A Slug-Length Distribution Law for Multiphase Transportation Systems

Abstract

Summary A probabilistic approach is used to analyze the formation of slugs in nearlyhorizontal pipelines. Knowledge of the probability amplitude for a liquidbridge in the formation zone allows derivation of the slug-length distributionlaw. The method is suitable for simulating complex situations, includingunsteady flows. For horizontal steady flows, a simple model of thisprobability, expected to be valid in large-diameter pipes or for long slugs(tail of the distribution), yields a Gaussian law, which is distinct from thelog-normal law. Frequency, mean length, and variance depend on only oneparameter, which may be predicted from theoretical study (not included here). This new length distribution law is verified against experimental data obtainedwith an air/water test loop (42 mm in diameter and 400 m long) installed in the Beauplan production laboratory in France and is compared with the log-normaldistribution law. Introduction Two-phase flow occurs frequently. In offshore areas, the separation device often is located far from the extraction area, and large-diameter two-phaseflowlines are required over long distances. While pressure-loss predictions arequite acceptable, one needs more knowledge about the flow pattern at theflowline exit to design the terminal equipment correctly. For slug flow, theslug catcher must be able to handle the largest slugs. Therefore, it isnecessary to determine both the mean volume of the slugs and their statisticalvolume distribution. Little information about this problem has appeared so far in the literature. The few available theoretical models that describe slug formation aredeterministic and exclude statistical analysis of the distributions. From anexperimental point of view, length distributions look like skewed curves. From Prudhoe Bay data, Brill et al. proposed using a log-normal law. Thecorresponding parameters can simply be adjusted to fit the experimental data. Too few parameters can simply be adjusted to fit the experimental data. Too fewdata are available, however, so it is not possible to distinguish between thisproposition and others. Moreover, no theoretical prediction about the variancecan be made. prediction about the variance can be made. We present here aprobabilistic analysis of slug formation in a nearly horizontal straightsection of a pipeline. The prerequisite is knowledge of the probability for aslug to appear at a prerequisite is knowledge of the probability for a slug toappear at a given position and time. We show how all probabilistic data aboutthe slug pattern can be extracted if this probability function is known. Itstheoretical determination is rather delicate owing to the undeterministicnature of turbulent flow. Nevertheless, a simple model feasible. We discussthis hypothesis, which we expect to be valid at least for large-diameter pipesand for the tail distribution of the largest slugs. The resulting law is atruncated Gaussian one (truncated because of the necessity of having a minimumlength for a slug to be stable). The proposed law has to be tested against experimental data. The Beauplanproduction laboratory has an operational device, consisting of a 400-m-long,42-mm-diameter air/water test loop. Because of a lack of space, we do notpresent all our tests; instead, we focus only on a representative experimentthat shows good agreement with our statistical 1aw. The log-normal law is alsopresented for comparison. Formation Probabilities. Let us first discuss the main phases in slugformation (Fig. 1). Near the entrance, liquid and gas flow concurrently as twodistinct stratified phases, with the gas above the liquid. After some distance, a dynamic equilibrium is reached, yielding a nearly constant liquid height inthe pipe. Waves appear at the gas/liquid interface and eventually, at time tn, a wave is sufficiently high to bridge the pipe at location Xn, suddenlyblocking the onward gas flow. A new slug is born (Fig. 1a). The gas-flowblockage leads to a rapid acceleration of the liquid contained in the bridge. Consequently, the rapidly growing slug picks up all the slow-moving liquidahead of it. Behind it, some picks up all the slow-moving liquid ahead of it. Behind it, some liquid is shed and forms the slug tail at a level below themean level in the stratified zone (Figs. 1b and 1c). During this extension process, the slug moves away from the formation zone. This zone extends farther with a velocity v (here assumed to be constant) untila new bridge is formed at time tn+1 and location Xn+i. Then the entireprocedure begins again. The turbulent nature of the stratified flow does not allow a deterministicestimation of the point where a bridge will appear. Thus, it is necessary tointroduce the probability 1(x, t)dxdt for such a bridge to appear at a locationand time ranging from X to x+dx and from t to t+dt, respectively. Although thisfunction remains unknown, it is reasonable to assume that its shape is depictedat time t in Fig. 1c by the curve in Fig. 1d. The underlying assumptions are(l) that the formation zone is rather long (so that boundary effects can beneglected) and therefore the hydrodynamic conditions are the same at each pointand time in the formation zone and (2) that the instabilities giving birth toslugs develop almost locally in both space and time. Thus, the probabilitydensity 1 has a nearly constant plateau a. (We do not discuss here thedetermination of a.) A theoretical computation of this quantity would determinethe wavelength, lambda, of dynamic instabilities and their characteristic time, lambda, needed to form a bridge, yielding the estimate . Such considerationsare not developed further here. We now consider a a parameter to be adjusted. Let us consider the time interval tn less than t tn+1. As the formation plateauranges from x=O to x=xn+V(t-tn), one is led to model the probability density as(1) where O is the step function [for z >O, O otherwise]. Let (Pform(xn+1, tn+1 xn, tn) be the probability density for the (n+1)the slug toappear at(xn+1, tn+1);we assume that the nth slug appeared at (xn, tn). Takinginto account that no other slug should have been produced in the time interval, one readily obtains (2) The probability Q(x) for a bridge occurring at x, independent of the history of the preceding slug, satisfies the integralequation (3) which averages over all production time t' and over all possible locations of the previous slug birthplace, y. This equation has a unique solution if oneuses the normalization condition (4) We can now determine the frequency, f, from the mean period: (5) Finally, along with many other quantities, we want to display therepartition law for, where is the distance from the liquid bridge to the end of the formation zone. SPEPE P. 166