Page 161 - Fundamentals of Reservoir Engineering

CHAPTER 4

DARCY'S LAW AND APPLICATIONS

4.1 INTRODUCTION

Darcy's empirical flow law was the first extension of the principles of classical fluid
dynamics to the flow of fluids through porous media. This chapter contains a simple
description of the law based on experimental evidence. For a more detailed theoretical
treatment of the subject, the reader is referred to the classical paper by King Hubbert 1
in which it is shown that Darcy's law can be derived from the Navier-Stokes equation of
motion of a viscous fluid.

The significance of Darcy's law is that it introduces flow rates into reservoir engineering
and, since the total surface oil production rate from a reservoir is

dN
q res = p
dt

it implicitly introduces a time scale in oil recovery calculations. The practical application
of this aspect of Darcy's law is demonstrated in the latter parts of the chapter in which a
brief description is given of the fundamental mechanics of well stimulation and
enhanced oil recovery.

4.2 DARCY'S LAW; FLUID POTENTIAL

Every branch of science and engineering has its own particular heroes, one only has to
think, for example, of the hallowed names of Newton and Einstein in physics or Darwin
in the natural sciences. In reservoir engineering, our equivalent is the nineteenth
century French engineer Henry Darcy who, although he didn't realise it, has earned
himself a special place in history as the first experimental reservoir engineer. In 1856
2
Darcy published a detailed account of his work in improving the waterworks in Dijon
and, in particular, on the design of a filter large enough to process the town's daily
water requirements. Although fluid dynamics was a fairly advanced subject in those
days, there were no published accounts of the phenomenon of fluid flow through a
porous medium and so, being a practical man, Darcy designed a filter, shown
schematically in fig. 4.1, in an attempt to investigate the matter.

The equipment consisted of an iron cylinder containing an unconsolidated sand pack,
about one metre in length, which was held between two permeable gauze screens.
Manometers were connected into the cylinder immediately above and below the sand
pack. By flowing water through the pack Darcy established that, for any flow rate, the
velocity of flow was directly proportional to the difference in manometric heights, the
relationship being