  The analysis of the double-diffusion model and
$\mathbf{H}(\mathrm{div})$-conforming method introduced in [B\"urger, M\'endez,
Ruiz-Baier, SINUM (2019), 57:1318--1343] is extended to the time-dependent
case. In addition, the efficiency and reliability analysis of residual-based
{\it a posteriori} error estimators for the steady, semi-discrete, and fully
discrete problems is established. The resulting methods are applied to simulate
the sedimentation of small particles in salinity-driven flows. The method
consists of Brezzi-Douglas-Marini approximations for velocity and compatible
piecewise discontinuous pressures, whereas Lagrangian elements are used for
concentration and salinity distribution. Numerical tests confirm the properties
of the proposed family of schemes and of the adaptive strategy guided by the
{\it a posteriori} error indicators.
