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<title>Muskat Material Balance</title>






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<h1 class="title toc-ignore">Muskat Material Balance</h1>
<h4 class="author"><em>Vignette Author</em></h4>
<h4 class="date"><em>2017-11-07</em></h4>



<p>For the Muskat’s Material Balance equation:</p>
<p><span class="math display">\[\newcommand{\numD}{{\dfrac {S_o}{B_o B_g} \dfrac {dR_s}{dP} +  
\dfrac {S_o}{B_o} \dfrac {k_g}{k_o} \dfrac {\mu_o}{\mu_g} \dfrac {dB_o}{dP} + 
(1 - S_o - S_w) \dfrac {1}{B_g} \dfrac {dB_g}{dP} }}\]</span></p>
<p><span class="math display">\[\dfrac {dS_o}{dP} = \dfrac {\numD} {1 + \dfrac {k_g}{k_o} \dfrac {\mu_o}{\mu_g} }\]</span></p>
<p>All the terms on the right side are function of pressure (<span class="math inline">\(P\)</span>) and saturation (<span class="math inline">\(S\)</span>), the equation could be reduced to:</p>
<p><span class="math display">\[ \dfrac{dS}{dP} = f(P, S)\]</span></p>
<p>A first-order ordinary diferential equation (<em>ODE</em>).</p>
<div id="plot-a-range-of-values-for-a-given-step-size" class="section level2">
<h2>Plot a range of values for a given step size</h2>
<p>Given the ODE: <span class="math display">\[ \dfrac{dS}{dP} = (2P +  S)\]</span></p>
<p>Find the first saturation values if the initial conditions are <span class="math inline">\(P_o\)</span> = 0, and <span class="math inline">\(S_o\)</span> = 1.</p>
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