The Effect of Reservoir and Fluid Properties on Production Decline Curves R. W. Gentry, SPE-AIME, Edinger, Inc.
A. W. McCray, SPE-AIME, U. of Oklahoma
Introduction The extrapolation of production decline curves to predict future oil production has a long history. The word "decline" is a misnomer; "decline curve" is a descriptive term for a graphical presentation of some aspect of the performance history of a well or lease. The graph may show a decline, an incline, or may remain flat. Through the years, the most commonly used decline curve, and the curve usually maintained by oil companies, is the production rate vs time curve plotted on semilog graph paper.
The analysis of a decline curve provides two important items of information: (1) the remaining oil and gas reserves to be expected, and (2) the remaining productive life of a well or lease. In addition, an explanation of any anomalies that appear on the graph is also useful. In recent years, it has been increasingly important to determine the combination of these items so that an accurate yearly estimate of future production can be made. This information is obtained by an analysis of the past performance as shown on the production rate vs time curve. The curve then is extrapolated into the future, and estimated future yearly production is taken from the extrapolated portion of the curve.
Decline Curve Analysis Problems Three basic problems associated with the extrapolation process are connected with the historical development of decline curve analysis and have been considered basic assumptions since being defined by ArpS.l,2 These as-
sumptions are as follow. 1. The extrapolation procedure is strictly empirical,
and a mathematical expression of the curve based on physical considerations can be set up only for a few simple cases.
2. Whatever causes governed the trend of a curve in the past will continue to govern its trend in the future in a uniform manner.
3. The decline exponent b in the equations developed by Arps (Table 1) must have a value of 0.0 ~ b ~ 1.0.
Because of empirical extrapolation, a decline curve usually will have a wide range of interpretations. The range of interpretations depends on the production stage of the property. If there is limited prior production history (i. e., a new well), there is a wider range of interpretations possible than for a well or property in the stripper stage of production. Also, each specific interpretation is a function of the experience, integrity, and objective of the evaluating engineer.
Various controllable and uncontrollable influences or causes govern the production performance of a well or lease. Some of these influences are as follow.
Controllable 1. Prorated production. 2. Remedial work on producing wells. 3. Fluid or gas injection into the producing reservoir. 4. Production problems, shutdowns, etc. 5. Problems with scale, paraffin, etc. 6. Limitations of producing equipment. 7. Changes in operating personnel.
This paper investigates the effects that rock andfluid properties impose on the production history of a well or lease as revealed by a decline curve. In particular, the effects on the constants of the exponential, hyperbolic, and harmonic decline curve equations are reported. An improved method is presented for analyzing production histories, with two field examples illustrating the method.
SEPTEMBER, 1978 1327
Uncontrollable 1. Physical characteristics of the reservoir. 2. Characteristics of the reservoir fluids. 3. Primary reservoir drive mechanisms.
Controllable influences are caused by man when producing oil from the property. Uncontrollable influences are those controlled by nature. Any single or combination of these influences can change at any time during producing life. If controllable influences are eliminated from the analysis, the resulting curve should reflect the combined effect of the three uncontrollable influences.
In prior years, the extrapolation procedure was empirical, but with modern technology there is no reason for it to remain so. Also, a critical examination of the causes or influences that govern decline behavior show that it is not reasonable to expect those causes that governed in the past to continue to do so uniformly in the future. Investigators3,4 also have shown that there are many instances in nature where the decline exponent b is greater than 1. O.
Although this discussion of problems tends to discourage using decline durves, the extrapolation of the ratetime curve is still the most used and efficient method for evaluating oil properties.
Historical Development In 1908, Arnold and Anderson5 published the first explanation and mathematical development o{the exponential decline. Note that this is still the most popular method of decline curve analysis used by engineers. In 1924, Cutler6 proposed a method of extrapolating hyperbolic declines by plotting the data on log-log paper and shifting the curve to obtain a straight line. The basics of this method are still being used today. In 1945, Arps published a paperl that classified decline curves into four general types: (1) exponential, (2) hyperbolic, (3) harmonic, and (4) ratio declines. The first three are familiar to engineers, but the ratio decline is seldom used. The second Arps paper2 (1956) did not contain any new information concerning decline curves, but it did put the mathematical formulas into a much simpler form. These equations are shown in Cols. 1 through 4 of Table 1. In 1972, Gentry7 manipulated the decline equations to obtain two dimensionless equations for each type of de-
cline. These equations are shown in Cols. 5 and 6 of Table 1. These equations were used to prepare two graphs that can be used to extrapolate hyperbolic and harmonic decline curves quickly. SliderS proposed an improved method of hyperbolic decline curve analysis using transparent overlays.
Since 1945, when Ref. 1 was published, investigators have been trying to associate the decline exponent b with the physical characteristics' or the active drive mechanism (uncontrollable influences) of the reservoir. Mead9 presented a list of b values for various types of reservoir drive mechanisms. Matthews and Lefkovits10, 11 investigated gravity drainage reservoirs on theoretical and actual field performance bases. One of their conclusions stated that the production declines for homogeneous gravity drainage reservoirs where a free surface exists will be of the hyperbolic type with b = 0.5. When tested on field cases, several instances were founc:i where b values were greater than 1.0. Matthews and Lefkovits concluded that this happened when two or more layers of different permeabilities were producing into a common wellbore. They predicted that as the more permeable layers were exhausted, the value of b would decline to an ultimate value of 0.5. They also implied a physical significance to the initial decline rate, ai'
The next logical step was to determine why production curves exhibit certain values for the three decline constants qi' ai' and b. Fetkovich12 attempted to find possible connections between decline constants and characteristics of the reservoir and fluids by using the material balance equation, rate equations, etc., to develop a set of correlations. The effects of stratified reservoirs and backpressure changes also were investigated.
Theoretical Development The equations in Table 1 were derived from the differential equation,
dq
a = Kqb = - dt . . ...................... (1) q
The general solution for the rate-time relationship is
q - qj - (1 + ba;t)lIb '
(b =to 1, b =,0) ....... (2)
This can be integrated to obtain the equation for the
Similar equations can be obtained for the special cases of b = 0.0 and b = 1.0.
Although b values from 0.0 to 1.0 generally are used, the above equations can be calculated using b values less than 0.0 and greater than 1.0. In fact, there is a limiting case for b = 00. In this case, the dimensionless Eq. 6 would become
The dimensionless values for b = 0.0, b = 0.5, and b = 1.0 have been plotted as a background on several figures in this paper, and in some instances, the dimensionless value for b = 00 also has been included. By using these as a baCkground and plotting computer-generated or actual field data on the curve, it can readily be seen how these data compare with the Arps equations.
An interesting result occurs if b is set equal to 0.5. In this case,
~= (~)O.5, qi t qt
(b = 0.5),
which can be further manipulated as
Np = (qiq)O.5t. . ........................ (8)
One computer simulator run (discussed later) produced a curve that can be described by the equation
~= (L)O.4. qit qi
This can be written for the general form as
~ = (L)a or Np = (qi I-a qa)t. . ......... (9) qit qi
This equation may describe certain decline behavior better than the Arps equations.
To solve the Arps equations for any particular decline curve, three constants must be determined: (1) qi' the initial production rate; (2) ai' the initial decline rate; and (3) b, the decline exponent.
Current practice is to take the existing production history and determine the values of these constants. This is
SEPTEMBER, 1978
done by trial and error, using overlays, by mathematical analysis, or by graphical methods. Once these constants are determined, they are put into the equations to calculate future production. However, if we could define some of the effects of reservoir and fluid characteristics on these constants, we could determine the constants and calculate future production without prior production history.
To find some of these relationships, fluid production should be monitored from reservoirs with known physical quantities containing fluids of known compositions. Production then could be plotted as a decline curve, and the three constants determined for each particular reservoir and fluid.
Reservoir Simulation Model The production of known fluids from a reservoir of known characteristics was done with the aid of a computer reservoir model. The model was constructed using assumed reservoir conditions of volume, porosity, temperature, pressure, water saturation, and absolute and relative permeability relationships. The reservoir was assumed to be filled with fluids of known characteristics of formation volume factor, solution gas content, oil and gas viscosities, and deviation factors for the reservoir gas. The reservoir model was constructed assuming the following limitations.
1. The reservoir is closed, horizontal, circular, and of constant thickness. It is homogeneous with constant permeability and relative permeability characteristics.
2. The material balance method for predicting performance histories of solution gas reservoirs permits a reasonable approximation of reservoir pressure and GOR as a function of cumulative oil production.
3. Ths solution of Darcy's radial flow equation for a bounded reservoir adequately describes the flow of fluid through the reservoir into the wellbore.
4. The reservoir driving mechanism is solution gas at or below the saturation pressure of the oil.
5. There are no capillary pressure gradients or gravity drainage effects.
6. Individual zones are separated by impermeable strata so that pressure communication between zones exists only at the well bore .
This computer model was used for calculations of all simulated production histories. Reservoir engineering equations for the material balance calculations were obtained from Ref. 13. Two sets of relative permeability relationships were obtained from Ref. 14 with some slight modifications. One set typifies a consolidated sand reservoir, and the other represents a dolomite reservoir. Two sets of fluid data (oil samples) taken from actual field data were used in the model. One oil sample was taken from a Bartlesville sandstone reservoir in Oklahoma County, OK, and the other was from a Hunton dolomitic reservoir in Kingfisher County, OK.
Experimental Procedure One relative permeability relationship and one fluid analysis were loaded into the computer model. The program then was run and the production histories were printed for that particular fluid-permeability system for absolute permeabilities of 1, 3, 5, 7, 9,11,13, and 15 md. The dimensionless values of ait, Np/qit, and qi/q
1329
were calculated and plotted on semilog paper with the solutions of the Arps equations in the background. This showed how b behaved in relation to the production ratio, qi/q, and to the solutions to the Arps equations. Also, the. results from two or more absolute permeabilities were summed to simulate the effect of two or more zones producing into the wellbore. These results were plotted to observe the effects of stratification.
This procedure was conducted for four fluidpermeability systems using all possible combinations of two fluid analyses and two relative permeability relationships. Additional computer runs were made to investigate the effect of drainage area, zone thickness, and porosity on the decline constants.
The initial computer run for each fluid-permeability system used the following standard set of reservoir data.
Porosity, % 18 Zone thickness, ft 20 Radius of well bore, ft 0.333 Radius to external boundary,
ft (acres) 745 (40) Reservoir water saturation, % 15 Initial reservoir Saturation pressure
Later, a search was made through actual production histories thought to have some characteristics of the simulated fluid-permeability systems.
Results of Study The results from the computer simulation model are summarized in Table 2. The graphical results of these computer runs are shown in Figs. 1 through 4.
Fluid-Permeability Sy~tem 1-1 The data used in these model runs were from the fluid analysis (Sample 1) of a fluid sample taken from a consolidated sandstone reservoir and relative permeability curves (Sample 1) for a typical sandstone reservoir. Fig. 1 shows the decline curve histories for three absolute permeabilities, and Fig. 2 shows the plot of dimensionless quantities Np/qil vs q/q. Also shown in the backc ground of Fig. 2 is the solution of the Arps equations. Note that the trend of the plotted values is not affected by absolute permeability. The histories from the lower permeabilities define the curve in the lower ranges of q;!q , and the histories of the higher permeabilities define the curve for the larger values of q;!q. The equation for the average curve plotted in Fig. 2 can be represented by
Np = (q) 0.4 qi l --q: .
In the lower range of q;!q, the plotted curve closely approximates the Arps curve for b = 0.0. If this is the case, then
N p = .!l.i-=-!l., aj
which can be manipulated to yield
!L= 1 - !!iNp-qj qj
1330
This means that a plot of q/qi vs Np will be a straight line with an intercept of q/qi = 1 when Np = 0; the slope of the line will be -a;!qi' This plot is shown in Fig. 3. From this curve we found that
a. = _q_i_ time-I. I 97,000
Therefore, ai was caiculated for each absolute permeability by simply knowing the initial producing rate. Knowing ai' the values of ail were calculated and plotted as shown in Fig. 4. Fig. 4 also shows the solutions of the Arps equations in the background.
One important observation is that the b value for the plotted curve does not remain constant. As shown in Fig. 2, the value of b at a q;lq ratio of 5 is less than 0.0, while at a ratio of 100 the b value is approaching 0.2.
The next run was made using the same load data, except the external drainage radius was increased to 1,053 ft (80 acres) and the thickness of zone was reduced to 10 ft,. When compared with the fIrst run, the data plotted almost identically on the dimensionless curves.
A third run was made to determine the effect reservoir volume would have on the curves. All load data were the same as the initial run, except porosity was loaded at 9%. This gave a reservoir volume of one-half the value of the initial run. The dimensionless curves again plotted nearly identically with the other computer runs. However, the relationship for ai changed to
a. = qi I 49,000'
Fluid-Permeability System 1-2
This computer simulation used Relative Permeability Relationship 2 to calculate production histories. These histories are plotted in Fig. 1. Using 'these relative permeability curves, we noticed that permeability to gas does not occur until oil saturation has been reduced from 85 to 55 %. This means that oil recovery at this reservoir should be more efficient. This is confirmed by the results obtained. Cumulative production from this reservoir amounted to 250,000 bbl of oil, while recovery from the previous system amounted to 100,000 bbl.
Fig. 2 shows the dimensionless plot of qJq vs Np/(qil) and indicates that the early life of this system will be harmonic (b = 1). Therefore,
Np=~: In(~j),
and a plot of Np vs q/qj on semilog paper should produce a straight line with a slope of -q;laj. This plot is shown in Fig. 3, and the initial decline relationship was
aj = 112~040 The dimensionless a;f relationship is plotted in Fig. 4.
The relationships indicate that this reservoir efficiently produces oil until a q;lq ratio of 6 is reached. At this point, significant quantities of gas are being produced, and as production continues, the relative permeability to gas increases rapidly, resulting in even larger quantities of produced gas. As this gas is released from the reservoir, it is expended without producing oil; oil production suddenly becomes inefficient. This can be seen in the abrupt changes on the curves. This simulated behavior
JOURNAL OF PETROLEUM TECHNOLOGY
indicates that relative permeability characteristics do exert an influence on production histories.
through 4. Production histories followed a hyperbolic decline with a b value of 0.3. In fact, these histories could have been predicted accurately using the Arps hyperbolic formula. Fluid-Permeability System 2-1
The fluid analysis used in these computer runs was from a fluid sample (Sample 2) taken from a well in Kingfisher County, OK, and was obtained from the dolomitic Hunton formation. This sample had a much higher formation volume factor and a greater solution gas content than Sample 1. This reservoir contains less oil and more gas than a reservoir filled with Fluid 1.
Fluid-Permeability System 2-2
Two computer runs were conducted using this fluidpermeability system. Again, one run was made using a 40-acre drainage area and a 20-ft zone thickness. The other run was made using an 80-acre drainage area and a lO-ft zone thickness. Drainage area and zone thickness had little effect on the dimensionless relationships. Also, a; was calculated for both runs to be
The data obtained from this system were similar in character to those obtained for Fluid-Permeability System 1-2. However, this system will recover less oil and more gas. The Np/q;t vs q;lq curve (Fig. 2) was shifted slightly to the right as a result of using Fluid Analysis 2. The value of b during early life was determined to be larger than 1.0, and the initial decline relationship was
a; = 71QOOO . ,
Stratified Reservoir Thus far, all data have assumed the reservoir is homogeneous with constant absolute permeability throughout its volume. Calculations were made using Computer Run 1, which combined data from two or more
a·=~ I 60,000·
Graphical analysis of this system is shown in Figs.
TABLE 2-RESULTS OF COMPUTER SIMULATION MODEL
Relative Permeability
Fluid Analysis
Sample 1 from Bartlesville Sandstone
Characteristics Ps = 2,520 psi Bo = 1.42 Rs. = 853 cu ft/bbl /L~ = 0.61 cp
Sample 2 from Hunton Dolomite
Characteristics Ps = 2,962 psi Bo = 1.779 Rs. = 1,395 cu ft/bbl /L~ = 0.233 cp
SEPTEMBER, 1978
Relative Permeability Relationship 2 from typical dolomite. 14
Critical gas saturation is 30% (gas starts flowing) krylk = kr/k = 0.11 at 45% gas saturation
First model run was for 40-acre drainage and 20-ft thickness. Analysis of the dimensionless curves revealed the following equations.
The decline exponent b varied from about 0.0 at lowq;lq rates to about 0.2 atq;lq = 100.
An additional computer run was made using 80-acre drainage and 10-ft thickness. The dimensionless curves plotted identically with the previous 40-acre case.
A third computer run was made using 40-acre drainage, 10-ft thickness, and 9% porosity instead of 18% used in the previous two runs. The dimenSionless curves again plotted identically with the two previous cases. However, since the oil volume in the drainage area is one-half the previous values, a new equation was found for a i = q;l(49,000).
First model run was for 40-acre drainage and 20-ft reservoir thickness. Analysis of the curves revealed the following:
a.=~ , 60,000
qi = 0.215kroh(pe - p,,J /LoBo [In (rJrw) - 0.5]
The decline exponent b maintained a value of 0.3 throughout the qJq range of 1 to 100.
An additional model run was made using 80-acre drainage and 10-ft reservoir thickness. The dimensionless curves plotted identically with the previous 40-acre case.
Relative Permeability Relationship 1 from typical consolidated sandstone. 14
Critical gas saturation is 5% (gas starts flowing) krglk = krolk = 0.12 at 27% gas saturation
Model run was for 40-acre drainage and 20-ft reservoir thickness. Analysis of the curves revealed the following equations:
The decline exponent b approximates 1.0 with low q;lq ratios that reduced to about 0.2 when q;lq = 100.
Model run was for 40-acre drainage and 20-ft reservoir thickness. Analysis of the curves revealed the following:
a.=~ , 71,000
q, = 0.215kroh(Pe -Pw) /LoBo [In(relrw) - 0.5]
The decline exponent b varied from a value of 1.0 at 10wqJq ratios to a value greater than 1.0at q;lq ratios offrom 5 to 10. The value of b reduced to about 0.4 at aqJq ratio of 100.
1331
-w W N
...... 0 c::: ;oc
~ t"" 0 'Tl
'" ~ ;oc 0 t"" tIl c::: ;:::
~ ::z:: z 0 t""
8 -<
Relative Permeabi 1 ity
System
Fluid Sample
OJ c_ o ..., Vl
"C c ttl
V>
E 0 ...
I.J...
"**'
OJ ..., E 0
0 Cl
E 0 ...
I.J...
N
"**'
q
Jill. Mo.
10,000
q
BbJ Mo.
1000
100 0
#1 Typical Sandstone
20
t - Years
'0 15 20
- Years
q
illlL Mo.
q
BbJ Mo.
#2 Typical Dolomite
t - Years
t - Years
Fig. 1-0ecline curve histories for four fluid-permeability systems.
Fig. 5-Decline curve histories for reservoirs with two layers of different absolute permeabilities.
o Homogeneous Formot ion
GD Imd+3md -3md+9md -5md+15md
6 Imd+ 5md - 3md + 15md
o Imd+7md x Imd+ IImd
• Imd+15md
•
0.9 0.8 0.7
• GD GD
0.6 0.5 0.4 0.3
N./Qit
16
Fig. 6-Dimensionless Np/qit curve histories for reservoirs with two layers of different absolute permeabilities.
17 18
• • x
0.2
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absolute permeabilities. Fig. 5 shows the production histories of different permeability combinations. Fig. 6 shows the behavior of the N p/q;t vs q;/q curve for various permeability combinations in comparison with the standard curve for a homogeneous reservoir.
It can be seen in Fig. 6 that heterogeneity tends to push the curve to greater corresponding b values as the degree of heterogeneity is increased. It is also apparent that for each curve a maximum b will be reached and then as q;lq becomes larger the b value will again approach the homogeneous value. We also found that the dimensionless relationship of the 1- to 3-md curve plotted the same as the 3- to 9-md and the 5- to 15-md curves. This can be seen in Fig. 6, and the conclusion can be drawn that any permeabilities with a 1:3 ratio would follow this same curve.
To determine more fully the effect of heterogeneity, a calculation was made on a hypothetical reservoir constructed of2 ftofl5-md sand and 18 ft ofl-md sand. The graphical analysis of this case is shown in Fig. 7. The production history shows a bend in the rate-time curve after about 1 year of production. The remainder of the production history appears to be almost exponential (straight line). During the early production history (first year), the 15-md sand is controlling the curve by contributing the largest part of production. After the first year, the 15-md sand is essentially depleted and the curve thereafter is controlled by the I-md sand since it is contributing the largest part of the production. The bend in the curve is the transition between the portion of production history controlled by the 15- and I-md sands.
Fig. 7 shows the effect of the heterogeneity on the Np/q;t vs q;lq dimensionless curve. This curve indicates that the b value will be a maximum of about 2 when the q;lq ratio is about 4:4.5. This is one of the reasons that decline curves can and do calculate b values that are greater than 1. Fig. 7 shows the a;t vs q;/q dimensionless relationship for this 1- to 15-md reservoir. Note that the points on this curve form a straight line for q;/q ratios greater than 3.0. This indicates that the later portion of the curve will be exponential, which confirms the observation of the rate-time curve. In fact, using the correct scales, the rate-time curve would exactly overlay the plot on Fig. 7.
In summary, separate zones producing into a common wellbore can affect the slope and the analysis of production decline curves. This effect can cause b values to be greater than 1. Also, the analysis technique presented here of using performance plotted on the dimensionless curves with the solution to the Arps equations as a background can be helpful when determining actual reservoirs with severe heterogeneity problems.
Summary Changing the fluid systems resulted in a greater change in the initial decline rate (a;). Also, a; is very sensitive to the volume of the reservoir. In both instances, where fluid characteristics were interchanged, the change in a; was 200 to 400% as compared with a change of only 15"10 18% when the relative permeability characteristics were changed.
Comparison of the production histories of the different fluid-permeability systems shows that changing the fluid characteristics had a smaller effect on the exponent b than
SEPTEMBER, 1978
changing the relative permeability characteristics. Comparison of Systems 1-1 with 2-1 and 1-2 with 2-2 indicate that changing the fluid characteristics also changed b from a value of 0.0 to 0.3 in the first instance and from a value of 1.0 to a value slightly larger than 1.0 in the second instance. However, the value of b was changed significantly when the relative permeability characteristics were changed. The value of b was increased from 0.0 to 1.0 when System 1-1 was compared with 1-2, and the value of b also was increased from 0.3 to a value greater than 1.0 when System 2-1 was compared to System 2-2.
The initial production rate (q;) depends on the permeability of the formation at the initial water saturation. The magnitude of q; also depends on the fluid characteristics as depicted by Darcy's flow equation.
In short, it appears that the relative permeability characteristics of the reservoir will have a greater effect on the decline curve constant b while the fluid characteristics will have a greater influence on the constants a; and q;. Also, permeability variation or reservoir heterogeneity will have a predictable effect on the production history.
Field Examples Field data from two wells were refined and plotted on dimensionless curves to show how the field data compared with the Arps solutions for hyperbolic curves.
Field Example 1
This is the production history of the well from which Fluid Sample 1 was obtained. The well is producing from the Bartlesville sandstone and is located in Oklahoma County, OK. The reservoir should have relative permeability characteristics similar to Relative Permeability Relationship 1. The history of this well is shown in Figs. 8 and 9. Not much can be learned about the reservoir from these curves, but some important effects of human manipulation on the production curve can be learned from this example. In Fig. 8, we can see that production had just started to establish a significant decline in 1948, when the formation was shot with nitroglycerin. This increased production, and the well had again established a decline when the reinjection of produced gas began. In 1961, several off-setting wells were shut in because of excessive GOR production. This, in effect, increased the drainage area of the well and flattened the decline. In 1966, the well was allowed to produce at high GOR 's to "blow down" the reservoir. These effects can be seen in Fig. 9.
The production history of this well shows the effect of man-induced stimulation. An estimate of production history in Fig. 8 shows that without the shot of nitroglycerin and gas reinjection, the ultimate production from this well would have been about 95,000 bbl of oil. This estimate was made using the results from Computer Run 1-1.
Fig_ 8-Production decline curve history for Field Example 1.
60
50
40
30
20
10
1.0
6Np/Qlt
Fig. 9-Dimensionless Np/q;f curve history for Field Example 1.
SEPTEMBER, 1978 1339
o 10,000 :=;; ';;; :Q
'" w f<t
'" z o f'-' ::> o o
'" I>-
I-YE ARS
sion above the bubble point.
Field Example 2
This example is the type of production history that prompted this study. This history exhibits a b value greater than 1.0. Figs. 10 and 11 show the history of this well.
The well is completed in the Mississippi limestone formation over a 300-ft interval. The producing formation is fractured with a tight matrix. Also, production is obtained from zones scattered throughout the pay section. This would indicate that the production history should resemble the curves in Fig. 7 where 18 ft of I-md formation was simulated as producing into the same bore as 2 ft of 15-md formation. This behavior can be seen in Fig. 11 for values of NP/(qit) from 1.0 to 0.3. At this point, the choke setting was changed to increase the producing rate of the well. This same behavior was repeated several times. The result of changing the choke size always increased or maintained the producing rate by reducing the back pressure on the producing formation. In each instance, this moved the Np/(qit) vs qJq curve to the right and maintained a b value greater than 1.0. In the future it will be impossible to increase production by resetting the choke. At that time, the curve probably will become almost vertical and again will move into the range of b values ofless than 1.0.
Fig. 1 O-Production decline curve history for Field Example 2. The dashed line in Fig. 11 shows an estimate of the
resulting curve if the choke had not been adjusted on the well, and the back pressure had remained constant. This
so
50
40
30
20
0.8 0.7
Np/q;1
Fig. 11-Dimensionless Np/q,t curve history for Field Example 2.
1340 JOURNAL OF PETROLEUM TECHNOLOGY
o
curve also shows the effect of permeability variation. One other observation can be made. The results of
human manipulation of the choke is not apparent in Fig. 10 (the rate-time curve), but these changes are easily seen and interpreted in Fig. 11. Plotting Np/( qjt) vs q/q can be a helpful diagnostic tool for evaluating the production history of a well or lease. Also, production curves exhibit b values greater than 1.0 due to human manipulation to maintain the highest production rate possible.
Conclusions 1. The dimensionless curves Np/ajt vs qh and ait vs
q;lq for a particular fluid-permeability system are not affected by the absolute permeability or size of the reservoir. The behavior of these plots is determined by (l) the characteristics of the contained fluid, (2) the relative permeability characteristics of the reservoir rock, (3) the reservoir drive mechanism, (4) reservoir heterogeneity, and (5) manual manipulation of production.
2. Reservoir heterogeneity tends to increase the magnitude of b as the degree of heterogeneity is increased. It also is apparent that b for a heterogeneous system will increase to a maximum value and then as the ratio q;lq becomes large, b will decrease and approach its homogeneous value.
3. Reservoir heterogeneity can and does cause b values to be greater than 1.0.
4. Manual manipulation of production can and does cause b values to be greater than 1.0.
5. The dimensionless plots for heterogeneous systems of 1 and 3 md, 3 and 9 md, and 5 and 15 md all plotted the same curve. This indicates that heterogeneous systems in the ratio of 1:3 will plot congruous dimensionless curves.
6. It appears that the relative-permeability characteristics of the reservoir have the greater effect on the decline exponent b, while the fluid characteristics have a greater influence on the constants aj and qj.
7. The equation Np/(ajt) = (q/qj)a may better define certain decline curves than do the Arps equations.
8. The plotting of production data on the Np/(ajt) vs q/q curve can be a helpful diagnostic tool for evaluating the production history of a well or lease.
Acknowledgment We thank Ward M. Edinger for permitting the use of facilities and office equipment of Edinger, Inc., when preparing this study.
Nomenclature a = decline as a fraction of pro<,iuction rate aj = initial decline b = decline exponent
Bo = oil formation volume factor
SEPTEMBER, 1978
BOj = initial oil formation volume factor h = formation thickness
krg = relative permeability to gas kro = relative permeability to oil K = constant
Np = cumulative oil production !1Np = oil production during an interval
Pe = reservoir pressure at external drainage radius
Pw = wellbore terminal pressure q = production rate qj = initial oil production rate r e = radius of drainage r w = wellbore radius
RSj = initial gas in solution t = time
a = arbitrary exponent 11-0 = oil viscosity
References 1. Arps, J. J.: "Analysis of Decline Curves," Trans., AIME (1945)
160,228-247. 2. Arps, J. J.: "Estimation of Primary Oil Reserves," Trans., AIME
(1956) 207,182-191. 3. Ramsey, H. J., Jr., and Guerrero, E. T.: "The Ability of Rate
Time Decline Curves to Predict Production Rates," J. Pet. Tech. (Feb. 1969) 139-141.
4. Shea, G. B., Higgins, R. V., and Lechtenberg, H. J.: "Decline and Forecast Studies Based on Performances of Selected California Oilfields," J. Pet. Tech. (Sept. 1964) 959-965.
5. Arnold, R. and Anderson, R.: "Preliminary Report on Coalinga Oil District," Bull., USGS (1908) 357, 79.
6. Cutler, W. W., Jr.: "Estimation of Underground Oil Reserves by Well Production Curves," Bull., USBM (1924) 228.
7. Gentry, R. W.: "Decline Curve Analysis," J. Pet. Tech. (Jan. 1972) 38-41.
8. Slider, H. C.: "A Simplified Method of Hyperbolic Decline Curve Analysis," J. Pet. Tech. (March 1968) 235-236.
9. Mead, H. N.: "Modifications to Decline Curve Analysis," Trans., AIME (1956) 207, 11-16.
10. Lefkovits, H. C. and Matthews, C. S.: "Application of Decline Curves to Gravity Drainage Reservoirs in the Stripper Stage," Trans.,AIME(1958) 213, 275-280.
11. Matthews, C. S. and Lefkovits, H. C.: "Gravity Drainage Performance of Depletion-Type Reservoir in the Stripper Stage," Trans., AIME (1956) 207, 265-274.
12. Fetkovitch, M. J.: "Decline Curve Analysis Using Type Curves, " paper SPE 4629 presented at the SPE-AIME 48th Annual Fall Meeting, Las Vegas, Sept. 30-0ct. 3, 1973.
13. Craft, B. C. and Hawkins, M. F.: Applied Petroleum Reservoir Engineering. Prentice-Hall, Inc., Englewood Cliffs, NJ (1962).
14. Amyx, J. W., Bass, D. M., Jr., and Whiting, R. L.: Petroleum Reservoir Engineering, McGraw-Hili Book Co., Inc., New York (1960) 204. .
15. Brons, Folkert: "On the Use and Misuse of Production Decline Curves," Prod. Monthly (Sept. 1963) 22. JPT
Original manuscript received in Society of Petroleum Engineers office Jan. 11,1977. Paper accepted for publication Nov. 16, 1977. Revised manuscript received June 13, 1978. Paper (SPE 6341) first presented at the SPE-AIME Economics and Evaluation Symposium, held in Dallas, Feb. 21-22,1977.
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