Transient Fluid and Heat Flow Modeling in Coupled Wellbore/Reservoir Systems

Summary This paper presents a transient wellbore simulator coupled with a semianalytic temperature model for computing wellbore-fluid-temperature profiles in flowing and shut-in wells. Either an analytic or a numeric reservoir model can be combined with the transient wellbore model for rapid computations of pressure, temperature, and velocity. We verified the simulator with transient data from gas and oil wells, where both surface and downhole data were available. The accuracy of the heat-transfer calculations improved with a variable-earth-temperature model and a newly developed numerical-differentiation scheme. This approach improved the calculated wellbore fluid-temperature profile, which, in turn, increased the accuracy of pressure calculations at both bottomhole and wellhead. The proposed simulator accurately mimics afterflow during surface shut-in by computing the velocity profile at each timestep and its consequent impact on temperature and density profiles in the wellbore. Surrounding formation temperature is updated in every timestep to account for changes in heat-transfer rate between the hotter wellbore fluid and the cooler formation. The optional hybrid numerical-differentiation routine removes the limitations imposed by the constant relaxation-parameter assumption used in previous analytic-temperature models. Both forward and reverse simulations are feasible. Forward simulations entail computing pressure, temperature, and velocity profiles at each wellbore node to allow matching field data gathered at any point in the wellbore. In contrast, reverse simulation allows translating pressures from one point to another in the wellbore, such as wellhead to bottomhole condition. Introduction Modeling of the changing pressure, temperature, and density profiles in the wellbore as a function of time is crucial for the design and analysis of pressure-transient tests, particularly when data are gathered off-bottom or in a deepwater setting, and the identification of potential flow-assurance issues. Other applications of this modeling approach include improving the design of production tubulars and artificial-lift systems, gathering pressure data for continuous reservoir management, and estimating flow rates from multiple producing horizons. A coupled wellbore/reservoir simulator entails simultaneous solution of mass, momentum, and energy balance equations, providing pressure and temperature as a function of depth and time for a predetermined surface flow rate. Almehaideb et al. (1989) studied the effects of multiphase flow and wellbore phase segregation during well testing. They used a fully implicit scheme to couple the wellbore and an isothermal black-oil reservoir model. The wellbore model accounts only for mass and momentum changes with time. Similarly, Winterfeld (1989) showed the simulations of buildup tests for both single and two-phase flows in relation to wellbore storage and phase redistribution. The Fairuzov et al. (2002) model formulation also falls into this category. Miller (1980) developed one of the earliest transient wellbore simulators, which accounts for changes in geothermal-fluid energy while flowing up the wellbore. In this model, mass and momentum equations are combined with the energy equation to yield an expression for pressure. After solving for pressure, density, energy, and velocity are calculated for the new timestep at a well gridblock. Hasan and his coworkers presented wellbore/reservoir simulators for gas (Kabir et al. 1996), oil (Hasan et al. 1997), and two-phase (Hasan et al. 1998) flows. Their formulation consists of a solution of coupled mass, momentum, and energy equations, all written in finite-difference form, and requires time-consuming separate matrix operations. In all cases, the wellbore model is coupled with an analytic reservoir model. Fan et al. (2000) developed a wellbore simulator for analyzing gas-well buildup tests. Their model uses a finite-difference scheme for heat transfer in the vertical direction. The heat loss from the fluid to the surroundings in the radial direction is represented by an analytical model.