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.

Summary This paper presents a physical model for predicting flow pattern, void fraction, and pressure drop during multiphase flow in vertical wells. The hydrodynamic conditions giving rise to various flow patterns are first analyzed. The method for predicting void fraction and pressure drop is then developed. In the development of the equations for pressure gradient, the contribution of the static head, frictional loss, and kinetic energy loss are examined. Laboratory data from various sources show excellent agreement with the model. Introduction A number of correlations are available for predicting pressure drop during multiphase flow. Because most of these correlations are entirely empirical, they are of doubtful reliability. The calculation procedures involved are also rather complicated. Therefore, a better approach is to attempt to model the flow system and then to test the model against actual data. Proper modeling of multiphase flow requires an understanding of the physical system. When cocurrent flows of multiple phases occur, the phases take up a variety of configurations, known as flow patterns. The particular flow pattern depends on the conditions of pressure, flow, and channel geometry. In the design of oil wells and pipelines, knowledge of the flow pattern or successive flow patterns that would exist in the equipment is essential for choosing a hydrodynamic theory appropriate for that pattern. The name given to a flow pattern is somewhat subjective. Hence, a multitude of terms have been used to describe the various possible phase distributions. In this paper, we will be concerned only with those patterns that are clearly distinguishable and generally recognized. The major flow patterns encountered in vertical cocurrent flow of gas and liquid are listed in standard textbooks and in the classic works of Orkiszewski,1 Aziz et al.,2 and Chierici et al.3 The four flow patterns---bubbly, slug, churn, and annular---are shown schematically in Fig. 1. At low gas flow rates, the gas phase tends to rise through the continuous liquid medium as small, discrete bubbles, giving rise to the name bubbly flow. As the gas flow rate increases, the smaller bubbles begin to coalesce and form larger bubbles. At sufficiently high gas flow rates, the agglomerated bubbles become large enough to occupy almost the entire pipe cross section. These large bubbles, known as "Taylor bubbles," separate the liquid slugs between them. The liquid slugs, which usually contain smaller entrained gas bubbles, provide the name of the flow regime. At still higher flow rates, the shear stress between the Taylor bubble and the liquid film increases, finally causing a breakdown of the liquid film and the bubbles. The resultant churning motion of the fluids gives rise to the name of this flow pattern. The final flow pattern, annular flow, occurs at extremely high gas flow rates, which cause the entire gas phase to flow through the central portion of the tube. Some liquid is entrained in the gas core as droplets, while the rest of the liquid flows up the wall through the annulus formed by the tube wall and the gas core. In an oil well, different flow patterns usually exist at different depths. For example, near bottom hole we may have only one phase. As the fluid moves upward, its pressure gradually decreases. At the point where the pressure becomes less than the bubblepoint pressure, gas will start coming out of solution and the flow pattern will be bubbly. As pressure decreases further, more gas may come out of solution and we may see the whole range of flow patterns shown in Fig. 2. Here we discuss the hydrodynamic conditions that give rise to the various flow-pattern transitions. The method for estimating pressure drop in each flow regime is then developed. In developing the equations for pressure gradient, we note that for vertical flow of gas/liquid mixtures, 90 to 99% of the total pressure drop is usually caused by the static head. Accurate estimation of the in-situ gas void fraction is therefore of great importance. Flow Pattern Transition The often chaotic nature of multiphase flow makes it difficult to describe and to classify flow patterns and hence to ascribe criteria for flow-pattern transitions correctly. In addition, although flow patterns are strongly influenced by such parameters as phase velocities and densities, other less important variables---such as the method of forming the two-phase flow, the extent of departure from local hydrodynamic equilibrium, the presence of trace contaminants, and various fluid properties---can influence a particular flow pattern. Despite these deficiencies, a number of methods have been proposed to predict flow pattern during gas/liquid two-phase flow. Some of these methods could be extended to liquid/liquid systems with less accuracy. One method of representing various flow-regime transitions is in the form of flow-pattern maps. Superficial phase velocities or generalized parameters containing these velocities are usually plotted to delineate the boundaries of different flow regimes. Obviously, the effect of secondary variables cannot be represented in a two-dimensional map. Any attempt to generalize the map requires the choice of parameters that would adequately represent various flow-pattern transitions. Because differing hydrodynamic conditions and balance of forces govern different transitions, a truly generalized map is almost impossible. Still, some maps are reasonably accurate. Among these, the map proposed by Govier et al.4 has found wide use in the petroleum industry. The flow-pattern map of Hewitt and Roberts5 has also been widely accepted in academia and the power-generating industry. An alternative, more flexible approach is to examine each transition individually and to develop criteria valid for that specific transition. Because this approach allows physical modeling of individual flow patterns, it is more reliable than the use of a map.

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Wellbore fluid temperature is governed by the rate of heat loss from the wellbore to the surrounding formation, which in turn is a function of depth and production/injection time. We present an approach to estimate wellbore fluid temperature during steady-state twophase flow. The method incorporates a new solution of the thermal diffusivity equation and the effect of both conductive and convective heat transport for the wellbore/formation system. For the multiphase flow in the wellbore, the Hasan-Kabir model has been adapted, although other mechanistic models may be used. A field example is used to illustrate the fluid temperature calculation procedure and shows the importance of accounting for convection in the tubing/casing annulus. A sensitivity study shows that significant differences exist between the predicted wellhead temperature and the formation surface temperature and that the fluid temperature gradient is nonlinear. This study further shows that increased free gas lowers the wellhead temperature as a result of the Joule-Thompson effect. In such cases, the expression for fluid temperature developed earlier for single-phase flow should not be applied when multiphase flow is encountered. An appropriate expression is presented in this work for wellbores producing multiphase fluids.

Heat loss from the wellbore fluid depends on the temperature distribution in the formation. Formation temperature distribution around a well was modeled by Ramey (1962) by assuming a vanishingly small wellbore radius. The assumption, while robust for some cases, could lead to unrealistic predictions of early-time behavior. This paper uses a rigorous model of heat transfer developed with consideration for the appropriate boundary conditions; that is, the heat transfer at the formation/wellbore interface is represented by the Fourier law of heat conduction.The superposition principle is used to account for the gradual change in heat transfer rate between the wellbore and the the formation.The results of the new analysis are in agreement with the classical work of Ramey for large times (dimensionless time, to > 10). However, significant differences are noted between the proposed solution and that of Ramey's loglinear approximation at small (to < 1.0) times.References and figures at the end of paper.These differences may have considerable effect on certain applications, such as static earth temperature estimation from temperature logs, and bottomhole temperature estimation for cyclic steam injection.We also present an approximate algebraic expression for the rigorous integral solution of dimensionless formation temperature, To. This simplified expression is accurate for most engineering calculations. Use of this solution is shown in the second part of the paper where the wellbore fluid temperature distribution is computed during two-phase flow.