Flow in a Porous Medium: Bernoulli Equation and Permeability Calculation
Explore the dynamics of flow in a porous medium, including the application of the Bernoulli equation, Darcy's law, and permeability calculations. Understand the transition from laminar to turbulent flow and solve examples to calculate coefficients of permeability.
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Review of Last Lesson Assume = ? ?,? such that : ? = ??= ? ?? ??? ? = ??= ? ?? d =? ???? +? ???y= -? dx - ? dy = ? ???? + ? ???y + C = ? ?? - ? dy + ? Or ?? polar coordinate = ? ?,? such that: With ??= ? ? ??? ?? ??? ??= ? ?????? +? ? ?????? + ? ????= -????? - ??dr d = ???? + C = ????? - ?? dr + C =
Flow in a porous Medium For one-dimentional flow of an incompressible real fluid in a streamtube, the Bernoulli equation written in differential form is: ? ? +?? ??+? = ??? ------------- 1 ? ?? ??may be neglected Suppose that V is small so that ? ? ???? ------------- 2 ???= ? ? +? = ? ???? ? ? ??? ? +? ? ??? +? or ? = ? ------------- 3 = ? ? Let ? = ? +? ? ? = ??? ??= ? ? (Darcy s law) ------------- 4 If this will be extended to the two dimensional case: ? ?? ? ? +? = ??? ??= ? ? = ?? ------------- 5 ? ? ?? ? ? +? = ??? ??= ? ? = ?? ------------- 6 ? Or in general ? = ??? ??= ? ?? ------------- 7 ? ? +? K is a coefficient of permeability has a dimensions of velocity and ranges from: 3 x 10-11 m/s for clay to 0.3 m/s for gravel = is seen to be the velocity potential of such a flowfield where ?
Flow in a porous Medium This is satisfied for laminar condition through a homogeneous porous medium. ?? ---------------8 ? = R= Reynolds number for porous media V = apparent velocity = Q/A d= effevtive or mediam grain size in sand When R < 1 , the flow is surely laminar and Darcy s linear law is valid. If R >>>1 , it is likely that the flow is turbulent and that ?? not negligible and equation 4 is not valid. Actual velocity vp The apparent velocity V= Q/A is not the actual vgelocity in the pores. The average velocity in the pores is : ??=? ?? is ? --------9 with ? =?????? ?? ????? --------10 ????? ?????? Where n is porosity
Flow in a porous Medium Example: Calculate the coefficient of permeability K for the following problem if the flowrate is 2.8 m/s Solution: from Eq.7 ? = ??? ??= ? ??= ? ? ?? ------------- 7 ?= ??? ?? ? = 1 Or ? = ??? ? note that ??= ??+?? ? and ??= ??+?? ?= ? ? h1 h2 = ??+?? ? - ??+?? ? = 2 2 ?- 1 ?= - 0.3 ? = ??? ?? ?.?? ?.? = ? ? ?=?.? ??? ? But ? =? ?(?.??)?= ?.???????? ? m/s ?.?? ?.? , ?.???????? ?= ? K = ?.?????? ? m/s
Flow in a porous Medium Example: For the following problem, calculate the flowrate using flow net method. Solution: 1 = ???= ? ??+?? ? 2 2 = ???= ? ??+?? ? = 2 1 =? 2 ? 1 ? ? = ?? ? ? = no. of equipotential lines ?= 1+ ? ?( ? x 1) ? )= q = v ? = q= = ( ?? ?? ? ?? ? ? = q = ?? q =?? ?? = no. of stream lines
