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

Abstract

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 $${\Lambda }_{n}\left({S}_{n}\right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mfenced> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mfenced> </mml:mrow> </mml:math> with mean 1 and no other parameters (regardless of wettability, saturation functions, viscosities, etc.). 5500 realistic functions $${\Lambda }_{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> were generated using (mixed-wet and strongly water-wet) relative permeabilities, capillary pressure and mobility ratios. The variation in $${\Lambda }_{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> appears limited, and the generated functions span most/all relevant cases. The scaled diffusion equation was solved for each case, and recovery vs time $$RF$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>RF</mml:mi> </mml:mrow> </mml:math> was analyzed. RF could be characterized by two (case specific) parameters $$RFtr$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>RFtr</mml:mi> </mml:mrow> </mml:math> and $$lr$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>lr</mml:mi> </mml:mrow> </mml:math> (the correlation overlapped the 5500 curves with mean $${R}^{2}=0.9989$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>0.9989</mml:mn> </mml:mrow> </mml:math> ): Recovery follows exactly $$\mathrm{RF}={T}_{n}^{0.5}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>RF</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0.5</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> before water meets the no-flow boundary (early time) but continues (late time) with marginal error until $$RFtr$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>RFtr</mml:mi> </mml:mrow> </mml:math> (highest recovery reached as $${T}_{n}^{0.5}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0.5</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> ) in an extended early-time regime. Recovery then approaches 1, with $$lr$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>lr</mml:mi> </mml:mrow> </mml:math> quantifying the decline in imbibition rate. $$RFtr$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>RFtr</mml:mi> </mml:mrow> </mml:math> was 0.05 to 0.2 higher than recovery when water reached the no-flow boundary (critical time). A new scaled time formulation $${T}_{n}=t/\tau {T}_{\mathrm{ch}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>t</mml:mi> <mml:mo>/</mml:mo> <mml:mi>τ</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>ch</mml:mi> </mml:msub> </mml:mrow> </mml:math> accounts for system length $$L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> and magnitude $$\overline{D }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> of the unscaled diffusion coefficient via $$\tau ={L}^{2}/\overline{D }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> , and $${T}_{\mathrm{ch}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>ch</mml:mi> </mml:msub> </mml:math> separately accounts for shape via $${\Lambda }_{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> . Parameters describing $${\Lambda }_{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> 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.

Topics & Concepts