The contact between well walls and permeable reservoir is called Well Reservoir Contact (WRC).
Specific flow rate (production or injection) through the differential element of WRC is proportional to delta pressure:
\frac{dq_{sf}}{dh} = \frac{dV}{dt \ dh} = T_h \cdot M \cdot (p_{e} - p_{wf}) |
where – is called specific productivity (or injectivity) of well-reservoir contact (see below),
– single-phase fluid mobility,
– formation pressure at external drainage boundary
(defined by the flow regime around element
),
– sandface bottomhole pressure across element
.
Surface flow rates at separator (or tubing head of injector well) can be found as integration along the full length of WRC :
q(t) = \int_{\Gamma_{WRC}} \ \bigg( \frac{1}{B^S} \frac{dq_{sf}}{dh} \bigg) \, dh = \int_{\Gamma_{WRC}} \bigg(
\frac{M \, (p_e - p_{wf})}{B^S}
\bigg) \, T_h \, dh |
where – formation volume factor at separator.
WRC specific producvity depends on flow reghime around well.
The most popular model is given by stationary (steady-state or pseudo steady-state) flow:
T_h = \frac{2 \pi \ k_{\perp} }{ \ln \frac{r_e}{r_w} - \epsilon + S} |
where
| geometric average permeability in transversal plane to WRC | |
| transvercal pertmeabilities | |
| drilling bit well radius | |
| external boundary of drainage area | |
| near-reservoir zone skin-factor | |
for steady-state flow regime (constant pressure at | |
for pseudo-state flow regime (no flow at |
Effective drainage radius can be approximated by Peaceman model: