There are a number of ways in which production decline data can be shown graphically. In Figure 1 , we see the production rate, q, plotted against time.

Figure 1

The rate is constant during the early life of the well, when either proration or maximum efficient rate limits are placed on it. Thereafter, as the reservoir pressure is reduced, the rate begins to decline.In Figure 2 , we see the production rate, q, plotted against the cumulative oil production, Np.

Figure 2

As expected the profiles of the curves in Figure 1 and Figure2 are somewhat similar, with the production rate declining as more fluids are produced from the reservoir.We will often plot the percentage of oil or water-cut versus cumulative production in situations where the ultimate production rate is controlled by the amount of water production we can handle on the lease. In Figure 3 , we have plotted the percentage oil production versus cumulative production.

Figure 3

When the value of the oil produced becomes equal to the cost of disposing of the produced water, we have reached the point of abandonment.In bottom-water drive fields, we might plot the location of the oil-water contact in the formation against cumulative oil production. As the contact reaches the top of the sand, we know that we have recovered the crude reserves for this well. A typical plot is shown in Figure 4 .

Figure 4

In a similar manner, we might plot the cumulative gas produced, Gp, versus the cumulative oil produced, Np ( Figure 5 ). This plot is particularly useful in cases where we know the expected total gas to be produced. This information provides an indication as to when the well will reach its abandonment point.

Figure 5

One of the more important plots which is incorporated into our estimation of reserves is that of the average reservoir pressure, pavg, versus time ( Figure6 ). Either pressure build-up tests or observation well data are used for this plot.

Figure 6

Finally, we should remember that for fixed volume gas reservoirs, the average reservoir pressure, pavg, over the compressibility factor, z, will plot as a declining straight line function of the cumulative gas produced on linear coordinate paper. This relationship is shown in Figure 7 and follows from the application of the gas law to a fixed volume container.

Figure 7

Production rate is by far the most popular dependent variable used in projecting future performance of wells and reservoirs. It has the advantage of being readily available and easily recorded. When no major changes in operating procedures are made and no stimulation treatments are applied, the production rate curves normally show a fairly smoothly declining trend over extended periods. This trend lends itself to extrapolation.This smoothness of decline is one important requirement for extrapolation. A second requirement is that there be a known end point; that is, a production rate at which a well or field begins to lose money if production continues. This point is referred to as the economic limit production rate. If we incorporate this value into our rate versus time and rate versus cumulative production curves, we can extrapolate each trend line to this cut-off point. Thereby, we can determine, for the first curve, the number of years the well or field will produce profitably prior to abandonment ( Figure 8 ), and for the second curve, the cumulative production at the time of abandonment.

Figure 8

The end point may also be a minimum oil-cut. In this case, we use the oil-cut versus cumulative production curve in order to calculate the total production at the time of abandonment ( Figure 9 ).

Figure 9

Likewise, if the known end point is the movement of the oil-water contact to the top of the oil sand, then we would use a plot of the oil-water contact elevation versus cumulative oil production to find the total production prior to abandonment ( Figure 10 ).

Figure 10

If the total expected gas production has been calculated with a degree of accuracy and serves as our cut-off point, we would plot the cumulative gas production versus the cumulative oil production in order to find the total oil to be produced ( Figure 11 ).

Figure 11

Finally, for a gas well or field, the minimum producing rate is determined by the back-pressure imposed on the well with or without surface compression. When the value or this limiting back-pressure is converted to a value or average reservoir pressure, pavg, divided by z and plotted against cumulative production, an estimate of the ultimate predicable gas reserves may be obtained ( Figure 12 ).

Figure 12

But a word of caution. There are two important limitations on our projections. First, the total reserves that are estimated to be produced must be reasonable; and second, the total time during which those reserves are to be produced must also be reasonable. if the projected total reserves greatly exceed the volumetric estimates, then we have made an error in our calculations. We may be producing from a much larger drainage area than the drainage area or normal spacing. if this is the case, it will not be long before an offset well is drilled, thereby diminishing the reserve estimates and shortening the life or the well.

CONSTANT PERCENTAGE DECLINE The nominal production decline rate equation shows the change in production rate per unit of time as a function of the production rate.(1.1)With constant percentage decline, the change in production rate per unit of time is a constant function of the production rate, q. This means that the decline rate, D, is constant. A negative sign is added to the right side of the equation so that D will have a positive value. If the variables of this relationship are separated, and integration takes place between the appropriate limits of flow rate and time, we obtain the rate-time relationship for the constant percentage decline curve.qt = qie-Dt (1.2)This is the rate-time relationship for the constant percentage or exponential decline curve.To obtain the rate-cumulative production relationship, we must integrate the flow rate, qt, over the time period, t.(1.3)qi is the initial production rate and qt is the production rate at some later time, perhaps at the time of abandonment if we are estimating future production. Note that D is a fraction and not a percentage. This is the rate-cumulative production equation for constant percentage or exponential decline curves.As mentioned above, the decline rate is constant for constant percentage decline. For convenience, we state that:D = K (1.4)

HYPERBOLIC AND HARMONIC DECLINE For hyperbolic decline, the decline rate is proportional to a fractional power of q. This may be expressed as:D = Kqn (1.5)n lies between zero and one, (0