Predicting Liquid Gradient in a Pumping-Well Annulus A. Rashid Hasan, SPE, U. of North Dakota C. Shah Kabir, SPE, Schlumberger Perforating & Testing Center Rehana Rahman"', U. of North Dakota Summary. This work proposes a hydrodynamic model for estimating gas void fraction, f g , in the bubbly and slug flow regimes. The model is developed from experimental work, involving an air/water system, and from theoretical arguments. The proposed model suggests that prediction of 1: , and hence the bottomhole pressure (BHP), is dependent on such variables as tubing-to-casing- diameter ratio, densities of gas anJ liquid, and surface tension. Available correlations do not include these variables as flexible inputs for a given system. Computation on a field example indicates that slug flow is the most dominant flow mechanism near the top of liquid column at the earliest times of a buildup test. As buildup progresses, transition from slug to bubbly flow occurs in the entire liquid column. Beyond the afterflow-dominated period, the effect of bubbly flow diminishes as gas flow becomes negligibly small. Comparisons of BHP's are made with the proposed and available correlations. Because the proposed model predicts fg between those of the Godbey-Dimon and Podio et al. correlations, BHP is predicted accordingly. Introduction Acoustic well sounding has become a well-established method for estimating BHP in a pumping oil well. The method involves deter- mining the gas/liquid interface in the tubing/casing annulus. From the knowledge of the lengths of gas and liquid columns, BHP can be estimated by adding the pressures exerted by these'columns to the casinghead pressure. Although simple in concept, this indirect BHP calculation presents potential problems in two areas: resolution of the acoustic device measuring the gas/liquid interface and esti- mation of the gas-entrained-liquid-column density. Significant progress has been made in the acoustic device's ability to monitor the movement of the liquid column as a function of time during a buildup test. Estimation of the changing liquid-column den- sity, however, is still fraught with uncertainties. The dead-liquid gradient needs to be adjusted by the so-called gradient correction factor, F gc (=I-f g ) to reflect the true column density. Three F gc correlations proposed by Gilbert, I Godbey and Dimon, 2 and Podio et al. 3 have found wide use in th,e petroleum industry, Relative merits of these correlations were addressed re- cently, 4 Research in two transfer in gas/liquid sys- tems and two-phase flow-has produced a wealth of information for predicting gas void fractions in stagnant liquid columns. None of these correlations, however, account for the effect of casing and tubing diameters on F gc . Reliable F gc prediction is critically im- portant because afterflow, during which the gas continues to bubble through the liquid column, dominates a buildup test in a typical pumping well. In most cases, semilog period beyond afterflow is seldom reached to allow conventional analysis for estimating permeability-thickness product, skin, and static pressure. Thus, anal- ysis of afterflow-dominated transients provides a viable alternative to the conventional semilog analysis. The purpose of this paper is to explore the relevant literature and to develop a hydrodynamic model for a practical range of flow con- ditions encountered in a pumping-well annulus through theoretical considerations and experimental work. Theoretical and Experimental Models Concerning Bubbly Flow. Researchers of multiphase flow have usually correlated the void fraction with drift flux, u. Drift flux is a way of expressing the difference between the in-situ velocities of the two phases, v g and vi-Le., "slip"-and is defined by the following expression: u=(v g -vN g (1-f g ). ,,, .. ,., ,, .. , (1) 'Now with Northeastern U, Copyright 1988 Society of Petroleum Engineers SPE Production Engineering, February 1988 Eq. I may be written in terms of the measurable superficial ve- locities of the phases, v gs and Vis' by noting that v gs =fgvg and Vis =(I-fg)VI: u=vgs(1-fg)-v/sfg' , '" " ', ,(2) For ideal bubbly flow, Wallis 5 suggests the following semitheo- retical relationship between drift flux, gas void fraction, and ter- minal rise velocity of a gas bubble, V eo : u=v eo f g (1-f g )n. ' .. , , (3) For a stagnant liquid column when Vis =0, substituting U from Eq. 2 into Eq. 3 gives There are a number of correlations 6,7 available for the terminal rise velocity, V eo , in an infinite medium. The equation proposed by Harmathy, 7 which has been supported by data from indepen- dent sources,8 gives V eo = 1.53[gu(PI-P g )]O,25/./P7 . ., , (5) Wallis 9 suggested that the value of n=2 be used along with Eq. 5 for V eo ; therefore, Eq. 4 becomes V gs = l.53f g (1-f g )[ga(PI-P g )]O.25 /./P7 c::.1.53f g (1-f g )(gu/p/)O.25, ., , , (6) There are disagreements 10-13 over the exact value of to be used in Eq. 4. The value of n is affected by impurities in the liquid, 9 the way bubbles are introduced into the column, 12 and the distance these bubbles travel from the point of injection. 9 Many workers have used a negative value for the exponent n and expressed fg/(1-fg)nl as a function of superficial gas velocity, For example, Mersmann 14 proposes the following relationship: fg (PI)!4(Plu3)V,'(PI)7;2 (l-f g )4 =0.46vgs ag gvt Pg'" , ........ (7) Akita and Yoshida 15 propose a similar equation with slightly different groupings of properties and values of constants. A number 113