A Mass-Conservative Single-Phase Proppant Transport Scheme for Planar-3D Hydraulic Fracturing

Abstract

Reliable prediction of proppant placement remains the chief uncertainty in hydraulic-fracture design, yet most open-source simulators either neglect solids entirely or accumulate percent-level mass errors that undermine forecast quality.  We present a lightweight, single-phase proppant module that extends PyFrac, an open-source planar-3D fracture code, while leaving its verified elasticity and pressure solvers untouched.  The new algorithm (i) couples Krieger–Dougherty apparent viscosity, (ii) incorporates Richardson–Zaki hindered settling, and (iii) advances suspended and deposited volume fractions on the existing finite-volume mesh with an explicit operator-split update that closes global mass balance to machine precision (≤ 2 × 10⁻¹²).A structured verification–validation campaign confirms: first-order spatial and temporal convergence across three mesh refinements and two Courant regimes; exact compliance with analytic viscosity and settling correlations; and unconditional stability under concentrations approaching 90 % of the random close-packing limit.  In a penny-shaped fracture case study representative of slick-water shale treatments (5 % vol sand, 300µm grains, 30s injection), the solver predicts a radial half-length growth from 0.32 m to 1.20 m, retains 19.9 kg.m-1of solids, and maintains suspended/deposited inventories that match analytic settling times within 0.1%. Peak volumetric concentration reaches 0.55 without numerical oscillations, and pressure evolution captures toughness-induced oscillatory behaviour.