## 1. Introduction

[2] In many porous natural and industrial systems, variable density effects play an important role, as they drive the flow patterns and thus strongly influence the associated mass or heat transfer processes. Several scientific areas are concerned by these phenomena such as hydrology, geophysics, oil reservoirs study, nuclear waste storage, CO2 sequestration, etc. [*Diersch and Kolditz*, 2002]. The problems studied in this context are typically related to the flow of brines with variable concentration in deep or near-surface rock formations. Due to its practical importance for groundwater resources management, the problem of seawater intrusion into coastal aquifers (first modeled by *Henry* [1964]) can be considered as the first problem that interested the hydrogeologic community in this field. Despite the relatively low concentration gradients in this configuration, the questions raised about the mathematical formulation of the physical processes, the geometry of the domain, the parameterization, the boundary conditions and the numerical methods to be applied in this case, have been widely discussed since then and until the 1980s [*Pinder and Cooper*, 1970; *Caltagirone et al.*, 1987; *Herbert et al.*, 1988; *Hassanizadeh and Leijnse*, 1988; *Diersch*, 1988]. New steps in this research field have been connected with the development of new technologies, as the nuclear waste storage in deep geological formations of low permeability and the CO2 sequestration in deep saline aquifers. The new scientific questions thus raised concern the possibility of high density and viscosity contrasts, the presence of reactive multispecies, dissolution-precipitation phenomena which affect the solid matrix [*Bouhlila*, 1999], or the presence of two phases [*Liu et al.*, 2011].

[3] In a context of nonreactive mass transfer, the density-dependent flows in porous media are induced by the spatiotemporal variations of the fluid density depending on a scalar field (the temperature or the solute concentration), which affect the buoyancy term in the Darcy equation or other more sophisticated momentum conservation equations. To solve the nonlinear equations that finally govern the coupled processes of flow and scalar field transport in porous media, sophisticated numerical methods are required. Besides, multiple solutions can exist and only one is obtained during a given time-evolution simulation [*Diersch and Kolditz*, 2002]. A typical case mentioned in the review of *Diersch and Kolditz* [2002] is the Horton-Rogers-Lapwood (HRL) problem which is the analog of the Rayleigh-Bénard convection for porous media. Although it is a simplified situation, the related results, often expressed in terms of critical Rayleigh numbers , can shed light on other more complex and more real configurations. *Weatherill et al.* [2004] also propose to use this problem as a test case for density-dependent groundwater flow and solute transport simulators.

[4] A linear stability analysis of the pure conduction base state in the HRL problem was carried out by *Sutton* [1970]. The critical thresholds for a square cavity, given by , correspond to perturbation eigenmodes with *m* horizontal cells and *n* vertical cells. The well known first critical Rayleigh number is given by and it gives rise to the development of a one-cell convective regime ( ). This one-cell regime has been shown to be stable up to where a fluctuating regime appears [*Caltagirone*, 1975]. Although the multicellular flows ( horizontal cells, *n* = 1) are unstable at onset from the base state, they have been obtained numerically for different values of , even in the range ( ) where a one-cell oscillatory flow exists [*Horne and O'Sullivan*, 1974]. The flow structure obtained depends on the initial conditions, the rate of heating, the perturbations applied. Partial heating of the lower boundary forces the solution as a unique mode of flow [*Horne and O'Sullivan*, 1974], whereas a temperature perturbation locally applied at this lower boundary can drive the oscillatory solution to either symmetric or asymmetric one-cell convection [*Horne and Caltagirone*, 1980]. Straus, Schubert and coworkers [*Straus and Schubert*, 1979; *Schubert and Straus*, 1979, 1982; *Kimura et al.*, 1986, 1989] considered both the square and cubic configurations. For a cubic box, they discussed the possible occurrence of either two-dimensional or three-dimensional flows depending on the value of the Rayleigh number [*Straus and Schubert*, 1979; *Schubert and Straus*, 1979]. For a square cavity, they studied the time-dependent single-cell convection [*Schubert and Straus*, 1982; *Kimura et al.*, 1986]. According to their results, with the increase of the Rayleigh number convection evolves from the initially periodic flow to a quasiperiodic flow, then followed by a second periodic flow, before the occurrence of a chaotic flow. Similar sequences of bifurcations are found in the case of a cubic cavity [*Kimura et al.*, 1989].

[5] In our paper, we want to revisit the Horton-Rogers-Lapwood problem in a porous square box using nonlinear numerical analyses appropriate for the study of multiple solution problems. A nice paper on this problem, using similar analyses, has been published by *Riley and Winters* [1989]. They studied the complex bifurcation structure of the problem and, using the dynamical system theory, they elucidated the interactions between the different branches of solutions for cavities with an aspect ratio between 1 and 2. Our objective is to extend this study (limited to Rayleigh numbers roughly below 400) to a much larger range of Rayleigh numbers in order to capture all the different stable steady solutions and to precisely determine their existence range. A continuation method allowing to follow the solution branches, to calculate their stability and to precisely locate the different bifurcation points is used for that, and the results obtained are shown as bifurcation diagrams. This method is described in section 3, just after the presentation of the model in section 2. The results are then given in section 4. Four stable steady solutions have been found corresponding to one to four rolls along the horizontal. These solutions are eventually destabilized at Hopf bifurcation points beyond which oscillatory solutions have been computed.