  A detailed non-linear analysis of the internal structure of spherical,
charged black holes that are accreting scalar matter is performed in the
framework of the Brans-Dicke theory of gravity. We choose the lowest value of
the Brans-Dicke parameter that is compatible with observational constraints.
First, the homogeneous approximation is used. It indicates that mass inflation
occurs and that the variations of the Brans-Dicke scalar inside the black hole,
which could in principle be large in the absence of mass inflation, become
small when mass inflation does occur. Then, a full non-linear numerical study
of the black hole interior perturbed by a self-gravitating massless uncharged
scalar-field is performed. We use an algorithm with adaptive mesh refinement
capabilities. In this way, the changes in the internal structure of the black
hole caused by mass inflation are determined, as well as the induced variations
of the Brans-Dicke scalar, confirming, qualitatively, the indications given by
the homogeneous approximation.
