Foam computer model helps in analysis of underbalanced drilling

Gefei Liu, George H. Medley Jr.
Maurer Engineering Inc.
Houston

• Equations
• Equations of State and Coefficients

A new mechanistic model attempts to overcome many of the problems associated with existing foam flow analyses. The model calculates varying Fanning friction factors, rather than assumed constant factors, along the flow path. Foam generated by mixing gas and liquid for underbalanced drilling has unique rheological characteristics, making it very difficult to accurately predict the pressure profile.

A user-friendly personal-computer program was developed to solve the mechanical energy balance equation for compressible foam flow. The program takes into account influxes of gas, liquid, and oil from formations. The pressure profile, foam quality, density, and cuttings transport are predicted by the model.

A sensitivity analysis window allows the user to quickly optimize the hydraulics program by selecting the best combination of injection pressure, back pressure, and gas/liquid injection rates.

This new model handles inclined and horizontal well bores and provides handy engineering and design tools for underbalanced drilling, well bore cleanout, and other foam operations.

Foam has been used extensively in the petroleum industry for decades. It has been proven effective and economic as a circulating fluid in hole cleanout and drilling operations. Advantages of foam drilling over conventional mud drilling include high penetration rates, a high cuttings transport ratio, and less formation damage. In areas with low bottom hole pressures, the use of a lighter fluid, such as foam, is required.

The complex and unique flow mechanism involved in foam operations often confuses drilling operators about the optimum combination of liquid and gas injection rates. Other questions remain, such as how to predict the bottom hole pressure and how to combine different controllable variables to optimize results. Existing foam drilling design methods largely depend on field operation charts or on calculations using a mainframe computer. During the past 20 years, extensive study of foam rheological behavior and factors affecting foam circulation in oil wells has made it possible to develop a comprehensive computer program to meet the demands of foam drilling design and analysis.

From existing foam rheology models and the steady-state mechanical energy balance equation, Maurer Engineering Inc. has developed a Windows-styled computer model (FOAM) to help operators with the design and analysis of underbalanced drilling. The program simulates underbalanced drilling operations using foam as a circulating medium and can be used to evaluate and develop operational guidelines.

This article presents the foam flow equations and explains how to numerically solve compressible non-Newtonian flow in a three-dimensional well bore. Equations of state describing pressure, volume, and temperature interactions of compressible foam are presented. Flow regimes ranging from laminar to turbulent are covered.

Rheological models

Foam can be treated as a homogeneous fluid with variable density and viscosity. During foam operations, foam quality depends on the pressure and temperature in the tubing or annulus. The pressure has to be determined through the mechanical energy balance equation, in which the frictional pressure drop term relies on the foam rheological model. It is therefore important to have an accurate rheological model describing dynamic foam behavior.

Theoretical approaches to the rheology of foam were presented by researchers in the early 1900s.1-3 Mitchell demonstrated that foam behaves as a Bingham plastic fluid, based on his experimental work in capillary tubes, and he empirically derived a set of equations for foam viscosity.4 It should be noted that these equations do not apply at the limiting case of 100% foam quality.

Krug presented plastic viscosities and yield strengths of foam as a function of foam quality.5 Beyer, et al., first formulated a foam rheological model from laboratory and pilot-scale experimental data.6 Their observations suggested that foam behaves as a Bingham plastic fluid. Their study did not demonstrate a dependence of yield point on liquid volume fraction or foam quality.

Sanghani and Ikoku experimentally studied foam rheology with a concentric annular viscometer that closely simulated actual hole conditions.7 They concluded that foam is a power-law pseudoplastic fluid with flow behavior index n and flow consistency K, both functions of foam quality.

A review of the literature shows varied opinions on foam rheological models. Some researchers found that the power-law model was statistically superior to the Bingham plastic model in correlating data, while others found that foam more closely obeys the Bingham plastic model. The computer model described here includes both power-law and Bingham plastic models and allows the user to choose which one to use.

Foam flow equations

In the special case of a two-phase system such as foam, where gas is finely and uniformly dispersed in the liquid phase, homogeneous fluid can be assumed, and no equation is required for the phase interface.

Foam consists of a compressible component (gas) and an incompressible component (liquid). The incompressible component is easier to handle because of its constant density. The compressible gas requires much more attention because its density depends on temperature and pressure.

Pressure is coupled with gas volume fractions through a friction factor. An improved version of Lord's pressure drop equation and Spoerker's method are used in the following equation derivation.8-9 The friction factor is calculated along the well bore rather than assumed constant.

Equations of state

The relationship between the variation of density of a fluid with pressure and temperature is termed the equation of state. For engineering purposes, the most practical form of the equation of state for a real gas is given by Equation 1.10

Equations of state for both downward and upward foam flow can be expressed as Equation 2. The coefficients a and b are defined in the accompanying box and Table 1.

The expressions of a and b take different forms for downward flow inside drill pipe and upward flow in the annulus. For upward flow in the annulus, the foam is mixed with rock cuttings. There are three phases present in the annular mixture in which liquid and cuttings are incompressible, whereas the gas phase is compressible.

Mechanical energy equations

Once the equations of state for foam have been established, the next step is to use the momentum and energy equations to analyze the dynamic foam behavior. The mechanical energy equation may be considered either a consequence of the momentum equation or a reduced form of the total energy equation.

For downward flow inside the drill pipe, the differential mechanical energy balance equation is given by Equation 3.

The average velocity of the foam, u, can be obtained using the continuity equation. In terms of specific volume, it can be expressed as Equation 4. The coefficient c is defined in the accompanying box.

After substituting Equation 4 into Equation 3, the differential mechanical energy balance takes the form in Equation 5. For upward flow in the annulus, the differential mechanical energy balance equation takes the form in Equation 6.

The average velocity of the foam in the annulus is also described by Equation 4. The variable c for upward annular flow is different, however (see accompanying box).

Substituting annular velocity into Equation 6 will yield the differential mechanical energy balance in upward annular flow (Equation 7).

Equations 5 and 7 can be solved numerically. The back pressure, which is known, serves as a boundary condition for Equation 7. Numerical techniques are used to calculate a sequence of pressure values corresponding to discrete values of the measured depth. The expressions of Fp and Fa are given in the accompanying box.

Pressure drop across nozzles

To calculate the pressure drop through a short constriction such as a bit nozzle, it generally is assumed that the change in elevation, the velocity upstream of the nozzle, and the frictional pressure loss across the nozzle are negligible (Fig. 1 [18482 bytes]). Thus, Equation 3 becomes Equation 8.

Substituting Equations 2 and 4 into Equation 8 and integrating yields Equation 9 in field units. Nozzle velocity Un is defined by Equation 10.

Equation 9 can be solved numerically to obtain the pressure upstream of nozzle P1. The bottom hole pressure P2 is calculated from Equation 7 beforehand.

Influx modeling

One advantage of foam drilling is a lower bottom hole pressure, which helps increase the rate of penetration. Influxes of gas, water, or oil can occur as a result of low bottom hole pressure, however. These influxes will change the existing foam system, resulting in a change in the pressure profile inside the drill pipe as well as in the annulus.

The total liquid density can be calculated from the rates and densities of the injected liquid and those of water/oil influxes (Equations 11 and 12).

The final liquid viscosity can be calculated in a similar fashion.

The molecular weight of the mixture of injected gas and influx gas can be calculated using weighting factors similar to those used for calculating liquid density and viscosity (Equations 13 and 14).

Equations of state for gas and upward annular foam flow should use these adjusted parameters for annular positions above the influx points.

Program operation

The FOAM model uses four sets of input data to organize well and drilling data and rheological parameters. Each of the four sets of input data is stored in a separate file. The well data input file stores well and field names and other documentation to identify the specific case being calculated. The well bore directional profile is described in the survey data input file. Inclination and azimuth are recorded at the corresponding survey point measured depth.

The third FOAM input file, tubular data input, includes information on the drillstring, casing, hole size, and bit nozzles. Additionally, locations and rates of influxes of gas, water, and oil from one or more zones can be specified. The final input is the parameter data input file, in which the user specifies gas and liquid injection rates and properties, drilling rate, cuttings size, rock density, temperature gradients, back pressure, and gas and fluid rheological data.

After the four data input files are completed, the calculations are performed. FOAM then tiles up the output screens, allowing the user to click on the individual output screens of interest.

Fig. 2 [16659 bytes] shows a foam pressure profile where the pressure increases from 665 psi at the compressor to 1,180 psi at the hole bottom and then decreases as aerated fluid flows up the annulus and expands. This pressure profile is useful in ensuring that pressures do not exceed frac pressure and formation pressure.

Fig. 3 [15987 bytes] shows that foam density for this example increases from 2.4 ppg at the surface to 3.3 ppg at the hole bottom, then increases to 3.7 ppg from the addition of cuttings at the bit and then decreases to air density as the gas expands in the annulus.

Accurate calculation of cuttings lifting velocity is critical for conducting safe and efficient drilling operations because poor hole cleaning is a major problem with air, gas, and mist drilling. Cuttings lifting problems are most critical at the top of the drill collars where the velocities are lowest.

Fig. 4 [15904 bytes]shows an example where the cuttings lifting velocity is only 56 fpm at the top of the drill collars because of the reduction in pipe diameter at this point. Large cuttings often cannot be lifted beyond this point and remain there until they are reground to a smaller size. This explains why air cuttings are often fine powder, while foam cuttings are much larger (1/8-1/4 in. diameter) because of the better lifting capacity of foams.

FOAM's output section also includes a sensitivity analysis feature that allows drilling engineers to observe the effects of changes in air and liquid injection rates and choke pressure on various downhole parameters. These variables, which can be adjusted in the field if problems occur, have been combined into a separate screen for ease of use in planning and troubleshooting wells. As changes are made in these variables, the effects of those changes can be viewed immediately, allowing quick optimization of the variables.

Comparison to other models

The FOAM computer model was validated by comparing it to existing test-well measurements and field measurements.

Comparisons were initially made with Chevron Petroleum Technology Co.'s Foamup computer program. Foamup output was derived from data provided by Chevron. In 1972, Chevron ran extensive laboratory tests for developing its model using a test well.

Foamup serves as the current industry standard for foam predictive models, even though it was developed in the early 1970s. The model runs in a mainframe environment.

Foamup is based on Bingham plastic fluid rheology, with a constant set value for fluid yield point. A constant yield point may not accurately model the rheology of foam fluids because, as the pressure changes at different depths, the foam quality also changes, resulting in changes in fluid viscosity and yield point. However, good results have been obtained with the Chevron model over the years.

The FOAM computer model includes the option of using the same basic fluid rheology model used by Chevron. FOAM also includes an option to change the initial yield point of the fluid used in the calculations. The Chevron rheology option was used as the basis of comparison between FOAM and Foamup. Twenty-three different cases were run to compare FOAM and Foamup using data from Chevron's tests.

Fig. 5 [16045 bytes]shows that the difference in the surface injection pressures predicted by the two programs was 0.7-21.2% for foam qualities of 74-100%. Foam qualities were all calculated at the surface in the annulus. The average difference in calculated injection pressure is 8.2%. This level of agreement is acceptable, assuming that one of the models can be demonstrated to be accurate.

The FOAM program, as currently designed, is based on theoretical assumptions that are valid for true foams only. If foam quality is greater than 96-97%, the calculations are not necessarily valid. Fig. 5 [16045 bytes] shows that FOAM agrees best with Chevron's Foamup at foam qualities of 99% and higher, however.

The average difference in bottom hole pressure predicted by the two programs in the true foam region (i.e., <97% foam quality) is only 9.9%. This level of agreement is acceptable, again assuming that either model is capable of accurate pressure prediction. FOAM is designed just for this application.

Two other models were identified in the available technical literature. The first of these was a Bingham plastic model developed by Krug and Mitchell.11 The other is a power-law model developed by Okpobiri and Ikoku.12

A comparison of FOAM's Bingham plastic model with the Krug/Mitchell model shows that the difference in predicted surface pressures between the two models ranges 10.3-18.2%. The average difference for the 11 test cases was 13.9%. In every case, the FOAM computer model predicted surface pressures lower than those predicted by the Krug/Mitchell model.

The bottom hole pressures predicted by the models differed from 4 to 16.2%. The average difference for all 11 cases was 9.5%. Again, all predictions by FOAM were lower than those made by the Krug/Mitchell model. A difference of less than 10% is acceptable, again assuming that either model is accurate.

A comparison of results from FOAM's power law fluid model option with the Okpobiri/Ikoku model found the closest agreement of any model considered. The differences in surface pressure predictions range from 1.3 to 13.6%. Predicted bottom hole pressures differed by 8.5-18.8% for the cases shown.

The average difference in surface pressures predicted by FOAM and Okpobiri/

Ikoku for all 11 cases run was only 5.2%; the average difference for bottom hole pressures was 8.6%. Both of these differences indicate acceptable agreement between these models.

Comparison to lab data

Results made available by Chevron from 1972 unpublished test well measurements were the best source of data available to gauge the accuracy of FOAM. Chevron's test well had a plugged-back total depth of 2,904 ft with 95/8-in. casing from surface to total depth. A string of 27/8-in. tubing was run to 2,809 ft. There were recording pressure gauges installed behind the tubing in the tubing-by-casing annulus at depths of 2,809 ft, 1,887 ft, and 953 ft.

Air and liquid (i.e., foam) were injected at 23 different rates and mixtures, and pressures were recorded by the gauges and at the surface. Each of these 23 data points was used to validate the FOAM model up to a foam quality of 97% because foams become unstable above this value. The pressures predicted by the model at each gauge depth and at the surface injection point were all compared to the measured pressures. All results were plotted as a function of foam quality at the surface.

Pressures predicted at the surface matched the measured pressures more closely than at any other location. The error in the surface pressure predictions made by FOAM ranged from 0.3% to a maximum of 30.3% (Fig. 6 [15879 bytes]). The average error was 11.1%.

This level of agreement (11.1%) is an acceptable level of accuracy. The measured data may have been no more accurate. None of the tests was repeated to gauge repeatability or precision of the measurements.

The pressures predicted at the bottom of the hole by FOAM were in close agreement with the measurements (Fig. 7 [15134 bytes]). The error ranged 1.1-25.8%.

The average disagreement between FOAM and the measurements for bottom hole pressure was 10.0%.

Comparison to field data

To collect field data under specific conditions, Maurer Engineering personnel visited a drilling location while foam drilling was under way. The operation was a horizontal reentry, and the visit occurred during the kickoff operation.

Accurate data were collected for two separate sets of conditions. One set of data was recorded when the bit was at the kickoff point; another set of measurements was taken when the hole was about 130 ft deeper at an inclination of 28. At both depths, surface pressure and liquid and gas injection rates were recorded.

The accuracy of the FOAM program ranged from +3.1 to 24.8%. Foam qualities for the two cases, calculated at a depth of 100 ft in the annulus, were 95.1 and 96.0%, respectively. The most accurate prediction was made at the lower foam quality.

The calculated surface pressures differed from actual measured pressures by only 20-40 psi.

In the first case, actual injection pressure was 610 psi with 800 scfm of air and 31 gpm of foamer solution. The second data point was measured with an injection pressure of 750 psi with 800 scfm of air and 24 gpm of foamer. The FOAM model predicted injection pressures of 629 psi and 714 psi, respectively, for the two cases.

Results

The FOAM model predictions compared favorably with Chevron's and other published foam models.

The model predictions also correlated well with field data from Chevron well data and a foam-drilled well in Kansas.

FOAM handles only foam fluids and will not handle cases where the foam quality is larger than 0.97.

The FOAM model is being expanded to include the transition from foam to air/mist in the annulus. This expanded model called Mudlite, being developed on the Drilling Engineering Association DEA-101 project, handles all cases from pure air to pure foam drilling.

Acknowledgment

The authors thank Harry Dearing of Chevron Petroleum Technology Co. for his advice and help in comparing the FOAM model to existing data. The authors also thank John Duda of the U.S. Department of Energy for his support and project guidance in developing the model.

References

1. Einstein, A., "Eine Neue Bestimmung der Molekuldimensionen," Annalen der Physik, Vol. 19, No. 5, 1906, p. 289.

2. Hatschek, E., "Die Viskositt der Dispersoide. I. Suspensoide," Kolloid Z., Vol. 7, 1910, p. 301.

3. Hatschek, E., "Die Viskositt der Dispersoide. II. Suspensoide," Kolloid Z., Vol. 8, 1910, p. 34.

4. Mitchell, B.J., "Viscosity of Foam," PhD dissertation, University of Oklahoma, 1969.

5. Krug, J.A., "Air and Water Requirements for Foam Drilling Operations," MS thesis, Colorado School of Mines, 1971.

6. Beyer, A.H., Millhone, R.S., and Foote, R.W., "Flow Behavior of Foam as a Well Circulating Fluid," Society of Petroleum Engineers paper 3986, presented at the SPE 47th Annual Fall Meeting, San Antonio, Oct. 2-5. 1972.

7. Sanghani, V., and Ikoku, C.U., "Rheology of Foam and Its Implications in Drilling and Cleanout Operations," American Society of Mechanical Engineers paper AO-203, presented at the Energy-Sources Technology Conference and Exhibition, Houston, Jan. 30-Feb. 3, 1983.

8. Lord, D.L., "Analysis of Dynamic and Static Foam Behavior," Journal of Petroleum Technology, January 1981.

9. Spoerker, H.F., Trepess, P., Valk, P., and Economides, M.J., "System Design for the Measurement of Downhole Dynamic Rheology for Foam Fracturing Fluid," SPE paper 22840, presented at the SPE 66th Annual Meeting, Dallas, Oct. 6-9, 1991.

10. Grovier, G.W., and Aziz, K., The Flow of Complex Mixtures in Pipes, Robert E. Krieger Publishing Co., Malabar, Fla., 1987.

11. Krug, J.A., and Mitchell, B.J., "Charts help find volume pressure needed for foam drilling," OGJ, Feb. 7, 1972, pp. 61-64.

12. Okpobiri, G.A., and Ikoku, C.U., "Volumetric Requirements for Foam and Mist Drilling Operations," SPE Drilling Engineering, February 1986.

13. Graham, R.L., "Air Drilling Technology Needs Assessment," Final Report under Contract No. GRI-95/0039 for the Gas Research Institute, Chicago, Reuben L. Graham Inc., October 1995.

14. Mitchell, B.J., "Test data fill theory gap on using foam as a drilling fluid," OGJ, Sept. 6, 1971, pp. 96-100.

Based on a presentation at Energy Week Conference & Exhibition, Houston, Jan. 29-Feb. 2.

The Authors