North Sea initially however can be modified for other locations.The term BB quantifies the conductance of the clay exchange cation as a specific temperature and a specific cation concentration. Juhasz (1981) published the expression linking B with Salinity and temperature as:B = (-1.28 + 0.225 * T - 0.0004059 * T^2) / (1.0+RW*0.045* T - 0.27) EQ 20WhereB Is expressed in mho.m-1/meq.cm-3RW Formation water resistivity at formation temperatureT Temperature in DegCB is automatically computed as a Zone Parameter in PETROLOG whenever BHT or RW is changed in a zone. B can also be modified manually by the user in the zone control file. Gravestock (1991)The term F1 = (Waxman-Smits Formation factor)According to Waxman and Thomas F1 is temperature independent.The reciprocal of the slope Co/CW line is F1It is essential to determine the slope on the linear section of Co/CW (see Fig 5) because the term B accounts for the curved portion.The method to determine F1 is as follows1. Saturate a core is a brine of a given salinity and measure Co2. Flush the core and replace with a brine of different salinity and measure Co again3. Repeat above as many time as necessary to trace the Co vs. CW plot and derive F1This method is often referred to as The Waxman-Smits multiple salinity method.This is an expensive and time consuming exercise.The term F1 (Hoyer & Spann method)Hoyer and Spann (1975) in their paper Comments on Obtaining Accurate Electrical Properties of Cores established this simple relationship of F1 vs. F where F is determined from a single measurement of Co vs. CW.F1 = F * (1.0 -RW*B*QV) = F * (1.0 - B*QV/CW) EQ 21WhereF Formation resistivity factor from a single measurement of CW/CoF1 Waxman-Smits Formation resistivity factorThe values of B and QV used in EQ 21 should be determined from the same core at the same temperature that was used to measure F in the laboratory.The terms a1, m1 (Waxman-Smits intercept and cementation exponent)Once a few F1 have been calculated using cores with different porosities, the values of a1 and m1 can be determined using the same conventional method to determine a and n. See figure 1.The porosity values used to determine a1 and m1 are total porosity PHIT which must be obtained by a dehumidified core analysis procedure.The terms n1 (Waxman-Smits saturation exponent)n1 should be measured with high conductivity water.The practice is to saturate a core with 100% brine (SW=1.0) and then remove gradually a percentage of the brine for partial saturations. See "Saturation Exponent" above.Modified Waxman Smits EquationThis equation is based on the Juhasz model and makes do without the BQV and uses the apparent bound water resistivity. Rwb Rwa where VCLAY = 100%Rwb is the bound water resistivity using PHICP from the D-N Z-Plot.Two sets of equations are used:1- When RW > RwbRo = F * Rwb * RW / (Rwb * (1 - VCLAY) + Vclay * RW)SWT = (Ro/RT)^(1/n)2- When RW 1.0) (see Figure 5).This problem is further enhanced by the fact that calcium and magnesium anions frequently occur in high concentration in fresh where RW > 1.0 in both the clay associated water and the free formation water.More studies and data need to be supplied in fresh formation water regarding the derivation of B for the use in the Waxman-Smits equation.QV vs. CEC (Juhasz approach)The Juhasz paper The central role of QV and Formation water salinity in the evaluation of shaly sands provides a table of average CEC and density values for the more common clay minerals and suggest an equation to derive QV from logs and CEC QV = (VCLD * RHOCLD * CEC)/PHIT EQ 22 Where PHIT: Total Porosity which includes clay associated water RHOCLD: Dry clay density taken from tables or lab measurements. CEC: Average CEC for clay minerals known to be in the formation. (For 100% clay of this type) VCLD: Volume of dry clay from logs. The proposed method to determine VCLD is: VCLD = (PHIN - PHID)/HICLD EQ 23 WherePHIN: Neutron porosity corrected for lithology and hydrocarbons PHID: Density porosity HICLD Hydrogen index of the average dry clay mineral mixture in the formation.

TABLE 2Table 2 is a typical table with the hydrogen index for the major clay mineralsTables of saturation equations

TABLE 3

TABLE 4Table 3 an 4 lists a few saturation equations expressed with resistivity input

TABLE 5

TABLE 6Table 5 an 6 lists a few saturation equations expressed with conductivity input References:G.E. Archie 1942 The electrical resistivity log as an aid in determining some reservoir characteristics Trans AIMI 146 54-62 Paul. E. Worthington: The Evolution of Shaly-Sand Concepts in Reservoir Evaluation, The Log Analyst, Jan-Feb 1985Charles R. Berg: Effective-Medium Resistivity Models for Calculating Water Saturation in Shaly Sands, The Log Analyst May-Jun 1996I. Juhasz: The Central Role of Qv and Formation Water Salinity in the evaluation of Shaly Formations., The Log Analyst Jul-Aug 1979D.I. Gravestock Behaviour of Waxman-Smits Parameter B in High Rw, High Temperature Reservoirs, The Log Analyst, Sep-Oct 1991H. Crocker K. Kutan: Log Interpretation in the Malay Basin , SPWLA 25th Annual logging symposium July 8-11 1980E. C. Thomas: Determination of Qv from membrane Potential measurements on Shaly Sands, SPE AIME Sept 1986Fertl W.H. and Hammack: G.W> 1971 A comparative look at water saturation computations in shaly pay sands, Trans SPWLA 12th Annual Logging Symposium R1-17Comparative results of saturation equations:We have constructed a pseudo data set containing a constant PHI, RT (water = 1 and Rclay = 60.0) and a variable GR as shown in Figure 5

Figure 6Figure 6 shows the pseudo log with the GR from 20 to 120 API in track 1. RT and the compute Rwa are in track 3 PHIT = 0.35 is in track 4.

Figure 7Figure 7 is the Rwa vs GR Z=Plot with the Vclay, Vsand PHIE and SW shown on he right hand track.

Figure 8 - 8aFigure 8 shows the SW computed using the Indonesia equation. The 100% is fine at Vclay = 0.0 however this equation add oil due to the presence of clays. There are insufficient corrections due to the presence of high resistivity clays in this example. Figure 8A shows the SW results using the Laminar equation. SW is over corrected in this case.

Figure 98, 9aFigure 9 shows SW computed using the Simandoux equation Figure 9a shows SW computed using the Fertl & Hammack equation

Figure 10Figure 10 shows the Modified Waxman Smit SW equation results Figure 10a shows the Waxman Smits results using a, m, n and F1 instead of a1, m1, n1 and F1. High resistivity clays can appear as oil if the BQV is incorrectly calculated.

Figure 11Figure 11.0 shows the SW computed using the Charlebois equation. It gives the perfect fit since this equation has been designed for logs that displays a typical Rwa vs GR as shown in figures 5 and 6

Figure 12Figure 12 shows the different SW calculated in an example where PHIT = 0.316 and RT = 10 Using a = 1, m = 2 and n = 2 we get in the clean sand: With Rw = 0.01 we get: PHIT = PHIT = 0.316 and F = 1.0 / (0.316^2) = 10.0 SW = SQRT ( F* RW /RT) = 0.10 Ass saturation equations give similar results where Vclay < 30%.