Decline Curve Analysis

1. Introduction

“Oil wells usually reach their maximum daily output shortly after they are completed. From that time they decline in production, the rapidity of declining depending on the output of the wells and on other factors governing their productivity. The production curve of a well shows the amount of oil production per unit of time for several consecutive periods; if the conditions affecting the rate of production are not changed by outside influences, the curve will be fairly regular, and if projected, will furnish useful knowledge as to the future production of the well.” The quote is taken from an article by J.O. Lewis and Carl H. Beal in 1918. Predicting the production of a well is a key factor in determining the value of a well. An over prediction of the well’s performance can cost a company tens to hundreds thousands of dollars. Decline curves are the most common means of forecasting production. Decline curves use data which is easy to obtain, they’re easy to plot, they yield results on a time basis, and they’re deceptively easy to analyze. Decline curve analysis is a graphical procedure used for analyzing declining production rates, and forecasting future performance of oil and gas wells. A curve fit of the past production performance is done using certain standard curves. This curve fit is then extrapolated to predict future performance. Decline curve analysis is a basic tool for estimating recoverable reserves. Conventional or basic decline curve analysis can be used only when the production history is long enough that a trend can be identified. Decline curve analysis is not grounded in fundamental theory but is based on empirical observations of production decline. Three types of decline curves have been identified, namely, exponential, hyperbolic and harmonic. There are some theoretical equivalents to these decline curves (for example, it can be demonstrated that under certain circumstances, such as constant well back-pressure, equations of fluid flow through porous media under "boundary dominated flow" conditions are equivalent to "exponential" decline). However, by and large, decline curve analysis is fundamentally an empirical process based on historical observations of well performance. Because of its empirical nature, decline curve analysis is applied, as is deemed appropriate for any particular situation, on single fluid streams or multiple fluid streams. For example, in some instances, the oil rate may exhibit an exponential decline, while in other situations it is the total liquids (oil + water) that exhibits the exponential trend. Thus in some instances the analysis is conducted on one fluid, sometimes on another, sometimes on the total fluids, sometimes on the ratio (for example Water-Oil-Ratio (WOR) or even (WOR + 1)). Since there is no overwhelming justification for any single variable to follow a particular trend, the practical approach to decline curve analysis is to choose the variable (gas, oil, oil + water, WOR, WGR etc..) that results in a recognizable trend, and to use that decline curve to forecast future performance.

It is implicitly assumed, when using decline curve analysis, that the factors causing the historical decline continue, unchanged, during the forecast period. These factors include both reservoir conditions and operating conditions. Some of the reservoir factors that affect the decline rate are pressure depletion, number of producing wells, drive mechanism, reservoir characteristics, saturation changes and relative permeability. Examples of operating conditions that influence the decline rate are: separator pressure, tubing size, choke setting, workovers, compression, operating hours and artificial lift. As long as these conditions do not change, then the trend in decline can be analyzed and extrapolated to forecast future well performance. If these conditions are altered, for example through a well workover, then the decline rate determined pre-workover will not be applicable to the post-workover period. When analyzing rate decline, two sets of curves are normally used. The flow rate is plotted either against time or against cumulative production. Time is the most convenient independent variable because extrapolation of rate-time graphs can be directly used for production forecasting and economic evaluations. However, plots of rate versus cumulative production have their own advantages. Not only do they provide a direct estimate of the ultimate recovery at a specified economic limit, but they will yield a more rigorous interpretation in situations where the production is influenced by intermittent operations. Good engineering practice demands that, whenever possible, decline curve analysis should be reconciled with other indicators of reserves, such as volumetric calculations, material balance and recovery factors. It should be noted that decline curve analysis results in an estimate of Recoverable Hydrocarbons, and NOT in Hydrocarbons-in-Place. Whereas the Hydrocarbons-in-Place are fixed by nature, the Recoverable hydrocarbons are affected by the operating conditions. For example a well producing from a reservoir containing 1BCF of gas-in-place may recover either 0.7 BCF or 0.9 BCF, depending on whether or not there is a compressor connected at the wellhead.

2. Theory

Decline curve analysis is not derived from theoretical first principles of fluid flow through porous media, but from empirical observations of the production performance of oil and gas wells. Three types of decline have been observed historically, namely: exponential, hyperbolic and harmonic. Decline curves represent production from the reservoir under “boundary dominated flow” conditions. This means that during the early life of a well, while it is still in “ transient flow” and the reservoir boundaries have not been reached, decline curves should NOT be expected to be applicable. Typically, during transient flow, the decline rate is high, but it stabilizes once boundary dominated flow is reached. For most wells this happens within a few months of production. However for low permeability wells (tight gas wells, in particular) transient flow conditions can last several years, and strictly speaking, should not be analyzed by decline curve methods until after they have reached stabilization.

All decline curve theory starts from the definition of the instantaneous or current decline rate, D, as follows:

qt

qD

∆∆−=

D, the decline rate, is “the fractional change in rate per unit time”, frequently expressed in “% per year”. In the diagram, D = Slope/Rate.

Exponential decline occurs when the decline rate, D, is constant. If D varies, the decline is considered to be either hyperbolic or harmonic, in which case, an exponent “b” is incorporated into the equation of the decline curve, to account for the changing decline rate. Exponential decline is given by:

where D is the decline rate (%) and is constant. Hyperbolic decline is described by:

where Di is the decline rate at flow rate qi, and b is an exponent that varies from 0 to 1. The decline rate D is not constant, so it must be associated with a specified rate, hence Di at flow rate qi. This is the most general form of decline equation. When b equals 1, the curve is said to be Harmonic. When 0 < b < 1, the curve is said to be Hyperbolic. When b=0, this form of the equation becomes indeterminate, but it can be shown that it is equivalent to Exponential decline. A comparison of Exponential, Hyperbolic and Harmonic declines is shown in the following diagram.

Decline curve analysis is usually conducted graphically, and in order to help in the interpretation, the equations are plotted in various combinations of “rate”, “log-rate”, “time” and “cumulative production”. The intent is to use the combination that will result in a straight line, which then becomes easy to extrapolate for forecasting purposes. The decline equations can also be used to forecast recoverable reserves at specified abandonment rates. 2.1. Exponential Theory All decline curve theory starts from the definition of the decline rate, D, as follows:

For exponential decline, D is constant. Integrating the above formulation yields:

which illustrates that a plot of log-rate versus time will yield a straight line of slope D/2.303. Cumulative Production is obtained by integrating the rate-time relationship. It can be shown that the flow rate is related to the cumulative production by:

which shows that a plot of rate versus Cumulative Production will be a straight line of slope D. Extrapolation of this straight line to any specified abandonment rate (including zero) gives the recoverable reserves. Cumulative production between time t1 and t2 can be obtained from

or, in terms of time

3. Nominal Decline Rate

When production follows an exponential decline (also known as constant percentage decline), there are two different ways of calculating the rate forecast. They both give the same answer. One is more suited to tabular data manipulation, and uses a “nominal decline rate”; the other is the classical graphical approach based on the traditional exponential decline equation. This section attempts to show the inter-relationships of the nominal and exponential decline rates. As discussed in exponential theory the rate (q2) at time (t2) is related to the rate (q1) a time (t1) by:

In practice, especially when rates are being calculated by hand (or calculator), and knowing that exponential decline is a constant percentage decline, some people like to calculate the rate sequence by simply multiplying the previous rate by a constant, equal to 1 minus the decline rate. (Some people refer to this procedure as constant percentage decline, but this is a misnamer because constant percentage decline is identical to exponential decline, and refers to a decline relationship and not to a method of calculation). In effect the procedure that is being applied results in the equation:

Note the use of two different constants, “D” and “d” in Equations1 and 2. Note also that the time interval is included in the “d” of Equation 2, but is explicit in Equation 1. Obviously these two equations are different from each other, yet there exists a value of “d” which when used in Equation 2 will give the same answer for q2 as that obtained by using a different value of “D” in Equation 1, provided the time intervals between q1 and q2 are the same. Because the two different procedures use two different percentage declines, and yet they can produce the same rate sequence, this has caused some confusion. To clarify the situation, “d” is called the nominal decline rate, in contrast to “D” the (true) exponential decline rate. (Remember that both “d” and “D” refer to the same exponentially declining rate - also called constant percentage decline!). The nominal decline rate, d is defined as: The relationship between the two is given by:

whereas the (true) exponential decline rate, D is, by definition equivalent to:

The difference between the two is illustrated below. In effect, “D” is related to the slope of the line, whereas “d” derives from the chord segment approximating that slope. Also, the rate by which the slope is divided is different - in the one case, it is the instantaneous rate, q; in the other case, it is the preceding rate, q1.

Usually, the nominal decline rate, d, arises when dealing with flow rate data in tabular rather than graphical format. The equivalence of the (true) exponential decline rate, D (as used in the exponential decline equation) and the “nominal” decline rate, d (as used in many hand calculations) is shown below:

The difference between d and D is usually very small, and is often within the accuracy of the data. So, even if they are misapplied, the error is usually not critical.

4. Exponential Analysis

The following steps are taken for exponential decline analysis, and for predicting future flow rates and recoverable reserves.

1. Plot flow rate vs. time on a semi-log plot (y-axis is logarithmic) and flow rate vs. cumulative production on a Cartesian (arithmetic coordinate) scale.

2. Allowing for the fact that the early time data may not be linear, fit a straight line through the linear portion of the data, and determine the decline rate “D” from the slope (-D/2.303) of the semi-log plot, or directly from the slope (D) of the rate-cumulative production plot.

3. Extrapolate to q = qE to obtain the recoverable hydrocarbons. 4. Extrapolate to any specified time or abandonment rate to obtain a rate forecast

and the cumulative recoverable hydrocarbons to that point in time.

5. Practical Decline Equations

The following diagram demonstrates rate-cumulative plots for oil and gas wells.

The following equation can be used for either oil or gas, provided the units are as specified below.

S.I. oil gas Q = 103 m3 Q = 106 m3 q = m3/day q = 103 m3/day Imperial oil gas Q = MSTB Q = Bscf q = BOPD q = MMscf/day The following diagram demonstrates semi-log rate-time plots for oil and gas wells.

The following equation can be used for either oil or gas, provided the units are as specified below.

S.I. oil gas q = M ln3TB/day q = 103 m3/day t = years t = years Imperial oil gas q = BOPD q = MMscf/d t = years t = years * Note: Time is typically measured in years. If "Time" (x-axis) on the rate-time plot is specified in days, the Dt in the decline equation is equal to: Dt(days)/365.25. "D" from the rate-time plot (DRT) should theoretically be equal to "D" from the rate-cumulative plot (DRC) however, practically speaking, DRT ≤ DRC. If DRT > DRC the analysis is NOT correct! When forecasting remaining reserves, extrapolate the decline curve to intercept with the economic rate (qEC). To calculate the RGIP or ROIP using either rate-time or rate-cumulative;

For gas specify units Bscf or 106 m3, and for oil specify units MSTB or 103 m3.

6. Effect of workover

A well treatment will affect the well’s production performance. Usually, an immediate increase in rate is realized. However, this rate increase will not be sustained and a rate decline will set in again, after an initial period of stabilization. The fact that there has been a rate increase due to the workover does not necessarilly mean that more reserves will be recovered. Whether the recoverable reserves will be increased by the workover depends on many factors. Three examples of different workover effects are presented: Example A, Re-perforation, hydraulic fracture, acidizing a single zone: The rate increases; the decline increases; the movable reserves are unchanged; and the economic reserves are increased. The skin near the wellbore is improved; the flow rate at the well is increased; No additional hydrocarbon reserves are added. Therefore the movable reserves remain the same. However, because of the economic abandonment limit, more reserves may be produced before abandonment.

Example B: Re-perforation or hydraulic fracture of additional zone: The rate increases; the decline increases; and the recoverable reserves increase. The skin near the wellbore is improved; the flow rate at the well is increased; Additional hydrocarbon reserves are added because a new zone is opened to flow at the well. Therefore the recoverable reserves are increased.

Example C: Adding Compression (or Pump): The rate increases; the decline stays the same; and the recoverable reserves increase Lowering the back-pressure on a producing well will increase the recoverable reserves. This can be achieved by installing a compressor on a gas well (or lowering an existing compressor’s suction pressure) or by installing a pump (or lowering an existing pump) on an oil well. Consider the case of a gas well producing against a back-pressure of 1000 psi. When the reservoir’s pressure approaches 1000 psi, the flow rate will be very small. If the back-pressure were reduced to say 200 psi (by adding compression), the flow rate would increase, and considerably more reserves could be recovered. This is illustrated below:

7. Values of b > 1

Hyperbolic decline follows the relationship:

where 0 < b < 1 A value of b = 0 corresponds to exponential decline; A value of b = 1 corresponds to harmonic decline. Values of b >1 are not consistent with decline curve theory, but they are sometimes encountered, and their meaning is explained below. Decline curve analysis is based on empirical observations of production rate decline, and not on theoretical derivations. Attempts to explain the observed behavior, using the theory of flow in porous media, lead to the fact that these empirically observed declines are related to “boundary dominated flow”. When a well is placed on production, there will be transient flow initially. Eventually, all the reservoir boundaries will be felt, and it is only after this time, that decline curve analysis becomes applicable, and the value of “b” lies in the range of 0 to 1, depending on the reservoir boundary conditions and the recovery mechanism. Occasionally, decline curves with values of b > 1 are encountered. Below are some reasons that have been presented to explain this:

1. The interpretation is wrong, and another value of b < 1 will fit the data.

2. The data is still in transient flow and has not reached “boundary dominated flow”.

3. Gentry and McCray, using numerical simulation showed that reservoir

layering can cause values of b > 1. (JPT 1978, 1327-41)

4. Bailey showed that some fractured gas wells showed values of b > 1, (and sometimes as high as 3.5 ! ). (OGJ Feb. 15, 1982, 117-118.)

Fetkovich believes that the data have been wrongly interpreted. More on this issue is presented when the relation between recovery mechanism and b is discussed.

8. Recovery Mechanism and b

Single-phase liquid production, high-pressure gas, tubing-restricted gas production, and poor waterflood performance lead to b = 0 (Fetkovich). Under solution gas drive, the lower the gas relative permeability, the smaller is the quantity of gas produced; hence the decline in reservoir pressure is slower, and accordingly the decline rate is lower (higher value of b). Simulation studies for a range of gas and oil relative permeability values have indicated 0.1 < b < 0.4 for different krg/kro curves, with the average curve resulting in b = 0.3. Note that the production data above the bubble point should not be analysed along with data below the bubble point. As pointed out in the introduction, decline analysis is valid when the recovery mechanism and the operating conditions do not vary with time. Above the bubble point pressure, b is 0 (exponential decline), while below the bubble point b increases as discussed above for solution gas drive. Typical gas wells have b in the range of 0.4 – 0.5. Conventional (light oil) reservoirs under edge water drive (effective water drive) seem to exhibit b = 0.5 (field experience). If there is present a mechanism that maintains reservoir pressure, the production rate would essentially remain constant (under constant producing pressure) and the decline would tend towards zero. Examples of such mechanisms could be: gas or water injection, a very active water drive or gas cap drive. Since the decline in reservoir pressure is very small, the production driving force remains large, and the decline in the producing rate is correspondingly smaller. For such cases, there is no theoretical reason why the decline coefficient could not be greater than 1. Much later in the life of these reservoirs, when the oil column thins, the production rate would decline exponentially. Hydrocarbon production is replaced by water. Arps’ original study indicated less than 10% of wells with b > 0.5, however a later study by Ramsay and Guerrero indicated about 40% of leases with b > 0.5. Commingled layered reservoirs lead to 0.5 < b <1.0 (Fetkovich). It is possible, under certain production scenarios that, at first, the rate does not decline. In such a case, decline analysis should be initialised in time from the start of declining rate. Field experience presented by Arps (149 fields) and by Ramsay and Guerrero (202 leases) indicate 0.1 < b < 0.9.

Exponential decline seems to be a rare occurrence! , yet it is probably the most common interpretation (no doubt due to the ease of analysis)

9. Hyperbolic Decline

With hyperbolic decline, the decline rate, D, is not constant (in contrast to exponential decline, when D is constant). Empirically, it has been found that for some production profiles, D is proportional to the production rate raised to a power of b, where b is between zero and 1. A value of b = 0 corresponds to exponential decline. A value of b = 1 is called harmonic decline. Sometimes, values of b > 1 are observed. These values do not conform to the traditional decline curves, and their meaning is discussed in the section entitled values of b > 1.

Unfortunately, hyperbolic decline does not plot as a linear relationship on common graph papers. As shown above, plotting on semi-log scales does not straighten a hyperbolic decline. Nor does plotting the “Rate versus Cumulative Production”. Prior to the widespread use of personal computers, this lack of linearity was the main reason for the restricted use of hyperbolic declines. Whereas exponential decline is consistent with our understanding of single-phase, boundary dominated flow, hyperbolic decline is considered to be consistent with the performance of solution-gas drive reservoirs. Further discussion on this can be found under the topic recovery mechanism and b. As illustrated here, hyperbolic decline predicts a slower decline rate than exponential. As a result, it is sometimes used to calculate the “probable reserves” while exponential decline is used to obtain minimum “proved reserves”.

10. Hyperbolic Theory

All decline curve theory starts from the definition of the decline rate, D, as follows:

For hyperbolic decline, D is NOT constant; D varies with the rate according to:

Where K is a constant equal to Di / (qi

b), and b is a constant with a value between 0 and 1.(It can be shown that the decline rate D at any time, t, is related to Di and b by:

Values of b > 1 and the relationship between reservoir recovery mechanism and b are discussed in the appropriate sections. When b = 0, D becomes a constant, independent of the flow rate, q, and the hyperbolic decline becomes identical with exponential decline. When b = 1, the hyperbolic decline becomes harmonic decline. Combining the equations above, and integrating, gives the hyperbolic decline equation:

Where qi and Di are the initial flow rate and the initial decline respectively, corresponding to the time, t = 0.

There is no simple way of re-casting this equation to obtain a straight line. Hence, when analyzing production data using hyperbolic decline, a non-linear regression must be performed to find the values of the constants b, Di, qi that best fit the data. In order to obtain the flow rate at any future time, the cumulative production up to that time, or the total recoverable reserves, the production decline curve must be extrapolated using the hyperbolic decline equation :

Having obtained the constants b, Di, qi from a curve fit of the production data, The flow rate at any time, t, is given by:

The cumulative production, Q, at any time, t, is obtained is from;

or in terms of q from:

Thus at an abandonment rate, q2 = 0, the total recoverable oil can be read from an extrapolation of the graph based on the hyperbolic decline equation, or it can calculated from:

11. Modified Hyperbolic Decline

Extrapolation of hyperbolic declines over long periods of time frequently results in unrealistically high reserves. To avoid this problem, it has been suggested (Robertson) that at some point in time, the hyperbolic decline be converted into an exponential decline. Thus, assume that for a particular example, the decline rate, D, starts at 30% and decreases through time in a hyperbolic manner. When it reaches a specified value, say 10%, the hyperbolic decline can be converted to an exponential decline, and the forecast continued using the exponential decline rate of 10%.

Robertson has presented a decline equation, which incorporates the transition from hyperbolic to exponential decline:

Where D is the asymptotic exponential decline rate, and 0 £ £ 1 is responsible for transition from hyperbolic behaviour to exponential. Cumulative production using the above equation is given by:

and , for b = 1,

The difficulty with this procedure is that the choice for the value of the final exponential decline rate cannot be determined in advance, and must be specified from experience.

12. Harmonic Decline

Harmonic decline is a special case of hyperbolic decline, with b = 1, i.e. the decline rate, D, is proportional to q. This means that the decline rate, D, goes to zero when q approaches zero. This type of performance is expected when very effective recovery mechanisms such as gravity drainage are active. Another example of harmonic decline is the production of high viscosity oil driven by encroaching edge-water. Due to unfavourable mobility ratio, early water breakthrough occurs and the bulk of the oil production will be obtained at high water cuts. If the total fluid rate is kept constant

then the increasing amount of water in the total fluid will cause the oil production to decline. This decline in oil rate may follow a harmonic decline.

13. Harmonic Theory

All decline curve theory starts from the definition of the decline rate, D, as follows:

For harmonic decline, D is NOT constant; D varies with the rate according to :

where : K is a constant equal to Di / qi Harmonic decline is a special case of hyperbolic decline,

with b = 1. This results in:

The cumulative production between t1 and t2 corresponding to the two flow rates, q1 and q2, can be calculated from:

Or in terms of time, from:

The ultimate production (at zero flow rate) can not be determined. From the above equations, it can be seen that the way to obtain a straight line for harmonic decline is to plot log-rate versus cumulative production. (Because harmonic decline is such a rare occurrence, this plot is not shown here.) The constants Di and qi can be determined by regression, or from a plot of log-rate versus cumulative production. The flow rate at any future time, the cumulative production until that time, and recoverable reserves at a specified abandonment rate can be found either from extrapolation of the curves or from the equations above.

14. Multiwell Pools

It is common practice to apply decline-curve analysis to aggregated production from a lease or pool. The extension of decline analysis from a single well to aggregated production from a number of wells is sometimes difficult to justify theoretically. For example, the sum of the flow rates from two wells with exponential decline is not exponential in general, unless both wells have the same decline. However, this concern is lessened, when there are sufficient wells to result in a statistical distribution. Purvis has written two papers in which the decline performance of a pool is studied in a statistical manner. Most of the difficulty in extending the single well analysis to an aggregate of wells is often due to the inevitable variation in the number of producing wells over time. If the wells have reasonably similar declines, it is suggested that the decline analysis be performed on an “average well per operating day”, and that the pool forecast be obtained from the performance of the average-well combined with a forecast of the number of producing wells, and adjusted by a factor to account for downtime. The economic abandonment rate for an aggregate of wells can be misleading. Consider, for example, the case of three producing wells, two of which are at a rate below the economic limit and one is producing at a high rate. It is very possible that the total rate of the three wells would be higher than three times the economic limit of any one well. Analysis of the aggregate production would result in continuation of the operation of all three wells (because their total flow rate is larger than the aggregate economic abandonment rate). Yet analysis of the individual well rates would clearly show that two of the wells should be abandoned. Another consideration in multiwell pools is to initialize the production rate of each well to a common start time. This makes it easier to arrive at the “average well” performance.

15. Type Curves - Arps

If the hyperbolic decline equation:

is presented in graphical form it can be seen to encompass the whole range of conditions from exponential decline (b=0) to harmonic decline (b=1). Each value of qi, Di and b will produce its own unique curve. Arps generalized these curves by making the equation dimensionless He defined a dimensionless rate as qDd = q / qi, and a dimensionless time as tDd = Dt. The resulting dimensionless equation is :

If this equation is plotted using dimensionless time and rate as the axes, a unique set of curves is obtained that represents ALL decline conditions for ALL wells. Notice that the obtained curves are independent of Di and qi when plotted on these dimensionless co-ordinates.

When these dimensionless curves are plotted on log – log scales as shown below, they are called Type Curves, and can be used for graphical analysis of actual production data. There are other type curves like Fetkovitch, Blasingame and Agarwal-Gardner that are used in decline rate analysis. In the following sections Fetkovich and Blasingame type curves will be introduced.

Type curves can be used for graphical interpretation of a well’s production decline. The production rate of the well is plotted on transparent graph paper of the same scale as the type curve, and overlain on the type curve (keeping the axes parallel). The data plot is moved over the type curves until the best fitting curve is determined. This gives the value of b. The value of qi and Di is determined by selecting any point on the data plot and the corresponding point on the type curve. The value of qi is obtained by dividing the “match point” values off the data plot and the type curve, qi = qt(match) / qDd(match). The value of Di is obtained by dividing the “match point” values off the type curve and the data plot, Di = tDd(match) / t(match). The production forecast is obtained by tracing the matched curve onto the data plot and extrapolating.

16. Type Curves - Fetkovich

Fetkovich presented a new set of type curves that extends the Arps type curves into the transient flow region. He recognized that decline curve analysis was applicable only during the time period when production was in boundary dominated flow, i.e. during the depletion period. This meant that the early production life of a well was not analyzable by the conventional decline curve methods. Fetkovich used analytical flow equations to generate type curves for transient flow, and he combined them with the empirical decline curve equations of Arps. Accordingly, The Fetkovich type curves are made up of two regions, which have been blended to be continuous, and thereby encompass the whole production life from early time (transient flow) to late time (boundary dominated flow). The right-hand side of the Fetkovich type curves is identical to the Arps type curves. The left-hand side of the Fetkovich type curves is derived from the analytical solution to the flow of a well in the centre of a finite circular reservoir, producing at a constant wellbore flowing pressure. Fetkovich showed that, for all sizes of reservoirs, when transient flow ended, the boundary-dominated flow could be represented by an exponential decline as shown below. A well producing at constant pressure will follow one of these curves. One reason for the success of Fetkovich type-curves is that most of the oil wells are produced under wide-open conditions i.e. at the constant lowest possible pressure. Combining the Fetkovich transient curves with the Arps decline curves, and blending them where the two sets of curves meet, results in the Fetkovich Decline type curves shown below.

Fetkovich noted that sometimes, the hyperbolic decline coefficient, b, determined using the Arps decline curves was larger than 1 (it is normally expected to be between 0 and 1). He postulated an explanation for this. He suggested that the data being analysed was still in the transient regime, and had not reached the boundary-dominated regime. If data-+ still in the transient flow regime are forced to match a hyperbolic decline, then values of b larger than 1 will result.

17. Type Curve Matching-Fetkovich

The actual rate-time data are plotted on a log-log scale of the same size as the type curves. This plot is called the “data plot. Any convenient units can be used for rate or time because a change in units simply causes a uniform shift of the raw data on a logarithmic scale. It is recommended that daily operated-rates be plotted, and not the monthly rates. Especially when transient data are analysed. However, for very noisy and cyclic data, Fetkovich suggests 6-month averaging of the data. In addition, a log-log plot of the cumulative production data versus time, if matched simultaneously with the rate data, should help in achieving a more unique match of the data with the type curves. The data plot is moved over the type curve plot, while the axes of the two plots are kept parallel, until a good match is obtained. The rate and the cumulative data should both fit the SAME corresponding type curve. Several different type curves should be tried to obtain the best fit of all the data. The type curve that best fits the data is selected and its “re/rwa” and “b” vales are noted.

To make a forecast, the selected type curve is traced on to the data plot, and extrapolated beyond the last data points. The future rate is read from the data plot, off the traced type-curve. Type curve analysis is done by selecting a match point, and reading its co-ordinates off the data plot (q and t)match, and off the type curve plot (qDd and tDd)match. At the same time the stem values (“re/rwa” and “b”) of the matching curve are noted From the right-hand-set of type curves, the Decline Curve Parameters can be obtained:

b = value of type curve stem (right-hand-set of type curves) (Note that for Exponential Decline Di=D and b=0) With these, we can now calculate the EUR (Expected Ultimate Recovery)

The ‘f’ subscript denotes conditions at the beginning of the forecast period. Now, we can estimate fluid-in-place and drainage area. Exponential

Hyperbolic

Harmonic

Oil

From left-hand-set of type curves, the Reservoir Parameters can be obtained, if pi, pwf, β, µ, h, ct, φ, and rw are known.

value of type curve stem (left-hand-set of type curves) Oil:

k is obtained from rearranging the definition of

Gas: k is obtained from rearranging the definition of

Solve for rwa from the definition of

18. Blasingame et al Decline Analysis

Background The production decline analysis techniques of Arps and Fetkovich are limited in that they do not account for variations in bottomhole flowing pressure in the transient regime, and only account for such variations empirically during boundary dominated flow (by means of the empirical depletion stems). In addition, changing PVT properties with reservoir pressure are not considered, for gas wells.

Blasingame and his students have developed a production decline method that accounts for these phenomena. The method uses a form of superposition time function, that only requires one depletion stem for typecurve matching, the harmonic stem. One important advantage of this method is the typecurves used for matching are identical to those used for Fetkovich decline analysis, without the empirical depletion stems. When the typecurves are plotted using Blasingame’s superposition time function, the analytical exponential stem of the Fetkovich typecurve becomes harmonic. The significance of this may not be readily evident until considering that pseudo-steady state depletion at a constant flow rate follows a harmonic decline, if the inverse of the flowing pressure is plotted against time. In effect, Blasingame’s typecurves allow depletion at a constant pressure to appear as if it were depletion at a constant flow rate. In fact, Blasingame et al have shown that boundary dominated flow with both declining rates and pressures appear as pseudo-steady state depletion at a constant rate, provided the rate and pressure decline monotonically. Blasingame’s improvements on the Fetkovich style of production decline analysis are further enhanced by the introduction of two additional typecurves, which are plotted concurrently with the normalized rate typecurve. The ‘rate integral’ and ‘rate integral derivative’ typecurves aid in obtaining a more unique match. The derivation of these will be discussed in a later section.

Recall that the Fetkovich type curves were based on combining the analytical solution to transient flow of a single-phase fluid at a constant wellbore flowing pressure, with the empirical Arps’s equations for boundary dominated flow. Fetkovich was of the belief that the exponent b could vary between zero and 1, and was correlatable with fluid properties as well as recovery mechanism. For example, single-phase flow of oil would result in an exponent of zero, while single-phase gas flow would exhibit b > 0, because of changes in gas properties. Later, Fraim and Wattenberger showed that if the changes in gas fluid properties were taken into account, i.e. with the use of pseudotime, boundary-dominated gas flow against a constant back pressure exhibits the same behaviour that an oil reservoir would do; the decline would follow the exponential curve, b = 0. The above findings relate to flow at constant wellbore pressure. The subsequent development by Blasingame et al was to account for changing wellbore flowing pressures, by defining a superposition time function, which they called material-balance-time. They showed that if the material-balance-time were used instead of actual producing time, what was previously an exponential decline would follow the harmonic decline stem instead.

More importantly, data obtained when both the rate and the flowing pressure are varying, can now be analyzed if material-balance-time is used. For example, if the production rate from a well is monotonically declining, and at the same time, its flowing bottomhole pressure is on continuous decline , a plot of q/Dp versus Q(t)/q(t) would follow the harmonic curve. (For a gas well, the changing gas properties should also be accounted for by using pseudotime). Blasingame-McCray-Palacio developed type-curves which show the analytical transient stems along with the analytical harmonic decline (but with the rest of the empirical hyperbolic stems absent). In addition, they introduced two other functions, the rate integral function, and the rate integral derivative function, which help in smoothing the often-noisy character of production data, and in obtaining a more unique match. The Blasingame suite of typecurves are very similar to the Fetkovich typecurves for constant pressure production. The only real difference is the absence of the empirical depletion stems on the Blasingame typecurves. These are not required because the usage of material balance time forces the boundary dominated data to fall only on the analytical harmonic stem.

Models The Blasingame suite of type curves consists of a number of different models:

1. Vertical well, radial flow model, 2. Vertical well, hydraulic fracture model, 3. Horizontal well model, 4. Waterflood model, and 5. Well interference model (declining reservoir pressure).

All models assume a circular outer boundary.

19. Blasingame Type Curve Matching

Data Preparation In Blasingame typecurve analysis, three rate functions can be plotted against material balance time. Material Balance time is defined as follows: Oil Wells:

Gas Wells:

The three rate functions are as follows: 1. Normalized Rate Oil Wells:

Gas Wells:

2. Rate Integral The rate integral is defined at any point in the producing life of a well, as the average rate at which the well has produced until that moment in time. The normalized rate integral is defined as follows: Oil Wells:

Gas Wells:

3. Rate Integral Derivative The rate integral derivative is defined as the semi logarithmic derivative of the rate integral function, with respect to material balance time. It is defined as follows: Oil Wells:

Gas Wells:

Calculation of Parameters The actual normalized rate, rate integral, and rate integral derivative are plotted vs. material balance time data on a log-log scale of the same size as the type curves. This plot is called the data plot. Any convenient units can be used for normalized rate or time because a change in units simply causes a uniform shift of the raw data on a logarithmic scale. It is recommended that daily operated-rates be plotted, and not the monthly rates, especially when transient data are analyzed. Any one of the curves can be used individually but often using more than one curve helps in achieving a more unique match of the data with the type curves. The data plot is moved over the type curve plot, while the axes of the two plots are kept parallel, until a good match is obtained. The rate, rate integral, and rate integral derivative data should all fit the SAME corresponding type curve. Several different type curves should be tried to obtain the best fit of all the data. The type curve that best fits the data is selected and its “re/rwa” (re/Xf for fractured case) value is noted. To create a forecast, the selected type curve is traced on to the data plot, and extrapolated beyond the last data points. The future rate is read from the data plot, off the traced type-curve. Type curve analysis is done by selecting a match point, and reading its co-ordinates off the data plot (q/DP and tca)match, and off the type curve plot (qDd and tDd) match. At the same time the stem value “re/rwa” of the matching curve is noted. Given a curve match, the following Reservoir Parameters can be obtained, if, β, µ, h, ct, φ, and rw are known: k, s(Xf), Area, GIP(OIP).

20. References

Arps, J.J.: "Analysis of Decline Curves," Trans., AIME (1945), 160, 228.

Arps, J.J.: "Estimation of Primary Oil Reserves," Trans., AIME (1956), pp. 182-191.

Agarwal, R.G; Gardner, D.C; Kleinsteiber, S.W; and Fussell, D.D.: "Analyzing Well Production Data Using Combined Type Curve and Decline Curve Analysis Concepts," paper SPE 57916 presented at the 1998 SPE Annual Technical Conference and Exhibition, New Orleans, 27-30 September.

Al-Hussainy, R; Ramey, H.J. Jr.;"Application of Real Gas Flow Theory to Well Testing and Deliverability Forecasting," JPT (1966) 18, 624-636.

Blasingame, T.A; McCray, T.L; Lee, W.J: "Decline Curve Analysis for Variable Pressure Drop/Variable Flowrate Systems," paper SPE 21513 presented at the SPE Gas Technology Symposium, 23-24 January, 1991

Blasingame, T.A; Lee, W.J: "Variable-rate Reservoir Limits Testing," paper SPE 15028 presented at the Permian Basin Oil & Gas Recovery Conference of the Society of Petroleum Engineers, Midland, 13-14 March, 1986.

Doublet, L.E; Pandie, P.K; McCollum, T.J; and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves Analysis of Oil Well Production Data Using Material

Balance Application to Field Cases," paper SPE 28688 presented at the 1994 Petroleum Conference and Exhibition of Mexico, Veracruz, 10-13 October.

Duong, AN: "A New Approach for Decline-Curve Analysis," paper SPE 18859 presented at the SPE Production Operations Symposium, Oklahoma City, 13-14 March, 1989.

Edwardson, M.J; Girner, H.M; Parkison, H.R; Williams, C.D; Matthews, C.S: "Calculation of Formation Temperature Disturbances Caused by Mud Circulation," paper SPE 124 presented at the 36th Annual Fall Meeting of SPE, Dallas, 8-11 October.

Fetkovich, M.J.: "Decline Curve using Type Curves," JPT (June 1980), 1065.

Fetkovich, M.J; Vienot, M.E; Bradley, M.D; and Kiesow, U.G.: "Decline Curve Analysis Using Type Curves - Case Histories," SPEFE ( December 1987), 637.

Fetkovich, M.J; Fetkovich, E.J; and Fetkovich, M.D.: "Useful Concepts for Decline Curve Forecasting, Reserve Estimation, and Analysis," SPERE (February 1996), 13.

Fraim, M.L; and Wattenbarger, R.A,: "Gas Reservoir Decline Curve Analysis Using Type Curves with Real Gas Pseudopressure and Normalized Time," SPEFE (December 1987), 671 .

Fraim, M.L; Lee, W.J: "Determination of Formation Properties From Long-Term Gas Well Production Affected By Non-Darcy Flow," paper SPE 16934 presented at the 1985 Annual Technical Conference and Exhibition , Dallas, 27-30 September.

Palacio, J.C; and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves Analysis of Gas Well Production Data," paper SPE 25909 presented at the 1993 Joint Rocky Mountain Regional and Low Permeability Reservoirs Symposium, Denver, 26-28 April.

Purvis, R.A.: "Analysis of Production Performance Graphs," JCPT (July-August 1985), 44.

Ramey, H.J; and Guerrero, E.T. : "The Ability of Rate-Time Curves to Predict Production rates" JPT (month and year missing), 103.

Robertson, S.: "Generalized Hyperbolic Equation," Unsolicited paper SPE 18731 (August 1988)

Spivey, J.P; Gatens, J.M; Semmelbeck, M.E; and Lee, W.J: "Integral Type Curves for Advanced Decline Curve Analysis," paper SPE 24301 presented at the Mid-Continenet Gas Symposium, Amarillo, 13-14 April, 1992.

Van Everdingen, A.F and Hurst, W: "The Application of the Laplace Transformation to Flow Problems in Reservoirs," Trans.AIME (1949), 186, 305-324.