Injectivity Outlet (Forward Flux→Pressure)

The standalone-wellbore and analytically-coupled modes close the bottom of the well with an injectivity outlet boundary condition (outlet_bc_type = sacconi), following Sacconi & Mahgerefteh (2020). The near-wellbore flow resistance relates the bottom-hole flowing pressure $P_\mathrm{BHF}$ to the reservoir pressure $P_\mathrm{res}$ and the sandface mass rate $\dot{m}$ through a quadratic model (their Eq. 40):

$$P_\mathrm{BHF}^{\,2} = P_\mathrm{res}^{\,2} + A + B\,\dot{m} + C\,\dot{m}^{\,2}$$

where $A$, $B$, $C$ are empirical injectivity coefficients. At zero flow ($\dot{m} = 0$) the bottom-hole pressure reduces to $\sqrt{P_\mathrm{res}^2 + A}$ (i.e. $P_\mathrm{res}$ when $A = 0$).

Forward (flux→pressure) construction

The equation is applied in the forward direction. The outlet does not invert the injectivity to read a rate off the bottom-cell pressure. Instead it takes the mass rate the interior actually delivers to the sandface,

$$\dot{m} = \rho\,u\,A_\mathrm{outlet},$$

with the velocity $u$ extrapolated zero-gradient from the bottom cell, and computes $P_\mathrm{BHF}$ forward from that delivered flux. The outlet ghost cell is set at $P_\mathrm{BHF}$ — hydrostatically continued half a cell down to the ghost centre in the cell-centred finite-volume scheme — with zero-gradient density, and the Riemann/AUSM face flux closes the outlet. The rate is therefore a result of the interior hydrodynamics, not an imposed inverse of the bottom-cell pressure.

This forward, finite-volume construction follows the Sacconi-group (R. Samuel) thesis formulation of the same injectivity equation.

Why forward matters — acoustic BHP response

When the wellhead feed rate changes, the resulting pressure/velocity wave reaches the sandface at the acoustic timescale (≈ seconds for a 2.5 km well). With the forward construction the delivered flux $\dot{m} = \rho u A$ rises as soon as the wave arrives, so $P_\mathrm{BHF}$ — and hence the recorded bottom-hole pressure — responds at the acoustic timescale. The earlier inverted form bled any incipient pressure rise straight into the reservoir, so the bottom-hole pressure could only climb at the much slower wellbore-storage rate; this is corrected by the forward construction.

At steady state the sandface rate equals the wellhead rate, so $P_\mathrm{BHF}$ reduces to exactly the value the injectivity equation gives for that rate — the steady result is unchanged.

Backflow

When the delivered flux is negative — for example during shut-in or ramp-down, when the column becomes lighter than the reservoir — fluid is drawn into the well. The outlet ghost is then built at reservoir conditions $(P_\mathrm{res},\,T_\mathrm{res})$ (with the same half-cell hydrostatic continuation), so the inflow enters at the reservoir temperature.

References

• Sacconi, A. & Mahgerefteh, H. (2020). Modelling start-up injection of CO₂ into highly-depleted gas fields. Energy, 191:116530.
• R. Samuel, PhD thesis (Sacconi group) — finite-volume forward injectivity outlet formulation.