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# Production forecasting decline curve analysis

Decline curve analysis (DCA) is a graphical procedure used for analyzing declining production rates and forecasting future performance of oil and gas wells. Oil and gas production rates decline as a function of time; loss of reservoir pressure, or changing relative volumes of the produced fluids, are usually the cause. Fitting a line through the performance history and assuming this same trend will continue in future forms the basis of DCA concept. It is important to note here that in absence of stabilized production trends the technique cannot be expected to give reliable results.

The technique is not necessarily grounded in fundamental theory but is based on empirical observation of production decline.

Three types of declines are observed:

- Exponential
- Hyperbolic
- Harmonic

There are theoretical equivalent to these decline processes. It can be demonstrated that under conditions such as constant well back pressure, equation of fluid flow through porous media under boundary dominated flow are equivalent to exponential decline. However for our purpose it is the empirical nature of this term which has a greater significance since it allows the technique to be applied to multiple fluid streams even ratios!

## Golden rule of decline curve analysis (DCA)

The basic assumption in this procedure is that whatever causes controlled the trend of a curve in the past will continue to govern its trend in the future in a uniform manner

## Decline curve analysis (DCA) history

J.J. Arps collected these ideas into a comprehensive set of equations defining the exponential, hyperbolic and harmonic declines. His work was further extended by other researchers to include special cases. Following section gives a historical perspective of work done on the subject;

- Arps 1945 and 1956.
- Brons 1963 and Fetkovitch 1983 applied constant pressure solution to diffucisivty equation and demonstrated that exponential decline curve actually reflects single phase, incompressible fluid production from a closed reservoir. DCA is more than a empirical curve fit.
- Fetkovitch 1980 and 1983 developed set of type curves to enhance application of DCA.
- Doublet and Blasingame 1995 developed theoretical basis for combining transient and boundary dominated flow for the pressure transient solution to the diffusivity equation.

## Decline curve analysis (DCA) today

The major application of DCA in the industry today is still based on equations and curves described by Arps. Arps applied the equation of Hyperbola to define three general equations to model production declines.

In order to locate a hyperbola in space one must know the following three variables.

- The starting point on Y axis, (qi), initial rate.
- Initial decline rate (Di)
- The degree of curvature of the line (b).

Arps did not provide physical reasons for the three types of decline. He only indicated that exponential decline (b=0) is most common and that the coefficient b generally ranges from 0 to 0.5.

Arp’s Equation for General Decline in a Well ..........................(1)

Clearly all wells do not exhibit exponential behavior during depletion. In many cases a more gradual hyperbolic decline is observed where rate time performance is better than estimated from exponential solutions implying that hyperbolic decline results from natural and artificial driving energies that slow down pressure depletion. Hyperbolic decline is observed when reservoir drive mechanism is solution gas cap drive, gas cap expansion or water drive. It is also possible where natural drive is supplemented by injection of water gas. The type of decline and its characteristic shape is a major feature of DCA. We shall be talking more about this as we go further. The various types of declines experienced by a well are documented in the **Fig 1 and Fig 2**.

INSERT FIGURE 1 q vs. Time showing various types of declines on Cartesian plot. (b value for hyperbolic curve =0.5)

INSERT FIGURE 2 Log q vs. Time showing various types of declines on Semilog plot. (b value for hyperbolic curve =0.5). Note change in shapes of curves.

Observe the change in Shapes of curve from Cartesian to logarithmic; this is very helpful in identification of type of decline.

Two sets of curves are normally used while analyzing production decline.

- Flow rate is plotted against Time:

- Very convenient since it provides future profiles directly.

- Flow rate against cumulative production:

- Able to incorporate impact of intermittent operations that impact production.
- Provide recovery estimates at a specific economic limit.

INSERT FIGURE 3 Rate verses Time and Rate verses Cum Oil

### Exponential Decline

Since b=0, Equation 1 can be re arranged as:

- -------------------------------------------------(2)
- Variables
- q = current production rate
- q i = initial production rate (start of production)
- d i = d = dt = nominal decline rate (a constant)
- t = cumulative time since start of production
- The most conservative and simplest equation of the decline curve family.

- Effective decline rate D remains constant over time.
- Log rate vs. time is a straight line on semi-log plot.
- Rate vs. cumulative is a straight line on a linear plot as shown below:

-------------------------------------(3)

Applies to a well producing at constant bottom hole pressure.

INSERT FIGURE 4 Rate vs. Time – Exponential Decline

INSERT FIGURE 5 Rate vs. Cum Oil – Exponential Decline

Reservoir types with exponential declines^{[1]}:

- Oil reservoirs
- Above the bubble point
- Down dip wells with gravity drainage
- Solution gas drive with unfavourable kg/ko

- Gas reservoirs
- High reservoir pressure (liquid-like compressibilities)
- Wells with liquid-loading problems

- Both oil and gas reservoirs
- Produced with small drawdown relative to reservoir pressure
- Tubing limited wells.

### Nominal and Effective decline

- There are two types of decline factors (often called the decline rate).
- The nominal decline factor d is defined as the negative slope of the curve representing the natural logarithm of the production rate q vs. time t or :
- Nominal decline is a continuous function and it is the decline factor that is used in the various mathematical equations relating to decline curve analysis. For exponential decline it is a constant with time.
- The effective decline factor D is a stepwise function that is in better agreement with data recording practices. It is the drop in production rate from qi to q1 over a specific time period.
- It is defined as
- D is the effective decline rate = the decline rate over a time period.
- This is the decline often quoted in e.g. commercial software decline graphs. Such software may, at users discretion, report nominal decline.
- It is the proportion by which the production rate reduces over a given time period.
- D is a constant only for constant percentage or exponential decline.
- D decreases with time for hyperbolic and harmonic decline
- (1> b > 0)

- It is easy to convert from a nominal decline factor to an effective decline factor and vice versa.
- Thus an ‘effective’ decline of 10 % per year is equivalent to a nominal decline of 10.54% per year and vice versa

INSERT Figure 6 Effective and Nominal Decline, Shape and Relationship

### Hyperbolic Decline

#### Flowrate

------------------------------------------------------(4)

#### Cumulative production

#### Variables

- q = current production rate
- q i = initial production rate (start of production)
- d i = initial nominal decline rate at t = 0
- t = cumulative time since start of production
- N p = cumulative production being analyzed
- b= hyperbolic decline constant (0 < b < 1)
- This is the most general formulation for decline curve analysis. Exponential (b=0) and harmonic (b=1) decline are special cases of this formula.

#### Three constants

The mathematical equation defining hyperbolic decline has three constants

- The initial production rate
- The initial decline rate (defined at the same time as the initial production rate)
- The “hyperbolic exponent” b.
- For most conventional analysis, 0 <b <1
- However for some cases b > 1 has also been found.

#### Decline rate is not a constant

The decline rate is not a constant but changes with time, since the data plots as a curve on semi-log paper

#### Hyperbolic exponent

The hyperbolic exponent ( b) is the rate of change of the decline rate with respect to time. This means that “b” is the second derivative of production rate with respect to time.

-----------------------------------(6)

#### Hyperbolic decline constant

The hyperbolic decline constant at some future time, t, is defined by the following equation:

- High b exponents give small values of d, i.e. (= flat decline curves)—WATCH OUT!
- Unconstrained hyperbolic “curve fits” can severely overestimate future production
- Often useful (and safe) to use some value of minimum effective decline to avoid over-flattening the curve (say 5 % per annum)
- Hyperbolic curve fits with a decline constant (b) greater than 1 usually imply production is being influenced by transient behavior. For example b=2 corresponds to transient linear flow and is commonly found with unconventional reservoirs. However, be very careful with these cases – you should build limits into your forecast to capture the eventual transition from transient to boundary dominated flow.

## Definition of b

The term ‘b’ has no units and is normally known as hyperbolic constant. Generally ‘b’ can range in value from 0 to 1 in the context of DCA for oil and gas wells. It is evident from **Eq 1** that a large value of b ( close to 1) has a dominant effect on shape of the curve q vs. t as t becomes large. This causes and maintains the shape of the curve during this time to be essentially flat. For a given set of values for q and b the short term shape for the curve is not largely effected by the value of b but the long terms shape is. This implies that in short term all decline curves; exponential, hyperbolic and harmonic give similar results. However due the very same reasons make it extremely difficult to determine the value of b. The problem is aggravated if the data is noisy ( which is often the case) making it possible to fit a wide range of b values to the same dataset. However since the value of b has large impact in the late time, it will lead to different estimates of EUR. Reliability in estimation of b increases with maturity of production data. The value of b captures a large number of physical events and processes. A large body of publications are dedicated to this topic.

## Minimum decline

- Need to use some value of minimum decline slope to avoid over-flattening of curve
- Convert to exponential decline when dt< dmin
- Dmin obtained from most mature wells in the field, analogous fields of “experience”

- Forecast of ultimate recovery should give a reasonable recovery factor based on estimated volumes of hydrocarbons in place.

## References

- ↑ Fetkovich, M. J., Fetkovich, E. J., & Fetkovich, M. D. (1996, February 1). Useful Concepts for Decline Curve Forecasting, Reserve Estimation, and Analysis. Society of Petroleum Engineers. http://dx.doi.org/10.2118/28628-PA

## Noteworthy papers in OnePetro

## Noteworthy books

Society of Petroleum Engineers (U.S.). 2011. Production forecasting. Richardson, Tex: Society of Petroleum Engineers. WorldCat or SPE Bookstore

## External links

## See also

Production forecasting glossary

Sandbox:Production forecasting building blocks

Sandbox:Production forecasting expectations

Sandbox:Production forecasting flowchart

Sandbox:Production forecasting in the financial markets