waxman smits

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Understanding the Waxman-Smits Equations (for version 2010.1 subject to change by Techsia) Errors in the Techlog Help File have now been corrected. The Waxman-Smits equations for the interpretation of electric logs for water saturation in oil-bearing shaly sands was first proposed in 1966 and verified in 1972. Until these equations were proposed the industry was saddled with a myriad of equations based roughly on some measure of Vshale. This seminal work demonstrated to the world that the correct parameter that controlled electrical response in shaly sands was Qv, the cation exchange capacity of the clay minerals surface per unit pore volume. The scientific community of the oil patch quickly recognized the usefulness of the approach and it quickly became an industry standard. However, the laboratory science was ahead of the logging industry and to this date there is no logging tool that can be used to determine Qv in situ. Core to log correlations are the best available, while in lieu of core measurements indirect calculation of Qv from log shale indicators is the normal practice. The most widely accepted empirical methods to use Qv (besides Waxman- Smits) are those proposed by Juhasz (which applies the Waxman-Smits approach utilizing logs only) and secondly, the Dual Water method. An earlier, but overlooked method that applies the Waxman-Smits approach utilizing logs only was published by Thomas and Haley. 1 But to apply this concept, we should understand its genesis. A MODEL TO EXPLAIN CONDUCTIVITY OF SHALY SANDS

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Page 1: Waxman Smits

Understanding the Waxman-Smits Equations(for version 2010.1 subject to change by Techsia)

Errors in the Techlog Help File have now been corrected.

The Waxman-Smits equations for the interpretation of electric logs for water saturation in oil-bearing shaly sands was first proposed in 1966 and verified in 1972. Until these equations were proposed the industry was saddled with a myriad of equations based roughly on some measure of Vshale. This seminal work demonstrated to the world that the correct parameter that controlled electrical response in shaly sands was Qv, the cation exchange capacity of the clay minerals surface per unit pore volume. The scientific community of the oil patch quickly recognized the usefulness of the approach and it quickly became an industry standard.

However, the laboratory science was ahead of the logging industry and to this date there is no logging tool that can be used to determine Qv in situ. Core to log correlations are the best available, while in lieu of core measurements indirect calculation of Qv from log shale indicators is the normal practice. The most widely accepted empirical methods to use Qv (besides Waxman-Smits) are those proposed by Juhasz (which applies the Waxman-Smits approach utilizing logs only) and secondly, the Dual Water method. An earlier, but overlooked method that applies the Waxman-Smits approach utilizing logs only was published by Thomas and Haley.1

But to apply this concept, we should understand its genesis.

A MODEL TO EXPLAIN CONDUCTIVITY OF SHALY SANDS

When Waxman and Smits presented their shaly sand conductivity model at the Annual Fall SPE Meeting in 1966, it was but one in a long string of models that had been proposed to explain the curious resistivity behavior of shaly sands first brought to our attention by Gus Archie in the late 1930’s. The names of the earlier model authors were 1)Patnode & Wyllie, 2)de Witte, 3)Winsauer & McCardell, 4)Poupon, Loy and Tixier, 5)Hill & Millburn, and 6)Simandoux. All of these workers recognized the problem and attempted a solution in a slightly different way and each built on previous work and on advancements in the industry and science in general. All were trying to provide help to the engineer interpreting resistivity logs and thus all fell into the same trap. They were used to seeing resistivity logs and thus expressed all the data in terms of resistivity that resulted in some awkward log-log or exponential-log equations, particularly since they had to be solved on slide rules. Many of the authors had pity on the engineers and presented graphical solutions. It took two physical chemists to look at the problem a different way and present the resistivity data as conductivity. (Historical note: Induction logs actually measure conductivity, but it is converted into resistivity to be presented so that it could easily

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be compared to the older electric logs and guard logs whose data were recorded as resistivity.) When the resistivity behavior of rocks is looked at in terms of conductivities, the simple relationship is obvious. To demonstrate this let me show you the relationship presented by Waxman and Smits.

Monroe Waxman working at the Shell Development research labs in Houston and Lambert Smits working at the Royal Dutch/Shell research labs in Rijswijk, The Netherlands painstakingly made conductivity measurements on hundreds of small core samples in the laboratory. The Smits samples are referred to as Group I samples, while the Waxman set of cores are referred to as Group II samples. Each core was carefully cleaned and characterized as to porosity, permeability, grain density and cation exchange capacity. The Group I samples used membrane potential to determine cation exchange capacity, while Group II used a wet chemistry method to determine cation exchange capacity. Each core was vacuum saturated with brine and its conductivity monitored until stable. This process was repeated with different salinity brines ranging from very dilute to near saturated. All measurements were carried out at 77 F. Waxman and Smits were building upon the earlier work of Hill and Milburn that had conclusively demonstrated that resistivity and SP behavior of shaly sands could be explained by accounting for the effect of cation exchange capacity. Unfortunately, Hill and Milburn never quantified the exchange capacity parameter independently, and just fit the resistivity data with a term called “b” and fit “b” to core data, all empirically; the resulting equations were cumbersome, e.g.,

Eq. (1)and

Eq. (2)

When the conductivity of the 100% brine-filled core, C0 , is plotted vs. the conductivity of the brine in the pores of the core, Cw , most of the data falls on a linear trend. This trend is shown below.

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Figure 1. Shaly Sand Core Conductivity Behavior, after Waxman & Smits

When the data are viewed in this format, the relationships are evident. Let’s discuss some important points. Firstly, the dotted line, labeled “Clean Sand” shows the behavior that one would expect for rock without any extra conductivity paths. It is the classical case that obeys Archie’s relationship. All the conductivity of the rock is due to the conductivity of the hydrated ions in the brine, thus when Cw goes to zero, C0 also goes to zero. Interestingly, if we had plotted these data as resistivities, then it would take a log-log plot to make the data appear linear. Thus for the dotted line, we can write Archie’s equations:

Eq. (3)

where 1/F is the slope of the dotted line, and

Eq. (4)where is the fractional porosity and “m” is a fitting parameter.

Secondly, the solid line, labeled “Shaly Sand” shows higher core conductivity at any given value of brine conductivity than the “Clean Sand.” Thus there is some mechanism causing increased conductivity other than hydrated ions in the brine.

Thirdly, the “Shaly Sand” line has a linear portion at high values of brine conductivity, the saltier part, but exhibits non-linear behavior in the low values of brine conductivity, the fresher part. Also, even when the brine conductivity goes to zero, the core has measurable conductivity. Interestingly, when we dry out the core,

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it does not have any conductivity. Thus the mechanism that explains the observed increased core conductivity requires water molecules.

Fourthly, the non-linear part of the curve, indicates that the “increased conductivity” depends upon the brine salinity in a non-linear fashion, and as the salinity of the brine increases, the “increased conductivity” reaches a maximum amount then becomes independent of the brine salinity.

Thus any shaly sand conductivity model must explain all of these observations to be credible. Monroe and Lambert also required that all terms in their equations be measurable in the laboratory and that they would limit their relationship to one additional adjustable parameter beyond what Archie had used. Waxman and Smits proposed the following explanation:

Eq. (5)

where Ce is the conductivity resulting from the clay exchange ions that are producing the increased conductivity, and 1/F* is the slope of the linear portion of the C0 vs. Cw plot. Monroe chose to designate the Archie “F” with an asterisk to signify that the observed slope has a contribution of clay exchange ions; the slopes of the two lines, constructed one measurement at a time, are not parallel, i.e., F F*, except at the unrealistic limit of infinite brine conductivity. (NOTE: Monroe chose the “*” since it was the only unused symbol left on his typewriter. To honor Monroe, I always use the asterisk in typewriter position rather than make it a superscript.) The fact that 1/F* is applied equally to the brine ions and the clay mineral counterions explicitly assumes that the current path (cell constant) is the same for the two conductivity components.

It is obvious from Figure 1 above that Ce is complex in its behavior, starting out at a low value and exponentially growing to a maximum value. The maximum value can be computed from Eq. (5), i.e., when C0 = 0 , then Ce = -Cw , and we can see this graphical construction in Figure 1; we extrapolate the slope of the linear part of the line back to the X-axis, which is -Cw , and is therefore , Ce(Max). Since each core has a different amount and type of clay minerals, each core has a different value of Ce(Max). Now the only problem remaining is how to determine Ce in the dilute region when Ce < Ce(Max).

From basic electrochemical behavior, we can decompose Ce , a conductivity, into two parts, an equivalent conductance times a concentration, or

, Eq. (6)where

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B = ionic equivalent conductance of the exchange cation, (Mhos/Meter)/(Eq/L),

Qv = concentration of exchange cations per unit pore volume, Eq/L .

Since we can determine the value of Qv experimentally for each core plug in the sample set, we have one adjustable parameter, B that we can fit to the data set. Thus the first parameter that can be computed is Bmax [since we know both Qv and Ce(Max)]. For the Group I samples, Bmax = 4.6, while for the Group II samples Bmax = 3.8. For many reasons explained in the original paper, the Group II samples are believed to be more accurate.

When the data are presented thusly, F* is temperature independent, because the temperature dependence is carried by Rw for the brine and B for the exchange ions.

When all the dilute region data are regressed, an empirical relationship for the exponential rise of B is developed:

Eq. (7)at 77 F, for the Na+ .

Monroe Waxman continued to work on gathering laboratory data on water-bearing shaly sands and extended his measurements to 200 C (392 F ) to satisfy the needs of engineers solving field problems at elevated temperature. These elevated temperature results were published in a joint article with E. C. Thomas in 1974 and the resulting B vs. Rw vs. Temp for Na+ is shown below as Figure 2. (NOTE: Monroe refit all his experimental data after he discovered an inconsistency and Figure 2 below is the reworked data set, published September 2008.)

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Figure 2. Waxman-Thomas B vs. Rw vs. Temperature for the Na+

Notice the large range of B as a function of temperature – it is almost one to one, i.e., Bmax increases 7-fold during an 8-fold Celsius temperature increase. Also notice how rapidly Bmax falls off in the dilute region, where Na+ concentration is less than

0.5 Normal ( that’s roughly 28 kppm Na+Cl- ). Also notice that Monroe showed the data in terms of Rw not Cw as recognition that the field engineers would have to convert Cw to Rw for their use and he wanted to save them time. From time to time, various individuals have developed a math model for the data in Figure 2 for use in digital computing. Arthur Purpich fit the data and his equations are used in Techlog as the default value.

Lastly note that the term B now contains all the statistical uncertainty of the directly measured parameters: Cw , Co , T , and Qv . Qv also has statistical uncertainty of its sub-components: grain density, , and cation exchange capacity. Monroe never published the regression errors, but they are substantial. Do not be lulled into believing that we know the value of B better than 20% at temperatures above room temperature.

Monroe always felt he met his objectives in producing an equation with the minimum number of parameters and that the parameters were verifiable in the lab. Once the form of the equation was proven to work, then the linear portion of the curves ( high salinity part ) could be refit with two independent parameters, i.e., since Cw and C0 are

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measured, we can solve for two unknowns; the slope of the line is 1/F* and the negative X-axis intercept is BQv, so we can determine B and Qv independently of any laboratory wet chemistry method for CEC. Comparison of Qv by the two methods showed excellent agreement in the Group II samples; this was not unexpected since they were carefully chosen to be homogeneous. The agreement between methods for the Group I samples was poorer due to the heterogeneous nature of many of the very shaly samples.

Monroe then extended the above relationship in the same form as did Archie, and set

Eq. (8)

Since C0 and Cw do not have the same temperature dependence, when F* is presented as Eq. (8) it is temperature dependent, and therefore so is m*. When one uses laboratory core measurements to determine the value of m*, then optimally they should be done at reservoir temperature using the same brine concentration and ionic make-up as the reservoir.

Of course, since we are really interested in quantifying and producing oil reserves, these water-bearing equations are not sufficient. They must be extended to include the case of hydrocarbon-bearing reservoirs. Monroe reluctantly agreed that an additional parameter would be needed to account for the effect of the oil saturation, but he reasoned that he could just manipulate the existing set of parameters to do the trick and not have to solve for another parameter other than oil saturation (water saturation, actually). Monroe’s reasoning is as follows: The hydrated clay counterions are in rapid exchange with the hydrated brine ions, thus the effective charge distribution is spread out over the entire pore network, so Qv is defined as CEC per unit pore volume. When oil in introduced into the pore network, it displaces the pore brine (water and hydrated ions), but not the hydrated clay counterions; we must maintain charge neutrality. Thus as oil volume increases, pore brine volume decreases and there are fewer hydrated brine ions left which can participate in exchange reactions with the hydrated clay counterions. Thus the effective concentration of hydrated clay counterions goes up because the pore water volume goes down, i.e., since Qv = CEC/PV, when PV decreases, Qv goes up. The question then becomes, what is the functional form of the change in volume. Monroe in classical form opted to try the simplest form, inversely proportional to water saturation (as for relative permeability), and he proposed the following equation:

Eq. (9)

where Qv’ is the new effective value of Qv when hydrocarbons are present, and Sw is

the fraction of pore volume value of water saturation.

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Monroe then used the same formulation as Archie, so let’s review Archie’s equations for “Clean Rocks”:

Eq. (10)

where F is defined by the relationship in Eq. (10) and R is resistivity, C is conductivity, the subscript 0 is for zero oil in the core or 100 % water saturation, and w is for water (brine).

When we now allow for the case that there may be some hydrocarbon in the pore space, then the expression becomes:

Eq. (11)

where the subscript “t” follows the field convention for “true” resistivity, and since Rt and Ct can vary with hydrocarbon saturation, “G” is not a constant as is “F”.

Archie then defined the Resistivity Index:

Eq. (12)

If we substitute Eq. (10) and Eq. (11) into Eq. (12), we get

Eq. (13)

Archie then related the Resistivity Index, I, to water saturation:

Eq. (14)

Combining Eq. (13) and Eq. (14), we get the following:

Eq. (15)

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Monroe then reasoned by analogy for a shaly sand, the following:

Eq. (16)where the asterisk is added to each term to signify that there is some contribution from hydrated clay counterions in the measurement.

Then Monroe rewrote Eq. (11) to account for hydrated clay counterion conductivity:

Eq. (17)

Substituting Eq. (16) into Eq. (17) yields

Eq. (18)

If we now convert conductivity into resistivity and solve for Sw, we get

Eq. (19)

Then if we substitute Eq. (8) into Eq. (19) and simplify, we get:

Eq. (20)

This is the equation that field engineers need to be able to solve. There are a lot of parameters that one needs to know: n* , m* , B , Qv , Rw, , and Rt. Standard Archie analysis requires five of the seven listed parameters, and only B and Qv have been added. The value of B is available if one knows temperature and brine resistivity (Figure 2), thus only Qv must be deduced by other means. A Qv log does not presently exist, thus numerous local correlations have been developed between clay sensitive parameters and Qv ; none are universally applicable. It is very much a reservoir-by-reservoir quest to be able to correlate Qv to some measurable log parameter. This need for knowing Qv led to self-consistent methods for determining BQv from logs alone, i.e., Thomas-Haley method, later re-invented by Juhasz as the “Normalized Qv” method.

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How do I know when it is important enough to go to all the trouble to make a Waxman-Smits correction? One simple screening method is based upon Eq. (20). The comparable Archie “Clean Rock” equation is

Eq. (21)

Thus when one compares term for term Eq. (20) and Eq. (21) one sees that they are the same except for a multiplier in the denominator:

and when the term,

then the value of Rt has been boosted by more than 10% and Sw is reduced by about 1%, assuming n* is near 2, or Sw is reduced by about 2% if n* is near 1.7. I do not bother with a Waxman-Smits correction unless Rw B Qv/Sw > 0.1 .

Please notice the clay conduction correction effect is a function of saturation, so the correction effect is larger the higher one goes up in the hydrocarbon column.

The fact that Eq. (20) is transcendental and one cannot solve for Sw in closed form should not be a deterrent to anyone with the ability to program a simple iterative loop; these programs usually converge in 4 or 5 iterative steps taking less than a second on even the smallest computer. Excel spreadsheet is quite adequate.

References

1. Thomas, E.C. and Haley, R.A. (1977) “Log Derived Shale Distribution in Sandstone and Its Effect Upon Porosity, Water Saturation and Permeability," Trans., Canadian Well Logging Society (1977) Paper N.

2. Waxman, M. H. and Smits, L. J. M.:"Electrical Conductivities in Oil-Bearing Shaly Sands," Soc. Pet. Eng. J. (June 1968) 107-22; Trans., AIME, 243.

3. Waxman, M. H. and Thomas, E. C.:"Electrical Conductivities in Shaly Sands - I. The Relation Between Hydrocarbon Saturation and Resistivity Index; II. The Temperature Coefficient of Electrical Conductivity," J. Pet. Tech. (Feb. 1974) 213-23; Trans., AIME, 257.

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4. Waxman, M. H. and Thomas, E. C.:"Electrical Conductivities in Shaly Sands - I. The Relation Between Hydrocarbon Saturation and Resistivity Index; II. The Temperature Coefficient of Electrical Conductivity," SPEJ. (Sept. 2007).

5. Jin, G., Torres-Verdin, C., Devarajan, S, Toumelin, E., and Thomas, E.C. (2007) “Pore-Scale Analysis of the Waxman-Smits Shaly-Sand Conductivity Model,” Petrophysics, 48; 2, P 104-120

Submitted by Dr. E. C. Thomas March 2010