A linear stability analysis for immiscible displacements

Alem�n, Miguel A.; Slattery, John C.

doi:10.1007/bf00138611

Cited by 7 publications

(2 citation statements)

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“…than on experiments (e.g. Chuoke et al 1959;Alemán and Slattery 1988;Chikhliwala and Yortsos 1988). One of the earliest studies showing immiscible fingering was in the work of van Meurs and van der Poel (1958) reproduced in Fig.…”

Section: Immiscible Viscous Fingering: Experimental Reviewmentioning

confidence: 94%

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

et al. 2020

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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 $$ f_{\rm w}^{*} $$ f w ∗ , and which is the chosen simulation input; (2) from the infinite choice of relative permeability (RP) functions, $$ k_{\rm rw}^{*} $$ k rw ∗ and $$ k_{\rm ro}^{*} $$ k ro ∗ , which yield the same $$ f_{\rm w}^{*} $$ f w ∗ , we choose the set which maximises the total mobility function, $$ \lambda_{\text{T}}^{{}} $$ λ T (where $$ \lambda_{\text{T}}^{{}} = \lambda_{\text{o}}^{{}} + \lambda_{\text{w}}^{{}} $$ λ T = λ o + λ w ), 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 $$ (\mu_{\text{o}} /\mu_{\text{w}} ) = 1600 $$ ( μ o / μ w ) = 1600 . Simulations are presented for two example cases, for different levels of water saturation in the main viscous finger (i.e. for 2 different underlying $$ f_{\rm w}^{*} $$ f w ∗ 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 ($$ f_{\rm w}^{*} $$ f w ∗ ) can give a chosen frontal shock saturation, $$ S_{\rm wf}^{*} $$ S wf ∗ , but at different frontal mobility ratios, $$ M(S_{\rm wf}^{*} ) $$ M ( S wf ∗ ) . 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 (kmax/kmin) ratio in the RCF goes from 100 to 3, and also under numerical grid coarsening.

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Section: Immiscible Viscous Fingering: Experimental Reviewmentioning

confidence: 94%

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

et al. 2020

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“…(a) Linear stability analysis of immiscible fingering: Linear stability analysis of two phase immiscible fingering was originally proposed to analyse the early time growth of unstable modes by Chuoke et al, (1959). This has been extensively developed by many workers (Huang et al, 1984;Yortsos and Huang, 1986;Alemán and Slattery, 1988;Chikhliwala et al, 1988;Chikhliwala and Yortsos, 1988;Yortsos and Hickernell, 1989;Yortsos, 1990;Riaz and Tchelepi, 2006a;Daripa and Pasa, 2008). The broad objective is to determine if there are some longer wavelength modes of the perturbation which will grow over time leading to instability.…”

Section: Viscous Fingering (Vf) Literaturementioning

confidence: 99%

Immiscible Viscous Fingering: the Effects of Wettability/Capillarity and Scaling

Beteta¹,

Sorbie²,

Skauge³

et al. 2023

Preprint

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Realistic immiscible viscous fingering, showing all of the complex finger structure observed in experiments, has proved to be very difficult to model using direct numerical simulation based on the two phase flow equations in porous media. Recently, a method was proposed by the authors to solve the viscous dominated immiscible fingering problem numerically. This method gave realistic complex immiscible fingering patterns and showed very good agreement with a set of viscous unstable 2D water ◊ oil displacement experiments. In addition, the method also gave a very good prediction of the response of the system to tertiary polymer injection. In this paper, we extend our previous work by considering the effect of wettability/ capillarity on immiscible viscous fingering, e.g. in a water ◊ oil displacement where viscosity ratio$\left({\mu }_{o}/{\mu }_{w}\right)$>> 1. We identify particular wetting states with the form of the corresponding capillary pressure used to simulate that system. It has long been known that the broad effect of capillarity is to act like a non-linear diffusion term in the two-phase flow equations, denoted here as $D\left({S}_{w}\right)$. Therefore, the addition of capillary pressure, ${P}_{c}\left({S}_{w}\right)$, into the equations acts as a damping or stabilization term on viscous fingering, where it is the derivative of the of this quantity that is important, i.e. $D\left({S}_{w}\right)\tilde\left(d{P}_{c}\left({S}_{w}\right)/d{S}_{w}\right)$. If this capillary effect is sufficiently large, then we expect that the viscous fingering to be completely damped, and linear stability theory has supported this view. However, no convincing numerical simulations have been presented showing this effect clearly for systems of different wettability, due to the problem of simulating realistic immiscible fingering in the first place (i.e. for the viscous dominated case where ${P}_{c}=0$). Since we already have a good method for numerically generating complex realistic immiscible fingering for the ${P}_{c}=0$ case, we are able for the first time to present a study examining both the viscous dominated limit and the gradual change in the viscous/capillary force balance. This force balance also depends on the physical size of the system as well as on the length scale of the capillary damping. To address these issues, scaling theory is applied, using the classical approach of Rapport (1955), to study this scaling in a systematic manner. In this paper, we show that the effect of wettability/capillarity on immiscible viscous fingering is somewhat more complex and interesting than the (broadly correct) qualitative description above. From a “lab scale” base case 2D water ◊ oil displacement showing clear immiscible viscous fingering which we have already matched very well using our numerical method, we examine the effects of introducing either a water wet (WW) or an oil wet (OW) capillary pressure, of different “magnitudes”. The characteristics of these 2 cases (WW and OW) are important in how the value of corresponding $D\left({S}_{w}\right)$ functions, relate to the (Buckley-Leverett, BL) shock front saturation, ${S}_{wf}$, of the viscous dominated (${P}_{c}=0$) case. By analysing this, and carrying out some confirming calculations, we show clearly why we expect to see much clearer immiscible fingering at the lab scale in oil wet rather than in water wet systems. Indeed, we demonstrate why it is very difficult to see immiscible fingering in WW lab systems. From this finding, one might conclude that since no fingering is observed for the WW lab scale case, then none would be expected at the larger “field” scale. However, by invoking scaling theory – specifically the viscous/capillary scaling group, ${C}_{VC1}$, (and a corresponding “shape group”, ${C}_{S1}$), we demonstrate very clearly that, although the WW viscous fingers do not usually appear at the lab scale, they emerge very distinctly as we “inflate” the system in size in a systematic manner. In contrast, we demonstrate exactly why it is much more likely to observe viscous fingering for the OW (or weakly wetting) case at the lab scale. Finally, to confirm our analysis of the WW and OW immiscible fingering conclusions at the lab scale, we present 2 experiments in a lab scale bead pack where $\left({\mu }_{o}/{\mu }_{w}\right)$=100; no fingering is seen in the WW case whereas clear developed immiscible fingering is observed in the OW case.

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Transport and Relaxation Phenomena in Porous Media

Hilfer

1996

Advances in Chemical Physics

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A linear stability analysis for immiscible displacements

Cited by 7 publications

References 19 publications

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

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

Immiscible Viscous Fingering: the Effects of Wettability/Capillarity and Scaling

Transport and Relaxation Phenomena in Porous Media

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