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<p>Fluid flow through rock occurs in many geological settings on different scales, at different temperature conditions and with different flow velocities. Depending on these conditions the fluid will be in local thermal equilibrium with the host rock or not. To explore the physical parameters controlling thermal non-equilibrium, the coupled heat equations for fluid and solid phases are formulated for a fluid migrating through a resting porous solid by porous flow. By non-dimensionalizing the equations, two non-dimensional numbers can be identified controlling thermal non-equilibrium: the Péclet number <span class="inline-formula"><i>Pe</i></span> describing the fluid velocity and the porosity <span class="inline-formula"><i>ϕ</i></span>. The equations are solved numerically for the fluid and solid temperature evolution for a simple 1D model setup with constant flow velocity. This setup defines a third non-dimensional number, the initial thermal gradient <span class="inline-formula"><i>G</i></span>, which is the reciprocal of the non-dimensional model height <span class="inline-formula"><i>H</i></span>. Three stages are observed: a transient stage followed by a stage with maximum non-equilibrium fluid-to-solid temperature difference, <span class="inline-formula">Δ<i>T</i>_max⁡</span>, and a stage approaching the steady state. A simplified time-independent ordinary differential equation for depth-dependent <span class="inline-formula">(<i>T</i>_f−<i>T</i>_s)</span> is derived and solved analytically. From these solutions simple scaling laws of the form <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M7" display="inline" overflow="scroll" dspmath="mathml"><mrow><mfenced close=")" open="("><mrow><msub><mi>T</mi><mi mathvariant="normal">f</mi></msub><mo>-</mo><msub><mi>T</mi><mi mathvariant="normal">s</mi></msub></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced close=")" open="("><mrow><mi mathvariant="italic">Pe</mi><mo>,</mo><mspace linebreak="nobreak" width="0.125em"/><mi>G</mi><mo>,</mo><mi>z</mi></mrow></mfenced></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="102pt" height="13pt" class="svg-formula" dspmath="mathimg" md5hash="ce3a4e1bdb7254f8175de16a58cd5ce5"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="se-13-1045-2022-ie00001.svg" width="102pt" height="13pt" src="se-13-1045-2022-ie00001.png"/></svg:svg></span></span> are derived. Due to scaling they do not depend explicitly on <span class="inline-formula"><i>ϕ</i></span> anymore. The solutions for <span class="inline-formula">Δ<i>T</i>_max⁡</span> and the scaling laws are in good agreement with the numerical solutions. The parameter space <span class="inline-formula"><i>Pe</i><i>G</i></span> is systematically explored. Three regimes can be identified: (1) at high <span class="inline-formula"><i>Pe</i></span> (<span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M12" display="inline" overflow="scroll" dspmath="mathml"><mrow><mo>&gt;</mo><mn mathvariant="normal">1</mn><mo>/</mo><mi>G</mi></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="32pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="87e59a37bc848b4a725bd4a32da136e7"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="se-13-1045-2022-ie00002.svg" width="32pt" height="14pt" src="se-13-1045-2022-ie00002.png"/></svg:svg></span></span>) strong thermal non-equilibrium develops independently of <span class="inline-formula"><i>Pe</i></span>, (2) at low <span class="inline-formula"><i>Pe</i></span> (<span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M15" display="inline" overflow="scroll" dspmath="mathml"><mrow><mo>&lt;</mo><mn mathvariant="normal">1</mn><mo>/</mo><mi>G</mi></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="32pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="e37904fa5af6ac67bbf5f74ceb5fd633"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="se-13-1045-2022-ie00003.svg" width="32pt" height="14pt" src="se-13-1045-2022-ie00003.png"/></svg:svg></span></span>) non-equilibrium decreases proportional to decreasing <span class="inline-formula"><i>Pe</i>⋅<i>G</i></span>, and (3) at low <span class="inline-formula"><i>Pe</i></span> (<span class="inline-formula">&lt;1</span>) and <span class="inline-formula"><i>G</i></span> of the order of 1 the scaling law is <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M20" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>T</mi><mo>max⁡</mo></msub><mo>≈</mo><mi mathvariant="italic">Pe</mi></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="58pt" height="13pt" class="svg-formula" dspmath="mathimg" md5hash="780d067a9b86fada4b87af3a088b1764"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="se-13-1045-2022-ie00004.svg" width="58pt" height="13pt" src="se-13-1045-2022-ie00004.png"/></svg:svg></span></span>. The scaling laws are also given in dimensional form. The dimensional <span class="inline-formula">Δ<i>T</i>_max⁡</span> depends on the initial temperature gradient, the flow velocity, the melt fraction, the interfacial boundary layer thickness, and the interfacial area density. The time scales for reaching thermal non-equilibrium scale with the advective timescale in the high-<span class="inline-formula"><i>Pe</i></span> regime and with the interfacial diffusion time in the other two low-<span class="inline-formula"><i>Pe</i></span> regimes. Applying the results to natural magmatic systems such as mid-ocean ridges can be done by estimating appropriate orders of <span class="inline-formula"><i>Pe</i></span> and <span class="inline-formula"><i>G</i></span>. Plotting such typical ranges in the <span class="inline-formula"><i>Pe</i></span>–<span class="inline-formula"><i>G</i></span> regime diagram reveals that (a) interstitial melt flow is in thermal equilibrium, (b) melt channeling such as revealed by dunite channels may reach moderate thermal non-equilibrium with fluid-to-solid temperature differences of up to several tens of kelvin, and (c) the dike regime is at full thermal non-equilibrium.</p>