Lecture 19 and 20 Flow through porous media

Lecture 19 and 20 Flow through porous media

History A lovely picture of Henry Darcy (1856)

Darcy’s Experiment water q 1 h 1 L h 2 2 atm

Hydraulic Head (h) Energy of a stationary liquid b y 1 Head = potential energy + pressure For this case: h 1 = h 2 Why? y 2

Head Loss Energy loss of liquid For steady moving viscous flow: Friction loss h 1 -h 2 = head loss = friction loss

Darcy’s Experiment 1 q L 2 A K is the hydraulic conductivity K [=] L/t Darcy’s law can be interpreted as a momentum, or as an energy equation

Generalization to nonvertical flow q Elevation datum L b A 2 1 z 2

Generalization to nonvertical flow

Definition of permeability, k Nutting (1930) found K ~ 1/ k is the intrinsic permeability k [=] L 2

Darcy’s Law q is the volumetric flow rate q [=] cm 3/s q/A = u is the flux velocity also called the linear velocity q/A [=] cm/s

Flux velocity and interstitial velocity Flux velocity Velocity of liquid averaged over total volume Interstitial velocity Velocity of liquid averaged over void volume (measures velocities in the pores)

How good is Darcy’s law? q potential kinetic Pay attention when you deviate from: • steady state • homogeneous material • incompressible fluid • average flow velocity Darcy flow Non-Darcy flow dp/ds

Units for k 1 cm 3 cube/s 1 darcy = 10 -12 m 2 1 atm =1 cp k=1 darcy 1 md = 10 -15 m 2

Oilfield units k in md is specific gravity in cp is dip angle q in STB/day dp/ds is psi/ft B in res bbl/STB

Horizontal Flow q x For horizontal flow s=x:

Vertical Flow through Core air 1 q 2 air p 1 = p 2 = patm= 0 psig

Vertical Flow with Driving Head b 1 L q 2

Upward Vertical Flow q L b

Radial Flow of a Production Well q rw re • s is the direction of flow • r is from well to the boundary

Radial Flow of a Production Well q rw re Can we now integrate?

Radial Flow of a Production Well A is a function of r r bb q re q constant for steady state

Radial Flow of a Production Well

Extension to heterogeneous systems Parallel beds q 1 k 1 b 1 q 2 k 2 b 2 q 3 k 3 b 3 L How to compute total q ? b w

Set up Darcy’s law for each layer q 1 k 1 b 1 q 2 k 2 b 2 q 3 k 3 b 3 L b w

Extension to heterogeneous systems Series of beds Δp 1 Δp 2 Δp 3 L 1 L 2 L 3 q Show that

Extension to heterogeneous systems Why are they so different? Δp 1 q 1 k 1=500 b 1=10 q 2 k 2=0. 01 b 2=1 q 3 k 3=10 b 3=5 Δp 2 Δp 3 b q k 1=500 L 1=10 k 3=10 k 2=0. 01 L 2=1 L 3=5 w L Does this seem realistic to you?