Numerical study of Darcy’s law of yield stress fluids on a deep tree-like network

Understanding the flow dynamics of yield stress fluids in porous media presents a substantial challenge. Both experiments and extensive numerical simulations frequently show a non-linear relationship between the flow rate and the pressure gradient, deviating from the traditional Darcy law. In this article, we consider a tree-like porous structure and utilize an exact mapping with the directed polymer with disordered bond energies on the Cayley tree. Specifically, we adapt an algorithm recently introduced by Brunet et al (2020 Europhys. Lett. 131 40002) to simulate exactly the tip region of branching random walks with the help of a spinal decomposition, to accurately compute the flow on extensive trees with several thousand generations. Our results confirm the asymptotic predictions proposed by Schimmenti et al (2023 Phys. Rev. E 108 L023102), tested therein only for moderate trees of about 20 generations.