Time discretization
Spatial discretization
Numerical well model
The contact between well walls and permeable reservoir is called Well Reservoir Contact (WRC).
Specific flow rate (production or injection) through the differential element dh of WRC is proportional to delta pressure:
| (1) | \frac{dq_{sf}}{dh} = \frac{dV}{dt \ dh} = T_h \cdot M \cdot (p_{e} - p_{wf}) |
where
T_h – is called specific productivity (or injectivity) of well-reservoir contact (see below),
M = \frac{1}{\mu} – single-phase fluid mobility,
p_{e}(t, h) – formation pressure at external drainage boundary r_e (defined by the flow regime around element dh),
p_{wf}(t, h) – sandface bottomhole pressure across element dh.
Surface flow rates at separator (or tubing head of injector well) can be found as integration along the full length of WRC \Gamma_{ WRC}:
| (2) | 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 B^{S} =\frac{V_{sf}}{V_S} =\frac{V_sf}{V^{\LARGE \circ}} \frac{V^{\LARGE \circ}}{V^S} = \frac{B(P,T)}{B(P^S,T^S)} – formation volume factor at separator.
WRC Specific Productivity
WRC specific producvity T_h depends on flow reghime around well.
The most popular model is given by stationary (steady-state or pseudo steady-state) flow:
| (3) | T_h = \frac{2 \pi \ k_{\perp} }{ \ln \frac{r_e}{r_w} - \epsilon + S} |
where
k_{\perp} = \sqrt{k_{\perp 1} \ k_{\perp 2}} | geometric average permeability in transversal plane to WRC |
\{ k_{\perp 1}, k_{\perp 2} \} | transvercal pertmeabilities |
r_w | drilling bit well radius |
r_e | external boundary of drainage area |
S | near-reservoir zone skin-factor |
\epsilon = 1/2 | for steady-state flow regime (constant pressure at r_e) |
\epsilon = 3/4 | for pseudo-state flow regime (no flow at r_e) |
Effective drainage radius can be approximated by Peaceman model:
where
{\bf D} = \{ D_{\perp 1}, D_{\perp 2} \} – dimensions of the grid cell around well in transversal plane to the well axis. Strictly speaking, the above formula is only valid in case well penetrates through the whole length of grid cell
\bf D perpendicular to the cell faces. There are many modifications and generalization of the Peaceman approximation but in the most practical cases it works very well when sufficiently fine LGR is applied.
(4)
r_e = 0.28 \ \frac{ \sqrt{
\Big( \frac{k_{\perp 2}}{k_{\perp 1}} \Big)^{1/2} D_{\perp 1} ^2
+
\Big( \frac{k_{\perp 1}}{k_{\perp 2}} \Big)^{1/2} D_{\perp 2} ^2
} }
{ \Big( \frac{k_{\perp 2}}{k_{\perp 1}} \Big)^{1/4} + \Big( \frac{k_{\perp 1}}{k_{\perp 2}} \Big)^{1/4}}