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

Abstract

AbstractSolutions 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 $${\Lambda }_{n}\left({S}_{n}\right)$$ Λ n S n with mean 1 and no other parameters (regardless of wettability, saturation functions, viscosities, etc.). 5500 realistic functions $${\Lambda }_{n}$$ Λ n were generated using (mixed-wet and strongly water-wet) relative permeabilities, capillary pressure and mobility ratios. The variation in $${\Lambda }_{n}$$ Λ 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 $$RF$$ RF was analyzed. RF could be characterized by two (case specific) parameters $$RFtr$$ RFtr and $$lr$$ lr (the correlation overlapped the 5500 curves with mean $${R}^{2}=0.9989$$ R 2 = 0.9989 ): Recovery follows exactly $$\mathrm{RF}={T}_{n}^{0.5}$$ RF = T n 0.5 before water meets the no-flow boundary (early time) but continues (late time) with marginal error until $$RFtr$$ RFtr (highest recovery reached as $${T}_{n}^{0.5}$$ T n 0.5 ) in an extended early-time regime. Recovery then approaches 1, with $$lr$$ lr quantifying the decline in imbibition rate. $$RFtr$$ RFtr was 0.05 to 0.2 higher than recovery when water reached the no-flow boundary (critical time). A new scaled time formulation $${T}_{n}=t/\tau {T}_{\mathrm{ch}}$$ T n = t / τ T ch accounts for system length $$L$$ L and magnitude $$\overline{D }$$ D ¯ of the unscaled diffusion coefficient via $$\tau ={L}^{2}/\overline{D }$$ τ = L 2 / D ¯ , and $${T}_{\mathrm{ch}}$$ T ch separately accounts for shape via $${\Lambda }_{n}$$ Λ n . Parameters describing $${\Lambda }_{n}$$ Λ 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.