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OA – DF - YB – LB: Thermal Simulation And Interpretation - Part 1
p 2/10
In single phase, the model uses the usual Black Oil PVT of Ecrin. In multi-phase a Peng
Robinson EOS with Volume Translation is used for the hydrocarbon phases, while the water is
treated as a single component.
The formulation of the thermal problem is a coupled system made of a mass balance equation,
and an energy balance equation in terms of enthalpy, h. In the reservoir, those equations can
be expressed in the general form below (written for single phase but easily extended to multi-
phase by a sum over the phases):
0
.
V
t
T
t
P
h
t
h
t
h
r
r
*
1
gv
v
*
is the average thermal conductivity of the rock + fluid medium.
*
is obtained with the
mixing model below, assuming constant fluid
f
and rock
r
thermal conductivities.
1
*
r f
In the wellbore the mass and energy balances take into account the exchanges within the
wellbore, and between the wellbore and the formation. After cutting the wellbore into
segments in front of the reservoir cells, the mass balance can be derived by equating the mass
variation in a segment with the sum of the mass inflow/outflow (subscript ‘b’ below =
wellbore; subscript ‘r’ = reservoir):
b
r
r
b
q
q
dt
d
V
Similarly, the energy conservation expresses the variation of energy
t
E
as the sum of the
exchanges with the segments above and below, and the exchanges with the reservoir:
r
r
p
b
b
p
c
TD q eh
q e eh
t
E
).
(
The terms e
c
and e
p
represent the kinetic and potential energies. The last term is the thermal
exchange due to conduction with the reservoir. The conductance ‘D’ is an overall transfer
coefficient accounting for the thermal properties of all the completion elements between the
wellbore and the formation.
t
E
- the transient expansion term - is considered in front of the reservoir in single phase
but not above. In multi-phase it is omitted everywhere in the wellbore. Note also that while
the wellbore energy equation considers convection in the wellbore (1
st
term) and conduction
with the formation (3
rd
term), longitudinal conduction is neglected in a wellbore segment.
Finally, it should be noted that the pressure drop within the wellbore is based on classical
correlations accounting for slippage, frictions, etc. as defined in the wellbore model.