Reservoir Engineering: Darcy’s Law
Henry Darcy experimentally demonstrated that volumetric flow rate through a saturated sand column remained directly proportional to (a) total head loss across the sand column; and (b) cross-sectional area; while it remained inversely proportional to (c) height of the saturated sand column.
1. Since Darcy’s law remains applicable, when the difference in the head is measured only at the two ends of the (vertical) column (as indicated by manometer readings in his original experiment); and, since, we are entitled to take into account the average pressure values, only @ (outer) reservoir boundaries (as per the original version of Darcy's law); how could we take into account any (average) reservoir pressure that remains associated within the reservoir boundary (i.e., within or inside a petroleum reservoir)?
By translating original Darcy’s algebraic equation into it’s equivalent ‘differential form’, which remains capable of describing spatial variations; then, Are we not, kind of, violating the original version of Darcy’s Law?
2. Having known that Darcy’s law is a simple algebraic expression that correlates volumetric flow rate as a function of total head loss measured over a finite length; how exactly, were we able to transform ‘Darcy’s equation in terms of potentials’ to a form ‘in terms of pressure and gravity’?
3. Having know that original Darcy’s law has a proportionality constant (hydraulic conductivity, which is a function of both aquifer’s rock and fluid properties), which is supposed to remain as a constant (for a particular column packing); how could we consider spatial dependence of any variable associated with the interior of a petroleum reservoir – while, using Darcian approach for estimating oil production?
4. Having known that ‘the value of the proportionality constant (hydraulic conductivity)’ in original Darcy’s law remains applicable only at the macroscopic-scale (or @ Darcian-scale); and it remains not applicable at any other smaller scale (as per the original version); then, how could we take into account any variable/parameter, which remains associated with a scale that happens to be smaller than Darcian/macrosocpic-scale – while, using Darcian approach for estimating oil production?
Are we not supposed to make averages (of the variables) over these micro-scale entities?
So, when we include parameters/variables from various scales (ignoring original version of Darcy); then, what does the resulting fluid flow equation using Darcian approach provide?
(a) Darcian/macro-scopic scale equation describe the changes of only macroscale variables with time and as a function of space?
OR
(b) Darcian/macro-scopic scale equation describe the changes of only micro-scale variables with time and as a function of space?
OR
(c) Darcian/macro-scopic scale equation describe the changes of both micro-scale and macro-scale variables with time and as a function of space?
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