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 $${\Lambda }_{n}\left({S}_{n}\right)$$ 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.

Simulation and Prediction of Countercurrent Spontaneous Imbibition at Early and Late Time Using Physics-Informed Neural Networks

“…The experimental results were reported by Fischer et al for different 1D COUCSI cases with different viscosity ratios. The interpreted saturation functions were provided by Andersen . The Λ n curves are visualized in Figure a and the results of the forward simulation in Figure b.…”

“…The interpreted saturation functions were provided by Andersen. 8 The Λ n curves are visualized in Figure 12a and the results of the forward simulation in Figure 12b. In all three cases, PINNs successfully regenerated the recovery profile of the imbibition process, closely matching the results obtained using FD and the experimental data.…”

“…By defining normalized spatial and temporal variables X , and T , and saturation S n the system of eqs and can be written (see Section A1 of the Supporting Information for details) into an equivalent dimensionless form ∂ normalT S normaln = ∂ X ( Λ n false( S normaln false) ∂ X S n ) , goodbreak0em2em⁣ ( 0 < X < 1 ) S normaln ( X = 0 , T ) = 1 , goodbreak0em2em⁣ ∂ X S n false| X = 1 = 0 , goodbreak0em2em⁣ S normaln ( X , T = 0 ) = 0 …”

“…Schmid & Geiger used McWhorter and Sunada’s solution to propose a universal time scale for COUCSI. Andersen showed that all one-dimensional (1D) COUCSI problems (regardless of wettability, saturation functions, and other parameters) could be normalized to the same form, which only depends on a normalized diffusion coefficient with a mean equal to one. By correlating the coefficient shape with recovery behavior, ET and LT recovery were predicted accurately without the need for numerical methods.…”

“…In ET, the advancement of the front is proportional to the square root of time, and there is a self-similarity behavior in the saturation front of fluid versus a similarity variable. , The recovery stays proportional to the square root of time during ET. However, it has been demonstrated comprehensively that recovery can follow this trend long into the LT regime. − The explanation is that the imbibition rate at the inlet, which controls recovery, acts as if it is still ET until the saturation profile has changed sufficiently from the self-similar ET saturation profile at the inlet …”