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

Abstract

Abstract Realistic immiscible viscous fingering, showing all of the complex finger structure observed in experiments, has proven 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 displacements where viscosity ratio $$\left( {\mu_{{\text{o}}} /\mu_{{\text{w}}} } \right) \gg 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:mrow> <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:mrow> </mml:mfenced> <mml:mo>≫</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . 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 nonlinear diffusion term in the two-phase flow equations, denoted here as $$D(S_{w} )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>w</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Therefore, the addition of capillary pressure, $$P_{c} (S_{w} )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>w</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , into the equations acts as a damping or stabilisation term on viscous fingering, where it is the derivative of this quantity that is important, i.e. $$D(S_{w} )\sim\left( {dP_{c} (S_{w} )/dS_{w} } \right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>w</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∼</mml:mo> <mml:mfenced> <mml:mrow> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>w</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>w</mml:mi> </mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> . 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 $$P_{c} = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ). Since we already have a good method for numerically generating complex realistic immiscible fingering for the $$P_{c} = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> 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 two cases (WW and OW) are important in how the value of corresponding $$D(S_{w} )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>w</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> functions, relate to the (Buckley–Leverett) shock front saturation, $$S_{wf}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow> <mml:mi>wf</mml:mi> </mml:mrow> </mml:msub> </mml:math> , of the viscous-dominated ( $$P_{c} = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ) 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, $$C_{{{\text{VC1}}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi>

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