# Migration rate calculation

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The rate of migration for oil or gas can be estimated using Darcy's law, the principal formula for calculating permeability. Darcy's law generally holds for rocks with tube-shaped pore systems; however, it is only an approximation for flow in rocks with high percentages of clays, like shales, due to the platey grain shape of the clays. The Kozeny–Carman correction estimates the permeability of rocks with high percentages of clays.

## Procedure

The procedure for calculating the migration rate of oil or gas is outlined in the table below.

1. Gather data, including permeability of carrier beds, viscosity of oil, fluid density, and pore pressure gradient.
2. Calculate the buoyancy pressure.
3. Calculate the rate of hydrocarbon migration.

## Calculating migration rate

Use the version of Darcy's law presented below to calculate the rate of migration for oil or gas:

${\mbox{R}}={-}({\mbox{k}}\times {\mbox{A/m}}\times [({\mbox{P}}_{{{\rm {grad}}}}+{\mbox{P}}_{{{\rm {c}}}})-\rho _{{{\rm {hc}}}}\times {\mbox{g}}])$

where:

• R = rate of migration (m3/sec)
• k = permeability to oil or gas at a given saturation (m2)
• A = cross-sectional area (m2)
• m = dynamic viscosity (Pa-sec) (use 0.01 Pa-sec for oil and 0.0001 for gas at 20°C293.15 K
68 °F
527.67 °R
; 0.001 Pa-sec for oil and 0.00001 for gas at 150°C423.15 K
302 °F
761.67 °R
• Pgrad = pore pressure gradient (Pa) (use 4.5 psi/ft if not available)
• Pc = capillary pressure gradient
• ρhc = hydrocarbon density (kg/m3)
• g = acceleration of gravity (~9.81 m/sec2)

## Correcting for clay-rich rocks

For rocks with high percentages of clay, use the Kozeny–Carman correction (k) as shown in the table below to obtain a closer approximation of permeability.

Porosity Use
> 10% ${\text{k}}={\frac {(0.2\times \phi ^{3})}{{\text{s}}^{2}}}\times {(1-\phi )^{2}}$
< 10% ${\text{k}}={\frac {(20\times \phi ^{3})}{{\text{s}}^{2}}}\times {(1-\phi )^{2}}$

where:

## Buoyancy pressure

Buoyancy pressure for a particular hydrocarbon must be calculated for its migration rate. Use the formula below to calculate buoyancy pressure:

${\mbox{P}}_{{{\rm {B}}}}={\mbox{g}}\times {\mbox{z}}\times (\rho _{{{\rm {w}}}}-\rho _{{{\rm {hc}}}})$

where:

• PB = buoyancy pressure
• z = height of hydrocarbon stringer
• ρw = water density
• ρhc = hydrocarbon density

## Minimum buoyancy pressure for migration

Migration upslope under a seal occurs when buoyancy is greater than capillary pressure, or

${\mbox{g}}\times {\mbox{l}}\times \sin {\mbox{Q}}\times (\rho _{{{\rm {w}}}}-\rho _{{{\rm {hc}}}})>2\gamma$

where:

• l = length of oil stringer
• Q = angle with the horizontal
• γ = interfacial tension (oil–water), dynes/cm

Each dip reversal in or near a flat hydrocarbon migration path will trap hydrocarbons and make continued hydrocarbon flow updip less likely.