Darcy’s Law

While the material balance can identify initial hydrocarbon in place, estimate production at various pressure, and provide identify the primary production mechanism, it is unable to answer the question of when the hydrocarbon will be produced. Darcy’s law complements the material balance by calculating flow rates. The combination of Darcy’s law and the material balance results in a model capable of predicting flow rates over time.

Henry Darcy formulated Darcy’s Law in 1856 as a result of his experimental studies on the flow of water through unconsolidated sand beds. Over time, it has been expanded to include the movements of other media and even two or more immiscible fluids.

There are four major influences on fluid flow. These are: number of phases present, the compressibility of the fluid, the geometry of the flow system, and finally the time characteristics of the flow system.

Let’s take a look at Darcy’s Law and discuss the various terms. u = the apparent velocity, bbls/day-ft2

k = permeability, millidarcies (md)

m = fluid viscosity, cp

p = pressure, psia

s = distance along flow path in ft

y = fluid specific gravity (always relative to water)

a = the angle measured counterclockwise from the downward vertical to the positive s direction

We’ll start with the driving force, the term in brackets. The term dp/ds represents the driving force caused by a fluid pressure gradient. The wellbore pressure will be lower than the reservoir pressure causing a pressure gradient to form along the flow path. This term is simply the difference in those pressures divided by the distance.

The second driving force is the hydraulic or gravitational gradient. This term includes a gravitational constat, the specific gravity of the fluid and the angle of the flow path relative to the force of gravity.

This driving force is adjusted by a constant, permeability and viscosity to calculate the apparent velocity of the fluid. This ‘apparent velocity’ is equal to the product of the flow rate and the formation volume factor divided by the total area of the rock perpendicular to the flow path. (v=q*B/A)

Terry, Ronald E., J. Brandon. Rogers, and B. C. Craft. Applied Petroleum Reservoir Engineering. Third ed. Massachusetts: Prentice Hall, 2014. Print.