Fluid Flow in the Subsurface (Darcy's Law)

The principle that governs how fluid moves in the subsurface is called Darcy's law.  Darcy’s law is an equation that defines the ability of a fluid to flow through a porous media such as rock.  It relies on the fact that the amount of flow between two points is directly related to the difference in pressure between the points, the distance between the points, and the interconnectivity of flow pathways in the rock between the points.  The measurement of interconnectivity is called permeability. 

In the subsurface, rock is deposited in layers.  Fluid flow within and between the rock layers is governed by the permeability of the rocks.  However, to account for permeability, it must be measured in both the vertical and horizontal directions.  For example, shale typically has permeabilities that are much lower vertically than horizontally (assuming flat lying shale beds).  This means that it is difficult for fluid to flow up and down through a shale bed but much easier for it to flow from side to side.  A good example of this characteristic is shown in the picture at left; which clearly indicates that it would be much easier for water to flow along the horizontal bedding planes in the shale where there are natural flow pathways instead of vertically where there are few flow pathways.

Ultimately, if the pressure difference between a hydraulically fractured zone and a fresh water aquifer is not great, the distance between the zones is relatively large, and there are rocks with low vertical permeabilities in between the deeper and the shallower zones, flow between the zones is unlikely to occur.  The exception to this is where there is a separate flow pathway such as an open borehole or a series of faults or joints that intersect both the fractured zone and the fresh water aquifer.  Under either of these circumstances, the pressure difference and distance will be the determining factors as to whether fluid can migrate from the lower to the upper zone.

For those with a greater interest in the mathematic principles behind fluid flow in the subsurface, the following is a description of Darcy's Law:

Darcy’s law is the equation that defines the ability of a fluid to flow through a porous media such as rock.  It relies on the principle that the amount of flow between two points is directly proportional to the difference in pressure between the points and the ability of the media through which it is flowing to impede the flow.  Here pressure refers to the excess of local pressure over the normal hydrostatic fluid pressure which, due to gravity, increases with depth like in a standing column of water.   This factor of flow impedance is referred to as permeability.  Put another way, Darcy's law is a simple proportional relationship between the instantaneous discharge rate through a porous medium and the pressure drop over a given distance.

In modern format, using a particular sign convention, Darcy's law is usually written as:

Q = -KA dh/dl

where:
Q =     rate of water flow (volume per time)
K =     hydraulic conductivity
A =     column cross sectional area
dh/dl = hydraulic gradient, that is, the change in head over the length of interest.

The following is a diagrammatic expression of Darcy's Law:



When calculating the possibility of fluid flow from a hydraulically fractured zone to a fresh water zone the application of Darcy’s law is critical because it sets out the specific conditions under which fluid could flow from one zone to another and will ultimately determine whether or not hydraulic fracturing fluids can reach a fresh water zone.

The darcy is referenced to a mixture of unit systems. A medium with a permeability of 1 darcy permits a flow of 1 cm³/s of a fluid with viscosity 1 cP (1 mPa·s) under a pressure gradient of 1 atm/cm acting across an area of 1 cm². A millidarcy (mD) is equal to 0.001 darcy.

 

 


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