document

Abstract

Nearly all the decline curve equations used today are based on the Arps hyperbolic equation¹, given as:

• Q(t)=Q₀×(1+b×D₀×t)^−1/b (1)

Using this equation, the production rate ‘Q’ at anytime ‘t’ can be calculated from the hyperbolic exponent ‘b’, the initial production rate ‘Q ₀’ and its corresponding decline rate ‘D₀’ at time zero.

Although Equation (1) is easy to use, the variation of the decline rate with time (except b=0) limits the applicability of the equation. For a hyperbolic decline curve (b>0), if a different production rate ‘Q _i’ on the curve is used an initial rate, a different corresponding decline rate ‘D _i’ needs to be identified for the equation to represent the same decline curve. Moreover, if there is a rate or reference time change in the production forecast period, the identified hyperbolic equation from the production history is no longer applicable.

In this paper, a generalized hyperbolic equation is derived to overcome the above limitations. Once a set of ‘Q₀’, ‘D₀ ’ and ‘b’ is identified from the production history, the equation can be used to predict the future rate regardless of the initial rate or time change.

Introduction

The decline curve analysis is one of the oldest and most popularly used techniques in the oil industry for predicting the future production rate. The earliest literature referring to a mathematical decline analysis approach was published in 1908². In 1944, Arps introduced a standard hyperbolic equation. Since then, the Arps hyperbolic equation has become a benchmark in the industry due to its simplicity and practicality^3,4.

The coefficient in the Arps equation, as shown in Equation (1), can generally be determined by matching the production history. When the hyperbolic exponent ‘b’ equals 0 or 1, the hyperbolic equation becomes an ‘exponential’ or ‘harmonic’ decline equation respectively. The exponential decline curve (b=0) has a constant decline rate, while the hyperbolic decline curve (b>0) has variable decline rate with time. The exponential decline equation is often used by engineers to predict the future rate because it is easy to apply. The applicability of the hyperbolic equation, however, is restricted by the difficulties in handling the variation of decline rate with time and the different initial production rate used in the production forecast.

In this paper, a generalized equation is derived based on the expanded Arps hyperbolic equation to overcome the above difficulties. The separation of variable technique and a time coordinate transformation are used to generalize the equation.

Exponential Decline Equation

Exponential decline is a special case of the hyperbolic decline. When the hyperbolic exponent ‘b’ equals zero, Equation (1) becomes an exponential decline equation:

• Q(t)=Q₀×e ^−(Do×t) (2)

In this equation, the decline rate ‘D’ at any time, defined by Equation (3), equals the initial decline rate ‘D ₀’:

• D=−(dQ/dt)/Q (3)

The advantage of having a decline rate ‘D₀ ’ in the exponential decline that is constant over time is that once an initial decline rate ‘D₀’ is identified, the decline rate can be used for the decline curve analysis regardless of changes in the initial rate or time. In this case, Equation (2) can be generalized as:

• Q(t)=Q_i×e ^−(Do×t)(Do t) (4)

Equation (4) can be viewed as a generalized exponential equation which allows an exponential decline curve to be expressed by using its constant decline rate ‘D₀’ and any initial production rate selected by the user.