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

Abstract

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}left({S}_{n}right) 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 analyzed. RF could be characterized by two (case specific) parameters RFtr and lr (the correlation overlapped the 5500 curves with mean {R}^{2}=0.9989): Recovery follows exactly mathrm{RF}={T}_{n}^{0.5} before water meets the no-flow boundary (early time) but continues (late time) with marginal error until RFtr (highest recovery reached as {T}_{n}^{0.5}) 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 {T}_{n}=t/tau {T}_{mathrm{ch}} accounts for system length L and magnitude overline{D } of the unscaled diffusion coefficient via tau ={L}^{2}/overline{D }, and {T}_{mathrm{ch}} 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 challenging interaction between inputs such as wettability, saturation functions and viscosities with time scales, early- and late-time recovery behavior.