Abstract Accurate prediction of the average plunger rise velocity is important to optimally unload liquid from a plunger lifted well. A modified Foss and Gaul model presented in this paper predicts the minimum build casing pressure criteria to open the motor controlled valve for a selected plunger rise velocity. Results from the modified Foss and Gaul model are contrasted with results from the original Foss and Gaul model. The minimum build casing pressures predicted using both models are compared to measured casing pressures at the same rise velocities measured from field data acquired on 10 different plunger lifted wells. During the shut-in time period the plunger falls through gas, liquid and then rests at bottom on the bumper spring. At the end of this time period when the surface valve opens sufficient pressure is required to build to enough magnitude to unload the accumulated liquid and conventional plunger from the bottom of the well to the surface. An industry rule-of-thumb load factor criterion is frequently used to determine at what casing pressure build, Pc, the well should be opened to unload the liquid to the surface. The best technique to predict the maximum casing build pressure, Pcmax, is to use a Foss and Gaul type model to predict the rise velocity within a range of 500–1000 feet per minute (fpm), more optimally at 750 fpm that will unload the plunger and liquid to the surface. The modified Foss and Gaul model favorably predicts a required casing operating pressure to bring the plunger and liquid to the surface at a specified average rise velocity. Using the modified model will allow an operator to determine the maximum shut-in casing pressure a plunger lifted well should be allowed to build before opening the valve and safely bring the plunger to the surface. Introduction The most common form of plunger lift is the conventional plunger lift method. This method includes an after-flow production time period with the plunger held at the surface. During the latter time of the production period, or when liquids are sensed to be accumulating in tubing of the well. The well is shut in for a period of time required for the plunger to fall through gas, through accumulated liquids, plus an additional time where the plunger will rest on the bottom hole bumper spring. The well is then opened to begin the unloading period and the plunger with liquids above, rises to the surface, delivers the liquid slug and then the production period begins again with the plunger held by differential pressure at the surface. Another form of plunger lift is continuous flow or a quick drop plunger cycle with minimum shut-in time. The type of plunger that opens to allow gas to pass through the plunger during the fall is not discussed in the paper. The emphasis of this paper is a model (compared to field data) to predict, during shut-in, when the casing operating pressure has reached a value of pressure that will bring the plunger and liquid slug up the well at a desired average velocity of rise. The desired velocity of rise, from industry experience, is 500–1000 fpm with be best value being 700–800 fpm average rise velocity. This paper presents a modified Foss and Gaul model to determine how casing operating build up pressure relates to average rise velocity of the plunger and liquid slug.

Abstract In this paper, the buckling equation and natural boundary conditions are derived with the aid of calculus of variations. The natural and geometric boundary conditions are used to determine the proper solution that represents the post-buckling configuration. Effects of friction and boundary conditions on the critical load of helical buckling are investigated. Theoretical results show that the effect of boundary conditions on helical buckling becomes negligible for a long pipe with dimensionless length greater than 5p. Velocity analysis shows that lateral velocity approaches infinity and lateral friction becomes dominant at the instant of buckling initiation. Thus, friction can significantly increase the critical load of helical buckling. However, once buckling is initiated, axial velocity becomes dominant again, and lateral friction becomes negligible for post-buckling behavior and axial load transfer analysis. Consequently, it is possible to seek an analytical solution for the buckling equation. To verify the proposed model and analytical results, the authors also conducted experimental studies. Experimental results support the proposed solutions. Introduction It is well known that when a pipe is subjected to an axial compressive load, the pipe will shorten due to axial compression. As the axial load increases, the pipe may change from a straight configuration to a sinusoidal wave-like or helical shape. These three different configurations are all static equilibrium states of a pipe subjected to axial compressive load. However, for a given axial load, only one configuration is stable. The critical load beyond which a pipe will change its configuration from a straight line into a sinusoidal wave-like shape is called the critical load for sinusoidal buckling. The critical load beyond which a pipe will change its configuration from a sinusoidal into a helical shape is called the critical load for helical buckling. One important issue of buckling analysis is to determine the critical load beyond which a pipe will change its configuration from one form to another. Another important issue is to obtain a solution that represents the post-buckling configuration when the axial load is greater than the critical load. Many researchers have been involved in the stability and post buckling analysis of tubing and drill pipe in vertical wells (A. Lubinski 1950, 1962, R.F. Mitchell, 1986, 2002), inclined wells (P.R. Paslay and D.B. Bogy, 1964, R. Dawson and P.R. Paslay, 1984, G. Gao, 1996, S. Miska et al, 1995, 1996, R.F. Mitchell, 1988, 1997), horizontal wells (Y. Chen, et al., 1990, G. Gao, 1996, G. Gao and S. Miska, 2008, S. Miska, et al, 1996, R.F. Mitchell, 2002, J. Wu and H. C. Juvkam-Wold, 1993(a), 1993(b)). Researches have also studied curved wells (G. Gao, 1996, X. He and A. Kyllingstad, 1993, W. Qiu, et al, 1998, R.F. Mitchell, 1999, 2006(a)), and tubing and pipe subjected to different kinds of loads including axial compressive load, torque (J.C. Cunha, 1995, G. Gao, 1996, S. Miska and J.C. Cunha, 1995, J. Wu, 1997), and axial frictional drag (G. Gao, 1996, X. He and A. Kyllingstad, 1993, S. Miska et al, 1996, R.F. Mitchell, 1986, 1995, 1996, J. Wu, H. C. Juvkam-Wold, 1993). The concept of helical buckling and the determination of the helical configuration of a buckled pipe proposed by A. Lubinski (1962) is one of the greatest break-through in modeling post-buckling of pipe constrained in a wellbore.

This paper was also presented as SPE 94156, "Does Vogel's IPR Work for Fractured Wells?" at the 2005 SPE Europec/EAGE Annual Conference held 13–16 June, in Madrid, Spain. Abstract It is a common practice in the oil industry that production engineers use Vogel's correlation to correct the IPR curve below the bubble pressure for unfractured and fractured wells. However, there has not been a comprehensive investigation to ensure if the Vogel's correlation can be applied for fractured wells. This paper presents a new correlation to build IPR curves or predict production performance below the bubble pressure for fractured wells. In order to investigate fractured well performance below bubble point, about 1,000 simulations runs were performed using well-refined size grid for several sets of relative permeability curves and PVT data. The simulation model has been validated against analytical solution. Those runs cover a big practical range of fracture penetration 0.1 to 1.0 and dimensionless fracture conductivities from 0.5 to 50. Steady state conditions were analyzed at this study.All the mechanisms that cause the difference between fractured well and radial flow performance below the bubble pressure has been also well studied and will be presented in this paper. It was found that Vogel's correlation underestimates fractured well performance below bubble point. Vogel suggests a correction of AOF by 45% meanwhile the simulation results and new correlation show that the correction should be only 22%. Therefore, engineer could have an error of 43% using Vogel for estimating AOF for a fractured well. Another finding of this study is that multiphase effect is dependent on fracture conductivity and almost independent on fracture penetration. Higher conductivity fractures has bigger gas banks therefore they are affected by multi-phase effect to greater extent than lower conductivity ones. The new correlation is now being used for different fields and better fits the data than Vogel's correlation. Introduction In 1968 Vogel1 developed a correlation to estimate IPR curves for two-phase flow. Vogel's study was based on a numerical simulation, assuming radial flow, initial reservoir conditions at the bubble pressure, undamaged well and pseudo-steady state. In his simulations, Vogel used 4 sets of PVT and relative permeability data and showed that at those conditions AOF decreases 1.8 times due to multi-phase effect. However, Vogel's correlation is often used in a wider range of conditions than it was developed, including fractured wells, reservoirs above bubble pressure and steady-state flow. Vogel's deliverability curve is described by the following equation.

Abstract This work presents a fractional flow theory analysis of gravity-dominated, immiscible displacement in porous media. This effort represents the first to extend Buckley-Leverett theory to account for the effects of countercurrent flow while fluid injection and production is ongoing. The driving force for production is ongoing. The driving force for countercurrent flow is gravity, phase density differences, and a non-horizontal flow dimension. Our analysis is limited by the usual fractional flow theory assumptions, most importantly one-dimensional flow. Countercurrent flows are possible if the ratio of gravity to viscous forces is sufficiently large. Our analysis characterizes the gravity to viscous force ratio by a dimensionless Gravity Number. Our effort yields a more general and unified theory of immiscible displacement. We present new graphical solution methods to predict displacement performance for arbitrary initial and injected conditions. Introduction The behavior of immiscible displacement in porous media depends on viscous and gravity forces. In general, one-dimensional flows may be uni-directional or countercurrent. Flow may also take place in either open or closed systems. Open-systems imply fluid injection into and/or fluid withdrawal from the system. Closed systems, in contrast, preclude injection or production. Viscous-dominated flows take place exclusively in open systems and are characterized by uni-directional flow. The theory of viscous-dominated, immiscible displacement was first introduced by Buckley and Leverett and Welge. Gravity-dominated flows, in contrast, may occur in either open or closed systems and may involve either uni-directional or countercurrent flow. The driving force for countercurrent flow is gravity, phase density differences, and a dipping reservoir. phase density differences, and a dipping reservoir. Gravity-dominated flows have been discussed by several investigators. Martin, Templeton et al. and Coats et al.' have discussed countercurrent flows; however, they limited their studies to systems precluding any fluid injection or production, i.e precluding any fluid injection or production, i.e closed systems. Richardson and Blackwell, Hagoort, and Joslin have discussed the performance of gravity-dominated floods; however, their applications were restricted to uni-directional flows in open systems. A unified theory accounting for the effects of countercurrent flow in both open and closed systems. i.e.. m systems with or without fluid injection, has not been presented to date. The purpose of this work is to present the general theory of immiscible displacement in porous media covering viscous- and gravity-dominated flows. This effort is the first to address such a wide range of displacements. ASSUMPTIONS The development presented herein is based on the following assumptions or idealizations:1. A one-dimensional, isothermal, homogeneous permeable medium.2. At most, there are three components: oil, water, and gas.3. At most, there are two flowing phases: either water and oil or oil and gas.4. The water component will not partition into the oil and gas phases.5. The oil component will not partition into the water or gas phases.6. The gas component will not partition into the water or oil phases.7. Darcy's law applies.5. No adsorption of components by the solidphase.9. No dissipation, i.e., no capillary pressure. diffusion, dispersion, or viscous fingering.10. The system is incompressible and the effects of pressure on the phase behavior and fractional flow are negligible.11. The initial fluid distributions are uniform.12. The relative permeability relationships are constant and not subject to hysteresis.13. If the system is open, the injected condition is constant. P. 197