  In this work, we illustrate the underlying mathematical structure of
mixed-dimensional models arising from the composition of graphs and continuous
domains. Such models are becoming popular in applications, in particular, to
model the human vasculature. We first discuss the model equations in the strong
form which describes the conservation of mass and Darcy's law in the continuum
and network as well as the coupling between them. By introducing proper
scaling, we propose a weak form that avoids degeneracy. Well-posedness of the
weak form is shown through standard Babu\v{s}ka-Brezzi theory. We also develop
the mixed formulation finite-element method and prove its well-posedness. A
mass-lumping technique is introduced to derive the two-point flux approximation
type discretization as well, due to its importance in applications. Based on
the Babu\v{s}ka-Brezzi theory, error estimates can be obtained for both the
finite-element scheme and the TPFA scheme. We also discuss efficient linear
solvers for discrete problems. Finally, we present some numerical examples to
verify the theoretical results and demonstrate the robustness of our proposed
discretization schemes.
