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A specific type of mathematical model of Decline Curve Analysis, based on the following equation for Production rate q(t)

(1) q(t) = q_0 \cdot \left( 1+b \cdot D_0 \cdot t \right)^{-1/b}


q_0 = q(t=0)

Initial production rate of a well (or groups of wells)

D_0 > 0

initial Production Decrement (the higher the D_0 is the stronger is decline)

0 \leq b \leq 1

defines the type of decline (see below)

It can be applied to any fluid production: water, oil or gas. 

The cumulative production is then given by:

(2) Q(t)=\int_0^t q(t) \, dt

The Production Decrement value can be a given an explicit form:

(3) D(t) = - \frac{dq}{dQ} = - \frac{d \, \ln q(t)}{dt} = \frac{D_0}{1+ b \cdot D_0 \cdot t}

The Recovery Pace is the inverse value to Production Decrement:

(4) \tau(t) = \frac{1}{D(t)} = - \frac{dQ}{dq} = - \left[ \frac{d \ln q}{dt} \right]^{-1} \quad \rightarrow \quad \tau(t) = \tau_0 + b\cdot t = \frac{1}{D_0} + b\cdot t

The ultimate recovery is defined as:

(5) Q_{\rm max}=Q(t=\infty)=\int_0^\infty q(t) \, dt =\frac{q_0}{D_0 \cdot (1-b)}

The RPR is defined as:

(6) RPR(t) = \frac{Q_{max}-Q(t)}{q(t)}

Arp's model is usually split into three types based on the value of b coefficient:

Exponential Production DeclineHyperbolic Production DeclineHarmonic Production Decline




(7) q(t)=q_0 \exp \left( -D_0 \, t \right)
(8) q(t) = \frac{q_0}{ \left( 1+b \cdot D_0 \cdot t \right)^{1/b} }
(9) q(t)=\frac{q_0}{1+D_0 \, t}
(10) Q(t)=\frac{q_0-q(t)}{D_0}
(11) Q(t)=\frac{q_0}{D_0 \, (1-b)} \, \left[ 1- \left( \frac{q(t)}{q_0} \right)^{1-b} \right]
(12) Q(t)=\frac{q_0}{D_0} \, \ln \left[ \frac{q_0}{q(t)} \right] =  \frac{q_0}{D_0} \ln q_0 + \frac{q_0}{D_0} \cdot \ln q(t)
(13) Q_{\rm max}=\frac{q_0}{D_0}
(14) Q_{\rm max}=\frac{q_0}{D_0 \cdot (1-b)}
(15) Q_{\rm max}=\infty
(16) D(t) = D_0 = \rm const
(17) D(t) =\frac{D_0}{1+ b \cdot D_0 \cdot t}
(18) D(t) = \frac{D_0}{1+ D_0 \cdot t}
(19) \tau(t) = \tau_0 = \rm const
(20) \tau(t) = \tau_0 + b \cdot t
(21) \tau(t) = \tau_0 + t
(22) \mathrm{RPR}(t) = \tau(t) = \tau_0 = \rm const
(23) \mathrm{RPR}(t) = \tau_0 \, \left[ 1 + \frac{b }{(1-b)} \cdot \frac{q_0}{q(t)} \right]
(24) \mathrm{RPR}(t) = \infty

The Exponential and Hyperbolic decline are applicable for Boundary Dominated Flow with finite reserves Q_{\rm max} \leq \infty while Harmonic decline is associated with production from the reservoir with infinite reserves Q_{\rm max} = \infty.

In other words the Harmonic decline is very slow.

Since all physical reserves are finite the true meaning of Harmonic decline is that up to date it did not reach the boundary of these reserves and at a certain point in future it will transit into a finite-reserves decline (possibly Exponential or Hyperbolic).

The Harmonic decline is also observed at the mature stage of waterflood projects.

Both Harmonic and Hyperbolic declines are empirical while Exponential decline has a physical meaning.
Exponential Production Decline has a physical meaning of declining production from finite drainage volume V_e with constant BHPp_{wf}(t) = \rm const (a specific type of Boundary Dominated Flow under Pseudo Steady State (PSS) conditions).

There are few approaches to match the Arps decline to the historical data which are covered in DCA Arps Matching @model.

The DCA Arps do not cover all types of production decline, but their application is quite broad and mathematics is quite simple which gained popularity as quick estimation of production perspectives.

See Also

Petroleum Industry / Upstream /  Production / Subsurface Production / Field Study & Modelling / Production Analysis / Decline Curve Analysis

Exponential Production Decline ][ Hyperbolic Production Decline ][ Harmonic Production Decline ][ Production Decrement ] 

[ DCA Arps Matching @model ]


Arps, J. J. (1945, December 1). Analysis of Decline Curves. Society of Petroleum Engineers. doi:10.2118/945228-G

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