Drilling fluids are in general complex heterogeneous mixtures of various types of base fluids and chemical additives that must remain stable over a range of temperature and pressure conditions. The properties of these complex mixtures, such as equivalent static density (ESD) and the rheological properties of the fluid mixture determine pressure losses in the system while drilling. It is often assumed that these properties and thus the equivalent circulating density (ECD) are constant throughout the duration of drilling activities. This assumption can prove to be quite wrong in cases where there

is large variation in the pressure/temperature conditions, such as in high pressure-high temperature (HPHT) wells, and deep-water drilling, where low temperature conditions are encountered very close to the sea bed.

In HPHT wells, as the total vertical depth increases, there is an increase in the bottom-hole temperature, as well as the hydrostatic head of the mud column. These two factors have opposing effects on equivalent circulating density. The increased hydrostatic head causes increase in the

equivalent circulating density due to compression. The increase in temperature on the other hand, causes a decrease in the equivalent circulating density due to thermal expansion. It is most often assumed that these two effects cancel each other out. This is not always the case, especially in high temperature, high pressure wells.

Large variations in equivalent circulating density can also occur when drilling in deep water environments where relatively cold temperatures are encountered in the riser, near the ocean bed. In deep-water wells, the seabed temperature can be as low as 30 F and hydrostatic pressures at the bottom

of the riser will be 2700 psi, with a mud density of 8.6 lb/gal and a water depth 6000-ft. The low temperature conditions can cause severe gelling of the drilling fluid, especially in oil-base muds (OBM). Failure to take this effect into account will result in underestimation of the equivalent circulating density of the drilling fluid.

Errors in the estimation of equivalent circulating density have an especially disastrous effect when drilling through formations with a small gap between pore pressure, and the pressure at which the formation will fracture. In such cases, the margin for error is very small and thus, the equivalent

circulating density must be estimated precisely. Disregarding pressure and temperature effects in this case can lead to greater probability for the occurrence of kicks, and blow-outs due to under-balanced pressure or fluid loss to the formation (lost circulation and formation damage) due to

overbalance pressure. Various experimental studies have also shown drilling fluid rheology to . Rheological parameters be very pressure and temperature dependent such as viscosity and yield stress affect frictional pressure losses during the

flow of drilling fluids. Failure to take into account the dependence of these parameters on temperature-pressure conditions can result in obtaining erroneous values for equivalent circulating density, which takes into account the hydrostatic head of the drilling fluid as well as the pressure loss it

experiences during flow.

The focus of this research is to study the effect of pressure and temperature on equivalent static density as well as equivalent circulating density of drilling fluids.

Numerous publications have dealt with the behavior of equivalent static density of drilling fluids in response to variations in pressure temperature conditions. Various models have been proposed in order to characterize this relationship, with some models being empirical in nature,

and some being compositional. The compositional model characterizes the volumetric behavior of drilling fluids based on the behavior of the individual constituents of the drilling fluid. Hence, prior knowledge of the composition of the drilling fluid is required for application of the compositional model.

In the compositional model, the density of any solids content in the drilling fluid is taken to be independent of temperature and pressure. It is assumed that any change in density is due to density changes in the liquid phases. It is also assumed that there are no physical and/or chemical interactions between the solid and liquid phases in the drilling fluid, or that the solid phase is inert. Hoberock et al proposed the following compositional model for equivalent static density of drilling fluids.

where,

ρο1, ρw1 = density of oil and water at temperature T1 and pressure P1, respectively

ρο2, ρw2 = density of oil and water at temperature T2 and pressure P2, respectively

f vo, fvw, fvs, fvc = fractional volume of oil, water, solid weighting material, and chemical additives, respectively .

P1, P2 = pressure at reference and condition “2”

T 1, T2 = temperature at reference and condition “2”

Application of the compositional model requires some knowledge of how the densities of each liquid phase in the mud, usually water and some type of hydrocarbon, change with changes in temperature and pressure. The static mud density at elevated pressure and temperature can be predicted

from knowledge of mud composition, density of constituents at ambient or standard temperature and pressure, and density of liquid constituents at elevated temperature and pressure.

Peters et al applied the Hoberock et al compositional model successfully to model volumetric behavior of diesel-based and mineral oil- based drilling fluids. In their study, they measured the density of the individual liquid components of each drilling fluid at temperatures varying from 78-350 F and pressures varying from 0-15,000 psi. Using this data in conjunction with Hoberock et al’s compositional model, they were able to predict the density of the drilling fluids at the elevated temperature-pressure conditions.

The model predictions yielded an error of <1 and="and" examined.="examined." of="of" over="over" p="p" pressure="pressure" range="range" temperature="temperature" the="the">

Sorelle et al proposed equations expressing the relationship between water and hydrocarbon (diesel oil No. 2) densities, and temperature/pressure for use with the compositional model with some success. Kutasov analyzed pressure-density-temperature behavior of water and proposed a similar

equation, which was reported to yield very low error in predicting water densities in the HTHP region.

Isambourg et al proposed a nine-parameter polynomial model to describe the volumetric behavior of the liquid phases in drilling fluids, which is applicable in the range of 14.5-20,000 psi and 60-400 F. This model characterizes the volumetric behavior of the liquid phases in the drilling fluid with respect to temperature and pressure, and is applied in a similar compositional model to that proposed by Hoberock et al. The model also assumes that all volumetric changes in the drilling fluid is due to the liquid phase, and application of the model requires a very accurate measurement of the reference mud density at surface conditions.

Kutasov proposed an empirical equation of state (EOS) model for drilling fluids to express the pressure-density-temperature dependent relationship. As is the case for the compositional model, mud density using Kutasov’s empirical equation of state is evaluated relative to its density at standard conditions (p= 14.7 psi, T = 60 oF). He applied the equation of state with a temperature-depth relationship in order evaluate hydrostatic pressure and equivalent static density as a function of depth.

Babu9 compared the accuracy of the two compositional models proposed by Sorelle et al4 and Kutasov8 respectively, and the empirical model proposed by Kutasov8 in predicting the mud weights for 12 different mud systems. The test samples consisted of 3 water based muds (WBM), 5

OBM’s formulated using diesel oil No. 2, and 4 OBM’s formulated using mineral oil. Babu9 found that the empirical model yielded more accurate estimates for the pressure-density-temperature behavior of a majority of the muds over the range of measured data more accurately than the compositional model. He also concluded that the empirical model has more practical application because unlike compositional models, it is not hindered by the need to know the contents of the drilling fluid in question.

Drilling fluids contain complex mixtures of additives, which can vary widely with the location of the well, and sometimes with different stages in the same well. This was especially apparent in the behavior of the drilling fluids prepared with diesel oil No. 2. Different oils available under the category of diesel oil No. 2 that were used in the preparation of OBM’s can exhibit different compressibility and thermal expansion characteristics, which were reflected in the pressure-density-temperature dependent behavior of the fluids prepared with them.

Research has also been reported on characterizing drilling fluid rheology at high temperature/high pressure conditions. Rommetveit et al approached their analysis of shear stress/shear rate data at high temperature and pressure by multiplying shear stress by a factor which depends on pressure, temperature and shear rate. Coefficients of this multiplying factor are fitted to shear stress/shear rate data directly without extracting rheological parameters such as yield stress first. This eliminates the need to characterize the behavior of each rheological parameter relative to pressure and temperature changes. In essence, they obtain an empirical model in which the effects of variation in all rheological parameters that describe fluid flow behavior are lumped together.

Another approach to the analysis of temperature and pressure effects on drilling fluid rheology is to consider the effect of temperature and pressure changes on each rheological parameter that describes the behavior of the fluid. The two most common models3 considered for such an analysis are the

Herschel-Bulkey/Power law model and the Casson model which is an acceptable description of oil based mud rheology. Of these two models, the Herschel-Bulkley model is the most robust, as it is a three parameter model as opposed to the Casson model which is a two parameter model. In the

analysis performed by Alderman et al on shear stress/shear rate data, the Herschel-Bulkley/Power and Casson models were considered. The behavior of each rheological parameter in these models with respect to changes in temperature and pressure was investigated. They studied a range of fluids covering un-weighted and weighted bentonite water-based drilling fluids with

and without deflocculant additives.

In order to estimate equivalent circulating density, it is important to take into account the effects of temperature and pressure on fluid rheology. Two methods are proposed to accomplish this by Rommetveit et al. They propose a stationary or static method and a dynamic method. In both methods, the contributions of hydrostatic and frictional pressure losses in high pressure/high temperature wells to the equivalent circulating density were considered. The variation in temperature vertically along the well bore is taken into account for both models, and drilling fluid properties are allowed to

vary relative to temperature.

The dynamic method however, also takes into account transient changes in temperature i.e. change in temperature over time. This effect is especially important in the case where circulation has been stopped for a significant amount of time. The drilling fluid temperature will begin to approach the temperature of the formation. Once circulation commences again as shown in Fig. 1.1, the lower part of the annulus will be cooled by cold fluid from the drill string and the upper part of the annulus will be warmed by hotter fluid coming from the bottom-hole. During this transient period, fluid

density and rheological characteristics can change rapidly due to rapid changes in temperature. Research on this effect is still at a very early stage and will not be taken into account during this study.

Figure 1.1- Schematic Diagram of Fluid in the Well bore at the Start of Circulation

Alderman et al performed rheological experiments on water based drilling fluids over a range of temperatures up to 260 oF and pressures up to 14,500 psi, using both weighted and unweighted drilling fluids. Rheograms were obtained for the water based drilling fluids, holding temperature constant

and varying pressure, and vice versa. It was found that the Herschel-Bulkley model yielded the best fit to the experimental data. Other models that were investigated are the Bingham plastic model, and the Casson model which some authors argue is the best model for characterizing oil-based drilling fluid

rheology.

For the Herschel-Bulkley model, it was found that the fluid viscosity at high shear rates increased with pressure to an extent, which increases with the fluid density, and decreases with temperature in a similar manner to pure water. Alderman et al found the yield stress to vary little with pressure- temperature conditions. The yield stress remained essentially constant with respect to temperature until a characteristic threshold temperature is attained.

This threshold temperature was found to depend on mud composition. Once this threshold is reached, the yield stress increases exponentially with 1/T. Alderman et al also found that the power law exponent increased with temperature, and decreased with pressure. This lead them to conclude that

the Casson model will become increasingly inaccurate at these two extremes, that is, at high temperature and low pressure.

The estimation of ECD under high temperature conditions requires knowledge of the temperatures to which the drilling fluid will be subjected to downhole. As the fluid is circulated in the wellbore, heat from the formation flows into the wellbore causing the wellbore fluid temperature to rise. This

process is more pronounced in deep, hot wells where the temperature difference between the formation and the well-bore fluid is greater. The process is very dynamic at early times, that is, at the commencement of circulation, with great changes in fluid temperature occurring over small

intervals of time.

There are two major methods for estimating the down-hole temperature of drilling fluid. The first is the analytical method. This method assumes constant fluid properties. Ramey solved the equations governing heat transfer in a well bore for the case of hot-fluid injection for enhanced oil recovery. His solution permits the estimation of the fluid, tubing and casing temperature as a function of depth. He assumed that heat transfer in the well bore is steady state, while heat transfer in the formation is unsteady radial conduction.

Holmes and Swift solved the heat transfer equations analytically for the case of flow in the drillpipe and annulus. They assumed the heat transfer in the wellbore to be steady state. However, they used a steady-state approximation to the transient heat transfer in the formation. They justified

this assumption by asserting that the heat transfer from the formation is negligible in comparison to the heat transfer between the drill pipe and annular sections due to the low thermal conductivity of the formation.

Arnold also solved the heat transfer equations analytically for both the hot-fluid injection case and the fluid circulation case. However, in circulation case, he did not assume steady state heat transfer in the formation. He represented the transient nature of heat flow from the formation with a dimensionless time function that is independent of depth16. Kabir et al also solved a similar set of equations, but for the case of flow down the annulus and up the drill pipe. They also assumed transient heat flow in the

formation, and evaluated a number of dimensionless time functions.

The second method of estimating fluid temperature during circulation involves allowing the fluid properties such as heat capacity, viscosity, and density to vary with the temperature conditions. This method involves solving the governing heat transfer equations numerically using a finite difference

scheme. Marshal et al created a model to estimate the transient and steady-state temperatures in a well bore during drilling, production and shut-in using a finite difference approach.

Romero and Touboul created a numerical simulator for designing and evaluating down-hole circulating temperatures during drilling and cementing operations in deep-water wells. Zhongming and Novotny developed a finite difference model to predict the well bore and formation transient temperature behavior during drilling fluid circulation for wells with multiple temperature gradients and well bore deviations.