Early and Late Time Analytical Solutions for Co-Current Spontaneous Imbibition and Generalized Scaling

2020 ◽  
Author(s):  
Østebø Pål Andersen
Keyword(s):  
Analytical Solutions ◽  
Spontaneous Imbibition ◽  
Late Time

scholarly journals ON ANALYTICAL SOLUTIONS OF f(R) MODIFIED GRAVITY THEORIES IN FLRW COSMOLOGIES

2013 ◽  
Vol 22 (02) ◽  
pp. 1350006 ◽  
Author(s):  
SILVIJE DOMAZET ◽  
VOJA RADOVANOVIĆ ◽  
MARKO SIMONOVIĆ ◽  
HRVOJE ŠTEFANČIĆ
Keyword(s):  
Power Law ◽  
Analytical Solutions ◽  
Modified Gravity ◽  
Cosmic Time ◽  
Equation Of Motion ◽  
Scale Factor ◽  
Ricci Scalar ◽  
Late Time ◽  
Modified Gravity Theories ◽  
Functional Forms

A novel analytical method for f(R) modified theories without matter in Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetimes is introduced. The equation of motion for the scale factor in terms of cosmic time is reduced to the equation for the evolution of the Ricci scalar R with the Hubble parameter H. The solution of equation of motion for actions of the form of power law in Ricci scalar R is presented with a detailed elaboration of the action quadratic in R. The reverse use of the introduced method is exemplified in finding functional forms f(R), which leads to specified scale factor functions. The analytical solutions are corroborated by numerical calculations with excellent agreement. Possible further applications to the phases of inflationary expansion and late-time acceleration as well as f(R) theories with radiation are outlined.

Analytical Solutions for Spontaneous Imbibition: Fractional-Flow Theory and Experimental Analysis

SPE Journal ◽  
2016 ◽  
Vol 21 (06) ◽  
pp. 2308-2316 ◽  
Author(s):  
K. S. Schmid ◽  
N.. Alyafei ◽  
S.. Geiger ◽  
M. J. Blunt
Keyword(s):  
Capillary Pressure ◽  
Relative Permeability ◽  
Analytical Solutions ◽  
Flow Theory ◽  
Spontaneous Imbibition ◽  
Fractional Flow ◽  
Matrix Block ◽  
Relative Permeability Curves ◽  
Fractional Flow Theory ◽  
Saturation Profiles

Summary We present analytical solutions for capillary-controlled displacement in one dimension by use of fractional-flow theory. We show how to construct solutions with a spreadsheet that can be used for the analysis of experiments as well as matrix-block-scale recovery in field settings. The solutions can be understood as the capillary analog to the classical Buckley-Leverett solution (Buckley and Leverett 1942) for viscous-dominated flow, and are valid for cocurrent and countercurrent spontaneous imbibition (SI), as well as for arbitrary capillary pressure and relative permeability curves. They can be used to study the influence of wettability, predicting saturation profiles and production rates characteristic for water-wet and mixed-wet conditions. We compare our results with in-situ measurements of saturation profiles for SI in a water-wet medium. We show that the characteristic shape of the saturation profile is consistent with the expected form of the relative permeabilities. We discuss how measurements of imbibition profiles, in combination with other measurements, could be used to determine relative permeability and capillary pressure.

scholarly journals Flow regimes for fluid injection into a confined porous medium

2015 ◽  
Vol 767 ◽  
pp. 881-909 ◽  
Author(s):  
Zhong Zheng ◽  
Bo Guo ◽  
Ivan C. Christov ◽  
Michael A. Celia ◽  
Howard A. Stone
Keyword(s):  
Porous Medium ◽  
Diffusion Equation ◽  
Governing Equation ◽  
Analytical Solutions ◽  
Nonlinear Diffusion ◽  
Flow Behaviour ◽  
Late Time ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Time Period

AbstractWe report theoretical and numerical studies of the flow behaviour when a fluid is injected into a confined porous medium saturated with another fluid of different density and viscosity. For a two-dimensional configuration with point source injection, a nonlinear convection–diffusion equation is derived to describe the time evolution of the fluid–fluid interface. In the early time period, the fluid motion is mainly driven by the buoyancy force and the governing equation is reduced to a nonlinear diffusion equation with a well-known self-similar solution. In the late time period, the fluid flow is mainly driven by the injection, and the governing equation is approximated by a nonlinear hyperbolic equation that determines the global spreading rate; a shock solution is obtained when the injected fluid is more viscous than the displaced fluid, whereas a rarefaction wave solution is found when the injected fluid is less viscous. In the late time period, we also obtain analytical solutions including the diffusive term associated with the buoyancy effects (for an injected fluid with a viscosity higher than or equal to that of the displaced fluid), which provide the structure of the moving front. Numerical simulations of the convection–diffusion equation are performed; the various analytical solutions are verified as appropriate asymptotic limits, and the transition processes between the individual limits are demonstrated. The flow behaviour is summarized in a diagram with five distinct dynamical regimes: a nonlinear diffusion regime, a transition regime, a travelling wave regime, an equal-viscosity regime, and a rarefaction regime.

scholarly journals Accurate early-time and late-time modeling of countercurrent spontaneous imbibition

2016 ◽  
Vol 52 (8) ◽  
pp. 6263-6276 ◽  
Author(s):  
Rafael March ◽  
Florian Doster ◽  
Sebastian Geiger
Keyword(s):  
Early Time ◽  
Spontaneous Imbibition ◽  
Late Time ◽  
Time Modeling
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