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<title>Leverett J-function</title>

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<span id="openzim-page-title" class="mw-page-title-main">Leverett <i>J</i>-function</span>

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<p>In <a href="Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a> and <a href="Geology" title="Geology">geology</a>, the <b>Leverett <i>J</i>-function</b> is a <a href="Dimensionless_quantity" title="Dimensionless quantity">dimensionless</a> function used to describe the <a href="Capillary_pressure" title="Capillary pressure">capillary pressure</a> required to force a fluid into the pores of a material like rock. Its primary purpose is to allow engineers and scientists to relate data from a specific rock sample to other similar rocks that may have different physical properties.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It provides a way to convert complex capillary pressure data from multiple rock types into a single, universal curve for a given reservoir or formation.

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<div class="mw-heading mw-heading2"><h2 id="Definition_and_formulation">Definition and formulation</h2></div>

<p>The function is based on the assumption that a porous rock can be modeled as a bundle of non-connecting capillary tubes. The <i>J</i>-function is defined as:

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<annotation encoding="application/x-tex">{\displaystyle J(S_{w})={\frac {p_{c}(S_{w}){\sqrt {k/\phi }}}{\gamma \cos \theta }}}</annotation>

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</math></span><img src="./f02c175e336253e439ac5750aaba275ccc1a5fb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.889ex; height:6.843ex;" alt="{\displaystyle J(S_{w})={\frac {p_{c}(S_{w}){\sqrt {k/\phi }}}{\gamma \cos \theta }}}" loading="lazy"></span></dd></dl>

<p>where:

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<annotation encoding="application/x-tex">{\displaystyle S_{w}}</annotation>

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</math></span><img src="./727e2945c3e0a54e22c86ed12ec1205a358961a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.834ex; height:2.509ex;" alt="{\displaystyle S_{w}}" loading="lazy"></span> is the water saturation, measured as a fraction</li>

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<annotation encoding="application/x-tex">{\displaystyle p_{c}}</annotation>

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</math></span><img src="./836639c3805ca867b1ff24dc6db7a6b24fc69158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.203ex; height:2.009ex;" alt="{\displaystyle p_{c}}" loading="lazy"></span> is the <a href="Capillary_pressure" title="Capillary pressure">capillary pressure</a> (in <a href="Pascal_(unit)" title="Pascal (unit)">pascals</a>)</li>

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> is the <a href="Permeability_(fluid)" class="mw-redirect" title="Permeability (fluid)">permeability</a> (in <a href="Square_metre" title="Square metre">m²</a>)</li>

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<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

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</math></span><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span> is the <a href="Porosity" title="Porosity">porosity</a> (dimensionless, 0–1)</li>

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math></span><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span> is the <a href="Surface_tension" title="Surface tension">surface tension</a> (in N/m)</li>

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<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {k/\phi }}}</annotation>

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</math></span><img src="./470e50a59fcba604adf99506d51ec880e9e8f1c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.083ex; height:4.843ex;" alt="{\displaystyle {\sqrt {k/\phi }}}" loading="lazy"></span> represents a characteristic length of the pore radii, allowing the function to normalize for differences in the rock's geological structure.

</p><p>The function is important in that it is constant for a given saturation within a reservoir, thus relating reservoir properties for neighboring beds.

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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Petroleum_engineering">Petroleum engineering</h3></div>

<p>The function is important in <a href="Petroleum_engineering" title="Petroleum engineering">petroleum engineering</a> because for a given fluid saturation (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{w}}">

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<annotation encoding="application/x-tex">{\displaystyle S_{w}}</annotation>

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</math></span><img src="./727e2945c3e0a54e22c86ed12ec1205a358961a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.834ex; height:2.509ex;" alt="{\displaystyle S_{w}}" loading="lazy"></span>), its value is considered constant for a specific reservoir. This allows engineers to measure capillary pressure on a small rock core in a laboratory and then extrapolate that data to understand fluid behavior across large areas of the reservoir, even where permeability and porosity may vary.<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Fuel_cells">Fuel cells</h3></div>

<p>This function is also widely used in modeling the two-phase flow of water and gas within <a href="Proton_exchange_membrane_fuel_cell" class="mw-redirect" title="Proton exchange membrane fuel cell">Proton exchange membrane fuel cells</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A high degree of hydration is needed for good proton conductivity, but excessive liquid water saturation in the pores of the catalyst layer or diffusion media can impede the transport of gas to the cathode, reducing performance. The J-function helps model and manage this balance.

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<ul><li><a href="Amott_test" title="Amott test">Amott test</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFM.C._Leverett1941" class="citation journal cs1">M.C. Leverett (1941). "Capillary behaviour in porous solids". <i>Transactions of the AIME</i> (142): <span class="nowrap">159–</span>172.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">J. M. LaManna, J. V. Bothe Jr., F. Y. Zhang, and M. M. Mench, <i><a href="J._Power_Sources" class="mw-redirect" title="J. Power Sources">J. Power Sources</a></i> <b>271</b>, 180 (2014).</span>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external free" href="http://www.ux.uis.no/~s-skj/ResTek1-v03/Notater/Tamu.Lecture.Notes/Capillary.Pressure/Lecture_16.ppt">http://www.ux.uis.no/~s-skj/ResTek1-v03/Notater/Tamu.Lecture.Notes/Capillary.Pressure/Lecture_16.ppt</a></li>

<li><a rel="nofollow" class="external free" href="http://perminc.com/resources/fundamentals-of-fluid-flow-in-porous-media/chapter-2-the-porous-medium/multi-phase-saturated-rock-properties/averaging-capillary-pressure-data-leverett-j-function/">http://perminc.com/resources/fundamentals-of-fluid-flow-in-porous-media/chapter-2-the-porous-medium/multi-phase-saturated-rock-properties/averaging-capillary-pressure-data-leverett-j-function/</a></li>

<li><a rel="nofollow" class="external text" href="http://perminc.com/resources/fundamentals-of-fluid-flow-in-porous-media/chapter-2-the-porous-medium/multi-phase-saturated-rock-properties/averaging-capillary-pressure-data-leverett-j-function/">Leverett J-Function in Multiphase Saturated Rocks</a></li></ul>

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