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

Solutions are investigated for 1D linear counter-current spontaneous imbibition (COUSI). It is shown theoretically that all COUSI scaled solutions depend only on a normalized coefficient lambda(n)(S-n) with mean 1 and no other parameters (regardless of wettability, saturation functions, viscosities, etc.). 5500 realistic functions lambda(n) were generated using (mixed-wet and strongly water-wet) relative permeabilities, capillary pressure and mobility ratios. The variation in lambda n appears limited, and the generated functions span most/all relevant cases. The scaled diffusion equation was solved for each case, and recovery vs time RF was ana-lyzed. RF could be characterized by two (case specific) parameters RFtr and lr (the cor-relation overlapped the 5500 curves with mean R-2 = 0.9989 ): Recovery follows exactly RF = T-0.5 (n) before water meets the no-flow boundary (early time) but continues (late time) with marginal error until RFtr (highest recovery reached as T-0.5 (n) ) in an extended early-time regime. Recovery then approaches 1, with lr quantifying the decline in imbibition rate. RFtr was 0.05 to 0.2 higher than recovery when water reached the no-flow boundary (critical time). A new scaled time formulation Tn = t/TTch accounts for system length L and magnitude D of the unscaled diffusion coefficient via a = L-2/D, and Tch separately accounts for shape via lambda(n). Parameters describing lambda(n) and recovery were correlated which permitted (1) predicting recovery (without solving the diffusion equation); (2) predicting diffusion coefficients explaining experimental recovery data; (3) explaining the challeng-ing interaction between inputs such as wettability, saturation functions and viscosities with time scales, early-and late-time recovery behavior.