Geoscience Reference
In-Depth Information
10
-5
10
-4
10
-3
0.01
0.1
1
10
100
1000
Metres
Fig. 4.4
Important length scales involved in reservoir modelling
densities, viscosities, and permeabilities. What
goes in must be balanced by what comes out,
and for a complex set of flow equations the
zero-sum constraint for each grid cell is essential.
Fluid flow in porous media is represented by
Darcy's Law (Sect.
4.3.2
) which relates the fluid
velocity, u, to the pressure gradient and two
terms representing the rock and the fluid:
where:
o and w refer to the oil and water phases,
k
rw
and k
ro
are the relative permeabilities of each
phase,
μ
are fluid viscosity and density,
P
c
is the capillary pressure,
∇
and
ˁ
P
o
is the gradient of pressure for the oil phase
This set of equations is non-linear as the k
rw
,
k
ro
and P
c
terms are all functions of phase satura-
tion, S
w,
which is itself controlled by the flow
rates. Thus, in order to solve these equations for a
given set of initial and boundary conditions,
numerical codes (reservoir simulators) are used,
in which saturation-dependent functions for k
rw
,
k
ro
and P
c
are given as input, and an iterative
numerical recipe is used to estimate saturation
and pressure. Figure
4.5
shows a typical set of
oil-water relative permeability curves with the
endpoint terminology.
Note that the total fluid mobility is
u
¼
k=μ:
∇
ð
P
þ ˁ
gz
Þ
ð
4
:
1
Þ
The pressure term comprises an imposed pres-
sure gradient,
(P), and a pressure gradient due
∇
to gravity,
gz). In Cartesian coordinates the
gradient of pressure,
(
ˁ
∇
P, is resolved as:
∇
dP
dx
þ
dP
dy
þ
dP
dz
P
¼
ð
4
:
2
Þ
∇
The rock (or the permeable medium) is
represented by the permeability tensor,
k
, and
fluid by the viscosity,
1 (mobil-
ity is the permeability/viscosity ratio for the
flowing phase). That is, the permeability of a
rock containing more than one phase is signifi-
cantly lower than a rock with only one phase.
Clearly the fluid viscosity is a key factor but the
fluid-fluid interactions also play a role. The
functions are drawn between 'endpoints,' which
are a mathematical convenience, but are also
based on physical phenomena - the point at
which the flow rate of one phase becomes insig-
nificant. However, the endpoint values them-
selves are not physically fixed. For example,
there exists a measurable irreducible water satu-
ration, but its precise value depends on many
things (e.g. oil phase pressure or temperature).
Many of the problems and errors in upscaling
<
.
When two or more fluid phases are flowing, it
becomes necessary to introduce terms for the
density, viscosity and permeability of each
phase and for the interfacial forces (both fluid-
fluid and fluid-solid). For two-phase immiscible
flow (oil and water), the two-phase Darcy equa-
tion and the capillary pressure equation are
used:
μ
u
o
¼
k
k
ro
=μ
o
:
∇
ð
P
o
þ ˁ
o
gz
Þ
ð
4
:
3
Þ
u
w
¼
k
k
rw
=μ
w
:
∇
ð
P
w
þ ˁ
w
gz
Þ
ð
4
:
4
Þ
P
c
¼
P
o
P
w
ð
4
:
5
Þ