Summary
We provide analytical solutions for the wellbore pressure during an injection/falloff-test problem under radial-flow conditions in homogeneous porous media where the injected fluid is carbonated water. For both the injection and falloff periods, we assume an isothermal process with thermodynamic equilibrium, a linear adsorption isotherm, and viscosities that depend only on the carbon dioxide (CO2) concentration. We also neglect CO2 diffusion, gravity effects, and capillarity effects. For the injection period, we first determine the saturation and concentration distributions with time in the reservoir by applying the method of characteristics to solve the appropriate system of hyperbolic conservation equations, where we assume incompressible fluids. In solving for water saturation and CO2 concentration in water, we neglect the change in water volume caused by the variation of the CO2 concentration in water. After solving for the saturation and concentration profiles, the pressure solution can be obtained by integrating Darcy's law, from the wellbore radius to infinity, while assuming an infinite-acting reservoir and invoking the Thompson-Reynolds steady-state theory (Thompson and Reynolds 1997b). Because Darcy's law does not assume incompressible flow, the pressure solution generated does not assume incompressible flow. To obtain an analytical expression for the wellbore pressure, however, we do assume that for injection and falloff, the total flow-rate profile in the reservoir is constant in a region from the wellbore to a radius greater than the radius of the flood front. The region within this radius increases with time and it is referred to as the steady-state region or zone (Thompson and Reynolds 1997b). During the falloff stage, it is assumed that there is no change in saturation in the reservoir, which is reasonable because we neglect capillary pressure, the gravity force, and fluid compressibilities when determining the saturation profile. Using these assumptions, we generate analytical solutions for a carbonated-water-injection (CWI)/falloff test and compare these solutions with those obtained with a commercial reservoir simulator using very fine spatial grids and very small timesteps. This comparison suggests that the analytical solutions presented can be used reliably to analyze pressure data obtained during CWI/falloff tests.
A Comparative Study on Wellbore Pressure Transmission of Water and Carbon Dioxide during Fracturing
