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Furuis IPR Model Correction and its Application in Horizontal Well Cresting Control & Inflow Control Device Installation Kareem Lateef, A. SPE* (King Fahd University of Petroleum and Minerals); & Omeke James Emeka, SPE.
Copyright 2012, Society of Petroleum Engineers Inc.
This paper was prepared for presentation at the 36th Annual SPE International Technical Conference and Exhibition in Abuja, Nigeria, August 6-8, 2012. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.
AbstractThe problem of reservoir heterogeneity has made the continued use of Inflow Performance Relation (IPR) models developed on the assumption of homogeneity impractical. Furui in 2003 (and previously Butler in 1994) solved the problem using fully penetrating well idea so that the model can be applied to different sections of the reservoir.This paper corrects two errors in Furuis model. The first error is the use of arithmetic mean of radii to evaluate the elliptic surface area of the well when an anisotropic reservoir is converted into an isotropic equivalent. This was corrected using an equivalent cylindrical surface area. The second error is the use of a planar isobaric surface at the end of the radial flow region instead of a circular one. These corrections gave the model a better performance-average of 99% against 96% by the original version, when compared with line source solution obtained by using Green function.Equating the hydrostatic and viscous pressure differences between the well and the water oil contact (WOC), a cresting equation that predicts the critical production rate and pressure drawdown for each zone needed to prevent water (or gas) encroachment was developed.Based the IPR and cresting equation, this work presents a mathematical concept on installation of Inflow Control Device (ICD) so that production from each zone is designed according to its requirements.
IntroductionCresting impacts the worth of a well in two ways. Besides lowering the oil production rate and eventually the recovery, it also impose additional cost of treating produced water and exposes the production tubing to the corrosive action of the water. As a result cresting has thus considerably drawn attention of production and reservoir engineers. Large amount of effort has been dispensed at developing a good model to accurately describe cresting in horizontal wells. All the methods rely on one idea- predicting the critical rate of production below which the pressure drop will not be sufficient to initiate the movement of unwanted fluids. Since the idea require a connection between production rate and pressure drop, the inflow performance relation becomes an important tool in this quest.
Literature reviewEfros (1963) proposed a critical flow rate based on the assumption that the critical rate is nearly independent of effective drainage radius. His correlation also neglected the effect of vertical mobility.Water cresting
Gas cresting
Where
Chaperson (1986) presented a simple estimate of critical rate under a steady or pseudo-steady state flow condition in an isotropic formation.Water cresting
Gas cresting
Karcher (1986) proposed a correlation that is very similar to Efros equation.Water cresting
Gas cresting
Where
Joshi (1988) suggest the following relationship for determining critical oil flow rate in horizontal wellsWater cresting
Gas cresting
Where
All these models relied on IPR equations derived by looking at the plan view of a horizontal well and as such failed to consider the upper and lower no flow boundaries on the steady state equations.Butler (1994) and Furui et. al. (2003) solved this problem by deriving a model from the side view of the well, looking in the axial direction of the well and assuming that fluids produced from the well travel horizontally from the outer boundaries and radially in the vicinity of the well.
Butlers IPR
Furuis IPR
Where
Sensitivity studies conducted with real well data have shown that Furui and Butlers models match actual IPR in most case.Modelling a new IPR.
This will give an elliptic isobaric line (equipotential surfaces when). To simplify, we substitute
Then
Hence,
Simplifying our substitution we get,
So, if we set
We have,
The effect of the transformation is that it replaces our reservoir with an isotropic equivalent with the vertical measurement scaled up by a factor of square root of Iani while the horizontal measurement is scaled down by the same factor.
The well bore is also affected and the shape is now an elliptic cylinder. Hence using the perimeter of an ellipse, the area open to flow at the well bore gives
Comparing this with the area open to flow for an equivalent cylindrical well bore we have,
Hence our equivalent well bore radius gives
For radial flow,
Giving
For the outer part of linear horizontal flow, we use two approaches1. Assume vertical plane surface at a distance from the well bore have the same pressure as the circular equipotential surface of radius.2. We take the pressure drop from the no-flow boundary to the equipotential surface of radius.For the first case,
Combining equation (4) and (5)
For the second case, we take a strip of the reservoir having thickness at a height above the centre plane.
Combining this with equation (5), we have,
Where
Recognizing that
We can write
Converting to field units we have,
To account for the effect of skin we have
Comparison with Butlers and Furuis modelsHorizontal Effective Well Length(Le),ft3343
Assumed Drainage Width (2Yb or W),ft8700
Wellbore Radius(rw),ft0.26
Effective Drainage Radius(re),ft3170
Net Pay Thickness(h),ft180
Reservoir Pressure(Pe),psi3000
Wellbore Flowing Pressure(Pwf),psi2666
Horizontal Permeability (kh),md784
0.62Vertical Permeability(kv),md2.5
Formation Volume Factor(Bo),resbbl/STB1.34
Skin Factor(s)-3.5
Fluid Viscosity (o), cp0.62
Actual Productivity Index (PI), bbl/day/psi2018.8
Using Butlers IPR,
Using Furuis IPR,
Using Equation 6b
Using Equation 8b
Applying the idea to crestingThe minimum pressure drop between the well bore and the WOC to initiate migration of water into the oil zone is the pressure drop between the well bore and the equipotential surface whose radius is so we have,
From Furui IPR,
Hence the maximum allowable pressure drawdown gives,
Similarly from Butlers IPR,
From our first IPR equation we have,
From the second IPR equation we have
From this we can determine the critical rate oil production and that will give
For Furui and Butler, we have
For our two equations, we have
Skin due to formation damage around the well bore also affects the oil critical rate and maximum allowable pressure drawdown. So taking skin into consideration we have,Equation 9 gives
Equation 10 gives
Equation 11 gives
Equation 12 becomes
Equation 13, gives
Equation 14 becomes
To use the equation for gas conning calculation, we simply replace the water- oil density difference with gas oil density difference.
The design of ICD(s) and ICV(s)
These equipments are used to control inflow from different sections of a horizontal well in order to achieve effective sweep of the drainage volume. Proper use of the equipments requires the knowledge of critical flow rate and maximum allowable pressure draw down in the different sections of the reservoir. While and ICV can be adjusted continually during the life of the well, an ICD cannot. Hence it is important that the device is properly designed from the beginning of production.
An ICD/ICV can be designed as follows:1. Determine the critical production rate per unit length of the well from different sections of the reservoir.2. Select about 80% of the minimum value of the rates calculated in (1.) in order to allow for inaccuracies in the delineation of water-oil and gas-oil contact.3. Using the value in (2), determine the corresponding production rate from other section using a uniform production rate per unit length (uniform flux).4. Convert this production rates to pressure drawdown(s) for other sections.5. Calculate the additional drawdown(s) required through the openings of the ICD/ICV.6. Calculate opening size needed for the pressure drop at the calculated flow rates.
Example:Horizontal section is 10,000'Permeability variation is as follow: Section 1: 500' Kh= 20 md Section 2: 500' Kh= 200 md Section 3: 1000' Kh= 50 md Section 4: 1500' Kh= 150 md Section 5: 1000' Kh= 40 md Section 6: 1500' Kh= 10 md Section 7: 200' Kh= 400 md Section 8: 800' Kh= 80 md Section 9: 400' Kh= 60 md Section 10: 600' Kh= 40 md Section 11: 300' Kh= 600 md Section 12: 700' Kh= 70 md Section 13: 250' Kh= 50 md Section 14: 450' Kh= 35 md Section 15: 300' Kh= 15 mdAnisotropic factor Kv/Kh=0.1The reservoir thickness = 200'The reservoir width =2500The wellbore radius is 6"Oil viscosity = 2cpFormation Volume Factor = 1.2STB/bblOil API = 45.4oSolution.Critical rate per unit length
Among the parameters needed to calculate the critical rate per unit length, only the horizontal permeability varies from one section to the other. The region with the smallest critical rate per unit length is the region with the lowest horizontal permeability.
The minimum value of occur at the section having 10md permeability
With 80% of this value, we calculate the pressure drop through the reservoir in other section when they are producing at the same rate per unit length.
We can now tabulate what each section will produce and the corresponding pressure drop.SectionLength(ft)Permeability (md)Allocated production rate (STB/day)
Pressure drop (psia)
15002018.2918.23
250020018.291.82
310005036.577.29
4150015054.862.43
510004036.579.11
615001054.8636.45
72004007.310.91
88008029.264.56
94006014.636.08
106004021.949.11
1130060010.970.61
127007025.605.21
13250509.147.29
144503516.4610.41
153001510.9724.30
Using section #6 with the highest pressure drop across the reservoir, we can now calculated the additional pressure drop required across the ICV/ICDSectionLength(ft)Permeability (md)Allocated production rate (STB/day)
Pressure drop (psia)
Required pressure drop across valves(psia)
15002018.2918.2318.23
250020018.291.8234.63
310005036.577.2929.16
4150015054.862.4334.02
510004036.579.1127.34
615001054.8636.450.00
72004007.310.9135.54
88008029.264.5631.89
94006014.636.0830.37
106004021.949.1127.34
1130060010.970.6135.84
127007025.605.2131.24
13250509.147.2929.16
144503516.4610.4126.04
153001510.9724.3012.15
The diameter of the valve required for the required pressure drop at the calculated production rates is
If the specific gravity of the fluid is known. In this case 0.8, we can calculate the opening sizesSectionAllocated production rate (STB/day)
Required pressure drop across valves(psia)
Required Valve opening d (inches)
118.2918.230.06639
218.2934.630.05655
336.5729.160.08347
454.8634.020.09837
536.5727.340.08483
654.860.00Full opening
77.3135.540.03552
829.2631.890.07302
914.6330.370.05226
1021.9427.340.06571
1110.9735.840.04342
1225.6031.240.06865
139.1429.160.04173
1416.4626.040.05761
1510.9712.150.05690
Even in a case where we are producing above the critical rate, this concept can also be applied to ensure uniform flux and simultaneous water breakthrough along the entire length of the horizontal section.RECOMMENDATIONS AND CONCLUSIONS1. The equations so derived are applicable to water and gas cresting for a well that is centrally located with respect to the reservoir boundary.2. The equations are based on steady state model, so gives the delta P between the wellbore and the boundary3. More works need to be done to account for eccentricity with respect to boundaries.4. Butlers IPR should be used in the form
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