Waxman-Smits –The Rolls-Royce of Shaly Sand Models

Alan JohnsonIntegrated Petrophysical Solutions Ltd

London Petrophysical Society Seminar – Petrophysics 101

7th March 2019

Why the title? Rolls-Royce versus assembly line. Rolls hand built by craftsmen; takes longer, costs more, higher quality..As you heard, I started my career in Shell. This was in the Hague and my first boss, although I didn’t appreciate it at the time, was Lambert Smits, the second half of the famous duo, the other being Munro Waxman who was based with Shell Oil in Houston

3/22/2019 9:13 AM

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Outline

• Introduction – Archie

• Introduce Shale Effects on electrical core measurements

• Development of the Waxman-Smits Model

• Practical Application of Waxman-Smits

• Clay Bound Water

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Core Resistivity Measurements

Water Resistivity

Rw ohmm

Sample Resistivity

Ro ohmm

Both Archie and Waxman Smits models based on detailed electrical measurements on core. The Dual Water model is based, in part, on the Waxman-Smits data but adopts a slightly different approach.

The history of Waxman Smits publication involves a few interesting twists and turns; I’ve included some references in the slides for your future reference.

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Core Sample Resistivity – Clean Sands (Archie)

If we saturate a set of core sanples from the same formation with water of different salinities (high salinity, low resistivity), and plot these versus porosity on log/log, the variation of the sample resistivity (Ro) with porosity follows a different line for each water salinity.

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Core Sample Resistivity – Formation Resistivity Factor

FRF or F

𝐹 =𝑅𝑜

𝑅𝑤

m

𝐿𝑜𝑔 𝐹 = −𝑚 × log(∅)

𝐹 =1

∅𝑚

𝑅𝑜

𝑅𝑤=

1

∅𝑚

Archie: The Electrical Resistivity Log as an Aid in Determining Some Reservoir Characteristics – Trans. AIME, 1942

Archie’s insight was to divide each Ro by the corresponding Rw, defining a new parameter FRF, connected to porosity by m.

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Varying Water Salinity – Clean SandsSingle Plug - Resistivity

𝐹 =𝑅𝑜

𝑅𝑤F

Another way to look at this, for a single sample, is to measure Ro for different values of Rw (different salinities) the (linear) plot now gives a line with slope F. Note when Rw = zero, Ro = 0.

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Varying Water Salinity – Clean SandsSingle Plug - CONDUCTIVITY

𝐹 =𝑅𝑜

𝑅𝑤=

𝐶𝑤

𝐶𝑜1

𝐹

𝐶𝑤 =1

𝑅𝑤& 𝐶𝑜 =

1

𝑅𝑜

𝐶𝑜

𝐶𝑤=

1

𝐹

𝐶𝑜 =𝐶𝑤

𝐹

𝐶𝑤

𝐹

Waxman & Smits’ insight was to convert this same plot to conductivity. C = 1/R. Same linear relationship with slope now 1/f. This is the basis of the so-called Co-Cwanalaysis. On this plot, Co = Cw/F for each Cw.

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Varying Water Salinity – SHALY SANDSingle Plug - Conductivity

𝐶𝑒𝑥𝑡𝑟𝑎

𝐶𝑤

𝐹

𝐶𝑜 =𝐶𝑤

𝐹𝐶𝑜 =

𝐶𝑤

𝐹+ 𝐶𝑒𝑥𝑡𝑟𝑎

With the addition of shale, sample conductivity goes up compared to a clean sample, this added conductivity being described in many models as Cextra.

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Varying Water Salinity – SHALY SANDSingle Plug - Conductivity

𝐶𝑜 =𝐶𝑤

𝐹𝐶𝑜 =

𝐶𝑤

𝐹+ 𝐶𝑒𝑥𝑡𝑟𝑎

Waxman-Smits

𝐶𝑜 =𝐶𝑤

𝐹∗+

𝐵𝑄𝑣

𝐹∗1

𝐹∗

𝐶𝑤

𝐹∗

𝐵𝑄𝑣

𝐹∗

With the addition of shale, sample conductivity goes up compared to a clean sample, this added conductivity being described in many models as Cextra. Waxman & Smits defined theis additional conductivity as B times Qv divided by F*, where F* is the equivalent clean sand formation factor.

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Waxman Smits - Qv

Cation Exchange Capacity is the concentration of cations in a brine required to balance the negative charges on the surface of a clay usually given in meq/g.

Qv is the Cation Exchange Capacity of a rock, expressed in moles of exchangeable cations (milli-equivalents) per millilitre of pore volume (meq./mlpv).

Qv meq/mlpv

<0.1 Clean

0.1-0.2 Slightly Shaly

0.2-0.3 Moderately Shaly

0.3-0.5 Shaly

>0.5 Very Shaly

Can be measured by titration of a crushed sample using a salt solution with a suitable cation (Barium or Ammonium).

Cation-exchange capacity (CEC) is a measure of how many cations can be retained on soil particle surfaces.[1] Negative charges on the surfaces of soil particles bind positively-charged atoms or molecules (cations), but allow these to exchange with other positively charged particles in the surrounding soil water.[2]

Labs use Ammonium Acetate

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Waxman Smits - B

While Qv defines the number of the Cations in the pore space, B defines how much each Cation contributes to conductivity. • B is in units on mho/m/eq./litre and varies with water salinity and temperature.• It is independent of clay type, distribution and amount.

Ref. Waxman & Thomas: Technical Note: An addendum to Electrical Conductivities in Shaly Sands I. The Relation Between Hydrocarbon Saturation and Resistivity Index; II. The Temperature Coefficient of Electrical Conductivity - SPE Journal September2007

One relationship published in the original Waxman-Smits paper in 1978. An updated version was published by Waxman & Thomas in 1974 with a updated, “corrected” version published by Waxman & Thomas in 2007.

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Waxman Smits – B: Mathematical Description (Dacy)

Ref. Dacy & Martin: Practical Advances in Core-Based Water Saturation Analysis of Shaly Tight Gas Sands- PETROPHYSICS Vol 49, No1 February 2008

𝐵 = 1 − 0.83𝑒𝑥𝑝− −2.47+0.229 𝐿𝑛 𝑇

2+1311

𝑇2

2

𝑅𝑤× −9.2431 + 2.6146 × 𝑇0.5 mmho/meq/litre

T in °C; Rw ohmm

Dacy & Martin had originally developed a relationship based on Waxman & Thomas’ 1974 graph; he was about to publish when W&T published their 2007 update and he had to update before publishing.

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Varying Water Salinity – SHALY SANDSingle Plug - Conductivity

Waxman-Smits

𝐶𝑜 =𝐶𝑤

𝐹∗+

𝐵𝑄𝑣

𝐹∗

𝐶𝑤

𝐹∗

𝐵𝑄𝑣

𝐹∗

1

𝐹∗

1

𝐹∗

Decreasing Cw

Back to the Co Cw graph. The shale data are curved at low Cw (high Rw) due to the drop in B at high Rw. At high Cw (Low Rw) B is constant and a straight line can be fitted through the data points.

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Varying Water Salinity – SHALY SANDSingle Plug - Conductivity

Waxman-Smits

𝐶𝑜 =𝐶𝑤

𝐹∗+

𝐵𝑄𝑣

𝐹∗

𝑩𝑸𝒗

𝑭∗-𝑩𝑸𝒗

1

𝐹∗

BxQv = 2.7 mho/m

B = 3.8 mho/m/meq/ml

Qv = 0.7 meq/ml pv

The red line also has a slope of 1/F*. Its intercept with Y axis is Bmax*Qv/F*. The intercept with the X axis is Bmax*Qv.

Co Cw analysis should use (at least) three brines of high enough salinity so that B is constant.Ensure all points lie on a straight line – regression artefacts.

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Varying Water Saturation – SHALY SANDConductivity

When Hydrocarbon is added, Waxman Smits assumes that the number of shale-contributed cations in the formation water stays the same, they just become more concentrated in the smaller water volume.

The excess conductivity term, 𝐵×𝑄𝑣

𝐹∗becomes

𝐵×𝑄𝑣

𝑺𝒘×𝐹∗

N* is the equivalent clean sand saturation exponent,

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Varying Water Saturation – SHALY SANDSingle Plug 18% porosity - Resistivity

Archie Sw = 40%

Waxman-Smits SW = 20%

In resistivity terms , the impact of a shale correction on calculated Sws is clearly seen. At a resistivity of 70 ohm the clean sand model would indicate and Sw of 40%, while a shaly sand analysis would give 20% Sw.At higher water salinity, lower Rw, the Archie conductivity increases, reducing Rt, but the shale conductivity term stays the same so it’s proportional effect is reduced and the difference is Sw is less, possibly within the range of uncertainty.

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Waxman-Smits Water Saturation

𝑆𝑤𝑛∗ =𝑅𝑤

𝑅𝑡 × 𝜑𝑚∗ ×1

1 +𝑅𝑤 × 𝐵 × 𝑄𝑣

𝑆𝑤

Waman & Smits: Electrical Conductivities of Shaly Sands – Trans. AIME, 1968

This, at last is the Waxman Smits Sw equation.There are three parameters to be defined: m*, n* and Qv. Hence more hands on work – Rolls Royce-like.Also Sw appears on both sides of the relationship, requiring an iterative solution; but it does converge.

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Waxman-Smits Analysism* Determination #1

1

𝐹∗

As we saw earlier F* can be derived from the Co Cw experiment, used to derive Qv on a core sample. Determining these on a range of core samples and plotting the values against porosity will provide a line of slope m*, the shale-free cementation exponent.

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Waxman-Smits Analysism* Determination #2

Measure Conventional F ( = Ro/Rw). • Range of samples, different porosities• Water salinity close to formation value

Determine Qv for each sample

Calculate:

𝐹∗ = F × 1 + 𝑅𝑤 × 𝐵 × 𝑄𝑣

A separate solution is to measure formation factor (F) at an appropriate water salinity for each sample, and apply the correction relationship using Qv. (derived from earlier slides)

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Waxman-Smits Analysisn* Determination

Measure Conventional Resistivity Index (RI)• Single samples, different saturations• Water Salinity close to formation value

Determine Qv for each sample

𝑅𝐼∗ = RI ×1 +

𝑅𝑤 × 𝐵 × 𝑄𝑣𝑆𝑤

1 + 𝑅𝑤 × 𝐵 × 𝑄𝑣

𝑅𝐼 =𝑅𝑡

𝑅𝑜=

1

𝑆𝑤𝑛

Calculate:

Similarly you can derive a range of resistvity indices versus Sw for each sample, and again apply a correction relationship to get RI* and hence n*

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Waxman-Smits AnalysisQv Determination

You can determine Qv on a range of core plugs by Co Cw or core trims by titration (cheaper). A standard technique is to plot these values against plug porosity (lower porosity equals higher shaliness). In this example giving a rather poor correlation. In this field a much better relationship was obtained by correlating to log GR (normalised).

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Waxman-Smits AnalysisClay-Bound Water

Hill, Shirley & Klein: Bound Water in Shaly Sands – Its Relation to Qv and other Formation Properties (1961) Edited by Waxman & Thomas – The Log Analyst, May-June 1979

Juhasz: The Central Role of Qv and Formation –Water Salinity in the evaluation of shaly formations – The Log Analyst, July-August 1979

∅𝐶𝐵𝑊 = ∅𝑇0.6425

𝑆𝑎𝑙0.5+ 0.22 ×

𝑄𝑣

𝜌𝐶𝐵𝑊

φCBW = Fraction of Clay Bound WaterφT = Total PorositySal = Salinity in g/l NaCl eq.ρCBW = Clay Bound Water Density = 1 g/cc(?)

Although Waxman & Smits did not assume a volume of clay bound water (anions being assumed to disperse throughout the pore water volume), their work partly built on earlier work by Hill, Kline and Shirly who recognise CBW. Their work was only published in 1979, possibly at the instigation of Mr Juhasz, who needed to be able to reference their work in his subsequent paper. H, K & S derived a relationship between Qv, formation water salinity and clay-bound water volume.

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Waxman-Smits AnalysisEffective Porosity

∅𝐸 = ∅𝑇 1 −0.6425

𝑆𝑎𝑙0.5+ 0.22 × 𝑄𝑣

SandClay

Solids

Clay Bound Water

Pore Space

Effective Porosity (φE)

Total Porosity (φT)

This relationship can be used to derive one version of effective porosity, where PhiE is defined as PhiT minus PhiCBW.

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Waxman-Smits AnalysisBrine Permeability

∅𝐸∅𝑇

= 1 −0.6425

𝑆𝑎𝑙0.5+ 0.22 × 𝑄𝑣

Juhasz: The Central Role of Qv and Formation –Water Salinity in the evaluation of shaly formations – The Log Analyst, July-August 1979

Swanson: Improved Estimates of Permeability from Capillary Pressure Measurements - SPE 54th Annual Technical Conference, Las Vegas, September 1979

𝐾𝑏𝑟𝑖𝑛𝑒 = 𝐾𝑎𝑖𝑟 ×∅𝐸∅𝑇

3

K Air > 750 mD

K Air ≤ 750 mD

𝐾𝑏𝑟𝑖𝑛𝑒 = 0.292 × 𝐾𝑎𝑖𝑟1.186 ×

∅𝐸∅𝑇

3

In his paper Juhasz also published a relationship between conventional air and brine permeability, depending on Qv and salinity. Not that even if there is no shale effect Kbrine is less than Qv for permeabilities less than 750 mD, which may reflect the standard Klinkenberg correction to air permeabilities.

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Waxman-Smits AnalysisMercury-Air Capillary Pressure

∅𝐸∅𝑇

= 1 −0.6425

𝑆𝑎𝑙0.5+ 0.22 × 𝑄𝑣

𝑃𝑐𝐶𝑜𝑟𝑟 = 𝑃𝑐𝑅𝑎𝑤 ×∅𝐸∅𝑇

−0.5

𝑆𝑤𝐶𝑜𝑟𝑟 = 𝑆𝑤𝑅𝑎𝑤 ×∅𝐸∅𝑇

In mercury-air cap curves, the wetting phase is represented by air and so no account is taken of clay bound water effects. In reality brine water saturations will be higher in the presence of clay bound water, due to the combined effects of reduced mobile water and constriction of the effective pore throats lined with clay bound water. The net results is seen as an increase in corrected capillary pressure and a reduction in corrected Sw. Juhasz reports the following relationships.

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Waxman-Smits AnalysisMercury-Air vs Dean-Stark Saturations

Mercury Air Data selected from the

equivalent height in the reservoir

(approximate) corresponding to the Dean

Stark data

Mercury Air Data corrected for Clay-Bound

Water

This is a real example to illustrate to CBW correction. We have dean stark water saturations measured on a range of core samples in a hydrocarbon bearing reservoir, plotted against the permeability of those plus, together with saturations predicted from raw mercury air cap curves at appropriate heights in that same reservoir. After applying the corrections to the mercury air data the two data sets overlay. This is one reason why I believe in the applicability of Waxman-Smits.

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Conclusions

• Waxman-Smits extends the clean-sand Archie model to include the effects of clay

• Like Archie, it is based on extensive laboratory core measurements

• Its effective application relies on core analysis data from the reservoir concerned

• The concept of Qv-based clay bound water can be extended to the analysis of porosity, permeability and capillary water saturations

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Questions?

alan@ips-petrophysics.co.ukwww.ips-petrophysics.co.uk

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