  We study homogenization of a locally periodic two-scale dual-continuum system
where each continuum interacts with the other. Equations for each continuum are
written separately with interaction terms (exchange terms) added. The
homogenization limit depends strongly on the scale of this continuum
interaction term with respect to the microscopic scale. In J. S. R. Park and V.
H. Hoang, {\it Hierarchical multiscale finite element method for multicontinuum
media}, arXiv:1906.04635, we study in details the case where the interaction
terms are scaled as $O(1/\ep^2)$ where $\ep$ is the microscale of the problem.
We establish rigorously homogenization limit for this case where we show that
in the homogenization limit, the dual-continuum structure disappears. In this
paper, we consider the case where this term is scaled as $O(1/\ep)$. This case
is far more interesting and difficult as the homogenized problem is a
dual-continuum system which contains features that are not in the original two
scale problem. In particular, the homogenized dual-continuum system contains
extra convection terms and negative interaction coefficients while the
interaction coefficient between the continua in the original two scale system
obtains both positive and negative values. We prove rigorously the
homogenization convergence. We also derive rigorously a homogenization
convergence rate. Homogenization of dual-continuum system of this type has not
been considered before.
