Applying the continuous-time random walk model to non-fickian dispersion in miscible displacement through carbonate rock

Abstract

In this article, we analyze mass transport in a miscible displacement process using a heterogeneous porous medium, applying both continuous-time random walk (CTRW) and advection-dispersion equation (ADE) models. We include a series of theoretical tests using the CTRW scheme against curves of the total concentration model (TCM), referred to here as blind tests. CTRW is compared with ADE based on two experimental displacement tests that are carried out through carbonate rock from a Brazilian outcrop. Experimental tests consist of (1) injecting fresh water into the rock sample that is saturated with a brine solution and (2) inverting the fluid injection sequence, with brine displacing fresh water. We obtain transport concentration curves using x-ray computed tomography (CT). This technique also provides in-situ concentration profiles at different positions along the sample. All experimental curves are analyzed using CTRW and ADE models. We calculate dispersion coefficients for ten different sections of carbonate rock and confirm variation of the dispersion coefficient. The CTRW model fits transport curves more closely than the classical ADE model. Mathematically, CTRW obtained an optimal result of the dispersion coefficient by an error of fitting. However, with the beta parameter, CTRW was ineffective for characterizing local heterogeneity of the rock. In theoretical blind tests, the CTRW model does not effectively reproduce the Peclet number of the original flow, although concentration curves are a reasonable match at low mass-transfer regimes. We found that after increased mass transfer, neither of the curves matched well