Well control model analyzes unsteady state, two-phase flow

• Equations [.pdf file]

A dynamic two-phase well control model accurately analyzes the behavior of kick fluids based on a realistic assumption of unsteady state two-phase mixture flow.

Two new sets of finite difference equations are developed to account for the effect of changing flow geometry. Therefore, the model is applicable for realistic two-phase well control simulation in wells with variable flow geometry.

This two-phase well control model works for onshore and offshore wells with water-based muds. Pressure responses during well control operations are analyzed by the solution of conservation of mass and conservation of momentum equations. The model also includes the effect of well control method, formation influx, and bubble rise velocity.

Well bore geometry and gas slip velocity are important factors in the prediction of the pressure behavior of kick fluids. To analyze any field well control choke pressure, pump pressure or bottom hole pressure (BHP) should be included.

Well control simulation has received attention in recent years because of its applicability and flexibility, and many computer models have been developed to analyze kick behavior.^2-7 The main objective of a kick simulator is to predict pressure and volume behavior of kick fluids as a function of time.

One of the earliest mathematical models was published by LeBlanc, et al., assuming a known volume of gas kick as a single phase.² They ignored effects of frictional pressure loss, gas-mud mixture, gas slip velocity, and reservoir parameters.

Nickens presented a dynamic two-phase computer model of a kicking well.³ He set up finite difference equations for uniform well bore geometry. Nickens also applied those equations for variable well bore geometry by changing the gas and liquid velocities at the locations of changes in area. He did not modify the conservation of momentum equation for uneven well bore geometry. The same equations were used by Starrett, et al., and Santos, but their research was limited to uniform well bore geometry.^{4 5}

Santos studied well control in horizontal wells. He considered bubble rise velocities at the top of the two-phase region, but he ignored the gas slip velocity at the bottom of the two-phase region.⁵

The dynamic two-phase well control model described in this article accurately analyzes the behavior of kick fluids based on a realistic assumption of unsteady state two-phase mixture flow. Two new sets of finite difference equations are developed to account for the effect of changing flow geometry. Therefore, the model is applicable for realistic two-phase well control simulation in wells with variable flow geometry.

The model also investigates the numerical problems associated with a two-phase model and their management. The simulation results are compared with results from field well control observations, a full-scale rig simulator, and a single-phase model.

The appropriate assumptions and governing equations are critical to simulate realistic two-phase well control operations. This two-phase model is based on the following assumptions:

• Unsteady state two-phase flow

• One-dimensional flow along flow path

• Water-based mud, in which gas solubility is negligible

• Incompressible mud

• Known mud temperature gradient with depth

• Kick occurs at the bottom of the well during drilling.

Two-phase region

Eight variables describe the two-phase flow system completely: pressure, temperature, gas and liquid fractions, gas and liquid densities, and gas and liquid velocities. Several different forms of the governing equations have been observed in the literature because of different assumptions, different notations, typographical errors, or even misunderstanding of the equations.

Starrett, et al., started with incorrect governing equations for their momentum balance equation for a two-phase mixture.⁴ However, they ended up with correct finite difference equations for the momentum balance equation, which was taken directly from the Nickens paper.

White, et al., considered the flow area change in their governing equations, but they disregarded the flow area effect in the momentum balance equation.⁶ That resulted in inconsistent units in their momentum balance equation. They did not provide their finite difference equations.

There are still five unknowns such as gas and liquid velocities, gas fraction, pressure, and gas density based on the above assumptions. Therefore, five equations are required to calculate the unknown variables with boundary conditions.

The conservation of mass equation for mud is given by Equation 1, and the conservation of mass equation for gas is given by Equation 2. The conservation of linear momentum equation for the mud-gas mixture is given by Equation 3. Equation 4 is the equation of state to compute gas density. The two-phase correlation to calculate in situ gas velocity is shown in Equation 5.

Equations 1-3 have consistent units, and the same form of equations are observed in Reference 7.

Single-phase region

Single-phase flow exists inside the drillstring and in part of the annulus. The annulus could have four regions: a single-phase region above the two-phase mixture, the two-phase mixture region, a single-phase region for the old mud, and a single-phase region for kill mud below the two-phase mixture. Only one or two of them exist in the beginning and at the end of well control simulation.

For the single-phase flow, Equations 1-5 are written in simple discrete form because of zero gas fraction and incompressible mud.

Equation 6 is a general pressure gradient equation including hydrostatic pressure gradient, frictional pressure loss gradient, and acceleration loss gradient.

Mathematical models

Even though frictional pressure loss is small in the annulus for a large well diameter at low kill rate, frictional pressure loss is critical for slim-hole wells or inside the choke line for offshore wells. Frictional pressure loss is considered to achieve more realistic simulation of kick behavior for all flow geometry and flow rates. The power-law fluid model is assumed.⁸

The estimation of two-phase frictional pressure loss is required to calculate the two-phase mixture momentum balance equation. The two-phase well control model utilizes the Beggs and Brill correlation.⁹

Even though the absolute value of gas density is small compared to that of drilling mud, the correct evaluation of gas density is essential to calculate the hydrostatic pressure of the two-phase mixture and to predict bubble rise velocities. The gas compressibility factor is calculated from the equation proposed by Dranchuk, et al.¹⁰

Gas viscosity is obtained using the Lee, et al., correlation.¹¹ Surface tension, which is necessary to estimate bubble rise velocity, is determined by the Katz, et al., method.¹²

Gas slip velocity is one of the parameters to describe a two-phase system. It also affects initial gas distribution and kick migration velocity during well shut in. The Hasan and Kabir model was chosen after an intensive literature survey.¹³

Solution procedures

Drilling. The two-phase model starts the simulation by taking a kick during drilling. The gas inflow rate is obtained from Equation A1.14

All parameters in the two-phase mixture region are evaluated at the middle point of the two-phase mixture weighted by the effective gas fraction. Because initial gas kick volume is relatively small, this approximation gives excellent results to compute pressure of the kick and the flowing bottom hole pressure. The effective flow rate for the single-phase region is the summation of mud circulation rate and gas inflow rate.

Pump off. One of the primary kick warning signs is increased mud return rate. The next step is to confirm that the well is flowing with the pump off. This is the same as during the drilling stage except reduced flow rate without circulating mud. The same calculations are repeated here.

Shut in. The next important step after detecting a kick is to shut the well in to prevent more influx from the formation. However, there is some flow from the formation as long as the BHP is less than the formation pressure. Because the total system volume is the same after well shut in, further inflow increases BHP. If BHP rises up to the formation pressure, the system has reached pressure equilibrium. At this point, shut in drill pipe pressure (Sidpp) and shut in casing pressure (SICP) are recorded. The amount of pressure buildup for the given duration is calculated from Equation 7.

The two-phase region is analyzed in discrete sections after well stabilization with the length specified by input data. An additional grid is given wherever flow area changes to make the calculation of grid block volume easy and to reduce complexity of finite difference equations. All grid information is assigned for each grid point as initial conditions.

Circulation. A constant BHP and kill rate are pressure and flow rate boundary conditions at the bottom of the well, respectively. If there is a single-phase region below the two-phase mixture, the pressure at the bottom of the two-phase mixture is calculated from Equation 6. For the two-phase region, a fully implicit finite difference method is applied for each discrete grid segment.

The following are the calculation procedures for two-phase flow when the annular area increases or stays the same:

1. Calculate new gas fraction (Hg) at i-th cell from Equation B2.

2. Compute new liquid fraction.

3. Estimate new liquid velocity at the i-th cell from Equation B1.

4. Obtain new gas velocity from the Hasan and Kabir correlation.

5. Determine the gas fraction with updated values from Equation B2.

6. Check the convergence of gas fractions. If they are close enough, go to Step 7, or repeat Steps 1-5.

7. Evaluate new pressure from Equation B3.

8. Check the convergence of pressure.

9. If pressure does not converge, repeat the whole procedure until a converged pressure solution is obtained.

Equations B1-B3 are liquid velocity, gas fraction, and pressure equations, respectively.

The procedures are repeated for the adjacent downstream cell. The iteration carries on to the end of the two-phase region. If the flow area decreases, Equations B4-B6 should be employed instead of Equations B1-B3, respectively.

If there is a single-phase region above the two-phase mixture, Equation 6 is applied from the top of the two-phase mixture to the surface. After calculations from the bottom of the well to the surface are completed, all information for the current time step is saved as old time information for the next iteration.

Numerical problems

Many numerical problems have been observed while programming this dynamic two-phase well control simulation. One of them is a divergence problem. The Hasan and Kabir correlation uses a flow pattern map which has a distinct boundary with different gas slip velocity.¹³ Therefore, the iterations between gas fraction and gas velocity do not converge near the flow regime boundary.

A modification of the flow map based on the gas fraction is necessary to have a converged solution: Bubble flow if Hg < 0.25; slug flow if 0.55 < hg < 0.75; and annular flow if hg > 0.9.

The slip velocity between any two flow regimes is assumed to vary linearly with gas fraction. Nickens used the same concept with different flow regimes.³

Sometimes the liquid velocity from Equations B1 or B4 becomes negative and causes numerical problems for functions which require a positive argument. Negative velocity is possible physically because gas moves faster than liquid during mud circulation.

If the calculated liquid velocity is negative, two modifications are made. One is that gas velocity is assumed the same as the slip velocity at that grid point. The other is that friction pressure loss for that grid is negligible. As iteration carries on to the next downstream grids, the liquid velocity becomes gradually positive because of gas expansion.

Although the fully implicit method is unconditionally stable, a large time step increases material balance error. To solve the time step selection problem, the time step is chosen so that the top interface of the two-phase mixture moves exactly one grid spacing in one time step. The grid size selection is not as critical as the time step, but it should be consistent with the time step. The shorter the time step, the smaller the grid spacing. About 50-80 ft grid spacing gives good results.

Numerical dissipation is the phenomenon in which the magnitude of the numerical solution tends to decrease and to spread out over a wide range with a small magnitude. In other words, the gas holdup calculated numerically at the top of the two-phase region propagates faster than the actual physical velocity with a small gas fraction. Gas holdup at the bottom interface of the two-phase mixture has a small value that eventually becomes zero.

To solve the gas holdup dissipation problem, the movement of the top interface is limited to move one grid spacing according to the time step size determined before. The movement of the bottom interface is determined by a minimum gas holdup of 0.005 or one-tenth of the initial gas holdup, whichever is smaller. One side effect observed is that material balance error increases by limiting gas movement at the top and bottom interfaces. Therefore, additional iterations are repeated until the summation of absolute relative material balance error is less than 0.01%.

Model comparisons

The two-phase well control model has been compared with field well control data, a full scale rig floor simulator, and a single-phase model.

The first comparison is made with field well control data. Table 1 [12765 bytes] lists the well and kick information taken from the company's drilling reports for the day before the kick and the day of the kick.

It is important to interpret the shut-in pressure difference between SICP and Sidpp. From simple hydrostatic pressure balance, the kick fluid density at the bottom of the well is 8.52 ppg. A formation water kick is a reasonable guess from shut in pressure information; rig personnel assumed the same.

A formation water kick results in almost constant choke pressure as long as the well geometry is constant over a certain length and choke pressure decreases as annulus capacity increases. Although a formation water kick was assumed from the initial shut in pressures, some gas kick behavior was observed from the field choke pressure (Fig. 1 [17152 bytes]).

Therefore, the well control team must pay attention to choke and pump pressures during the entire well control operation and continually try to figure out what type of kick is present regardless of initial guesses.

The initial pit gain of 25 bbl gives a lot higher SICP and choke pressure than the field observation. The initial pit gain is reduced continuously to match observed field data. Fig. 1 [17152 bytes] shows a comparison of field data with model data for a 6-bbl pit gain. Although the choke pressure for the two-phase model looks higher than the field observation, the choke pressure applied for the actual field well control was lower than the required pressure. That is inferred from the pump pressure which was lower than the minimum pressure required to keep the BHP the same as the formation pressure.

Because the choke pressure for the field data does not maintain constant BHP, the field choke pressure is converted using pump pressure after the kill mud passes the bit nozzles. Fig. 2 [17129 bytes] displays the similar choke pressure match between the two-phase model and transformed field data. The BHP in the actual well control operation was lower than the formation pressure so that there was additional inflow while circulating kill mud. Therefore, field observation showed high choke pressure after the kick arrived at the surface.

More than 25 bbl of kick entered the well. This was a combination of formation water and gas. Formation water kick was assumed from the well shut-in information, but gas kick behavior was observed during kill mud circulation. The gas kick came out of the well at the end of the well control operation.

The two-phase model was also compared with a full-scale rig floor simulator, which simulates the kick as a single gas slug with gas slip velocity (Table 2 [10910 bytes]).

The two models predict the choke pressure very closely in the early stage, but there is significant difference for maximum choke pressure (Fig. 3 [13214 bytes]). Although the full-scale simulator considers gas slip velocity, there is a time delay. All single-phase models have difficulty in estimating gas slip velocity correctly because that is usually a function of gas holdup and flow pattern.

A single-phase model assumes that kick fluid enters into a well as a single phase and remains as a single slug throughout the well control operation. The pressure and volume of the kick are two major unknowns; those are determined from dynamic equilibrium with a specified BHP.

Fig. 4 [13519 bytes] shows the choke pressure comparison of the two-phase model results and the single-phase model results for an offshore well. All the input data are the same as in Table 2 except some data listed in the figure. Two distinct differences are the magnitude of the choke pressure and time delay due to two-phase mixture effects and gas slip velocity, both of which are ignored in the single-phase model.

Results

Three different formation permeabilities were used to see the effects of variable gas kick influx rate. The pit volume warning level was 15 bbl.

Fig. 5 [12093 bytes] displays pit volume gain from drilling to well shut in. More problems are expected for high formation permeability. For 50-md formation permeability, it takes only 1.35 min for a 16-bbl gas kick gain. It happens in a short time so that the rig working crew should watch all kick indicators very closely. There is an additional 4.3-bbl kick due to reaction delay from kick detection to well shut in. Pit volume gain at the surface remains constant after well shut in while the number of moles of gas kick adds on continuously until well stabilization.

For 1.0-md formation permeability, it takes about 9.13 min for 15.05 bbl of gas kick to enter. There are only 0.82 bbl of additional pit gain after kick detection. It also takes a longer time for well stabilization. Therefore, a well control team in the field has more time to prepare any necessary actions to bring the kicking well under control. Several minutes of time delay are also affordable.

Fig. 6 [11948 bytes] shows surface casing pressure from drilling to well shut in. Before closing a blowout preventer, surface casing pressure is zero. Shut in casing pressure at the surface increases until the BHP balances with formation pressure. The lower the formation permeability, the longer is the time to stabilize the well pressures. The high formation permeability gives higher choke pressure mainly because of larger initial pit gain in this case.

Sidpp buildup shows the same trends as SICP buildup in Fig. 6. However, the stabilized Sidpp is the same in each case regardless of initial pit volume gain because Sidpp is a function of formation pressure, mud density in the well, and well depth and is independent of kick size.

There are small ID choke and kill lines on offshore wells. These lines have small volume capacity and high flow friction during well control operations. Choke pressure increases rapidly to compensate for the hydrostatic pressure reduction when the gas kick starts to fill the choke line (Fig. 7 [14801 bytes]).

Two ways are suggested in the field to minimize operational problems for offshore well control due to the choke line. One is to practice very low pump rates to have more time to displace the choke line. The other is to use both lines as kick fluid paths which will enlarge volume capacity and displacement time.

A high flow friction effect is seen in Fig. 8 [11902 bytes]. A choke line with 3-in. ID requires a lot lower back pressure at the surface than that of 5-in. ID choke line before the kick fills the choke line. When the kick starts to fill the choke line, the effect of choke line capacity is dominant. High friction loss also justifies the use of the dynamic kill method, and the kill rate should be chosen carefully so as not to exceed the formation fracture pressure.

The following conclusions have been drawn from this two-phase well control study:

• A two-phase well control model with reservoir performance has been developed; it can handle onshore and offshore wells for the driller's method and the engineer's method.

• Two new sets of finite difference equations were derived and used to handle variable flow geometry.

• The typical numerical problems for two-phase well control modeling are numerical dissipation, difficulty of selecting time step and grid sizes, and divergence due to a distinct two-phase flow map.

• The single-phase model predicts higher pit volume gain and peak choke pressure for all the test cases in this study. Although the single-phase model is very conservative, it yields almost the same results in the beginning of mud circulation but a time delay is observed.

• The choke pressure highly depends on kick height in the annulus so that well and drillstring geometry should be considered in detail for the well control study.

• In addition to choke pressure, BHP or pump pressure should be available for realistic field well control choke pressure analysis.

• A kick with high formation permeability could result in a huge pit gain if the kick is not detected and reacted to quickly.

References

1. Danenberger, E.P., "Outer Continental Shelf Drilling Blowouts, 1971-1991," paper No. 7248 presented at the 25th Annual Offshore Technology Conference, Houston, May 3-6, 1993.

2. LeBlanc, J.L., and Lewis, R.L., "A Mathematical Model of a Gas Kick," American Institute of Mechanical Engineers, Transactions, Vol. 243, August 1968, pp. 888-898.

3. Nickens, H.V., "A Dynamic Computer Model of a Kicking Well," Society of Petroleum Engineers Drilling Engineering, June 1987, pp. 159-173.

4. Starrett, M.P., Hill, A.D., and Sepehrnoori, K., "A Shallow-Gas-Kick Simulator Including Diverter Performance," SPE Drilling Engineering, March 1990, pp. 79-85.

5. Santos, O.L.A., "Well-Control Operations in Horizontal Wells," SPE Drilling Engineering, June 1991, pp. 111-116.

6. White, D.B., and Walton, I.C., "A Computer Model for Kicks in Water- and Oil-Based Muds," paper No. 19975 presented at the 1990 SPE/International Association of Drilling Contractors Annual Drilling Conference, Houston, Feb. 27-Mar. 2, 1990.

7. Rommetveit, R., "Kick simulator improves well control engineering and planning," OGJ, Aug. 22, 1994, pp. 64-71.

8. Bourgoyne, A.T. Jr., Millheim, K.K., Chenevert, M.E., and Young, F.S. Jr., "Applied Drilling Engineering," SPE Textbook Series, Richardson, Tex., 1986, pp. 152-155.

9. Beggs, H.D., and Brill, J.P., "A Study of Two-Phase Flow in Inclined Pipes," Journal of Petroleum Technology, May 1973, pp. 607-617.

10. Dranchuk, P.M., and Abou-Kassem, J.H., "Calculation of Z Factors For Natural Gases Using Equations of State," Journal of Canadian Petroleum Technology, July-September 1975, pp. 34-36.

11. Lee, A.L., and Gonzalez, M.H., "The Viscosity of Natural Gas," AIME Transactions, Vol. 243, August 1966, pp. 997-1000.

12. Brill, J.P., and Beggs, H.D., Two-Phase Flow in Pipes, 3rd printing, Tulsa, 1984.

13. Hasan, A.R., and Kabir, C.S., "A Study of Multiphase Flow Behavior in Vertical Oil Wells: Part I-Theoretical Treatment," paper No. 15138 presented at the SPE 56th California Regional Meeting, Oakland, Apr. 2-4, 1986.

14. Lee, J.W., Well Testing, 1st printing, SPE Textbook Series, Richardson, Tex., 1982, p. 76.

The Authors