A comprehensive investigation was undertaken to study two-phase flow in a radial electric submersible pump (ESP) using diesel/CO 2 mixture data from an earlier study. Currently, no dynamic model is available for multistage pumps. A dynamic five-equation model was developed and verified. The model incorporates pump geometry, stage inlet pressure, inlet void fraction, fluid properties, and number of stages. The two-phase flow physics for the pump is found to be vastly different from that in pipe flow. Insights into a pump's surging tendency were obtained.


Within a 1D framework, single-phase pressure drop in pipes can be predicted to within about 2%. For two-phase flow in pipes, even with the thousands of papers written over the last 3 decades, errors of about 10% to 15% are not uncommon. For ESP's, even liquid-only flow behavior predictions can easily involve errors of 50%. Two-phase ESP behavior is obviously more complex and only recently began gaining attention. This results partly from new interest in offshore pipeline pumping. Also, more ESP's are being used in wells with some of free gas at pump intake, and gas ingestion occurs despite the use of efficient downhole separators.

The investigation reported here studied two-phase flow in a radial ESP (Fig. 1) using Les and Bearden's 1 diesel/CO2 mixture data.

Previous Work

Sachdeva2 gave a comprehensive description of all the previous studies.

The Russians did limited work on the problem of using ESP's in gassy wells as early as 1958. However, their studies were extremely simplistic, and the pump was treated as a black box. Nuclear industry studies constitute the bulk of literature described in Ref. 2. Apart from limited Russian literature, Lea and Bearden's1 study is the only one published for the petroleum industry. The diesel/CO2 data gathered for the radial I-42B stage was used to develop the model described here.

The nuclear industry models cannot be used in the petroleum industry for the following reasons.

  1. Most models rely on correlation rather than sound development based on flow physics.

  2. Nuclear industry pumps are volute-type, single-stage (as opposed to diffuser-type, multistage ESP's).

  3. Pump diameters are substantially higher (scaling effects are unknown).

  4. Most models are valid for very low inlet void fractions (<10%).

  5. Most studies have low inlet pressures (<10 psig).

  6. No pressure (multistaging) effects are considered.

The photographic evidence gathered in these studies has been valuable in the development of the present model. Also, papers by Patel and Runstadler,3 Zakem,4 and Furuya5 make for interesting reading.

Model Formulation

Fig. 2 represents the model formulation approach:

  1. derive an idealized, 1D liquid model,

  2. derive the same for two-phase flow, and

  3. use the difference (Step 1-Step 2) to get the multiphase head degradation.

Theoretical Liquid-Only Pressure Rise

The steady-state momentum equation for frictionless incompressible liquid flowing radially outward in an impeller channel is6

  • Equation 1

Where var, va?, and vaz= components of velocity in the radial, tangential, and axial directions, respectively, for a cylindrical coordinate system. We know that for the pump speed ranges involved, gr can be neglected along with (varz). Next, assume that var does not change in the ? direction, although fluid circulation does exist in the impeller (Figs. 3 and 4). Thus, Eq. 1 reduces to

  • Equation 2


  • Equation 3

Fig. 5 explains the nomenclature. The relationship between r and z is

  • Equation 4

Conservation of mass gives

  • Equations 5-7

And ß(r)= mr+c.

The pump manufacturer provides the blade angle profile. Combining Eqs. 2 through 7 yields (for the impeller)

  • Equation 8

Eq. 8 can be used for a diffuser if we note that d r/dz=-sinß and O=0. If the variation between ß1 and ß2 is assumed to be linear, r and z can be shown to be related:

  • Equations 9 and 10

Frictional losses are now incorporated into the above equation. The friction factor used in pipe flows cannot be used because velocity profiles in circular, rotating channels are quite different. The following steps are used to approximate the effects of curvature (for the diffuser) and both curvature and rotation (for the impeller).

  1. Assume a smooth channel surface (Blasius friction factor). This assumption is reasonable because manufacturers ensure that impellers and diffusers are very smooth so that losses are minimized.

  2. Calculate diffuser friction,

    • Equation 11

    where fc/fs= multiplicative factor to account for curved channels. Ito7 derived an expression for fc from the logarithmic velocity distribution law.

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