Erlend H. Vefring, Bjorn-Tore Anfinsen, Knut Steinar Bjorkevoll, Rolv RommetveitRF-Rogaland Research

Bergen, Norway

A simulator accurately predicts downhole pressures and temperatures in high pressure, high temperature (HPHT) wells by using a dynamic and two dimensional mathematical model.

The model accounts for heat transfer between the well bore fluid and surroundings. Results from the simulator compare reasonably well with results from actual offshore wells.

At HPHT conditions, traditional models for rheology and density are inaccurate. Advanced models for rheological properties and density have therefore been developed and are included in the simulator. The simulator can include HPHT laboratory data on rheology and density from actual muds used in the field. The governing equations are solved by a finite difference method.

Results from the simulator have been compared to experiments in an offshore well. The well had a bottom hole pressure of approximately 1,000 bar and a bottom hole temperature of approximately 170 C. The data used for comparison were pressure and temperature over time at several positions in the annulus. The simulation results compare well with the experimental data.

The simulator can be used for well design and well planning in relation to temperature effects on well control parameters. It can also be used to improve safety in deeper wells with small margins between pore pressure and fracture pressure.

Many HPHT wells have been drilled in the North Sea, and some operators have experienced more pressure control problems in these wells than in traditional wells. Some of these wells have also been characterized by small margins between pore pressure and fracture pressure, putting increased demands on pressure control.

In an HPHT well, both the temperature and pressure vary considerably. The mud density and rheology vary significantly with pressure and temperature within the ranges experienced in these wells. The prediction of pressures in HPHT wells therefore requires accurate description of the temperature profile in the well in addition to accurate models for density and rheology within the pressure and temperature ranges experienced.

In 1990, a project was begun with the objective of developing a simulator, called Presmod, for calculation of accurate well pressure in HPHT wells. The basis for the project was a one-dimensional well pressure model developed by RF-Rogaland Research and a two-dimensional temperature simulator developed by Elf Aquitaine.1-4

The first task was to develop improved models for mud properties at HPHT conditions.

The pressure model and the temperature model were then integrated. A user-friendly interface on a Unix/X Window/Motif basis was developed.

This interface improves the usefulness of the simulator by making it easily accessible to drilling engineers for design and planning purposes. The last task of this project was to compare results from this simulator with experimental data. Pressure and temperature were measured in an offshore HPHT well and were used for this comparison.

This article briefly presents the model basis and then describes the offshore experiments performed.

### PRESSURE AND VELOCITY

In the flowing region, the governing equations are given by Equations 1 and 2 for the conservation of mud mass and Equations 3 and 4 for the conservation of mud momentum.

In these equations, t is time, z is depth, A is flow cross sectional area, pit is mud density in the tubing, plt is mud density in the annulus, vlt is flow velocity of mud in the tubing, vla is flow velocity of mud in the annulus, pt is pressure in the tubing, pa is pressure in the annulus, ft is pressure loss in the tubing, and fa is pressure loss in the annulus.

Equations 5 and 6 show the factors affecting mud density. In these equations, pl is mud density, Tt is temperature in the tubing, and Ta is temperature in annulus, which are determined from the temperature model described in the next section.

The density can be calculated according to the model by Sorelle, the model by Kemp and Thomas, or by interpolation in tables of HPHT laboratory data.5-6 The model by Sorelle includes simple correlations for oil and water.5 These correlations are based on laboratory data for diesel oil and tables of physical properties of water. The model by Kemp and Thomas compensates water density for salt content.6 Interaction forces between different salt ions and water are calculated.

Frictional pressure loss is represented by Equations 7 and 8. The computation of pressure losses requires a rheology model, and the rheology models that can be used in the simulator are Bingham, power law, and Robertson-Stiff. Laboratory data for the rheology at various pressures and temperatures are given as input to the simulator. Equations 1-8 constitute a system of eight equations for eight unknowns: plt, Pla, Vlt, Vla, pt, pa, ft, and fa.

The boundary conditions are given by the following:

vlt(z = O,t) = Vlt(t)Pa(z = O,t) = Patm

## TEMPERATURE

The conservation of energy is given by Equation 9. In this equation, p is density, H is enthalpy, and qs is the heat generation in the system from mechanical (rotational) energy and hydraulic energy. Qf is the forced convective term, expressed by Equation 10. Qc is the conductive and natural conductive term which does not have a general expression. In the case of purely convective isotropic materials, Qc may be expressed by Equation 11. In this equation, lambda is thermal conductivity and T is temperature.

## OFFSHORE EXPERIMENTS

The well in which the experiments were performed had a bottom hole pressure of approximately 1,000 bar and a bottom hole temperature of approximately 170 C.

Pressure and temperature were measured continuously at five downhole positions during surge and swab tests, circulation tests, and gelation tests.

Also, a large number of surface parameters, like bit depth, rotational speed, mud flow rate in and out, mud density, and temperature, were monitored continuously.

## COMPARISON

Results from the Presmod simulator were compared to field data taken in an offshore HPHT well. The following comparisons of downhole pressures and temperatures are from a circulation test with water-based mud. The circulation rate was varied in steps of 200 l./min between 0 and 1,000 l./min (Fig. 1).

The mud density and rheological properties were measured at relevant HPHT conditions. The heat capacities and thermal conductivities of relevant materials were taken from published tables or derived from works by Guenot.7

The results of the calculations, in particular downhole temperatures, were very sensitive to mud properties (density, Theology, heat capacity, and thermal conductivity). It is essential to know how these parameters vary as functions of pressure and temperature to obtain reliable temperature calculations.

The initial downhole mud temperatures were given as input to Presmod, while later temperatures were calculated by the program. The temperature as a function of time is shown in (Fig. 2). The calculated temperature varies more than the measured temperature during the circulation test, but the discrepancy is at most 7 C.

Fig. 3 shows the measured and calculated equivalent circulating density values near the bottom. The calculated values are less than 0.5% off during the first half of the circulation test, while the measured equivalent circulating density drops faster than the calculated values during the last part.

Note that the measured temperature is almost constant during the test. This constant temperature indicates that the drop in equivalent circulating density is not a temperature effect, but rather a variation of mud density and rheology during the test.

Fig. 4 shows the pressure difference over the bottom hole assembly (BHA). The calculated values are less than 10% off during the whole test.

Fig. 5 shows the temperature profiles in the drill pipe and the annulus compared to the geothermal temperature, at 28.3 hr.

## RESULTS

A simulator for accurate prediction of downhole pressures and temperatures in HPHT wells has been developed. The simulator can be used for accurate prediction of pressures and temperatures in HPHT wells during drilling and work-overs.

To be able to predict pressures accurately in HPHT wells, accurate models for the temperature are needed in addition to accurate models for density and rheology within ranges of temperature and pressure experienced in HPHT wells.

Results from the simulator compare reasonably well with results from the offshore experiments. The discrepancies observed can be contributed to variation of mud density and rheology during the test.

## REFERENCES

- Ekrann, S., and Rommetveit, R., "A Simulator for Gas Kicks in Oil-Based Drilling Muds," Society of Petroleum Engineers paper 14182, 1985.
- Vefring, E.H., and Rommetveit, R., "An Advanced Kick Simulator Operating in a User-Friendly X-Window System Environment," SPE paper 22314, 1991.
- Rommetveit, R., and Vefring, E.H., "Comparison of Results from an Advanced Gas Kick Simulator with Surface and Downhole Data from Full Scale Gas Kick Experiments in an Inclined Well," SPE paper 22558, 1991.
- Corre, B., Eymard, R., and Guenot, A., "Numerical Computation of Temperature Distribution in a Wellbore While Drilling," SPE paper 13208, 1984.
- Sorelle, R.R., Jardiolin, R.A., Buckley, P., and Barrios, J.R., "Mathematical Field Model Predicts Downhole Density Changes in Static Drilling Fluids," SPE paper 11118, 1982.
- Kemp, N.P., and Thomas, D.C., "Density Modelling for Pure and Mixed Salt Brines as a Function of Composition, Temperature, and Pressure," Society of Petroleum Engineers/International Association of Drilling Contractors paper 16079, 1987.
- Guenot, A., "Regime Thermique Des Puits En Forage," report No. 99/85, Societe Nationale Elf Aquitaine, Pau, France, 1985.

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