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Unformatted text preview: Petrophysics MSc Course Notes Permeability Dr. Paul Glover Page 21 3. PERMEABILITY 3.1 Theory The permeability of a rock is a measure of the ease with which the rock will permit the passage of fluids. The fundamental physical law which governs this is called the Navier-Stokes equation, and it is very complex. For the purposes of flow in rocks we can usually assume that the flow is laminar, and this assumption allows great simplification in the equations. It should also be noted that the permeability to a single fluid is different to the permeability where more than one fluid phase is flowing. When there are two or more immiscible fluid phases flowing we use relative permeability, which will be introduced in this section, but covered in much more detail on the Formation Evaluation course later in the MSc. The fluid flow through a cylindrical tube is expressed by Poiseuille’s equation, which is a simplification of Navier-Stokes equation for the particular geometry, laminar flow, and uncompressible fluids. This equation can be written as (3.1) where: Q = the flow rate (cm 3 /s or m 3 /s) r = the radius of the tube (cm or m) P o = the outlet fluid pressure (dynes/cm 2 or Pa) P i = the inlet fluid pressure (dynes/cm 2 or Pa) m = the dynamic viscosity of the fluid (poise or Pa.s) L = the length of the tube (cm or m) About 150 years ago Darcy carried out simple experiments on packs of sand, and hence developed an empirical formula that remains the main permeability formula in use in the oil industry today. Darcy used the apparatus shown in Fig. 3.1, where he used a vertical sand pack through which water flowed under the influence of gravity while measuring the fluid pressures at the top and bottom of the pack by the heights of manometers. Here the difference in fluid pressures can be calculated from h 1 –h 2 providing the density of the fluid is known. It has since been validated for most rock types and certain common fluids. Darcy’s formula can be expressed as (3.2) where: Q = the flow rate (cm 3 /s or m 3 /s) P o = the outlet fluid pressure (dynes/cm 2 or Pa) P i = the inlet fluid pressure (dynes/cm 2 or Pa) m = the dynamic viscosity of the fluid (poise or Pa.s) L = the length of the tube (cm or m) k = the permeability of the sample (darcy or m 2 ) A = the area of the sample (cm 2 or m 2 ) L P P r Q o i m p 8 ) ( 4- = L P P A k Q o i m ) (- = Petrophysics MSc Course Notes Permeability Dr. Paul Glover Page 22 Figure 3.1 The Darcy apparatus (after Hubert, 1953). Note, either the first set of quoted units can be used (c.g.s system) as in the oil industry, or the second set (S.I. units) as in academic research, but one must use the units consistently. The units of permeability are the darcy, D, and m 2 , where 1 D = 0.9869 × 10-12 m 2 . One darcy is the permeability of a sample 1 cm long with a cross-sectional area of 1 cm 2 , when a pressure difference of 1 dyne/cm 2 between the ends of the sample causes a fluid with a dynamic viscosity of 1 poise to flow...
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This note was uploaded on 10/30/2011 for the course PETROLEUM Short cour taught by Professor Dr.paulglover during the Winter '11 term at University of Aberdeen.
- Winter '11