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<ul>
<li><a href="#create-decline-curves">Create Decline Curves</a></li>
<li><a href="#origins">Origins</a></li>
<li><a href="#rate-time-decline-curves">Rate-Time Decline Curves</a>
<ul>
<li><a href="#exponential-decline">Exponential Decline</a></li>
<li><a href="#hyperbolic-decline">Hyperbolic Decline</a></li>
<li><a href="#harmonic-decline">Harmonic Decline</a></li>
</ul>
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<li><a href="#rate-cumulative-production-decline-curves">Rate-Cumulative Production Decline Curves</a>
<ul>
<li><a href="#exponential-decline-1">Exponential Decline</a></li>
<li><a href="#hyperbolic-decline-1">Hyperbolic Decline</a></li>
<li><a href="#harmonic-decline-1">Harmonic Decline</a></li>
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<li><a href="#decline-rate">Decline Rate</a></li>
<li><a href="#references">References</a></li>
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<h1 id="decline-curve-analysis">Decline Curve Analysis</h1>
<p><a href="#R-image-f15229493bb15f7ec596280d832ee649" class="lightbox-link"><img class="noborder lazy lightbox noshadow figure-image" loading="lazy" src="/images/plot-window/DeclineCurves.png" style=" height: auto; width: auto;"></a>
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<h2 id="create-decline-curves">Create Decline Curves</h2>
<p><strong>Decline Curve Analysis</strong> (DCA) can be created from the right-click menu for a curve in the <strong>Plot Project Tree</strong>.</p>
<p>Three decline curves are created, and the values for the decline curves can be inspected visually in the plot and values can be displayed using <strong>Show Plot Data</strong> from the menu inside the plot window.</p>
<p><a href="#R-image-dd96a8ccfb2945c8be66bb4f5de6b6fc" class="lightbox-link"><img class="noborder lazy lightbox noshadow figure-image" loading="lazy" src="/images/plot-window/DeclineCurves_create.png" style=" height: auto; width: auto;"></a>
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<h2 id="origins">Origins</h2>
<p>J.J. Arps <a href="/plot-window/declinecurveanalysis/index.html#1">[1]</a> concluded that the decline in oil production rate ($q_i$) over time can be described by these equations:</p>
<p>$$\frac{ 1 }{ q_0 } \frac{\partial q_0}{ \partial t} = -D$$</p>
<p>where the decline rate, $D$ is a time-dependent function:</p>
<p>$$D = \frac{D_i}{1+bD_i t}$$</p>
<p>where:</p>
<ul>
<li>$q_0$ is the production rate (e.g. oil production) in a unit of choice (e.g STB/day)</li>
<li>$t$ is the time</li>
<li>$D$ is the time dependent decline rate</li>
<li>$D_i$ is the initial decline rate (constant)</li>
<li>$b$ is a dimensionless constant (typically used as a tuning parameter to match actual field data) and is in the range of $ 0 &lt;= b &lt;= 1 $.</li>
</ul>
<p>The equations can be used to forecast future reservoir and well production.</p>
<p>Based on the value of $b$ in the function, Arps classified the decline curves into three types:</p>
<ul>
<li>The exponential decline has $b = 0$.</li>
<li>The harmonic decline has $b = 1$.</li>
<li>The hyperbolic decline has $b$ ranges between 0 and 1.</li>
</ul>
<p>It is important to note that the decline curve is an empirical model and assumes a simplified representation of the complex physical and geological factors affecting production decline.</p>
<h2 id="rate-time-decline-curves">Rate-Time Decline Curves</h2>
<h3 id="exponential-decline">Exponential Decline</h3>
<p>Exponential decline is the production decline when $ b = 0 $. This gives a constant decline ($D_i = D$).</p>
<p>$$q_0 = q_i e^{-Dt }$$</p>
<h3 id="hyperbolic-decline">Hyperbolic Decline</h3>
<p>Hyperbolic decline is the generic case where $ 0 &lt; b &lt; 1 $.</p>
<p>$$q_0 = \frac{q_i}{ (1+bD_i t )^\frac{1}{b} }$$</p>
<h3 id="harmonic-decline">Harmonic Decline</h3>
<p>Harmonic decline is the production decline when $b = 1$:</p>
<p>$$q_0 = \frac{q_i}{ (1+D_i t ) }$$</p>
<h2 id="rate-cumulative-production-decline-curves">Rate-Cumulative Production Decline Curves</h2>
<h3 id="exponential-decline-1">Exponential Decline</h3>
<p>Exponential decline is the production decline when $b = 0$. This gives a constant decline ($D_i = D$).</p>
<p>$$N_p = \frac{q_i - q_0}{D}$$</p>
<h3 id="hyperbolic-decline-1">Hyperbolic Decline</h3>
<p>Hyperbolic decline is the generic case where $ 0 &lt; b &lt; 1 $.</p>
<p>$$N_p = \frac{q_i^b}{D_i(1-b)} [q_i^{(1-b)} - q_0^{(1-b)}]$$</p>
<h3 id="harmonic-decline-1">Harmonic Decline</h3>
<p>Harmonic decline is the production decline when $ b = 1 $:</p>
<p>$$N_p = \frac{q_i}{ D_i } * \ln(\frac{q_i}{ q_0} )$$</p>
<h2 id="decline-rate">Decline Rate</h2>
<p>The continuous decline rate ($D_i$) can be determined from production history data. Using production rate and time data the value is the slope of the straight line on a semi-log plot. Taking two points on from the data $(t_1, q_1)$ and $(t_2, q_2)$:</p>
<p>$$D_i = \frac{1}{t_2 - t_1} \ln(\frac{q_1}{q_2})$$</p>
<p>where:</p>
<ul>
<li>$t_1$ is the time of the first point.</li>
<li>$q_1$ is the production rate in the first point.</li>
<li>$t_2$ is the time of the second point.</li>
<li>$q_2$ is the production rate in the second point.</li>
</ul>
<h2 id="references">References</h2>
<p><a id="1">[1]</a>
Arps, J. J.: “Analysis of Decline Curves,” SPE-945228-G, Trans. of the AIME (1945)</p>
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