Archie, Hingle, and Pickett:

Denver Colorado USA

Archie, Hingle, and Pickett:Determining fluid saturation for fun and profit…

Dan Krygowski

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Where Dan hopes to go todayAnswer the question, “So, why do we care about fluid saturation, anyway?”A review of Archie’s equation.A look at some graphical solutions to Archie’s equation (Hingle and Pickett plots) which quickly provide saturation answers, and also predict some Archie parameters that we’d otherwise have to estimate.A few closing remarks (and a run for the door).

Hawking…

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Formation fluid saturationFormation fluid saturation is an important quantity:

It is necessary to determine the volume of fluids (water and hydrocarbons) present;It may give us some indication as to what will be produced.

ButWe often have to estimate several parameters that are

needed to make the saturation calculation.

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A reminder of history1927

Conrad and Marcel Schlumberger run the first log in Pechelbron field in France. “Electric coring” is born.

1942Gus Archie presents a paper at the AIME meeting in Dallas quantifying the relationship between fluid saturation and formation resistivity.

So it took 15 years to go from a qualitative response to a quantitative, but empirical, relationship.

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Archie’s equationDefining the terms:

nm

t

ww R

RaS1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∗∗

=φ

water saturation

tortuosity factorformation water resistivity

saturation exponent

cementation exponent

porosityformation resistivity

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Archie’s equationDefining the terms:

nm

t

ww R

RaS1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∗∗

=φ

water saturation

tortuosity factorformation water resistivity

saturation exponent

cementation exponent

porosityformation resistivity

Traditional sources:

electric logs,induction logs, laterologs

sonic, density, neutron, nuclear magnetic resonance,

resistivity logs

samples, SP logscore

core

core

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Another history lesson: pCNo, not preCambrian, preComputer:When big, burly guys (usually engineers) stood

over you while you looked at the log data, and said, “Hey, do I run pipe or call for cement? You’re costing me money here…”

So, we needed techniques to get quantitative answers quickly.

If they helped in other ways, so much the better…

A “pattern recognition” approach.

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So it’s not just equations……two examples for illustration:

1. Last Chance High Country #1A constructed well to illustrate the techniques.

2. Bill Barrett Last Dance 43C-3-792Mamm Creek, Piceance Basin, Colorado(a small section of the Mesaverde)

A well to bring us back to reality.

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Last Chance High Country #1

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Barrett Last Dance 43C-3-792Mamm Creek, Piceance Basin, Colorado

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Enter Tom HingleA graphical solution to Archie’s equation so that plotting resistivity against porosity will produce arrangements of the data (which are recognizable as patterns):

φ∗⎟⎟⎠

⎞⎜⎜⎝

⎛

∗=⎟⎟

⎠

⎞⎜⎜⎝

⎛ m

w

nwm

t RaS

R

111

Y-axis X-axisA family of lines from which information can be discerned.

+b (b=0)m*xy =

[1959, SEG 29th

Annual Meeting]

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Hingle plotThe y-axis is built from

but scaled in resistivity or

conductivity so the points can be plotted directly on the graph. The y-axis becomes (very) non-linear.

The x-axis is scaled so that porosity increases from left to right. Porosity, bulk density, or sonic traveltime can be plotted.

Water-bearing line

porosity

(data)

The water-bearing line is placed by the location of the data points.

Zero porosity (= RHOmatrixor DTmatrix)

Lines of decreasing water saturation

m

tR

11⎟⎟⎠

⎞⎜⎜⎝

⎛

The y-axis on this plot is constructed using m=2.0

Conductivity Resistivity, Rt

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So, with the Hingle plot…One can determine water saturation directly from the plot, without

knowing Rw, orhaving to calculate porosity.

In addition, the plot will predict matrix values of sonic or density, so porosity can be more confidently calculated.But, one has to assume values for a, m, n,And special paper must be used, which depends on the assumed value of m.

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Looking at our ideal data

1.0

2.0

1.5

2.5

3

5

10

1520

501005002000

2.2 2.02.42.8 2.6

Sw = 1.0

0.50

0.25

Hingle Plot; m = 2.0

Baker Atlas, 1985Bulk Density

Rt

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Looking at real data

Bulk Density

Vsh

ale

1

0

Sw = 1

0.25

02.8 2.2

(1/R

t)^(1

/m)

Hingle plot

2.65

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Enter Dick PickettA graphical solution to Archie’s equation so that plotting resistivity against porosity (both on logarithmic scales) will produce linear arrangements of the data:

( ) ( ) ( )wwt RaSnRm

⋅+−−= logloglog1)log(φ (graph paper format)

m*x

Y-axis X-axis A family of lines from which information can be discerned.

+ by =

Y-intercept

[1966, SPE (JPT)1973, SPWLA]

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Pickett plotBoth scales are logarithmic.To save calculations DT-DTma or RHOma-RHOB could be used on the y-axis instead of porosity.

Resistivity

Poro

sity

No “special” graph paper is needed.

(data)

The water-bearing line is placed by the location of the data points.

Lines of decreasing water saturation

Water-bearing line.Slope = -1/cementation exponent

The intercept of the water-bearing line at Phi = 1 is a*Rw.

The plot can be done in Excel.

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So, with the Pickett plot…One can determine water saturation directly from the plot, without

knowing Rw, orknowing m.

In addition, the plot will predict Rw and m.But, one has to assume matrix values if sonic or density is used for porosity.And the plot can be done on readily available graph paper, or more easily in MS Excel.

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Looking at our ideal data

Pickett Plot

0.01

0.1

1

0.01 0.1 1 10 100 1000

Rt

PhiD

Sw = 1 0.5 0.25

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Looking at our real data

Vsh

ale

1

0

Sw = 1

1

0.1

0.010.1 1 10 100

Den

sity

Por

osity

0.75 0.5 0.25

Pickett plot

Deep Resistivity

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So, what happens now?Well, we have two graphical methods to quickly

determine water saturation, and other parameters:

Hingle: Porosity matrix valuesPickett: Rw, and cementation exponent, m

Q: Can we use them in concert?A: Well, sort of…

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Pickett and Hingle in concertBassiouni (1974, SPE) shows

a method to use both to resolve several parameters,

BUT it is painful to do by hand.

AND, while interactive Pickett plots are common in many software packages, few packages have Hingle plots, and only one has linked interactive Pickett and Hingle plots.

SO it’s possible, but tedious.

Iterate until convergence

m

ρma

m m

ρma

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In conclusionPickett and Hingle plots are two slightly different graphical solutions of Archie’s saturation equation.While fluid saturation is needed for the calculation of volumetrics, it is less useful in predicting production.

And, are other quantities, like Bulk Volume Water, more helpful there?

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In conclusionPickett and Hingle plots also predict some other Archie or porosity parameters.Each can give a quick and useful view of the data, even when you have a computer to do the grunt calculation work for you.In the information age, pattern recognition is alive and well, and it may provide some insights to the subsurface that numbers won’t.