Development of Generalized Well Semi-analytical Coning Models

Development of Generalized Well Semi-analytical Coning Models

CHAPTER ONE

PREAMBLE

Quite often, oil deposits are associated with an underlying water aquifer and an overriding gas cap. In many situations, the oil reserve is desired at the surface while the associated fluid is preferred within the reservoir either because they are not valuable at the surface as in the case of produced water or the resources to harness the gas if produced to the surface are not readily available. The reservoir water or gas may also be required for pressure maintenance in production optimization within the reservoir. Whatever may be the intention of prefering to keep the water and/or gas within the reservoir, it is found in practice a difficult goal to achieve due to coning of the unwanted fluid(s). Coning is the tendency of the underlying water in contact with the oil to rise locally towards the producing well due to the greater pressure depletion near the producing well and the viscous drag the production of oil is having on the water-oil interface.

CHAPTER TWO

BASIC CONCEPT AND LITERATURE REVIEW

THE CONING MECHANISM

Within the reservoir, distinct fluid phases, if mobile, tend to take up positions relative to their density. Gravitational force induces the gas, if present, to occupy the top most part of the reservoir while water goes down the structure and the oil in between. A transition zone may occur at the interface boundary due to the effect of capillary pressure in which sharp fluid interface is only an engineering idealization. Neglecting capillary pressure is a common practice in petroleum reservoir

coning modelling^1,3,4. The state of fluid equilibration within the reservoir is disturbed by oil production. This introduces pressure differential around the producing well that is often referred to as viscous drag. The local pressure differential around the well means that fluid can more easily move towards the producing well. This tendency for local fluid migration towards the producing well is counterbalanced by the tendency of the fluid system to maintain gravitational segregation. The result is that the surface of the fluid interface is tensed and deformed into some characteristic bell shape⁶.

As long as production continues under circumstances and rate that allows for appreciable pressure differential around the well, the cone continues to grow towards the well. The height of the cone above or below the original or prevailing fluid interface for the case of water or gas coning respectively, determines the net gravitational pull on the interface. At some certain point, the amount of gravitational force may effectively balance the viscous force and the growth of the cone is halted. At this stage, the cone is said to be stable as it does not progress with time. When it does progress towards the well with time, the cone is said to be unstable. Unstable cone may eventually break into the well and multiphase production will ensue. The time period between production commencements from original condition to when the unwanted fluid cones into the well is referred to as the breakthrough time⁶.

If the production rate is such that the height of the developed cone is just about to break into the well before the cone becomes stable, the rate is called the critical oil production rate. It is theoretically the maximum production rate possible without cone breakthrough. In many situations, the critical rate is uneconomic and is often exceeded. Rates above the critical are termed super-critical. At such a rate, the cone at some time gets to the perforated interval. The determination of this time of breakthrough had been the subject of an appreciable amount of paper as it promotes production and surface facility optimization^3,4. Surface facilities utilization is dependent on the type and quantity of the produced fluid and hence on the time of cone breakthrough.

CHAPTER THREE

METHODOLOGY

PREAMBLE

The major questions that are associated with the coning of unwanted fluid in the course of oil production can be categorized into three: what is the maximum stable monophasic or critical rate of oil production, what is the breakthrough time for supercritical production rate, and what is the trend of production after breakthrough? The answers to these questions are found to be dependent on particular well configurations: vertical, horizontal or slanted. Answers to these questions have been pursued empirically and analytically as described in the previous chapter. This study intends to proffer an approximate analytical solution to the three basic coning challenges. These implies the generation of generalized expressions and procedures for the determination of critical rate of oil production, breakthrough time and after breakthrough behaviour prediction for the three basic well configurations.

CHAPTER FOUR

RESULT AND ANALYSIS

Implementing the methodology set out in the previous chapter allows for the computation of critical rate, breakthrough time and after breakthrough trends for vertical, horizontal and inclined wells. The computation requires the use of a computational software. Scilab,a free source mathematical computational software, was used in the calculation. Writing a simple program to carry out the computation ensures that several computation can be done within reasonable time.

Scilab codes developed for major computations are presented in Appendix C.

CHAPTER FIVE

SUMMARY, CONCLUSIONS AND FUTURE DIRECTION

SUMMARY

The stated objective of this study is to develop a general procedure for the calculation of the critical rate for oil production in the presence of an active water aquifer or gas cap, the determination of the time of breakthrough if a rate higher than the critical is maintained and the after breakthrough behaviour when supercritical production rate is allowed. It was stated that the industry lacked the complete set of approximate analytical approaches to cone evolution modelling for vertical, horizontal and inclined wells. Adhering to that target, a procedure for calculating the stated parameters for all three well configurations using fundamental analytical line source solutions for vertical and horizontal wells as developed in Ref. 3 was generated. Other suitable horizontal and vertical analytical model pair may be used as the solution kernel. This procedure provides a complete suite of semi-analytical solutions for the determination of critical rate, breakthrough time and after breakthrough trend for vertical, horizontal and inclined wells.

CONCLUSIONS

A semi-analytical procedure for computing inclined well breakthrough time have been developed.
The formulation was used in calculating critical rate of oil production for inclind well following a procedure demonstrated by Ozkan and Raghavan³for vertical and horizontal wells.
A novel approach in semi-analytically generating post-breakthrough prediction for inclined wells was presented.
This approach was extended to vertical and horizontal wells.
Simplified correlations for the determination of inclined well critical rate of oil production, breakthrough time for vertical, horizontal and inclined wells were generated from the semianalytical formulation developed in this work.

FURTHER STUDIES

The natural extention of this work will be to employ other line source analytical solution pairs other than what was used in this work. It will be interesting to note how the method perform with analytical solutions that could relax the assumption of unit mobility ratio and steady state flow. These two assumptions may not always be attained in practice and their inclusion reduces the accuracy of the procedure for common practical well conditions. Effort geared towards developing a correcting correlation for mobility ratio could extend the utility of the current research. More research on the incorporation of other factors affecting phase flow at the perforation could substantially improve on this work. The solution of the fundamental diffusivity equation for other reservoir conditions such as simultaneous active water aquifer and gas cap and pseudosteady state flow for weak pressure support will increase the practicality of the procedure.

The post-breakthrough predictions from this study could be used as backbone in generating consistent post-breakthrough correlations.

REFERENCES

Chaperon I.: “Theoretical Study of Coning Toward Horizontal and Vertical Wells in Anisotropic
Formations: Subcritical and Critical Rates”, paper SPE 15377 prepared for presentation at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, Oct. 5-8
Hoyland, Lelf A. et al.: “Critical Rate for Water Coning: Correlation and Analytical Solution”, SPE Reservoir Engineering(November 1989) pp 495-502
Ozkan E. and Raghavan R.: “A Breakthrough Time Correlation for Coning Toward Horizontal
Wells”, paper SPE 20964 prepared for presentation at Europec 90, the Hague, Netherlands, October 22-24.
Papatzacos P. et al.: “Cone Breakthrough Time for Horizontal Wells”, SPE Reservoir Engineering(August 1991) 311-318 Joshi S. D.: “Water and Gas Coning in Vertical and Horizontal Wells”, Horizontal Well Technology,
Pennwell Books of Tulsa (1991) 252-325
Ahmed Tarek: Reservoir Engineering Handbook, Gulf Publishing Company, Houston (2000) 569635
Walter J. Karplus: “Water Coning Before Breakthrough – An Electronic Analog Treatment”, Petroleum Transactions, AIME (1956) 240-6
Karcher, B. J.: “Some Practical Formulas to Predict Horizontal Well Behaviour”, paper SPE 15430 prepared for presentation at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, Oct. 5-8
Russell T. J. et al.: “Prediction of Capillary Fluid Interface during Gas and Water Coning in Vertical Wells”, SPE Journal(December 2005) 440-448

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