Effect of Capillary Heterogeneity on Buckley-Leverett Displacement

Abstract

Summary The Buckley-Leverett (BL) equation has been routinely used for thedescription of immiscible displacement in laboratory operations and forparameter estimation and prediction. Recent advances, however, have led to aserious challenge of the validity of many of the traditional theories. For thecase of BL displacement, most studies have examined heterogeneity effects asthey relate to viscous fingering. In this paper, we investigate the effects ofcapillary heterogeneity induced by variations in permeability in the directionof displacement. We study a variety permeability in the direction ofdisplacement. We study a variety of heterogeneity profiles, both uncorrelatedand correlated spatially. We find that capillary heterogeneity significantlyaffects the saturation distributions, which closely follow the heterogeneityvariation. The saturation response is stronger at lower rates, at moreunfavorable mobility, under drainage conditions, and for smaller spatialcorrelations. The results are in agreement with available experimental data forprimary drainage and secondary imbibition. They may be useful in theinterpretation of saturation profiles and the identification ofheterogeneity. Introduction There is an increasing recognition of the important role played by rockheterogeneity on the performance of an oil recovery process at various scales. At the pore level, the variability in pore size, connectivity, and spatialcorrelation has long been considered a fundamental issue. Such properties asabsolute permeability, relative permeabilities, capillary pressure, andpermeability, relative permeabilities, capillary pressure, and dispersion havebeen directly related to local heterogeneity. This connection is still beingactively pursued with the use of statistical theories. Larger-scaleheterogeneity is also common to natural rocks. Previous studies addressed howsuch issues affect mobility, flow velocity, and pressure, which are the mainvariables in miscible displacement. Although sharing several common features, immiscible displacements are subject to different considerations. At the porelevel, drainage or imbibition processes involve the displacement of aninterface. Low-rate secondary imbibition proceeds by a different mechanism. Relative permeabilities introduce well-known fractional-flow effects, whilecapillarity acts over length scales that are considerably larger than those ofmass dispersion. For example, the scale at which capillarity can be comparablewith viscous effects is macroscopic (e.g., of the order of the core diameter), particularly for lower permeabilities or lower flow velocities. At particularlyfor lower permeabilities or lower flow velocities. At such conditions, capillarity acts over macroscopic scales. Any variation in permeability, orsuch other parameters as wettability, would impart a macroscopic heterogeneityon capillarity. With the exceptions of the traditional end effect at the coreexit and the well-known capillary effects under static, no-flow conditions(e.g., see Ref. 10), the studies on capillary heterogeneity, particularly underflow conditions, have been very limited. Also sparse are experimental resultson capillary heterogeneities, although recent investigations report importanteffects for immiscible displacement in eolian sandstones and glass-bead packs. It is the objective of this paper to address capillary effects at conditions oftransient two-phase flow in a 1D constant-rate process. Similar transient 1Ddisplacements in heterogeneous cores were studied by Watson et al. However, they analyzed overall pressure drops in the limit of high flow rates, wherecapillarity is negligible. The present study differs in that capillary effectsare emphasized and the focus is on local, rather than overall, profiles. Atconditions of spatial homogeneity, the process to be discussed below would beindistinguishable from a standard BL displacement. When spatial variations incapillarity are considered, however, a position-dependent contribution to thefractional flow arises, leading to an augmented BL formulation. This effect isnovel and, as shown below, substantially alters the saturation profiles at the Position of heterogeneity. The process entails, as a special case, the endeffects typically present in displacements in laboratory cores. We examine theprocess sensitivity to amplitude, variation, spatial correlations, flow rate, mobility ratio, and rock wettability. Our analysis is juxtaposed with that of Watson et al. Finally, the results are compared with published experimentaldata. The approach used relies on conventional continuum equations. Werecognize that this approach may not be adequate for porous media withspatially sharp changes in properties. Under such conditions, finitepore-network models may be the preferred alternative. We are currentlyinvestigating this possibility. On the other hand, direct comparison withexperimental results in heterogeneous systems would allow delineation of theextent of applicability of the continuum models. Such work is also currently inprogress. In the time span between the first edition of this paper and dxpresent revision, we completed a separate study of the paper and dx presentrevision, we completed a separate study of the same effects but at steady-stateflow and for the case of adiabatic vapor/liquid flow. The results obtained in Ref. 13 help explain many of the features presented below.