Dykstra-Parson's Method for Isolated Layers

TPG4150 Reservoir Recovery Techniques 2013
Handout note 6 - Dykstra-Parson´s Method
page 2 of 5
where the end-point mobilities are defined as
and
λ oi ′ =
λ wi ′ =
⎛ k ro
′ ⎞
⎜ ⎟
⎝ ⎠
µ o
i
⎛ k rw
′ ⎞
⎜ ⎟
⎝ ⎠
µ w
i
. (4)
(3)
For an incompressible system, the two velocities are equal, i.e. u i = u oi = u wi . These Darcyvelocities
are average velocities over the total flow area. Actual front velocity (derived used
mass balance at the front) may be expressed as:
dx i
dt
=
u i
ΔS i
φ i
. (5)
In addition, the sum of the pressure drops ahead and behind the front is equal to the total
imposed pressure drop across the layer:
ΔP = ΔP w + ΔP o ,
By combination of the Equations (1), (2) and (5), we get the expression for the frontal
velocity in a layer:
dx i
dt
= − k i
φ i
ΔS i
ΔP
x i
+ L− x i
λ wi ′ ′
λ oi
(6)
As will be shown below, we are interested in finding expressions for relative front positions,
and therefore we will use index R to denote a reference layer. Then, taking the ratio of frontal
velocities in layers i and R, we obtain the following relationship:
d˙ x i
˙ x
= F R
+ M R
(1− x ˙ R
)
d˙ x i . (7)
R
x ˙ i
+ M i
(1− x ˙ i
)
Here, the end point mobility ratio is defined as
⎛
M i =
k rw
µ o
⎞
⎜ ⎟
⎝ µ w ⎠
k ro
the heterogeneity factor as
i
, (8)
F i = k φ ΔS λ ′
i R R wi
, (9)
k R
φ i
ΔS i
λ ′ wR
and the dimensionless distance as
˙ x i
=
x i
L . (10)
Norwegian University of Science and Technology
Professor Jon Kleppe
Department of Petroleum Engineering and Applied Geophysics 19.9.13