© 2010 Elsevier Inc. All rights reserved.
Doi: 10.1016/C2009-0-30429-8
CONING
Coning is primarily the result of movement of reservoir fluids in the direction of least resistance, balanced by a tendency of the fluids to maintain gravity equilibrium. The analysis may be made with respect to either gas or water. Let the original condition of reservoir fluids exist as shown schematically in Figure 9-1, water underlying oil and gas overly- ing oil. For the purposes of discussion, assume that a well is partially penetrating the formation (as shown in Figure 9-1) so that the production interval is halfway between the fluid contacts.
Production from the well would create pressure gradients that tend to lower the gas-oil contact and elevate the water-oil contact in the immedi- ate vicinity of the well. Counterbalancing these flow gradients is the ten- dency of the gas to remain above the oil zone because of its lower den- sity and of the water to remain below the oil zone because of its higher density. These counterbalancing forces tend to deform the gas-oil and water-oil contacts into a bell shape as shown schematically in Figure 9-2.
There are essentially three forces that may affect fluid flow distribu- tions around the well bores. These are:
• Capillary forces
• Gravity forces
• Viscous forces
Figure 9-1.Original reservoir static condition.
Capillary forces usually have a negligible effect on coning and will be neglected. Gravity forces are directed in the vertical direction and arise from fluid density differences. The term viscous forces refers to the pres- sure gradients’ associated fluid flow through the reservoir as described by Darcy’s Law. Therefore, at any given time, there is a balance between gravitational and viscous forces at points on and away from the well completion interval. When the dynamic (viscous) forces at the wellbore exceed gravitational forces, a “cone” will ultimately break into the well.
We can expand on the above basic visualization of coning by introduc- ing the concepts of:
• Stable cone
• Unstable cone
• Critical production rate
If a well is produced at a constant rate and the pressure gradients in the drainage system have become constant, a steady-state condition is reached. If at this condition the dynamic (viscous) forces at the well are less than the gravity forces, then the water or gas cone that has formed will not extend to the well. Moreover, the cone will neither advance nor recede, thus establishing what is known as a stable cone.Conversely, if the pressure in the system is an unsteady-state condition, then an unsta- ble cone will continue to advance until steady-state conditions prevail.
If the flowing pressure drop at the well is sufficient to overcome the gravity forces, the unstable cone will grow and ultimately break into the
Figure 9-2.Gas and water coning.
well. It is important to note that in a realistic sense, stable system cones may only be “pseudo-stable” because the drainage system and pressure distributions generally change. For example, with reservoir depletion, the water-oil contact may advance toward the completion interval, thereby increasing chances for coning. As another example, reduced productivity due to well damage requires a corresponding increase in the flowing pressure drop to maintain a given production rate. This increase in pres- sure drop may force an otherwise stable cone into a well.
The critical production rate is the rate above which the flowing pres- sure gradient at the well causes water (or gas) to cone into the well. It is, therefore, the maximum rate of oil production without concurrent pro- duction of the displacing phase by coning. At the critical rate, the built- up cone is stable but is at a position of incipient breakthrough.
Defining the conditions for achieving the maximum water-free and/or gas-free oil production rate is a difficult problem to solve. Engineers are frequently faced with the following specific problems:
1. Predicting the maximum flow rate that can be assigned to a completed well without the simultaneous production of water and/or free-gas 2. Defining the optimum length and position of the interval to be perfo-
rated in a well in order to obtain the maximum water and gas-free pro- duction rate
Calhoun (1960) pointed out that the rate at which the fluids can come to an equilibrium level in the rock may be so slow, due to the low perme- ability or to capillary properties, that the gradient toward the wellbore overcomes it. Under these circumstances, the water is lifted into the well- bore and the gas flows downward, creating a cone as illustrated in Figure 9-2. Not only is the direction of gradients reversed with gas and oil cones, but the rapidity with which the two levels will balance will differ.
Also, the rapidity with which any fluid will move is inversely propor- tional to its viscosity, and, therefore, the gas has a greater tendency to cone than does water. For this reason, the amount of coning will depend upon the viscosity of the oil compared to that of water.
It is evident that the degree or rapidity of coning will depend upon the rate at which fluid is withdrawn from the well and upon the permeability in the vertical direction kvcompared to that in the horizontal direction kh. It will also depend upon the distance from the wellbore withdrawal point to the gas-oil or oil-water discontinuity.
The elimination of coning could be aided by shallower penetration of wells where there is a water zone or by the development of better hori- zontal permeability. Although the vertical permeability could not be less- ened, the ratio of horizontal to vertical flow can be increased by such techniques as acidizing or pressure parting the formation. The application of such techniques needs to be controlled so that the effect occurs above the water zone or below the gas zone, whichever is the desirable case.
This permits a more uniform rise of a water table.
Once either gas coning or water coning has occurred, it is possible to shut in the well and permit the contacts to restabilize. Unless conditions for rapid attainment of gravity equilibrium are present, restabilization will not be extremely satisfactory. Fortunately, bottom water is found often where favorable conditions for gravity separation do exist. Gas coning is more difficult to avoid because gas saturation, once formed, is difficult to eliminate.
There are essentially three categories of correlation that are used to solve the coning problem. These categories are:
• Critical rate calculations
• Breakthrough time predictions
• Well performance calculations after breakthrough
The above categories of calculations are applicable in evaluating the coning problem in vertical and horizontal wells.
CONING IN VERTICAL WELLS Vertical Well Critical Rate Correlations
Critical rate Qocis defined as the maximum allowable oil flow rate that can be imposed on the well to avoid a cone breakthrough. The critical rate would correspond to the development of a stable cone to an eleva- tion just below the bottom of the perforated interval in an oil-water sys- tem or to an elevation just above the top of the perforated interval in a gas-oil system. There are several empirical correlations that are com- monly used to predict the oil critical rate, including the correlations of:
• Meyer-Garder
• Chierici-Ciucci
• Hoyland-Papatzacos-Skjaeveland
• Chaney et al.
• Chaperson
• Schols
The practical applications of these correlations in predicting the criti- cal oil flow rate are presented over the following pages.
The Meyer-Garder Correlation
Meyer and Garder (1954) suggest that coning development is a result of the radial flow of the oil and associated pressure sink around the well- bore. In their derivations, Meyer and Garder assume a homogeneous sys- tem with a uniform permeability throughout the reservoir, i.e., kh=kv. It should be pointed out that the ratio kh/kvis the most critical term in eval- uating and solving the coning problem. They developed three separate correlations for determining the critical oil flow rate:
• Gas coning
• Water coning
• Combined gas and water coning Gas coning
Consider the schematic illustration of the gas coning problem shown in Figure 9-3.
Figure 9-3.Gas coning.
Meyer and Garder correlated the critical oil rate required to achieve a stable gas cone with the following well penetration and fluid parameters:
• Difference in the oil and gas density
• Depth Dtfrom the original gas-oil contact to the top of the perforations
• The oil column thickness h
The well perforated interval hp, in a gas-oil system, is essentially defined as
hp=h −Dt
Meyer and Garder propose the following expression for determining the oil critical flow rate in a gas-oil system:
where Qoc=critical oil rate, STB/day
ρg, ρo=density of gas and oil, respectively, lb/ft3 ko=effective oil permeability, md
re, rw=drainage and wellbore radius, respectively, ft h=oil column thickness, ft
Dt=distance from the gas-oil contact to the top of the perforations, ft
Water coning
Meyer and Garder propose a similar expression for determining the crit- ical oil rate in the water coning system shown schematically in Figure 9-4.
The proposed relationship has the following form:
where ρw =water density, lb/ft3 hp =perforated interval, ft
oc
4 w o
e w
o o o
Q = 0.246 10 p
(r /r ) k
B h h
× ⎡ −
⎣⎢
⎤
⎦⎥
⎛
⎝⎜ ⎞
⎠⎟ −
− ρ ρ
μ
ln ( 2 2) (9-22)
oc
4 o g
e w
o o o
2
Q 0.246 10 t
r r k
B h h D
= × −
( )
⎡
⎣⎢
⎤
⎦⎥
⎛
⎝⎜ ⎞
⎠⎟⎡⎣ − −
− ρ ρ
μ
ln / ( )2⎤⎤⎦ (9-1)
Simultaneous gas and water coning
If the effective oil-pay thickness h is comprised between a gas cap and a water zone (Figure 9-5), the completion interval hpmust be such as to permit maximum oil-production rate without having gas and water simultaneously produced by coning, gas breaking through at the top of the interval and water at the bottom.
This case is of particular interest in the production from a thin column underlaid by bottom water and overlaid by gas.
For this combined gas and water coning, Pirson (1977) combined Equations 9-1 and 9-2 to produce the following simplified expression for determining the maximum oil flow rate without gas and water coning:
oc
4 o
o o
2 p e w
w o
o g
w
Q = 0.246 10 k B
h h
r r
( )
× ⎡
⎣⎢
⎤
⎦⎥
−
( )
× − −
−
μ
ρ ρ ρ ρ
2
2
ln /
ρρ ρ ρ ρ ρ ρ
ρ ρ
−
⎛
⎝⎜
⎞
⎠⎟ + − − −
−
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢
⎢
⎤
⎦⎥
⎥
g
o g
o g
w g
( ) 1
2
(9-3)
Figure 9-4.Water coning.
Example 9-1
A vertical well is drilled in an oil reservoir overlaid by a gas cap. The related well and reservoir data are given below:
horizontal and vertical permeability, i.e., kh, kv =110 md oil relative permeability, kro =0.85
oil density, ρo =47.5 lb/ft3
gas density, ρg =5.1 lb/ft3
oil viscosity, μo =0.73 cp
oil formation volume factor, Bo =1.1 bbl/STB
oil column thickness, h =40 ft
perforated interval, hp =15 ft
depth from GOC to top of perforations, Dt =25 ft
wellbore radius, rw =0.25 ft
drainage radius, re =660 ft
Using the Meyer and Garder relationships, calculate the critical oil flow rate.
Solution
The critical oil flow rate for this gas coning problem can be deter- mined by applying Equation 9-1. The following two steps summarize Meyer-Garder methodology:
Figure 9-5.The development of gas and water coning.
Step 1.Calculate effective oil permeability ko: ko=krok =(0.85) (110) =93.5 md Step 2.Solve for Qocby applying Equation 9-1:
Example 9-2
Resolve Example 9-1 assuming that the oil zone is underlaid by bot- tom water. The water density is given as 63.76 lb/ft3. The well comple- tion interval is 15 feet as measured from the top of the formation (no gas cap) to the bottom of the perforations.
Solution
The critical oil flow rate for this water coning problem can be estimated by applying Equation 9-2. The equation is designed to determine the criti- cal rate at which the water cone “touches” the bottom of the well to give
The above two examples signify the effect of the fluid density differ- ences on critical oil flow rate.
Example 9-3
A vertical well is drilled in an oil reservoir that is overlaid by a gas cap and underlaid by bottom water. Figure 9-6 shows an illustration of the simultaneous gas and water coning.
oc 2 2
oc
Q = 0.246 63.76 47.5 l (660 / 0.25)
93.5
(0.73)(1.1) 40 15 Q = 8.13 STB/day
× ⎡ −
⎣⎢
⎤
⎦⎥
⎛
⎝⎜
⎞
⎠⎟ −
10−4 ( )
n [ ]
oc 2 2
Q = 0.246 47.5 5.1 l (660/0.25)
93.5
(0.73)(1.1) 40 (40 25)
= 21.20 STB/day
×10−4 − − −
n [ ]
The following data are available:
oil density ρo =47.5 lb/ft3
water density ρw =63.76 lb/ft3
gas density ρg =5.1 lb/ft3
oil viscosity μo =0.73 cp
oil FVF Bo =1.1 bbl/STB
oil column thickness h =65 ft
depth from GOC to top of perforations Dt =25 ft well perforated interval hp =15 ft
wellbore radius rw =0.25 ft
drainage radius re =660 ft
oil effective permeability ko =93.5 md horizontal and vertical permeability, i.e., kh, kv =110 md oil relative permeability kro =0.85
Calculate the maximum permissible oil rate that can be imposed to avoid cones breakthrough, i.e., water and gas coning.
Solution
Apply Equation 9-3 to solve for the simultaneous gas and water con- ing problem, to give:
Figure 9-6.Gas and water coning problem (Example 9-3).
Pirson (1977) derives a relationship for determining the optimum placement of the desired hpfeet of perforation in an oil zone with a gas cap above and a water zone below. Pirson proposes that the optimum dis- tance Dtfrom the GOC to the top of the perforations can be determined from the following expression:
where the distance Dtis expressed in feet.
Example 9-4
Using the data given in Example 9-3, calculate the optimum distance for the placement of the 15-foot perforations.
Solution
Applying Equation 9-4 gives
Slider (1976) presented an excellent overview of the coning problem and the above-proposed predictive expressions. Slider points out that Equations 9-1 through 9-4 are not based on realistic assumptions. One of the biggest difficulties is in the assumption that the permeability is the same in all directions. As noted, this assumption is seldom realistic.
Since sedimentary formations were initially laid down in thin, horizontal D = (65 15) 1t 47.5 5.1
63.76 5.1 = 13.9 ft
− − −
−
⎡
⎣⎢
⎤
⎦⎥
t p
o g
w g
D = (h−h ) 1− −
−
⎡
⎣⎢ ⎤
⎦⎥ ρ ρ
ρ ρ (9-4)
oc
2 2
Q = 0.246 93.5
(0.73) (1.1)
65 15 (660/0.25) (63
× ⎡ −
⎣⎢
⎤
⎦⎥
×
10−4
Ln ..76 47.5)
+ (47.5 5.1) 1 47.5 2
− −
−
⎛⎝⎜ ⎞
⎠⎟
⎡
⎣⎢
− −
47 5 5 1 63 76 5 1
. . 2
. .
−−
−
⎛⎝⎜ ⎞
⎠⎟
⎤
⎦⎥ = 5.1
63.76 5. 17.1 STB/day 1
sheets, it is natural for the formation permeability to vary from one sheet to another vertically.
Therefore, there is generally quite a difference between the permeabil- ity measured in a vertical direction and the permeability measured in a horizontal direction. Furthermore, the permeability in the horizontal direction is normally considerably greater than the permeability in the vertical direction. This also seems logical when we recognize that very thin, even microscopic sheets of impermeable material, such as shale, may have been periodically deposited. These permeability barriers have a great effect on the vertical flow and have very little effect on the horizon- tal flow, which would be parallel to the plane of the sheets.
The Chierici-Ciucci Approach
Chierici and Ciucci (1964) used a potentiometric model to predict the coning behavior in vertical oil wells. The results of their work are pre- sented in dimensionless graphs that take into account the vertical and horizontal permeability. The diagrams can be used for solving the follow- ing two types of problems:
a. Given the reservoir and fluid properties, as well as the position of and length of the perforated interval, determine the maximum oil produc- tion rate without water and/or gas coning.
b. Given the reservoir and fluids characteristics only, determine the opti- mum position of the perforated interval.
The authors introduced four dimensionless parameters that can be deter- mined from a graphical correlation to determine the critical flow rates.
The proposed four dimensionless parameters are shown in Figure 9-7 and defined as follows:
Effective dimensionless radius rDe:
The first dimensionless parameter that the authors used to correlate results of potentiometric model is called the effective dimensionless radius and is defined by:
r r
h k
De k
e h
v
= (9-5)
Meyer and Garder stated that the proposed graphical correlation is valid in the following range of rDevalues:
5 ≤rDe≤80
where h=oil column thickness, ft re=drainage radius, ft
kv, kh=vertical and horizontal permeability, respectively Dimensionless perforated length e:
The second dimensionless parameter that the authors used in develop- ing their correlation is termed the dimensionless perforated length and is defined by:
The authors pointed out that the proposed graphical correlation is valid when the value of the dimensionless perforated length is in the fol- lowing range:
0 ≤ ε ≤0.75
Dimensionless gas cone ratio dg:
The authors introduced the dimensionless gas cone ratio as defined by the following relationship:
ε =h hp/ (9-6)
Figure 9-7. Water and gas coning in a homogeneous formation. (After Chierici, Ciucci, and Pizzi, courtesy JPT,August 1964.)
with
0.070 ≤ δg≤0.9
where Dt is the distance from the original GOC to the top of perfora- tions, ft.
Dimensionless water cone ratio dw:
The last dimensionless parameter that Chierici et al. proposed in developing their correlation is called the dimensionless water cone ratio and is defined by:
with
0.07 ≤ δw≤0.9
where Db=distance from the original WOC to the bottom of the perforations, ft
Chierici and coauthors proposed that the oil-water and gas-oil contacts are stable only if the oil production rate of the well is not higher than the following rates:
where Qow =critical oil flow rate in oil-water system, STB/day Qog =critical oil flow rate in gas-oil system, STB/day ρo, ρw, ρg =densities in lb/ft3
ψw =water dimensionless function ψg =gas dimensionless function
kh =horizontal permeability, md
og
2
o g
o o
ro h g De g
Q = 0.492 h ( )
B (k k ) (r , , )
× − −
104 ρ ρ
μ Ψ ε δ (9-10)
ow
2
w o
o o
ro h w De w
Q = 0.492 h ( )
B (k k ) (r , , )
× − −
10 4 ρ ρ
μ ψ ε δ (9-9)
w= D /hb
δ (9-8)
g= D /ht
δ (9-7)
The authors provided a set of working graphs for determining the dimensionless function ψ from the calculated dimensionless parameters rDe, ε, and δ. These graphs are shown in Figures 9-8 through 9-14. This set of curves should be only applied to homogeneous formations.
It should be noted that if a gas cap and an aquifer are present together, the following conditions must be satisfied in order to avoid water and free-gas production.
Qo≤Qow and
Qo≤Qog
0.006 0.008 0.010 0.020 0.040 0.060 0.080 0.200 0.80
0.70 0.60
0.50
0.40 0.35
0.30 0.25
0.100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.07 0.10
0.15 4
3
2
0.50 0.25
ψ
0=0.90
0=0.20
ρog ρwo=1
DE=5
ε
Figure 9-8. Dimensionless functions for rDe= 5. (After Chierici, Ciucci, and Pizzi, courtesy JPT, August 1964.)
(text continued on page 602)
0.006 0.008 0.010 0.020 0.040 0.060 0.080 0.60
0.50 0.40
0.35 0.30 0.25
0.100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.07 0.10
0.15 4
3
2
0.50 0.25 ρog
ψ
ρog ρwo=1
0=0.20
ε
Figure 9-9.Dimensionless functions for rDe=10. (After Chierici, Ciucci, and Pizzi, courtesy JPT,August 1964.)
0.006 0.008 0.010 0.020 0.040 0.060 0.080 0.200
0.80 0.70
0.60 0.50
0.40
0.30 0.25
0.100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.07 0.10
0.15 4
3
2
0.50 0.25
ψ
0.35 0=0.90
0=0.20
ρog ρwo=1
DE=20
ε
Figure 9-10. Dimensionless functions for rDe = 20. (After Chierici, Ciucci, and Pizzi, courtesy JPT,August 1964.)
0.006 0.008 0.010 0.020 0.040 0.060 0.080 0.200
0.80 0.70
0.60 0.50
0.40 0.30 0.25
0.100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.07 0.10
0.15 4
3
2
0.50 0.25
ψ
0.35 0=0.90
0=0.20
ρog ρwo=1
DE=30
ε
Figure 9-11. Dimensionless functions for rDe = 30. (After Chierici, Ciucci, and Pizzi, courtesy JPT,August 1964.)
0.006 0.008 0.010 0.020 0.040 0.060 0.080 0.200
0.80 0.70
0.60 0.50
0.40 0.30
0.100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.07 0.10
0.15 4
3
2
0.50 0.25
ψ
0.35
0.25 0-090
0=0.20
ρog ρwo=1
DE=40
ε
Figure 9-12. Dimensionless functions for rDe = 40. (After Chierici, Ciucci, and Pizzi, courtesy JPT,August 1964.)
0.006 0.008 0.010 0.020 0.040 0.060 0.080 0.200
0.70 0.60
0.50 0.40
0.30 0.100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.07 0.10
0.15 4
3
2
0.50 0.25
ψ
0.35
0.25 0=0.90 0.80
0=0.20
ρog ρwo=1
DE=60
ε
Figure 9-13. Dimensionless functions for rDe = 60. (After Chierici, Ciucci, and Pizzi, courtesy JPT,August 1964.)
0.006 0.008 0.010 0.020 0.040 0.060 0.080 0.200
0.70 0.60
0.50 0.40
0.30 0.100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.07 0.10
0.15 4
3
2
0.50 0.25
ψ
0.35
0.25 0=0.90 0.80
0=0.20
ρog ρwo=1
DE=80
ε
Figure 9-14. Dimensionless functions for rDe = 80. (After Chierici, Ciucci, and Pizzi, courtesy JPT,August 1964.)
Example 9-5
A vertical well is drilled on a regular 40-acre spacing in an oil reser- voir that is overlaid by a gas cap and underlaid by an aquifer. The follow- ing data are available:
Oil pay thickness h =140 ft
Distance from the GOC to the top of perforations Dt =50 ft Length of the perforated interval hP =30 ft
Horizontal permeability kh =300 md
Relative oil permeability kro =1.00
Vertical permeability kv =90 md
Oil density ρo =46.24 lb/ft3
Water density ρw =68.14 lb/ft3
Gas density ρg =6.12 lb/ft3
Oil FVF Bo =1.25 bbl/STB
Oil viscosity μo =1.11 cp
A schematic representation of the given data is shown in Figure 9-15.
Calculate the maximum allowable oil flow rate without water and free- gas production.
Solution
Step 1.Calculate the drainage radius re: πre2=(40)(43,560)
re=745 ft
Step 2.Compute the distance from the WOC to the bottom of the perfo- rations Db:
Db=h −Dt−hp
Db=140 −50 −30 =60 ft
(text continued from page 598)
Step 3.Find the dimensionless radius rDefrom Equation 9-5:
Step 4.Calculate the dimensionless perforated length ε by applying Equation 9-6:
Step 5.Calculate the gas cone ratio δgfrom Equation 9-7:
Step 6.Determine the water cone ratio δwby applying Equation 9-8:
Step 7.Calculate the oil-gas and water-oil density differences:
Δρow= ρw− ρo=68.14 −46.24 =21.90 lb/ft3 Δρog= ρo− ρg=46.24 −6.12 =40.12 lb/ft3
w= 60
140 = 0.429 δ
g= 50
140 = 0.357 δ
ε= 30
140 = 0.214 rDe = 745
140 300
90 = 9.72
Figure 9-15.Gas and water coning problem (Example 9-5).
Step 8.Find the density differences ratio:
Step 9.From Figure 9-10, which corresponds to rDe = 10; approximate the dimensionless functions ψgand ψw:
for ε =0.214 and δg=0.357 to give ψg=0.051 and
for ε =0.214 and δw=0.429 to give ψw=0.065
Step 10.Estimate the oil critical rate by applying Equations 9-9 and 9-10:
These calculations show that the water coning is the limiting condition for the oil flow rate. The maximum oil rate without water or free-gas pro- duction is, therefore, 297 STB/day.
Chierici and Ciucci (1964) proposed a methodology for determining the optimum completion interval in coning problems. The method is basically based on the “trial and error” approach.
For a given dimensionless radius rDe and knowing GOC, WOC, and fluids density, the specific steps of the proposed methodology are sum- marized below:
Step 1.Assume the length of the perforated interval hp. Step 2.Calculate the dimensionless perforated length ε =hp/h.
Step 3.Select the appropriate family of curves that corresponds to rDe, interpolate if necessary, and enter the working charts with ε on the x-axis and move vertically to the calculated ratio Δρog/Δρow. Estimate the corresponding δand ψ. Designate these two dimen-
ow
2
og
2
Q = 0.492 140 (21.90)
(1.25) (1.11) (1) (300) STB day Q = 0.492 140 (40.12)
(1.25) (1.11) (1) (300) 0.051 = 426 STB/day
× =
×
−
−
10 0 065 97
10
4
4
[ ] . /
[ ]
2 Δ og/Δ ow= 40.12
21.90 = 1.83
ρ ρ