Page 35 - Principles of Catalyst Development

16 Principles of Applied Reservoir Simulation

2.3 Decline Curve Analysis

Arps [1945] studied the relationship between flow rate and time for
producing wells. Assuming constant flowing pressure, he found the relationship;

^ =-«?"" (2-10)

where a and n are empirically determined constants. The empirical constant n
ranges from 0 to 1,
Solutions to Eq. (2.10) show the expected decline in flow rate as the
production time increases. A fit of an equation of the form of Eq. (2.10) to flow
rate data is referred to as decline curve analysis. Three decline curves have been
identified based on the value of n.
The Exponential Decline curve corresponds to n — 0. It has the solution
q=q te' at (2.11)
where q i is initial rate and a is a factor that is determined by fitting Eq. (2.11)
to well or field data.
The Hyperbolic Decline curve corresponds to a value of n in the range
0 < n < 1. The rate solution has the form
n n
q~ = nat + q: (2.12)
where q i is initial rate and a is a factor that is determined by fitting Eq. (2.12)
to well or field data.
The Harmonic Decline curve corresponds to n - 1. The rate solution is
equivalent to Eq. (2.12) with n = 1, thus

q~ l = nat + q: 1 (2.13)
where q l is initial rate and a is a factor that is determined by fitting Eq. (2.13)
to well or field data.
Decline curves are fit to actual data by plotting the logarithm of observed
rates versus time t. The semi-log plot yields the following equation for
exponential decline:
tnq=tnq i-at (2.14)

Equation (2.14) has the form y - mx + b for a straight line with slope m and
intercept b. In the case of exponential decline, time / corresponds to the