On the Modelling of Immiscible Viscous Fingering in Two-Phase Flow in Porous Media

Abstract Viscous fingering in porous media is an instability which occurs when a low-viscosity injected fluid displaces a much more viscous resident fluid, under miscible or immiscible conditions. Immiscible viscous fingering is more complex and has been found to be difficult to simulate numerically and is the main focus of this paper. Many researchers have identified the source of the problem of simulating realistic immiscible fingering as being in the numerics of the process, and a large number of studies have appeared applying high-order numerical schemes to the problem with some limited success. We believe that this view is incorrect and that the solution to the problem of modelling immiscible viscous fingering lies in the physics and related mathematical formulation of the problem. At the heart of our approach is what we describe as the resolution of the “ M - paradox ”, where M is the mobility ratio, as explained below. In this paper, we present a new 4-stage approach to the modelling of realistic two-phase immiscible viscous fingering by (1) formulating the problem based on the experimentally observed fractional flows in the fingers, which we denote as $$ f_{\rm w}^{*} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> , and which is the chosen simulation input; (2) from the infinite choice of relative permeability (RP) functions, $$ k_{\rm rw}^{*} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mrow> <mml:mi>rw</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> and $$ k_{\rm ro}^{*} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mrow> <mml:mi>ro</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> , which yield the same $$ f_{\rm w}^{*} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> , we choose the set which maximises the total mobility function, $$ \lambda_{\text{T}}^{{}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mtext>T</mml:mtext> </mml:mrow> <mml:mrow/> </mml:msubsup> </mml:math> (where $$ \lambda_{\text{T}}^{{}} = \lambda_{\text{o}}^{{}} + \lambda_{\text{w}}^{{}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mtext>T</mml:mtext> </mml:mrow> <mml:mrow/> </mml:msubsup> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mtext>o</mml:mtext> </mml:mrow> <mml:mrow/> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mtext>w</mml:mtext> </mml:mrow> <mml:mrow/> </mml:msubsup> </mml:mrow> </mml:math> ), i.e. minimises the pressure drop across the fingering system; (3) the permeability structure of the heterogeneous domain (the porous medium) is then chosen based on a random correlated field (RCF) in this case; and finally, (4) using a sufficiently fine numerical grid, but with simple transport numerics. Using our approach, realistic immiscible fingering can be simulated using elementary numerical methods (e.g. single-point upstreaming) for the solution of the two-phase fluid transport equations. The method is illustrated by simulating the type of immiscible viscous fingering observed in many experiments in 2D slabs of rock where water displaces very viscous oil where the oil/water viscosity ratio is $$ (\mu_{\text{o}} /\mu_{\text{w}} ) = 1600 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mtext>o</mml:mtext> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mtext>w</mml:mtext> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1600</mml:mn> </mml:mrow> </mml:math> . Simulations are presented for two example cases, for different levels of water saturation in the main viscous finger (i.e. for 2 different underlying $$ f_{\rm w}^{*} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> functions) produce very realistic fingering patterns which are qualitatively similar to observations in several respects, as discussed. Additional simulations of tertiary polymer flooding are also presented for which good experimental data are available for displacements in 2D rock slabs (Skauge et al., in: Presented at SPE Improved Oil Recovery Symposium, 14–18 April, Tulsa, Oklahoma, USA, SPE-154292-MS, 2012. 10.2118/154292-MS , EAGE 17th European Symposium on Improved Oil Recovery, St. Petersburg, Russia, 2013; Vik et al., in: Presented at SPE Europec featured at 80th EAGE Conference and Exhibition, Copenhagen, Denmark, SPE-190866-MS, 2018. 10.2118/190866-MS ). The finger patterns for the polymer displacements and the magnitude and timing of the oil displacement response show excellent qualitative agreement with experiment, and indeed, they fully explain the observations in terms of an enhanced viscous crossflow mechanism (Sorbie and Skauge, in: Proceedings of the EAGE 20th Symposium on IOR, Pau, France, 2019). As a sensitivity, we also present some example results where the adjusted fractional flow ( $$ f_{\rm w}^{*} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> ) can give a chosen frontal shock saturation, $$ S_{\rm wf}^{*} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mi>wf</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> , but at different frontal mobility ratios, $$ M(S_{\rm wf}^{*} ) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mi>wf</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Finally, two tests on the robustness of the method are presented on the effect of both rescaling the permeability field and on grid coarsening. It is demonstrated that our approach is very robust to both permeability field rescaling, i.e. where the ( k max / k min ) ratio in the RCF goes from 100 to 3, and also under numerical grid coarsening.

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