While these models are simple, the equations employed to describe the respective phenomena of interest can be implemented in 3-dimensional models with more complex geometries. The different physics interfaces can in principle also be coupled and solved for within the same model — for example the coupling of fluid flow with heat transfer or matrix poroelasticity can provide insights valuable for field operations.
The embedded figures and movie were created with COMSOL.
COMSOL’s Application Gallery provides alternate examples created with the Subsurface Flow Module.
Water is injected into a sandstone aquifer through a gravel-packed, cased and perforated vertical well. The inertial effects for flow through the gravel pack are considered, and they are shown to become significant through the perforation tunnels at higher injection rates.
This model also highlights the flow rate distribution among individual perforation tunnels, as well as vertical pressure profiles in the vicinity of the wellbore.
The featured capabilities can be used to understand details of the flow near the wellbore-reservoir interface, and can also assist with well log analysis (whether for injection or production).
This example showcases capabilities relevant for downhole temperature and pressure monitoring. It builds on the model above, by extending its scope to describe thermal effects as well. The whole system is initially in thermal equilibrium, at the aquifer’s temperature and pressure. Water injection then begins, at constant pressure and with a lower fluid temperature. The transient temperature distribution is monitored throughout the reservoir and the wellbore. The injection and perforation tunnel velocities are also probed, and it is found that their values decrease in time, as the cold water front advances and the viscosity of the fluid behind it increases. Similar computational analysis can be performed for producer wells.
Another variation of the 2D injector well models above, this time coupling single-phase flow with convective solute transport. The whole system is initially at the reservoir pressure. Isothermic water injection then begins, containing trichloroethylene (TCE) tracer for the first ten minutes. The tracer concentration distribution is monitored throughout the aquifer and the wellbore.
Two penny-shaped hydraulic fractures extend the reach of a horizontal open hole well drilled into a low permeability reservoir. The hydraulic fractures are sand-packed and intersect a network of activated natural fractures, modeled here as a stack of “sugar cubes”. Inertial effects are considered within the sand pack. Their impact on the well production rate, as well as that of the natural fracture network, is assessed for various pressure drawdown values.
The incompressible Navier-Stokes equations are solved for water flow in the pore space of a hexagonal close pack of perfect spheres. This 3-dimensional model exemplifies the emergence of inertial effects in laminar flow through porous media. The empirical Darcy law and Forchheimer equation for porous media are thus naturally derived by modeling the fluid flow at the pore space level.
Water is injected from the bottom into a vertical core plug saturated with oil, and the water saturation Sw distribution is probed as the liquid displacement occurs. This model employs a synthetic capillary pressure Pc(Sw) curve and Corey-type correlations for the relative permeability functions of the two liquids.
The capabilities employed in this model can be used for analyzing laboratory experimental data, or for the modeling of flow in petroleum reservoirs–applications for the latter including water injection for secondary recovery or hydraulic fracturing.
A pressure pulse is applied to the upstream end of a sample holder containing a tight rock core plug and filled with nitrogen gas. The transient pressure response at the downstream end is compared to the input signal. The core plug considered corresponds to one of the samples reported in SPE-89867.
The equations employed by this model can be used not only to assist with laboratory data analysis, but also for larger scale modeling of gas flow in the reservoir and near-wellbore area.
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