This two dimensional model describes the temperature distribution as cold water is injected into a vertical well completed into a sandstone aquifer. The whole system is initially at equilibrium, at the reservoir temperature and pressure. Water injection occurs for 30 minutes, following which the system is monitored for another half an hour.
The well is cased and perforated, and water represents the single phase. The well is completed with a concentric screen, modeled here as a porous medium. The annular gap is filled with a sand pack (shown in green in Figure 1) up to three quarters of the considered well length. Water is injected at constant pressure through the top of the base pipe, which feeds into the screen pipe. The equidistant perforation tunnels are also filled with a sand pack having the same properties as that in the annulus.
The fluid flow equations solved for are the stationary Navier-Stokes in the free domains, and the Brinkman equations for the porous ones (screen, sand pack, and aquifer matrix). Inertial effects are considered for the sand pack, and described via the Ergun correlation for the beta factor. The sand pack permeability is also determined with the Ergun model. All materials are considered incompressible.
The convective heat transfer equations are fully coupled with those for fluid flow, thus allowing for thermally-induced fluid density and viscosity changes as the system evolves.
The fluid flow boundary conditions are constant injection pressure (Pinlet) at the base pipe inlet, constant zero pressure (Poutlet) at the aquifer far field outlet, and no flow on the other outer boundaries of the model. The wellbore connects hydraulically to the aquifer only through the sand-packed perforation tunnels. The heat transfer boundary conditions are T = Tinj = 20°C at the base pipe inlet and T = Taq = 75°C at the far boundaries of the aquifer.
The length of the well is 1.5 m, the outer and inner diameters of the screen and casing are 3.32 inch and 6.63 inch, respectively, and the modeled aquifer extends to 15 ft away from the axis of the well. The borehole has a diameter of 9.5 inch, while the seven perforation tunnels are 0.75 inch-wide. The screen assembly hydraulically connects the interior of the screen pipe to the well annulus.
The animated Figure 2 below depicts the temperature distribution throughout the modeled system (injection occurs only through the first 30 minutes), while Figure 3 shows the average values of the temperature in the screen pipe, the sand pack, and the near and far sections of the aquifer.
The dynamic plot in Figure 4 shows how the vertical temperature distribution at different radial distances from the well axis changes with time. The selected vertical edges are the well axis (blue curve), the screen outer diameter (cyan curve), the inner and outer walls of the cement-casing assembly (green and black curves, respectively), and the near-far aquifer division line (red curve, located 1.5 ft away from the well axis). The locations of the apparent peaks correspond to those of the perforation tunnels. The casing data would be similar to measurements obtained with a distributed temperature sensing (DTS) monitoring tool in a real system.
Another interesting aspect is the impact on the injection rate of the changing water viscosity, due to temperature variations. As the colder injected water invades more and more of the system, its viscosity increases accordingly, which translates into an augmented resistance to flow. This is indeed observed in our model: the blue line in Figure 5 shows how dramatically the injected fluid velocity (a proxy for injection rate) decreases with time, despite the applied pressure boundary conditions remaining unchanged. The other curves in the Figure show the temporal evolution of the average fluid velocity values through the top, middle and bottom perforation tunnels.