A Numerical Study of Viscous Instabilities: Effect of Controlling Parameters and Scaling Considerations
- Mohan Kelkar (U. of Tulsa) | Surendra P. Gupta (Amoco Production Co.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- February 1991
- Document Type
- Journal Paper
- 121 - 128
- 1991. Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 5.3.2 Multiphase Flow, 5.5 Reservoir Simulation, 5.1 Reservoir Characterisation, 1.6.9 Coring, Fishing, 5.2 Reservoir Fluid Dynamics, 4.3.4 Scale, 5.3.1 Flow in Porous Media, 5.6.5 Tracers, 5.7.2 Recovery Factors, 4.1.5 Processing Equipment, 1.2.3 Rock properties, 5.4.9 Miscible Methods
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An accurate finite-element simulator with a very fine 2D grid was used inthis study of viscous instabilities. The simulator has negligiblegrid-orientation and numerical-dispersion effects and treats longitudinal andtransverse dispersivities separately. The simulator was validated by comparingnumerical results with the analytical solution for unit-mobility-ratio miscibledisplacements under varying longitudinal and transverse dispersivities and wasfurther tested by simulating the results of published laboratory displacementsunder adverse-mobility-ratio conditions. A good match was obtained between thesimulated and experimental results for recovery and effluent-concentrationprofiles for adverse-mobility-ratio displacements. A permeability variance ofonly 0. 1 or an inlet concentration perturbation for a homogeneous system wasenough to initiate the effects of viscous fingers seen in laboratorydisplacements.
Results showed that the parameters that control unstable displacements arethe permeability variance (Dykstra-Parsons coefficient), the size ofheterogeneity (scale length), the mobility ratio, and the dimensionlesstransverse Peclet number, Npet. It was concluded that the instabilitiesincrease with increases in the permeability variance, the scale length, and themobility ratio. With increased instability, the recovery decreases and thebreakthrough time decreases. For Npet, less than 0.01) the displacement remainsunstable. In addition, the effect of longitudinal dispersivity is negligible.As long as the parameters (mobility ratio, permeability variance, and size ofheterogeneities, Npet, less than 0.01) permeability variance, and size ofheterogeneities, Npet, less than 0.01) were the same, the effect of the size ofthe modeled medium on recovery and effluent profiles was insignificant. Thisimplies that the effects of viscous instabilities can be scaled within therange of parameters investigated.
Oil displacement by miscible flooding processes, including CO2 flooding, isaffected by instability at the flood front caused by an adverse mobility ratio.An unfavorable-mobility-ratio displacement coupled with the heterogeneity ofthe porous medium results in viscous instability (fingering), which in turnaffects the displacement efficiency of multiple-contact-miscible processes,sweep efficiency, breakthrough time, and slug-size design.
Our previous investigations showed that the effect of small-scaleheterogeneity of the porous media on the displacement front can be representedby an effective dispersivity. The investigation, however, was restricted tounit-mobility-ratio displacements. The purpose of this investigation is toextend the previous analysis to purpose of this investigation is to extend theprevious analysis to adverse-mobility-ratio (greater than unity)displacements.
Extensive literature is available on the phenomenon of viscous fingering.The literature can be divided into three categories: analytical, experimental,and numerical. Analytical work involves the mathematical description of theinception and modeling of fingers and the method to scale fingers to fieldconditions. Experimental work involves laboratory core experiments underadverse-mobility-ratio conditions. The numerical work relates to differentsimulation techniques that are used to predict and simulate the physicalbehavior of the viscous fingers.
Most of the previous numerical work concentrated on matching theexperimental laboratory data for adverse-mobility-ratio displacements. Some ofit matched the data of Blackwell et al. in sandpacks; others matched their ownexperimental data. The numerical techniques used varied from finite-differencesimulations to the method of weighted residuals to random walk models. Most ofthese investigators were reasonably successful in matching experimentallaboratory data, indicating that the physical phenomenon of fingering can bemimicked by simulators if appropriate precautions are taken. Chief among theseare the use of a large number of gridblocks, minimization of grid-orientationeffects, and proper initiation of viscous fingers.
In addition to matching experimental data, some investigators conductedsensitivity studies to investigate the effect of various parameters on viscousfingering. Detailed simulations by Giordano et al. and Moissis et al. indicatethat mobility ratio and scale length of heterogeneity significantly affect thecharacteristics of viscous fingers. An increase in either of these twoparameters accentuates the instability of the viscous fingers. parametersaccentuates the instability of the viscous fingers. Although numericalinvestigations in the last few years have remarkably improved the physicalunderstanding of viscous fingering, some questions still remain unanswered.First, can the effects of laboratory-observed viscous fingering on displacementperformance be modeled? Second, how is a viscous instability affected byvarious parameters including mobility ratio, permeability variance, size ofheterogeneities, and rock dispersivity values? permeability variance, size ofheterogeneities, and rock dispersivity values? And third, can the viscousfingering be scaled? This paper addresses these questions.
To address these questions, several assumptions were made: the displacementwas contact miscible, the gravity effects were negligible (densities ofdisplacing and displaced phases were identical), and the permeabilitydistribution for small-scale heterogeneities was log-normal. In addition, asshown in Fig. 1, a linear (strictly rectilinear) displacement was conducted inwhich 2D flow effects were captured.
Amoco's finite-element simulator, modified to include both longitudinal andtransverse rock dispersivities, was used for this investigation.
A three-step approach was taken in this study. First, the finite-elementmodel was modified and validated with analytical solutions and the literatureexperimental results. Second, with the same model, mechanistic aspects ofviscous instabilities were investigated. Third, the effect of differentvariables on recovery and effluent profiles was analyzed to provide insightinto scaling of viscous fingers.
The effects of various parameters on viscous instabilities wereinvestigated. These parameters (Table 1) include longitudinal dispersivity(rock dispersivity), ratio of transverse to longitudinal dispersivity,permeability variance (Dykstra-Parsons coefficient), scale length (size ofheterogeneity), mobility ratio, spatial permeability distribution, and size ofthe modeled medium. The longitudinal values were chosen on the basis of theavailable literature evidence. The mobility-ratio values chosen represent atypical range of mobility ratios observed for miscible-displacementprocesses.
To define the permeability heterogeneity, it is necessary to characterizetwo parameters-permeability variance and scale length. The permeabilityvariance characterizes the extent of heterogeneity, whereas permeabilityvariance characterizes the extent of heterogeneity, whereas the scale lengthcharacterizes the size of heterogeneity. In this study, the permeabilitydistribution was assumed to be log-normal.
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