  We derive and analyze in the framework of the mild-slope approximation a new
double-layer Boussinesq-type model which is linearly and nonlinearly accurate
up to deep water. Assuming the flow to be irrotational, we formulate the
problem in terms of the velocity potential thereby lowering the number of
unknowns. The model derivation combines two approaches, namely the method
proposed by Agnon et al. (Agnon et al. 1999, J. Fluid Mech., 399 pp. 319-333)
and enhanced by Madsen et al. (Madsen et al. 2003, Proc. R. Soc. Lond. A, 459
pp. 1075-1104) which consists in constructing infinite-series Taylor solutions
to the Laplace equation, to truncate them at a finite order and to use Pad\'e
approximants, and the double-layer approach of Lynett & Liu (Lynett & Liu 2004,
Proc. R. Soc. Lond. A, 460 pp. 2637-2669) allowing to lower the order of
derivatives. We formulate the model in terms of a static Dirichlet-Neumann
operator translated from the free surface to the still-water level, and we
derive an approximate inverse of this operator that can be built once and for
all. The final model consists of only four equations both in one and two
horizontal dimensions, and includes only second-order derivatives, which is a
major improvement in comparison with so-called high-order Boussinesq models. A
linear analysis of the model is performed and its properties are optimized
using a free parameter determining the position of the interface between the
two layers. Excellent dispersion and shoaling properties are obtained, allowing
the model to be applied up to deep water. Finally, numerical simulations are
performed to quantify the nonlinear behaviour of the model, and the results
exhibit a nonlinear range of validity reaching deep water areas.
