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GATE-2019 (79)

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79) The Bulk resistivity of a carbonate formation having 10% porosity which is 75% saturated with hydrocarbons is 500m. The bulk resistivity of the formation when the porosity is doubled and 100% saturated with water is ___________m (round off to 1 decimal place).(Assume the tortuosity, cementation factor and saturation exponent to be 1, 2, and 2 respectively). (Thanks to Neeraja AU)
Solution

Water saturation Sw from Archies formula is
S_w=\sqrt{\frac{R_o}{R_t}}---(1)

Ro - Resistivity of the fully saturated rock

Sw – Water saturation of the uninvaded zone

Rw – Resistivity of the formation water

RtTrue Resistivity of the formation

Formation Factor

F=\frac{R_o}{R_w}

FR_w=R_o----(2)

Substitute eq.(2) in eq.(1)

S_w=\sqrt{\frac{FR_w}{R_t}}---(3)

Formation factor, F=\frac{a}{\phi ^m}---(4)

a = tortuosity factor = 1

ϕ – porosity

eq.(4) substitute in eq.(3)

S_w=\sqrt{\frac{a}{\phi^m}\frac{R_w}{R_t}}----(5)

Given that

Rt =500m

Porosity = 10% = 0.1

Hydrocarbon Saturation, Sh = 75% = 0.75

Sw+Sh=1
Sw = 1- Sh = 1 – 0.75 = 0.25

S_w=\sqrt{\frac{a}{\phi^2}\frac{R_w}{R_t}}

0.25=\sqrt{\frac{1}{(0.1)^2}\times\frac{R_w}{500}}

(0.25)^2={\frac{1}{(0.1)^2}\times\frac{R_w}{500}}

\frac{R_w}{500}.\frac{1}{(0.1)^2}=(0.25)^2

R_w=(0.25)^2\times500\times(0.1)^2

Rw = 0.3125 Ωm

Therefore,
Rt = ?

when ϕ = 20% = 0.2 (porosity doubled)

Sw = 100% = 1

S_w=\sqrt{\frac{a}{\phi^2}\frac{R_w}{R_t}}

1=\sqrt{\frac{1}{(0.2)^2}.\frac{0.3125}{R_t}}

\frac{0.3125}{R_t}.\frac{1}{(0.2)^2}=1

R_t(0.2)^2=0.3125

R_t= \frac{0.3125}{(0.2)^2}

Rt= 7.81 m


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