A NEW EFFECTIVE MEDIUM ALGORITHM FOR CALCULATING …



an effective medium algorithm for calculating water saturations at any salinity or frequency

(2003-6 by Charles R. Berg

Short Title: “Effective Medium Saturation Algorithm”

ABSTRACT

A dispersed-shale algorithmic approach combines sand, shale, and hydrocarbon components into the Hanai-Bruggeman equation. The new “incremental” model accommodates sand and shale porosity exponents, saturation exponent, and the associated grain conductivities. The method is applicable to all water salinities and tool frequencies. Its advantage over previous methods is that it works at any water resistivity or tool frequency while allowing saturation and porosity exponents to have values other than 1.5. The algorithm is general and is written to accommodate simultaneous mixing of up to three disperse components and the formation water, but it can be extended to accommodate any number of disperse elements. The hydrocarbon and sand elements have zero conductivity at low frequency, but can be nonzero for calculating saturations at higher frequencies.

The incremental model compares well with the Waxman-Smits model on multiple water-resistivity saturation data from two published experimental data sets. On a log example where the water resistivity is 3.6 ohm-m, it compares favorably to the dual-water/Juhasz model. In the same well, invaded-zone dielectric saturations are nearly identical to those calculated with CRIM.

The algorithm is adaptable to other rock conductivity problems such as vuggy porosity, vuggy water saturation, and clay-coated sand grains. The potential exists for the algorithm to be part of a comprehensive computer program for calculating rock conductivities and saturations using many different combinations of rock fabrics and compositions.

INTRODUCTION

Empirical saturation models

Low frequency

A large number of empirical models have been proposed for calculating water saturation (Sw) in shaly sand. Worthington (1985) summarized the evolution and concepts of many of these models. The models were broken into two major categories, those based on shale volume fractions (Vsh) and those based on the effective concentration of clay-exchange cations (Qv). In general, Vsh models are readily applicable to log analysis, while Qv models need to be calibrated to Vsh using core measurements in local areas. The two major Qv models are Waxman and Smits (1968) and Clavier, et al. (1984). These two models have both been used as the basis for the most popular log-analysis shaly sand model.

The dual-water/Juhasz model is the most widely used dispersed-shale model. It was introduced by Best, et al. (1978) as the “Cyberlook” equation for log analysis. The derivation was loosely based on the experimental dual water model of Clavier, et al., (1984). (The apparently later Clavier, et al., model was at that time widely distributed, but it had not yet been published in a peer-reviewed journal.) Dual-water derivations involve treating shale-bound water as a separate conductive element to the rest of the formation water. Later on, the Cyberlook model was modified into the actual dual-water log-analysis equation used today.

Juhasz (1981) developed a relationship similar to Cyberlook based on the concept of normalized Qv in the Waxman and Smits (1968) model. The current dual-water log-analysis formula is identical to the Juhasz formula when log analysis variables are substituted for their respective, specialized variables. Credit here is given to Juhasz for the equation because he published the final form of the equation first. In addition, his derivation is more detailed than Best’s, et al., (1978), especially in his explicit use of m and n as well as his description of what to use for m. Although the dual-water/Juhasz model is based on two Qv models, it is ultimately a Vsh model.

The laminated sand model of Poupon, et al., (1954) bears mentioning here. When all of the shale is contained in thin beds between clean sand beds, this model is the best to use. The concept of resistors in parallel used in its derivation is also used in many saturation models to “subtract” laminated shale effects before doing dispersed-shale Sw calculations. For theory purists, resistors-in-parallel is not theoretical on the order of Maxwell’s equations. However, it is widely accepted, whether explicit or not, that rocks are Ohmic substances, that is, they obey Ohm’s law. Thus for purposes of discussion in this paper, resistors in parallel will be considered valid theory. (Since the Poupon, et al., model uses the empirical Archie [1942] saturation equation it has to be considered empirical, unless one considers that Archie’s relationship is a result of the theory used here when matrix conductivity is set to zero.)

High frequency

Birchak, et al., (1974) published the basic CRIM model. The particular application was for detecting soil moisture and was published before widespread commercial use of dielectric logs; therefore, no application to log analysis was given. Wharton, et al., (1980) introduced the use of CRIM in the analysis of dielectric logs. In current log-analysis applications, it is customary to add a hydrocarbon conductivity term to the equation from Birchak, et al. Besides being very simple, an advantage of CRIM is that it can be solved directly for Sw. The main problem with CRIM or any other high-frequency method is estimation of complex conductivity or permittivity of water at a given frequency and temperature. In this paper, water complex conductivity was calculated using the method of Dahlberg and Ference (1984).

Theoretical models

The theory that has had the most success in describing the electrical behavior of rocks falls in the general category of differential effective medium (DEM) models. The Hanai-Bruggeman (HB) equation for complex conductivity (Bruggeman, 1935, Hanai 1960a, 1960b, and Bussian, 1983) of a two-component mixture of matrix and water is as follows:

[pic] (1)

where f = porosity, sw = complex water conductivity, s0 = complex mixture conductivity, sr = complex matrix (disperse phase) conductivity, and m = cementation exponent or geometric factor. The frequency dependence of the conductivities can be stated as

[pic] (2)

where s = complex conductivity, [pic] = real (dc) conductivity, i = [pic], w = angular frequency, e0 = permittivity of free space, and K = dielectric constant. In low frequency (resistivity) logs, the imaginary part of conductivity is insignificant. At MHz frequencies, the imaginary part becomes significant, and at GHz frequencies the imaginary part dominates.

In the original derivation of the HB equation, the porosity exponent m is a constant 1.5. This is a natural consequence of the derivation and involves the volume of spherical grains. If the grain shape, orientation, or packing is changed, the value of m changes. By induction, the same is true of the saturation exponent n. Any model for calculating Sw from rocks should take into account both the exponent and grain conductivity belonging to each respective component.

Note that f in equation 1 is a real number while the conductivity variables can be complex. Any computation scheme for calculating Sw at high frequency should take this into account unless the water conductivity is so low that the water dielectric constant is, for all practical purposes, real (in other words, fresh water). In that case, real dielectric constants can be used.

The main purpose behind the HB derivation was to extend the basic theory, sometimes referred to as the average t-matrix approximation (ATA), to higher concentrations, because ATA is only valid for dilute suspensions of particles in water. For the same reason, Sen, et al., (1981) used a different theory, the coherent potential approximation (CPA) to derive the identical equation. Sen, et al., claimed superior theory to Hanai because their “self similar” derivation used the water as host with the grains being added as spherical inclusions. They argued that because Hanai’s derivation (1960a) added inclusions of water to a rock host, it was therefore not self similar and therefore not like real rock. Such a model would be non-percolating at normal porosities. In other words, if the matrix had no conductivity, the rock would be non-conductive at porosities below about 50%. However, Sen, et al., were apparently unaware of the short paper by Hanai (1960b), in which he showed that his equation could indeed be derived through ATA with the rock grains as inclusions in water. In other words, Hanai’s equation was in fact “self similar”. Therefore, Hanai should get the credit for it, especially since he published first. The fact that the HB equation can be derived through different theory would seem to verify the equation itself.

CPA is actually somewhat superior to Hanai’s ATA for a couple of reasons. First, the theory is self consistent while ATA is not. “Self consistent” means that all components are treated equally. ATA is not self consistent because one of the elements has to be designated as host. Second, in CPA additional disperse elements (with different conductivities) can easily be added. Models derived from CPA will be discussed in the “mixing rules” section below.

In the past, HB models have had limitations that prevented their widespread use in log analysis. The main problem with the models has been how to mix the disperse components, rock grains and hydrocarbons, while honoring not only their grain conductivities (sr), but also m and n. Some models have attempted to honor all of the input variables, but are limited in their range of applicability, for instance, low water resistivity (Rw). Other models have a wide range of Rw but limit m and n to a value of 1.5. The computational technique developed here not only honors m, n and sr, but it also can be used in low Rw environments as well as high Rw environments and at all tool frequencies.

previous HB models

Mixing Rules

The main problem encountered when using the HB equation in the study of rock conductivity is that of mixing rules for the disperse components (rock grains and hydrocarbons.) For clarity, the initial discussion will deal with mixing only two disperse elements at a time. There are four basic approaches that have been used in the past to attack the problem.

The most “exact” approach is to derive, from theory behind the HB equation or the equation itself, new relationships for mixing the disperse elements and water simultaneously. In other words, the same type of approach used to derive the HB equation is used in formulating the mixing rule. This type of model will be called a “first principle” and not “exact”, because the HB equation is an approximation no matter what theory has been used to derive it. The new “incremental” model derived here is a first principle approach even though no equation is derived per se, but it is different from other first-principle models in that it uses the HB equation (1) directly instead of its theoretical antecedents.

In general, the most common type of mixing is to calculate the effective conductivity of the water and just one disperse element and then to use the new conductivity as the fluid conductivity for calculating the total conductivity with the remaining element. This type of model will be called “medium-in-medium”. A frequently-used medium-in-medium method is to calculate an effective conductivity of the water and hydrocarbons before calculating the whole rock conductivity. One reason that this approach is so desirable is that the effective fluid conductivity can be calculated by an Archie-type (1942) formula instead of the HB equation itself, because hydrocarbon conductivity is essentially zero at low frequencies. The following the “Archie” fluid relationship is taken from Lima and Sharma (1990):

[pic], (3)

Where sf is the effective fluid conductivity, sw is the water conductivity, and n is the saturation exponent. The resulting fluid conductivity can then be used in conjunction with other mixing schemes to find the total conductivity. This approach for incorporating the hydrocarbons into Sw calculations will be called “hydrocarbons first”. The “hydrocarbons first” method is used with many of the models discussed below.

Archie’s law (1942) is a natural consequence of setting matrix conductivity to zero in the HB equation (1). The “hydrocarbons first” method can be used to derive the Archie combined saturation equation from the HB equation with matrix conductivity set to zero:

[pic], (4)

where st is the total conductivity.

Another way of mixing matrix elements is to formulate a relationship to calculate a composite grain conductivity and then to use that grain conductivity along with a fluid conductivity to calculate the total conductivity. This will be called the “grain conductivity” approach.

The final type of model covered here has been to derive whole-medium approximations that fit a limited range of conditions. This type will be called “whole-medium”.

First Principle

Feng and Sen (1985), using CPA theory, developed high-frequency effective-medium approaches to saturation calculation, one assuming spherical grains and another one that allowed for spheroidal grains. Both models allowed for hydrocarbon conductivity to be nonzero, which is necessary at higher frequencies because hydrocarbon dielectric constants are significant. Neither relationship is easily applied to the shaly sand problem, the spherical one because it does not allow m or n other than 1.5 and the nonspherical one because the geometric variables analogous to m and n are related to the dimensions of the spheroidal shapes and not to the classical petrophysical variables m and n.

Lima and Sharma (1990) developed a first-principle model for mixing shale and sand grains in which the derivation was similar to the CPA-derived spherical saturation model of Feng and Sen (1985). The mixing of matrix conductivities was fully taken into account, but the derivation involved only spherical particles. The nature of the resulting equation makes it difficult to retroactively fit msa and msh into the relationship, and thus it not well suited for the shaly sand problem. However, because the matrix conductivity is taken into account, the equation was used to check the accuracy of the incremental algorithm developed in this paper.

Medium-in-Medium

Spalburg (1988) proposed a dispersed-shale model for determining shaly sand Sw. He used the “hydrocarbons first” approach described above to incorporate the hydrocarbons. Then, he added the shale into the hydrocarbon and water mixture to get a new “fluid” conductivity. This fluid conductivity was then used with the sand grains to calculate the total effective conductivity. Figure 1 illustrates how the matrix components were mixed. Although this model is well suited for the shaly sand problem at high water resistivity (Rw), it did not gain wide acceptance, perhaps because of the difficulty in calculating Sw and perhaps because of the limited distribution of its publication.

Grain Conductivity

Lima and Sharma (1990) investigated models other than the first-principle one discussed above. Primary among these other models was their grain-conductivity model. In this approximation, the composite grain conductivity was calculated by assuming that the shale was in the form of a coating around each sand grain. To calculate Sw, the “hydrocarbons first” method was used. This coated-grain model was later applied to log data in Lima and Dalcin (1995) and Lima, et al., (2005).

Berg (1995) developed a saturation equation that could be solved directly for Sw as a quadratic. The basic assumption was that composite grain conductivity could be calculated by a volumetric weighted average (resistors in parallel.) The equation worked well for calculating Sw from experimental data at all frequencies. Later in Berg (1996) the approach was extended to calculating composite sand/shale grain conductivity in shaly sands. The log analysis method worked well at Rw below about 0.4 ohm-m, but tended to calculate overly low water saturations in fresher formation water, as did the coated-grain model above. However, most of the basic principles involving the adaptation of the model to standard log analysis practices can be adapted to other effective medium models.

In both grain-conductivity approaches, m and n can be fully incorporated into the calculations. The main drawback of the parallel model is that the approximation affects the accuracy of Sw calculations at high Rw. In other words, the assumption that the hydrocarbons and shale will contribute in proportion to their respective conductivities is only true when the water has a much higher conductivity than both matrix and hydrocarbons. These errors are negligible at low Rw, but at high Rw errors become significant. On the other hand, the coated-grain approximation may yet be valid for a true coated-grain configuration at any Rw, but as a dispersed-shale approximation, it does not work well at high Rw. Interestingly, these two grain-conductivity approaches have similar results as discussed in the section on comparison to other models.

Whole-Medium Approximations

Bussian (1983) approximated the HB equation with a binomial expansion. This approximation was then used to develop a semi-empirical saturation model based on the manner in which Sw is calculated in some empirical models. This model worked only at low Rw and mixing rules were not developed for getting a composite sand/shale grain conductivity.

Lima and Sharma (1990) developed low-Rw and high-Rw expansions of their first-principle model described in the first-principle section above. These were based on binomial expansions similar to Bussian’s, and their formulation contained mixing rules for shale and sand. Unfortunately, msa and msh were not explicitly dealt with, and although it is possible to add them retroactively, the correctness of such substitutions is uncertain.

Sherman (1987) developed a high-frequency effective-medium saturation model. Saturation calculation was based on the common assumption that water saturation is the ratio of the calculated porosity to the measured porosity. This method honored porosity and saturation exponents as well as grain conductivity, but did not account for hydrocarbon complex conductivity, which is non-zero at high frequencies.

Recent advances

Since the advent of the dual-water/Juhasz equation, there has been less pressure to develop effective medium models for the shaly sand problem. Although the dual-water/Juhasz model works well in a large number of environments, some problems environments such as high-Rw and low fsh can stand some improvement. Although there have been some effective medium applications papers, there has been little activity in developing new log analysis models from the HB equation.

As mentioned before, the coated-grain model has been applied in Lima and Dalcin (1995) and Lima, et al., (2005). The latter paper used some new adjustments to fit a high-Rw log example. The parallel model of Berg (1995, 1996), was applied to a U.S. Gulf Coast well in Berg and Berg (1996).

Effective medium models have recently been used to study dual porosity systems. Berg (2006) developed vug and fracture models based on the HB equation. Kazatchenko, et al., (2006) modeled carbonate conductivity using effective medium theory to model pores and cracks.

The New Incremental Model

The incremental method breaks down the calculation into steps in which tiny but proportional amounts of the disperse elements are added to the mixture with each iteration (Figure 2). The mixture conductivity from the current iteration will be the fluid conductivity for the next iteration. The details of the computation will be described more fully in the section on model development and in the Appendix. The section that follows will illustrate the correctness of the computational technique. A good mathematical description of the method would be “discrete integration”.

Comparison to other models

Because most low-frequency electric logs are displayed in resistivity, much of the discussion that follows will be in terms of resistivity.

Effective-Medium Models

As discussed before, Lima and Sharma (1990, equation 17) developed a first-principle approach from CPA theory for mixing the sand and shale grain conductivities. In the derivation of the model, the sand and shale are added in infinitesimally small but proportional amounts, thus negating the order problem. Figure 3 shows a resistivity response of the Lima and Sharma model. Since m in this model is a constant 1.5 it is not easily adapted to log analysis, but it is still possible compare the results of this model to the new incremental model of the next figure. Figure 4 shows the resistivity response of the incremental method to the same input variables as in Figure 3. The curves are virtually identical to those of Figure 3 calculated by the Lima and Sharma (1990) model. The maximum relative error between the incremental method and Lima and Sharma equation 17 for the parameters shown on Figure 3 was 4x10-8 using 10,000 iterations. With 100 iterations, the maximum error increased to 4x10-4, which is still much smaller than measurement error (in logs) and fast enough to calculate saturations on hundreds of meters of log in a fraction of a second on a personal computer, even for complex dielectric calculations. Finding st is fairly fast, but finding Sw is somewhat slower because it has to be calculated using techniques that must repeatedly calculate st.

The previous discussion used m and n values of 1.5 in order to test the closeness of the incremental method to the Lima and Sharma model that was derived with m and n of 1.5. In addition, to keep things simple, sand and shale porosity were the same at 20%. To show resistivity response within the range of real log data, the following response curves in use msa of 2.0 and msh of 3.0, while sand porosity (fsa) is 20% and shale porosity (fsh) is 5%. Figure 5 shows the resistivity response of the incremental method to the more realistic variables. At low Rw, shale resistivity is higher than sand resistivity, a common occurrence at low Rw.

Figure 6 illustrates the effect that mixing order can have on calculations. Shale-first (Spalburg, 1988), sand-first, and incremental models are compared at a single Vsh value of 0.2. Clearly, mixing order makes a significant difference in calculated resistivities.

High Frequency

Berg (1995, Figure 7) compared the parallel matrix model to the spherical model of Feng and Sen, 1985. The correspondence was fairly close, and differed mainly at very low water saturations by a few percent. Figure 7 shows the negligible differences between calculations using the incremental model and the Feng and Sen spherical model. This should not be surprising because both are first-principle approaches to the mixing problem. In the other figures from Berg (1995) that compared the parallel matrix model to the models of Feng and Sen, substituting the parallel matrix calculations for incremental calculations yielded figures that were nearly identical. In other words, at high frequency the parallel matrix model is a good approximation for incremental model as well as the Feng and Sen model.

Empirical Models

Matrix Response

Because the following models will not calculate shale resistivity (Rsh), it was calculated using the HB equation with the same shale grain resistivity used above (Rrsh=1). Figure 8 shows the resistivity response of the laminated shale model of Poupon, et al., (1954). Agreement is not good with the incremental model, but that does not matter since it was designed for interlaminated sand and shale. In fact, the model has theoretical justification if that is indeed the nature of the shale distribution. (Remembering, of course, that Archie’s [1942] saturation equation has theoretical justification through effective medium theory.) If the amount of laminated shale versus dispersed shale is known, the effects of the laminated shale can be subtracted before the dispersed-shale calculations.

Figure 9 shows the resistivity response of the dual-water/Juhasz model. The resistivity response is quite different than the incremental model. The fsh of 5% may seem low to some, but it can occur frequently in older and more compacted rocks. The reason that this low value might seem unlikely is the practice of using porosity-calculation techniques that tend to boost shale and shaly sand porosities, such as the popular neutron-density “crossplot”. Although the higher crossplot porosities generally calculate more reasonable saturations with the dual-water/Juhasz model, it would seem to be better to use a model that could use porosities closer to core porosities.

Sw Response

Figure 10 shows the total resistivity (Rt) response of the incremental model to varying Sw. At the 0.15 Vsh used, the near linearity of the curves is striking. At higher Vsh values, the curves become more depressed toward the high-Rw end. Figure 11 shows the saturation resistivity response of the dual-water/Juhasz model. Of the models discussed here, only the Spalburg (1988) dispersed-clay model was close to the incremental model at high Rw. Both grain conductivity approximations, the Lima and Sharma coated grain model (1990) and the parallel matrix model (Berg, 1996) were even more depressed at the high-Rw end.

model development

A Brief Description

The algorithm first divides the volumes for k disperse elements into equal portions for each increment s. It then calculates initial porosities and volumes as well as those summations that remain constant. The HB routine is called [pic] times, each time adding the small volume portions to the mix. After the initial step, where fluid conductivity is actually sw, the total conductivity from the previous step is used as the fluid conductivity. Although the programming appears simple on the surface, it is fairly time-consuming, especially for Sw calculations, because Sw is found by repeatedly calculating total conductivity (st) using different values for Sw until calculated st matches the measured st within a given tolerance. The regula falsi type of techniques used to calculate Sw from resistivity logs usually take about 15 to 20 iterations until they converge on the answer when the calculation tolerance is 10-6. (See the Appendix for an example.) On the other hand, dielectric calculations must use minimization techniques to calculate Sw and can take 50 or 60 iterations to converge. Since dielectric calculations use complex numbers, which are about 4 times slower than real calculations, dielectric computation times can be slower. The regula falsi techniques that have been tried are the “Golden” and “Brent” routines from Press, et al., (1996). Minimizations discussed in this paper used mainly the downhill simplex routine from Press, et al., but occasionally Microsoft® Excel’s Solver add-in was used to check the calculations.

At high frequency, Sw was calculated by minimizing the absolute value of the difference between the measured and calculated complex st. In fact, the complex-number routines, when used to calculate shaly sand (low frequency) Sw yield identical results to the real-number routines. The dielectric minimization technique has been used on several effective-medium models as well as on CRIM with good results. Most of the effective medium models tried, including the Lima-Sharma (1990) coated grain, Berg (1995) parallel matrix, Feng and Sen (1985) spherical, and the Spalburg (1988) dispersed-shale models had nearly identical results. The main exception was Sherman’s (1987), which consistently calculated about 10% lower than the rest. This is most likely because in that model, hydrocarbon complex conductivity is not taken into account.

A Detailed Description

In order to develop the incremental algorithm, the procedure was written such that the only volumetric input was the bulk volume fraction of each disperse element. Porosity in such a system is the remaining volume fraction:

[pic],

Where V1, V2,…Vk are the bulk volume fractions of the disperse elements. Handling volumes as bulk volumes simplifies the calculations because all of the volume fractions can be treated the same and mixing order can be easily modified. The example discussed here will be the full incremental method where sand, shale, and hydrocarbons are added at the same time. The bulk volume fractions for the full incremental method are as follows:

[pic] (5)

[pic] (6)

[pic] (7)

Where V1 is the shale volume, V2 is the sand volume, and V3 is the hydrocarbon volume. These volumes are all relative to the total volume.

Shale volume (Vshg) here is the volumetric ratio of the shale grains to the sand grains. In some shaly sand models, the whole-rock sand versus shale is required. If the sand and shale porosities are the same then there is no distinction between the two volumes. However, there is commonly a large difference in sand and shale porosity. For the calculations discussed here, this distinction does not matter, because the measurement of shale volume from natural radioactivity is necessarily a grain to grain volumetric ratio as opposed to a whole rock ratio, since formation water most likely has little to do with the measurement. This brings up the question whether the gamma-derived Vsh should be converted to a whole rock ratio for those Sw methods that call for a whole-rock ratio. In the remainder of this paper Vsh will be used synonymously Vshg.

The basic calculations are fairly simple. The only parameters that change at each calculation for total conductivity are f and sw/s0. The HB equation is called k times for each step. For example, if the number of steps is set for 100 and the number of elements is 3, the HB equation will be called 300 times. The porosity for each increment i at disperse element j is

[pic], (8)

where ft = total water volume and the disperse volume at each step is

[pic]. (9)

In a computer program it is not necessary to do the full calculation for Vij for each increment—all that is needed is to add each incremental volume to the total each loop. The routine based on the HB equation (1) that calculates each incremental s0 can be represented as:

[pic]. (10)

Because equation 10 is transcendental with respect to s0i,j, traditional representation with that variable isolated on the left is not possible. Equation 11 shows equation 10 in the function-type representation

[pic], (11)

Where [pic]is used in place of sw in the each function call, except for the first call where the actual sw is used. In each successive call to the HB routine, calculated s0 from the previous step is used in place of sw. The Appendix shows the details of how st is calculated in C++ using the incremental algorithm. Any effective-medium technique with the same variable input can be used in the function call in equation 11.

Solving the HB equation for s0 at each increment was done using either the Newton-Raphson or Brent techniques for finding roots. In order to reduce time-consuming calls to the HB routine, when the matrix conductivity was essentially zero an Archie-type of relationship is used to calculate the total conductivity. Also, it was found that when the order of input of the volume fractions was reversed on each successive step, the error with respect to the Lima and Sharma (1990) results was smaller. By using the order reversal it is possible to gain accuracy without increasing the number of steps used.

application

Experimental data

Comparison to Waxman and Smits (1968)

Experimental data from Clavier, et al., (1984) and Hofman, et al., (1998) were used to compare the incremental model to the Waxman and Smits model. These experiments involved measuring Sw at multiple values of sw. Figure 12 shows Clavier, et al., sample 3279B in a nonlinear least-squares comparison of the incremental model to the Waxman and Smits (1968) model. (The Waxman-Smits calculations used the revised value of their variable B published in Waxman and Thomas, 1974). For a fair statistical comparison, Qv was added as a free variable in the Waxman-Smits calculations so that there would be three free variables (m, n, and Qv) to match the free variables used in the incremental calculations (m, n, and Rr). (Normally, Qv is measured experimentally as a shaliness indicator.)

For calculations run on all of the samples on both data sets, the total error squared of the incremental model was 0.00591 (S/m)2, while the total error squared of the Waxman-Smits model was 0.00720 (S/m)2. (The smaller the total error squared, the better the fit.) The difference between these values is not very significant. Calculations were performed using the downhill simplex technique.

Effective-medium comparisons

Since effective-medium calculations involving experimental data do not involve mixing of the matrix conductivities, the calculations differ only the way in which hydrocarbons and water are mixed. On the two data sets above, the hydrocarbons-first approach yielded an error squared of 0.00693 (S/m)2, the parallel-matrix model of Berg (1995) yielded an error squared of 0.00638 (S/m)2, and the Bussian (1983) model yielded error squared of 0.00573 (S/m)2. (Remember from above that the error squared of the incremental model was 0.00591 [S/m]2). As with the empirical data above the differences in the results are not significant.

In doing the statistical comparisons above, it was found that adding the hydrocarbons within the incremental calculations (the “full” incremental technique), came up with n that varied systematically with calculations using the water-first technique. Based on the above calculations, the relationship between saturation exponents is as follows:

[pic], (12)

where ni is n calculated from the incremental model. This empirical relationship had a coefficient of determination of 0.83. In the log example below, the incremental technique was used only for mixing of the matrix elements (sand and shale) so that the same value for n could be applied for both the incremental model and the dual-water/Juhasz model.

Log data

The majority of the relationships used here for calculating effective-medium Sw in logs were published in Berg (1996). The main way that the incremental calculations here are different from the parallel-matrix calculations in that paper is that it is unnecessary to calculate composite sr and m, because the mixing is done within the algorithm. The rest of the relationships are standard log-analysis relationships that can be used with any effective medium model with a couple of exceptions. It has been found in practice that the method for finding msa in Berg (1996) was somewhat unstable because it depended upon a calculated value for fsa. The value of msa is a clean-sand value that should be exactly the same as that used in Archie (1942) calculations and in the Juhasz (1981) normalized Qv method. In other words, use values typical of clean sand for msa. The other exception, to be discussed more later, is to use only the density porosity and not to use techniques that will weight the porosity toward the neutron curve. This is because effective-medium techniques require porosities closer to core porosity, especially in shales.

Another important input value that is needed is Rrsh. This can be found by using Rsh, Rw, fsh, and msh with the HB equation (Berg, 1996, equation 14). The true value of msh can vary, but 2.7 is a good value to use. Error in input msh is largely compensated in the calculation Rrsh.

The input data for the field example (Figure 13) were copied from Lima and Dalcin (1995). Table 1 lists the parameters used in calculation. Sw was calculated by the new incremental method and the dual-water/Juhasz method. Invaded-zone dielectric water saturation (Sxok) was calculated with both the incremental method and CRIM (Wharton, et al., 1980). These calculations compare well to measured core Sw published in Lima and Dalcin (1995) for the interval from 240 m to 298 m. Equations from Schlumberger (1989) were used to calculate complex dielectric permittivity, and the method described in Dahlberg and Ference (1984) to calculate water dielectric constant. The CRIM expression used was slightly different from the Schlumberger and Wharton, et al., equations in that a separate term for hydrocarbon permittivity was added. Salinities used with the incremental method were calculated from invaded-zone resistivity (Rxo) while the CRIM method calculates salinity along with Sxok.

Although the results of the incremental and CRIM methods are very close in this example, they diverge significantly when the formation water has higher salinities. The differences at higher salinities have been investigated and have to do with the calculation of dielectric permittivity from attenuation and propagation time and also with the calculation of water dielectric permittivity. It appears that in the CRIM method, calculation of salinity along with Sw can exaggerate the effects of measurement error. Because this is outside the scope of this paper, it will not be discussed here.

The dual-water/Juhasz Sw calculations seem reasonable at first, because the difference might be explained by moved hydrocarbons. However, with Vsh of about 15% to 20% in even the cleaner sands, the calculated saturations are possibly below irreducible. In addition, Sw calculations disagree by about the same amount in the wet sand at about 290 m, which would rule out moved hydrocarbons as being the cause. A better explanation would involve the differences in resistivity response demonstrated in Figs. 10 and 11.

For comparison purposes, the neutron-density crossplot technique has been used (Dewan, 1983, Asquith, 1990). This method and other similar ones tend to overestimate shale porosities. In this particular log example there was no problem, but in areas with low shale porosity this can cause problems in estimating grain resistivity as well as problems with having calculated porosities match core porosities in shaly reservoir rock. For that reason, it is recommended for all effective-medium calculations that porosities be calculated solely from the density log using a matrix density of 2.65 g/cc. The grain conductivity of 2.65 g/cc was also recommended by Juhasz (1981) for his model, although common practice with dual-water/Juhasz calculations is to use neutron-density “crossplot” porosities because of calculation problems at low shale porosities.

conclusions

An effective-medium mixing algorithm has been developed that works well at all salinities and frequencies. The new method compares well to the dual-water/Juhasz method at low frequency, low Rw, and low Vsh, and the results are nearly identical to CRIM at high frequencies and fresh water.

The potential exists to calculate Sw directly from complex conductivity in MWD logs. This would probably be more accurate than calculating saturations from an empirically adjusted resistivity log from the same data, but it is not known whether the differences would be significant enough to warrant doing so.

In shaly sands where Rw is less than 0.4 ohm-m, the dual-water/Juhasz method gives results fairly close to the incremental method. In cases where shale resistivity cannot be measured, the incremental method can use shale grain resistivity from another well or possibly even from core data. On the other hand, the dual-water/Juhasz method needs a whole-shale resistivity measurement. Of course, it may be possible to use the original Waxman and Smits (1968) or Clavier, et al., (1984) models, but these methods have to be calibrated to cores which are not always available. As demonstrated in the section on experimental data, effective-medium parameters can also be adapted to core data.

Another area where the dual-water/Juhasz method runs into difficulty is where fsh is low. This could be related to the fact that in the dual-water/Juhasz equation when Sw and Vsh are set to 1.0, the resulting Rt is not equal to Rsh unless f = fsh. The neutron/density “crossplot” porosity techniques seem to alleviate this problem somewhat, but at the expense of making calculated porosities higher than core (actual) porosities. The incremental routine needs only total porosity in the actual calculations and thus can be calculated from density porosity alone, which is much closer to core porosity, when using the correct matrix density. When calculating shaly sand porosities for any effective-medium method, it is recommended that only density porosity be used with a grain conductivity of 2.65 g/cc (or lower depending on the type and amount of clay present). It is also recommended that Rt be corrected for laminated shale and that linear Vsh calculations be used when doing so.

Additional work needs to be done in the intermediate frequencies where both the real and imaginary parts of complex conductivities are significant. Among the problems faced is that experimentally m and n approach 1.5 as frequencies approach the GHz range, while at low frequencies they commonly range from 1.3 to 3.0 and even higher. If the change in exponents is strictly a function of frequency, an empirical relationship might be derived to calculate exponents based on frequency. If it turns out that the dielectric constant is effectively 1.5 while the dc conductivity carries the higher values, then a comprehensive solution to the problem might involve either breaking down the calculations into real and imaginary parts to apply separate exponents or prorating the exponents according to the magnitudes of the real and imaginary parts.

With the incremental algorithm it is possible to calculate saturations for rocks with a wide variety of fabrics and matrix conductivity. In the past, problems dealing with shale and rock fabrics have largely been dealt with separately or small groups. For example, it is common to subtract the effects of layered shales and then to calculate Sw on the reservoir rock. This paper focused on the dispersed shale problem, but other configurations can be modeled. For example, coated-grain conductivity from Lima and Sharma (1990) could be calculated and then input as one of the components into the incremental algorithm, since it can be extended by any number of elements. Another example would be adding shale particles (consisting of both grains and water) to represent structural shale in a manner similar to the Spalburg (1988) structural shale model. In an application not directly related to shaly sand, particles of water could be added to represent vugs in a manner similar to the Berg (2006) dual-porosity vug model. Vuggy Sw could then be modeled by using the theory behind the Lima-Sharma coated-grain model to calculate the effective conductivity of a water-coated drop of oil and then incorporate that into the incremental algorithm to model water saturation in vuggy porosity. After the bulk-rock conductivities have been calculated, it would then be possible to incorporate the effect of fractures as in Berg (2006). To summarize, nearly any combination of rock fabrics could be modeled using a single algorithm, eliminating the need for “designer” models limited to specific rock types or fabrics. The shaly sand problem in particular has been prone to problems in specific areas when the normal models will not work. Currently, petrophysicists must choose whatever model happens to work best in a given problem area using one of the many models available. With this type of problem the incremental algorithm might provide a deterministic means of calculating Sw based on rock fabric and composition instead of simply using “what works best”.

The incremental model, and effective medium techniques in general, offer a unified approach to calculating water saturations at any frequency. In the past, separate and totally unrelated methods have, by necessity, been used on high and low frequency data, and intermediate frequency data have been empirically corrected to resemble low frequency data. There have been many complaints by petrophysicists that MWD “resistivity” (converted from MHz-frequency data) logs do not look like their corresponding wireline (low frequency) logs, thus bringing into question saturation calculations. The potential exists to unify calculation of water saturation at different frequencies.

A mathematical problem that needs to be investigated is whether the HB equation can be integrated symbolically to derive a comprehensive equation to replace the incremental method. If this is possible, even for just two disperse elements, it will speed up calculations considerably, especially for dielectric computations. The discrete method developed here may be fully adequate for calculations, but a continuous symbolic representation is desirable, even if only for esthetics.

Nomenclature

e0 Dielectric permittivity of free space = 8.854x10-12 F/m

f Frequency (cycles per second)

fa Volume fractions for the coherent potential approximation (CPA)

f Porosity

fsa Sand porosity

fij Porosity for increment i at disperse element j

fsh Shale porosity

ft Total water volume for incremental calculations. When mixing hydrocarbons simultaneously with matrix this is not f but is Swf.

i,j Subscripts for increments and disperse elements, respectively

k Number of disperse elements for the incremental model

K Relative dielectric permittivity (dielectric constant)

m Porosity (or cementation) exponent

msa Sand porosity exponent

msh Shale porosity exponent

n Saturation exponent

ni Saturation exponent for the full incremental model where hydrocarbons are

mixed in addition to sand and shale in the incremental algorithm

Qv Effective concentration of clay-exchange cations (equiv/l)

Rsh Shale resistivity (ohm-m). This is the composite resistivity of both water

and grains.

Rrsh Shale grain resistivity (ohm-m). This is the resistivity attributed to the

grains, including that caused by hydration water, but not “far” water.

R0 Whole-rock wet resistivity (ohm-m)

Rt Whole-rock resistivity including hydrocarbons (ohm-m)

Rw Water resistivity (ohm-m)

s Number of steps (increments) for the incremental model

Sw Water saturation

Sxok Invaded-zone water saturation

s0 Whole-rock wet conductivity (S/m)

sf Fluid conductivity (S/m)

sr Matrix conductivity (S/m)

sw Water conductivity (S/m)

st Whole-rock conductivity including hydrocarbons (S/m)

[pic] The real part of complex conductivity (S/m)

w Angular frequency (2pf)

Vsh Shale volume, volume of shale, including water, to the whole rock

Vshg Shale volume, grains of shale in relation to grains of the whole rock

V1, V2, V3, Vj Incremental bulk-volume fractions with order subscripts

Vij Disperse volume sum at increment i and disperse element j

acknowledgements

I would like to thank Andy May, Steven Krehbiel, Robert Berg and Anthony Gangi for their insights and suggestions. In addition I would like to thank Andy May for translating the code from C++ to Fortran and for testing his Fortran code on the example well and another well.

appendix—C++ Source Code

The program listing here was written in Borland ® C++ Builder Version 6. The code has been tested also in Borland ® C++ BuilderX, Borland ® C++ 5.2, and Microsoft ® Visual C++ 6.0. Source code in other programming languages is available from the author. These languages include Fortran 77 and Pascal (Borland ® Delphi version).

In the program, s is replaced by C. Otherwise, the variables closely approximate the variables in the text. The procedure Ctincremental2part is a water-first algorithm for calculating total conductivity given Cw, Crsh, Vshg, Sw, f, msh, msa, and n. It first calculates the fluid conductivity and passes the result to C0incrementalNpart along with the values for the remaining parameters. C0incrementalNpart is a general routine for calculating total conductivity using the incremental method. It can take up 3 disperse components and can easily be extended to more. This routine calls C0HB for each incremental addition of disperse material. C0HB is calculated using the zbrent algorithm from Press, et al. (1996). The zbrent routine is a regula falsi type of algorithm that searches for roots by of functions by trying different values of the dependent variable. In the case of C0HB, zbrent tries different values of C0 until the difference between calculated f and input f is within a given tolerance. A Newton-Raphson method for finding roots was more efficient, but it is not shown here to keep the code brief.

SwIncremental2part is a procedure for calculating Sw given Ct, Cw, Crsh, Vshg, f, msh, msa, and n. The function zbrent is used again, this time trying different values of Sw until calculated Ct, matches measured Ct,.

#include

#include "increm.h"

#include "zbrent2.h"

// The algorithm zbrent is a regula-falsi-type algorithm from Press, et al.

//(1996). It is used here both to find total conductivity (C0) in the HB

//(Hanai-Bruggeman) equation and also to find Sw for the incremental method.

// There were two modifications to the original routine. First, all float

//types were changed to double. Second, the variable ITMAX was made an input

//parameter instead of a defined constant. A Newton-Raphson algorithm for

//finding HB C0 was more efficient than using zbrent, but zbrent is used

//here for simplicity and brevity.

// The following variables are used in the calcRoot routine below. They must

//be defined outside of calcRoot because the zbrent routine (Press, et al.,

//1996) which uses calcRoot takes a function with only one parameter.

double Cw3, Cr2, phi3, m2;

int count2;

double calcRoot(double C0){

count2++;

return pow(Cw3 / C0, (m2 - 1) / m2) * (C0 - Cr2) / (Cw3 - Cr2) - phi3;

}

void C0HB(double Cw, double Cr, double phi, double m, double tolerance,

int maxIterations, double &C0, int &count){

count = 0;

if (fabs(Cw) < tolerance)

C0 = tolerance;

else if (fabs(phi - 1) < tolerance)

C0 = Cw; //all water

else{

if (fabs(Cr) < tolerance)

C0 = Cw * pow(phi, m); //Archie's law

else{

//setting the variables to be used by calcRoot

Cw3 = Cw;

Cr2 = Cr;

phi3 = phi;

m2 = m;

count2 = 0;

//calculating C0

C0 = zbrent(calcRoot, Cw, Cr, tolerance, maxIterations);

count = count2; //getting the count from calcRoot

}

}

}

//Arrays here are one element larger than needed in order to match indices in

//the text that start at 1.

typedef double Array3[4];

void C0incrementalNpart(double Cw, Array3 C, Array3 Vol, Array3 m, int s, int k,

double tolerance, int maxIterations, double &C0, int &maxCount,

int &totalCount){

//This routine takes up to 3 disperse elements, but could easily be modified to

//accept more.

double Vij, phiT, phiij;

Array3 V;

int i, j, count;

//initialize

count = 0;

maxCount = 0;

totalCount = 0;

Vij = 0;

phiT = 1;

for (int p = 1; p tolerance or SwCount>ITMAX, another regula falsi-type method

//could be tried, such as Ridders' or the false position methods from

//Press, et al.

Sw = zbrent(calcSw_RootCt, 2.0, 0.0, tolerance, ITMAX);

//getting output variables set by calcSw_RootCt

minCt = minCt2;

SwCount = SwCount2;

maxCount = maxCount2;

totalCount = totalCount2;

}

References

Archie, G. E., 1942, The electrical resistivity log as an aid in determining some reservoir characteristics: Petroleum Technology, 1, 55-62.

Asquith, G. B., 1990, Log evaluation of shaly sandstone reservoirs: a practical guide: American Association of Petroleum Geologists continuing education course note series 31, 59 p.

Birchak, J.R., C.G. Gardner, J.E. Hipp, J.M. Victor, 1974, High dielectric constant microwave probes for sensing soil moisture, Proceedings of the IEEE, 64, no. 1, p. 93-98.

Berg, C.R., 1995, A simple effective-medium model for water saturation in porous rocks, Geophysics, 60, no. 4, p. 1070-1080.

Berg, C.R., 1996, Effective-medium resistivity models for calculating water saturation in shaly sands, The Log Analyst, 37, no. 3, p. 16-28.

Berg, C.R., 2006, Dual Porosity Equations from Effective Medium Theory, to be presented at SPE Annual Technical Conference and Exhibition, SPE-101698-PP.

Berg, C.R., and R.R. Berg, 1996, Water saturations in a Wilcox shaly reservoir sandstone, Fordoche Field, Point Coupee Parish Louisiana, GCAGS Transactions, 46, p. 41-45.

Bruggeman, D. A., G., 1935, Berechnung verschiedener physikalischer konstanten von heterogenen Substantzen: Annalen der Physik, 24, 636-664.

Bussian, A. E., 1983, Electrical conductance in a porous medium: Geophysics, 48, 1258-1268.

Best, D.L., J.S. Gardner, and J.L. Dumanoir, 1978, A computer-processed wellsite log computation, SPWLA 19th Annual Logging Symposium, paper Z.

Clavier, C., G. Coates, and J. Dumanoir, 1984, Theoretical and experimental bases for the dual-water model for interpretation of shaly sands: Society of Petroleum Engineers Journal, 24, 153-168.

Dahlberg, K.E., and M. Ference, 1984, A quantitative test of the electromagnetic propagation (EPT) log for residual oil determination, SPWLA 25th Annual Logging Symposium Transactions, paper DDD.

de Lima, O.A.L., and C.L.R. Dalcin, 1995, Application of a new shaly sand model for interpreting resistivity and dielectric log measurements, The Log Analyst, 36, March-April, p. 29-41.

de Lima, O.A.L., and M.M. Sharma, 1990, A grain conductivity approach to shaly sandstones: Geophysics, 55, 1347-1356.

de Lima, O.A.L., M.B. Clenell, G.G. Nery, and S. Niwas, 2005, A volumetric approach for the resistivity response of freshwater shaly sandstones, Geophysics, 70, no. 1, F1-F10.

Feng, S., and P.N. Sen, 1985, Geometrical model of conductive and dielectric properties of partially saturated rocks: Journal of Applied Physics, 58, 3236-3243.

Dewan, J.T., 1983, Essentials of modern open-hole log interpretation, PennWell Publishing Co., Tulsa, Oklahoma, 361p.

Hanai, T., 1960a, Theory of the dielectric dispersion due to the interfacial polarization and its application to emulsions: Kolloid-Zeitschrift, 171, 23-31.

_____1960b, a remark on "Theory of the dielectric dispersion due to the interfacial polarization and its application to emulsions:" Kolloid-Zeitschrift, 175, 61-62.

Hofman, J.P., A. de Kuijper, R.K.J. Sandor, M. Hausenblas, R.J.M. Bonnie, and T.W. Fens, 1998, The Group III Shaly Sand Data Set, SPE Reservoir Evaluation and Engineering, 1, no. 6, SPE 39107.

Juhasz, I., 1981, Normalized Qv – the key to shaly sand evaluation using the Waxman-Smits equation in the absence of core data, SPWLA Transactions, Paper Z.

Kazatchenko, E., M. Markov, A. Mousatov, and E. Pervago, 2006, Simulation of electrical resistivity of dual-porosity carbonate formations saturated with fluid mixtures, Petrophysics, 47, no. 1, p.23-36.

Poupon, A., M.E. Loy, and M.P. Tixier, 1954, A contribution to electrical log interpretation in shaly sands: AIME Transactions, 201, 138-145.

Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, 1996, Numerical recipes in C[;] the art of scientific computing, Cambridge University Press.

Sen, P. N., Scala, C., and M. H. Cohen, 1981, A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads: Geophysics, 46, 781-795.

Schlumberger, 1989, Log interpretation principles/applications, Schlumberger Educational Services.

Sherman, M.M., 1987, A model for the determination of water saturation from dielectric permittivity measurements: The Log Analyst, 28, 282-288.

Spalburg, M., 1988, The effective medium theory used to derive conductivity equations for clean and shaly hydrocarbon-bearing reservoirs, Eleventh European Formation Evaluation Symposium, Paper O.

Waxman, M.H., and L.J.M. Smits, 1968, Electrical conductivities in oil-bearing shaly sand: Society of Petroleum Engineers Journal, 8, 107-122.

Waxman, M.H., and E.C. Thomas, 1974, Electrical conductivities in shaly sands: I. The relation between hydrocarbon saturation and resistivity index; II. The temperature coefficient of electrical conductivity: J. Petr. Tech., Trans. AIME, 257, 213-225.

Wharton, R.P., G.A. Hazen, R.N. Rau, and D.L. Best, 1980, Electromagnetic propagation logging: advances in Technique and interpretation, SPE 9267.

Worthington, P.F., 1985, The evolution of shaly-sand concepts in reservoir evaluation, The Log Analyst, 26, January-February.

tables

Table 1. Calculation parameters for the well in Figure 13.

[pic]

Figure captions

Figure 1. Schematic diagram showing the mixing of three disperse matrix elements using the HB equation (1). This was the concept behind Spalburg’s (1988) dispersed-clay model (his equation 10). There are two calculation steps. The first step adds the shale to the water to get a “fluid” conductivity for the next step. The second step combines the new fluid conductivity with the sand conductivity. In this step, because the sand has zero conductivity, Archie’s law (1942, equation 3) can be used to calculate the total effective conductivity. In each case, the “porosity” used at each step, f1and f2, is the ratio of the fluid volume divided by the respective bulk-volume fractions, V1 and V2.

Figure 2. A schematic diagram showing the steps used in the first increment in the new incremental modal. The sequence is very similar to Figure 1 except that the shale and sand bulk volumes have been divided by the number of increments (s) and the two steps are repeated s times. When s = 1, the results are identical to the shale-first results illustrated in Figure 1. With each repeat, the fluid conductivity is the final fluid conductivity from the previous sequence. The new porosity for each step is the ratio of the total volume for the step to the incremental rock volume. In this manner, the total volume at the end of the last increment is equal to 1.

Figure 3. A typical resistivity response of Lima and Sharma (1990), their equation 17. This model was a first-principle type of approach to mixing that took into account the sand and shale grain conductivities but kept m constant at 1.5. In the derivation, the sand and shale grains were added to the mixture in infinitesimally small but proportional amounts. Following are the input variables: shale grain resistivity (Rrsh) =1, shale and sand porosity exponent (msh and msa) = 1.5, Sw = 1, shale and sand porosity (fsh and fsa) = 0.2.

Figure 4. Resistivity response calculated using the new incremental method whereby the shale and sand particles are added in small increments in volumes proportional to Vsh. The results are nearly identical to the Lima and Sharma (1990) model of Figure 3, with results approaching the latter as the increments get smaller. These particular calculations used 100 increments, but at 10 increments the plots were still visually indistinguishable from each other. The calculation parameters are the same as in Figure 3. In addition, msa and msh were set to 1.5 to match the non-adjustable exponents in the Lima and Sharma model.

Figure 5. The resistivity response of the incremental method using input variables in which sand and shale f and m differ. Variables were: msa =2.0 and msh = 3.0, fsa = 20%, fsh = 5%, srsh = 1.0 ohm-m. Notice that at low values of Rw, shalier rocks paradoxically have higher resistivity than cleaner rocks. The effect is due partly to the lower porosity of the shale and partly due to the higher value of msh than msa. This counterintuitive relationship can found in many low-resistivity, low-contrast rocks.

Figure 6. Comparison of shale-first (Spalburg, 1988), sand-first, and incremental models at Vsh = 0.2. The other parameters are the same as in Figure 5. The order in which sand and shale are added makes significant differences at both high and low Rw.

Figure 7. The difference between calculations using the Feng and Sen (1985) spherical model and the incremental model. As in Berg (1995, Figure 7), the parameters are sw = 0.05 S/m, f = 0.22, water dielectric constant (Kw) = 80, matrix dielectric constant (Kr) = 4.7, hydrocarbon dielectric constant (Kh) = 2.2, m = 1.5, n = 1.5, and frequency (f) = 2.5 MHz. The number of steps used in the incremental model was set to 100. The differences are negligible except at Sw values far below irreducible Sw.

Figure 8. Resistivity response of the laminated shale model of Poupon, et al., (1954). The HB equation has been used to calculate input shale resistivity using the corresponding shale variables. At high Rw these response curves are similar to those of the parallel matrix and coated grain models. This empirical model has good theoretical basis if the interbedded sands are clean, since Archie’s relationships (the “empirical” part of the laminated shale model) can be derived from the HB equation. To be accurate, however, the shale distribution must fit the model, which in this case is interlaminated clean sand and shale.

Figure 9. Resistivity response of the dual-water/Juhasz model (Juhasz, 1981; Best, et al, 1978). Shale resistivity (at Vsh=1) has been calculated using the HB equation. Although the response here seems very different from the incremental model, saturations calculated on logs seem to be fairly good, especially when fsa and fsh are roughly equal. In actual examples similar to this one, the dual-water/Juhasz model response can be improved by using neutron-density crossplot porosities. However, doing this will make the calculated porosities higher than core porosities.

Figure 10. Response of total resistivity (Rt) with respect to Sw with the incremental method (hydrocarbons-first version). Input parameters are the same as in Figure 5 except that Vsh has been held constant at 0.15. Additionally, n has been set at 2. (The Sw= 0 curve starts just above 1000 ohm-m on the Rt axis.)

Figure 11. Rt response of the dual-water/Juhasz model. Input parameters are the same as for Figure 10. As in Figure 9 shale resistivity is calculated using the HB equation (1). Although the curves are fairly depressed at high Rw compared to Figure 10, the configurations of saturation curves for both the parallel matrix (Berg, 1996) and coated grain (Lima and Sharma, 1990) models were even more depressed. Of all the models discussed here, the Spalburg (1988) dispersed-clay model was closest to the incremental model at high Rw.

Figure 12. A comparison of least-squares fits of the incremental model and the Waxman and Smits (1968) model on sample 3279B published in Clavier, et al., (1984). In this data set, Sw was measured at four different values of sw. The free variables in the incremental model were m, n, and sr while the free variables in the Waxman-Smits calculations were m, n, and Qv. In the Waxman-Smits calculations, the Waxman and Thomas (1974) revised value for the variable B was used. The calculations were performed using the downhill simplex method (Press, et al., 1996) and Microsoft® Excel Solver add-in was used as a double-check.

Figure 13. Log from the Alto do Rodrigues Field copied from Lima and Dalcin (1995). “EMT Sw” calculations are from the incremental method and “DW Sw” calculations are from the dual-water/Juhasz model. “EMT Sxok” calculations are by the incremental method and “CRIM Sxok” calculations are from the CRIM method of Wharton, et al. (1980), except that water complex conductivity was calculated using the method of Dahlberg and Ference (1984). Sxok from the incremental and CRIM methods are so close that it is difficult to see the dashed line of the CRIM calculations. Wells with higher water salinity exhibit larger differences between CRIM and dielectric effective medium calculations. See Table 1 for calculation parameters.

Figures

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Figure 1.

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Figure 2.

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Figure 3.

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Figure 4.

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Figure 5.

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Figure 6.

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Figure 7.

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Figure 8.

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Figure 9.

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Figure 10.

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Figure 11.

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Figure 12.

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Figure 13.

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