Analytic Wellbore Temperature Model for Transient Gas-Well Testing

Summary Questions arise whether bottomhole pressures (BHPs), derived from their wellhead counterpart (WHP), lend themselves to transient analysis. That is because considerable heat exchange may affect the wellbore-density profile, thereby making the WHP translation a nontrivial exercise. In other words, gas density is dependent on both spatial locations in the wellbore and time during transient testing. Fully coupled wellbore/reservoir simulators are available to tackle this situation. However, they are not readily adaptable for their numeric formulations. This paper presents analytic expressions, derived from first principles, for computing time-dependent fluid temperature at any point in the wellbore during both drawdown and buildup testing. The simplicity of the analytic expressions for Tf (z, t) is profound in that one can compute flowing or shut-in BHPs on a spreadsheet. Two tests were considered to verify the new analytic formulae. In one case, measurements were available at both sandface and surface, and partial wellhead information was available in the other case. We explored a parametric study to assess whether a given wellbore/reservoir system will lend itself to wellhead measurements for valid transient analysis. Reservoir flow capacity (kh) turned out to be the most influential variable. Introduction Gas-well testing is sometimes conducted by measuring pressures at the wellhead. Both cost and circumstance (high pressure/high temperature, or HP/HT)often necessitate WHP monitoring or running the risk of having no tests at all. Methods for computing BHP from wellhead pressures for steady flow in gas wells are well established in the literature. For dry-gas wells, the widely used method of Cullender and Smith is most accurate, as confirmed by subsequent studies. For wet gas, either a two-phase model, such as the one offered by Govier and Fogarasi, or the modified Cullender-Smith approach appears satisfactory. However, these methods apply to steady-state gas flow and implicitly presuppose that the wellbore is in thermal equilibrium with the formation. These assumptions may be tested during a transient test. That is because unsteady-state wellbore heat transfer occurs even after the cessation of the wellbore-fluid-storage period. Steady-state fluid flow ordinarily implies the absence of wellbore effects from the viewpoint of transient testing. Consequently, one needs to develop working equations by conserving mass, momentum, and energy in the wellbore to capture physical phenomena. Earlier, we presented a forward model and showed its capability to reproduce BHP, WHP, and wellhead temperature (WHT) given reservoir and wellbore parameters. However, translation of WHP to BHP was not demonstrated clearly. The intent of this work is to present a framework for rigorous computation of BHP from WHP. To achieve this objective, we developed analytic expressions for depth- and time-dependent fluid temperature during both flow and shut-in tests. These temperature relations, in turn, allow computation of gas density and, therefore, pressure at any point in the wellbore.