The injection and production of fluids is realized by implementing Peaceman’s well model. In contrast of using a constant injection or production rate, Peaceman’s well model automatically adjusts the amount of injected or extracted mass to the reservoir response. This can for example avoid an unrealistic high pressure in the near wellbore area and takes furthermore the mobility of the occurring fluids into account (Chen
2007). Certainly the size of an invoked grid cell significantly exceeds the real well diameter which sophisticates the well impact on the reservoir. Grid refinement around the well position allows a better representation of the reality but indeed requires considerable numerical expanse. In contrast, the implementation of Peaceman’s well model ensures a more simple comprise between representing the reality and defining a mathematical representation. By introducing an equivalent radius, Peaceman was able to find a connection between the occurring grid cell pressure and the bottom-hole flowing pressure. It is possible to approach the quantity of the equivalent radius, by either solving the analytical well flow model, numerically solving the pressure equation or directly calculating the yielding pressure between the well and its neighboring grid cells. The quantity of the equivalent radius amounts to approximately one-fifth of the average grid cell length. The simplest expression of Peaceman’s well model is listed below and is valid for a homogeneous reservoir and single-phase fluid flow:
$$\begin{aligned} Q=\frac{2\rho K_{xy} h_{z}}{\mu \left( \ln \left( \frac{r_{\text {e}}}{r_{\text {w}}}\right) +s\right) }\ \times {(P_{\text {wf}}-P)} \end{aligned}$$
(3)
where
Q is the injection or production rate in (kg/s),
\(\rho\) is the fluid density in (kg/m
\(^3\)),
\(K_{xy}\) is the horizontal permeability of the grid cell containing the well in (m
\(^2\)),
\(h_{z}\) is the grid cell height in (m),
\({\mu }\) is the fluid viscosity in (Pa s),
\(r_{\text {e}}\) is the equivalent radius in (m),
\(r_{\text {w}}\) is the geometrical well radius in (m) and
s is the wellbore skin factor. Besides the physical behavior of the fluid density and viscosity, the difference of a defined bottom-hole flowing pressure
\({P_{\text {wf}}}\) and the actual reservoir pressure
P in the well grid cell adjusts the amount of the injected or extracted mass to the arising reservoir response. The multi-compositional two-phase flow formulation of our model for UHS requires the additional consideration of phase mobilities and concentrations of the components. Since our model is based on the balance of moles, a modification of the units is necessary. For injection, it is sufficient to consider the gas phase:
$$\begin{aligned} \hat{Q}^{k}=\frac{c_{\text {g}}^{k,\text {inj}}\rho _{\text {g}}k_{\text {rg}}}{\mu _{\text {g}}}\times \frac{2 K_{xy} h_{z}}{\ln \left( \frac{r_{\text {e}}}{r_{\text {w}}}\right) +s}\times ({P_{\text {wf}}-P_{g}}) \end{aligned}$$
(4)
where
\(\hat{Q}\) is the injection or production rate in (mol/s) and
\(c_{\text {g}}^{k,\text {inj}}\) is the composition of the injected gas. Molar density, dynamic viscosity and relative permeability are the actual values in the well grid cell. For production, both phases need to be considered:
$$\begin{aligned} \hat{Q}^{k}& = \frac{2 K_{xy} h_{z}}{\ln \left( \frac{r_{\text {e}}}{r_{\text {w}}}\right) +s}\nonumber \\&\quad \times \left( \frac{c_{\text {g}}^{{k}}\rho _{\text {g}}k_{\text {rg}}}{\mu _{\text {g}}}({P_{\text {wf}}-P_{g}})+\frac{c_{\text {w}}^{{k}}\rho _{\text {w}}k_{\text {rw}}}{\mu _{\text {w}}}({P_{\text {wf}}-P_{\text {w}}})\right) \end{aligned}$$
(5)
where
\(c_{\text {g}}^{{k}}\) and
\(c_{\text {w}}^{{k}}\) in this case are also the actual values in the grid cell containing the well. A scaling of the Dirac delta function is required to transform the point or line source to a volume source, where
\(V_{\text {c}}\) is the volume of the well grid cell (cf. Eq.
18).
$$\begin{aligned} q^{{k}}=\frac{\hat{Q}^{{k}}}{V_{\text {c}}} \end{aligned}$$
(6)