Early- and Late-Time Prediction of Counter-Current Spontaneous Imbibition and Estimation of the Capillary Diffusion Coefficient

Abstract

Abstract Solutions are presented to predict 1D counter-current spontaneous imbibition oil recovery at early and late times, i.e. before and after the imbibing water reaches the noflow boundary based on knowing the capillary diffusion coefficient. The system is presented in a normalized form where the space, time and saturation variables are scaled. The normalized capillary diffusion coefficient (NCDC) has an area of 1 when integrated over the saturation range with positive capillary pressure (where spontaneous imbibition occurs). The scaled solution of the system hence only depends on the shape of the NCDC function and not its magnitude (which by definition has an area of 1). Based on the semi-analytical solution by McWhorter and Sunada (1990) scaled recovery equals the square root of scaled time for early times. The time scale depends on one part τ related to known constants, and a part Tch related to the NCDC shape. The normalized critical time Tn,cr when the square root regime ends is known exactly from this solution. Recovery thereafter depends on the NCDC and must be calculated numerically. A dataset is generated based on combining 1000 sets of relative permeabilities and capillary pressure functions with mixed-wet to strongly water-wet states and end point mobility ratios into corresponding NCDCs. Recovery is studied in terms of how long the early square root of time behavior lasts, what the imbibition rate coefficient is in that period, and how recovery changes at late time. The former lasts longer than Tn,cr in practice and is instead characterized by a transition time Tn,tr. The parameters Tch, Tn,tr are well correlated with the CDNC shape, as quantified by one or two fractions zα,β denoting the fraction of the NCDC between normalized saturations α and β. Late time recovery was modeled using an extended Arps type decline curve where one parameter r was correlated with zα,β. We thus find relations between the shape of the NCDC and the behavior of the solution at early and late times. Next we interpret recovery data to estimate the absolute capillary diffusion coefficient. This is done by systematically determining RFtr and the Arps parameter from tuning or type curves and linking them to associated NCDC fractions zα,β. The methods are illustrated using literature experimental data. The predicted NCDCs could predict the original recovery curves. It was found that systems with high non-wetting phase mobility and strong water-wetness are likely to have most of the recovery occur as proportional to the square root of time.