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

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:

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

where

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body	q_0 = q(t=0)

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

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body	D_0 > 0

initial Production Decrement (the higher the

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body	D_0

is the stronger is decline)

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body	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:

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

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

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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:

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\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:

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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:

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

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

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coefficient:

Exponential Production Decline

Hyperbolic Production Decline

Harmonic Production Decline

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body	b=0

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body	0<b<1

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body	b=1

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q(t)=q_0 \exp \left( -D_0 \, t \right)

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

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q(t)=\frac{q_0}{1+D_0 \, t}

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Q(t)=\frac{q_0-q(t)}{D_0}

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Q(t)=\frac{q_0}{D_0 \, (1-b)} \, \left[ 1- \left( \frac{q(t)}{q_0} \right)^{1-b}  \right]

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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)

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Q_{\rm max}=\frac{q_0}{D_0}

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Q_{\rm max}=\frac{q_0}{D_0 \cdot (1-b)}

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Q_{\rm max}=\infty

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D(t) = D_0 = \rm const

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D(t) =\frac{D_0}{1+ b \cdot D_0 \cdot t}

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D(t) = \frac{D_0}{1+ D_0 \cdot t}

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\tau(t) = \tau_0 = \rm const

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\tau(t) = \tau_0 + b \cdot t

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\tau(t) = \tau_0 + t

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\mathrm{RPR}(t) = \tau(t) = \tau_0 = \rm const

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\mathrm{RPR}(t) = \tau_0 \, \left[ 
1 + \frac{b }{(1-b)} \cdot \frac{q_0}{q(t)}
\right]

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\mathrm{RPR}(t) = \infty

The Exponential and Hyperbolic decline are applicable for Boundary Dominated Flow with finite reserves

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body	--uriencoded--Q_%7B\rm max%7D \leq \infty

while Harmonic decline is associated with production from the reservoir with infinite reserves

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body	--uriencoded--Q_%7B\rm max%7D = \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

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with constant BHP:

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body	p_{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.

References

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

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Введение

Экспресс-Анализ Кривых Падения Дебитов (DCA = Decline Curve Analysis) ставит своей задачей оценить будущую динамику добычи скважины (или группы скважин) на основе известной предыстории.

Традиционные методы анализа (типа Арпс) основаны на эмпирических формулах и используют для анализа только информацию о дебитах скважин.

Современные методы, помимо данных о дебитах, вовлекают в анализ имеющуюся информацию о давлении и моделируют поведение кривых на основе решения уравнения диффузии давления в пласте, что часто побуждает относить эти методы к разделу ГДИ.

Математические модели

Метод Арпс (Arps) является исторически первым и до сих пор одним из самых популярных на практике методом предсказания динамики добычи без привлечения сведений о давлении в пластах.

В основе метода лежит следующая эмпирическая формула для дебита:

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q(t)=\frac{q_{i}}{[1+b \, D \, t]^{\frac{1}{b}}}

Коэффициент

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body	q_i = q(t=0)

имеет смысл начального дебита скважины (или группы скважин),

а коэффициент

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D=-\frac{1}{q}\frac{dq}{dt}

имеет смысл декремента падения добычи (чем больше

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тем сильнее будет падать добыча со временем).

Для анализа также используется накопленная добыча:

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

На практике различают четыре разновидности Арпс-режимов:

Экспоненциальный

b = 1

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q(t)=q_{i} \exp \big [ -D \, t \big ]

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Q(t)=\frac{q_{i}-q(t)}{D}

Гармонический

b = 0

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q(t)=\frac{q_{i}}{[1+D \, t]}

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Q(t)=\frac{q_{i}}{D}\ln (\frac{q_{i}}{q(t)})

Гиперболический

b = 0..1

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q(t)=\frac{q_{i}}{[1+b \, D \, t]^{\frac{1}{b}}}

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Q(t)=\frac{q_{i}}{D \, (1-b)}(q_{i}^{1-b}-q(t)^{1-b})

Power Loss

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D=D_{\infty} + \frac{t^{n-1}}{\tau^{n}}

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q(t)=q_{i} \exp \big [ -D_{\infty}t- \bigg(\frac{t}{\tau} \bigg)^{n} \big]

Хотя в целом такой подход является феноменологическим, конкретно экспоненциальный режим падения добычи имеет физическое обоснование, представляя собой режим псевдо-стационарной радиальной фильтрации в замкнутом резервуаре.

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