  We are concerned with a system of partial differential equations describing
internal flows of homogeneous incompressible fluids of Bingham type in which
the value of activation (the so-called yield) stress depends on the internal
pore pressure governed by an advection-diffusion equation. After providing the
physical background of the considered model, paying attention to the
assumptions involved in its derivation, we focus on the PDE analysis of the
initial and boundary value problems. We give several equivalent descriptions
for the considered class of fluids of Bingham type. In particular, we exploit
the possibility to write such a response as an implicit tensorial constitutive
equation, involving the pore pressure, the deviatoric part of the Cauchy stress
and the velocity gradient. Interestingly, this tensorial response can be
characterized by two scalar constraints. We employ a similar approach to treat
stick-slip boundary conditions. Within such a setting we prove long time and
large data existence of weak solutions to the evolutionary problem in three
dimensions.
