The term reservoir description is applied to characterizing a homogeneous system as opposed to a heterogeneous one. A reservoir description defines the reservoir at a level where a property changes sufficiently so that more than a single average must be used to model the flow. In this sense, a reservoir composed of a section of coarse gravel and a section of fine sand, where these two materials are separated and have significantly different permeabilities, is heterogeneous in nature. Defining dimensions, locating areas and establishing average properties of the gravel and sand constitutes a reservoir description, and is a satisfactory approach for reservoir-level type problems.
The governing flow equation describing flow through as porous medium is known as Darcy's law, which is a relationship between the volumetric flow rate of a fluid flowing linearly through a porous medium and the energy loss of the fluid in motion.
Darcy's law is expressed as:
The parameter, K, is a proportionality constant that is known as the hydraulic conductivity.
Darcy's law is considered valid for creeping flow where the Reynolds number is less than one. The Reynolds number in open conduit flow is the ratio of inertial to viscous forces and is defined in terms of a characteristic length perpendicular to flow for the system. Using four times the hydraulic radius to replace the length perpendicular to flow and correcting the velocity with porosity yields a Reynolds number in the form:
The hydraulic conductivity K depends on the properties of the fluid and on the pore structure of the medium. It is temperature-dependent, since the properties of the fluid (density and viscosity) are temperature-dependent. Hydraulic conductivity can be written more specifically in terms of the intrinsic permeability and the properties of the fluid.
where k is the intrinsic permeability of the porous medium and is a function only of the pore structure. The intrinsic permeability is not temperature-dependent.
In its differential form, Darcy's equation is:
A fx dx
The minus sign results from the definition of Ap, which is equal to p2 - p,, a negative quantity. The term q is known as the seepage velocity and is equivalent to the velocity of approach v„, which is also used in the definition of the Reynolds number.
Permeability is normally determined using linear flow in the incompressible or compressible form, depending on whether a liquid or gas is used as the flowing fluid. The volumetric flowrate Q (or QJ is determined at several pressure drops. Q (or Qm) is plotted versus the average pressure pm. The slope of this line will yield the fluid conductivity K or, if the fluid density and viscosity are known, it provides the intrinsic permeability k. For gases, the fluid conductivity depends on pressure, so that
where b depends on the fluid and the porous medium. Under such circumstances a straight line results (as with a liquid), but it does not pass through the origin; instead it has a slope of bK and intercept K. The explanation for this phenomenon is that gases do not always stick to the walls of the porous medium. This slippage shows up as an apparent dependence of the permeability on pressure.
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