Pore-network modeling of viscoplastic flows: Exploring the Hele-Shaw cell flow analogy

Abstract

We explore the relationship between a yield stress fluid flow in a Hele-Shaw cell with irregular walls, and a non-Darcy 2D porous medium flow with limiting pressure gradient. The continuum (Hele-Shaw) flow is a much-studied simplification of a cementing displacement flow model, solved via a variational formulation that leads to a convex streamfunction minimization problem. Our interest is with the analogy between the discretization used for the numerical solution and network flows that model the associated porous media flow, i.e. via the pore-throat approach. A staggered mesh, with pressure nodes at the cell corners and streamfunction nodes at the cell center is used for the continuum problem, which naturally separates into a network representation comprising primal and dual graphs, linking streamfunction and pressure nodes, respectively. We show explicitly how the continuum model defines a network model and vice versa. We develop the variational form of the network flow, including an appropriate (discrete) streamfunction minimization and a discrete version of the principle of virtual work.

Two network models are explored, based on different interpretations of the minimization problem. The network flow results are compared with analogous computed continuum flow results in 3 specific geometries. We find that our network model I, which is the most natural interpretation of the continuum model as a network flow using our discretization, generally under-predicts flow rates. This is problematic from the perspective of considering the network flow as an approximation to the porous media or Hele-Shaw flow. Network model II rectifies this situation, via a pressure interpolation method. In our examples we find that the network II flow converges to the continuum flow as the mesh and network are refined. This is not the usual comparison made, as in many pore-throat models the network is fixed according to the underlying pore-space geometry. Despite the differences, both network models have their own advantages and disadvantages. Network model I offers a more natural way of modeling the flow and is easier to apply, for example to complex meshes, e.g. unstructured triangular. Network model II gives the more accurate physical representation of the Hele-Shaw flow. Lastly, we have developed approximate algebraic relationships for the total flow rate as a function of the total applied pressure gradient, both close to the critical onset pressure and for large pressure gradients. These correlations align well with previous findings for Bingham fluid flows in porous media.