High-concentration-gradient dispersion in porous media: experiments, analysis and approximations

Abstract

Various experimental and theoretical studies have shown that Fick's law, based on the assumption of a linear relation between solute dispersive mass ux and concentration gradient, is not valid when high concentration gradients are encountered in a porous medium. The value of the macrodispersivity is found to decrease as the magnitude of the concentration gradient increases. The classical, linear theory does not provide an explanation for this phenomenon. A recently developed theory suggests a nonlinear relation between concentration gradient and dispersive mass ux, introducing a new parameter in addition to the longitudinal and transversal dispersivities. Once a unique set of relevant parameters has been determined (experimentally), the nonlinear theory provides satisfactory results, matching experimental data of column tests, over a wide range of density dierences between resident and invading uids. The lower limit of the nonlinear theory, i.e. very low (tracer) density dierences, recovers the linear f o rmulation of Fick's law. The equations describing high concentration brine transport are a uid mass balance, a salt mass balance in combination with a nonlinear dispersive mass ux equation, Darcy's law and an equation of state. We study the resulting set of nonlinear p a rtial dierential equations and derive explicit (exact) and semi-explicit solutions, under various assumptions. A comparison is made between mathematical solutions, numerical solutions and experimental data. The results indicate that the simple explicit solution can be used to simulate experiments in a wide range of density dierences, given a unique set of experimentally determined parameters. The analysis shows that enhanced ow due to the compressibility eect, which is caused by local uid density variations, is neglectable in all cases considered. The linear f o rmulation of Fick's law appears to give an upperbound for magnitude of the compressibility eect.1991 Mathematics Subject Classication: 35K65, 58G11, 76S05

Numerical analysis of stable brine displacements for evaluation of density-dependent flow theory

“…They presented a non-linear dispersion theory that was shown to predict both low and high concentration displacements for a unique set of parameter values. The theory was also found (Schotting et al, 1999) to fit well the experimental results for high concentration displacements obtained by Moser (1995). However, the fitted dispersion coefficient was found to be considerably lower than that determined by Hassanizadeh and Lcijnse (1995) for a porous medium of similar median grain size.…”

Viscous and gravitational contributions to mixing during vertical brine transport in water‐saturated porous media

“…Fick's law relates the dispersive flux to the concentration gradient. The experimental and theoretical contributions of Leijnse [1988, 1995], Schotting et al [1999], and Watson et al [2002] question whether the dispersive flux is linearly dependent upon the concentration gradient. They formulate a nonlinear dispersive flux expression that reduces to Fick's law at low salt concentrations, but as Schotting et al [1999] state, the formulation is not mechanistically linked to the physical processes causing dispersion.…”

“…The experimental data collected by Schotting et al [1999], Watson et al [2002], and Menand and Woods [2005] are included in a later analysis of brine properties on dispersion.…”

Modeling Transverse Dispersion and Variable Density Flow in Porous Media

“…Hassanizadeh (1990) and Hassanizadeh and Leijnse (1995) suggested a new nonlinear extension of Fick's laws. Their findings are confirmed by Schotting et al (1999), who analyzed the experiments of Moser (1995) in homogeneous media. They also derived an analytical solution based on the nonlinear theory of Hassanizadeh.…”