Implicit Incompressible Porous Flow using SPH

TL;DR

<3-5 sentence high-level summary> This paper addresses the challenge of stably simulating incompressible porous flow within an SPH framework by allowing fluid and solid overlap through porosity-aware density estimation and a unified implicit pressure solve. It introduces an implicit linear system that couples non-pressure forces (capillary adhesion, drag, buoyancy) with the pressure-driven incompressibility constraint, along with saturation-dependent elasticity adjustments. Key contributions include porosity-consistent density definitions for solid and fluid phases, an IISPH-based overlapped-pressure formulation, and a strongly coupled solver that remains stable across common porous-flow phenomena. The approach is validated through stability tests, dam-based validation, and parameter studies, demonstrating robust behavior and rich visual results for porous interactions with capillary and adhesive effects.

Abstract

We present a novel implicit porous flow solver using SPH, which maintains fluid incompressibility and is able to model a wide range of scenarios, driven by strongly coupled solid-fluid interaction forces. Many previous SPH porous flow methods reduce particle volumes as they transition across the solid-fluid interface, resulting in significant stability issues. We instead allow fluid and solid to overlap by deriving a new density estimation. This further allows us to extend SPH pressure solvers to take local porosity into account and results in strict enforcement of incompressibility. As a result, we can simulate porous flow using physically consistent pressure forces between fluid and solid. In contrast to previous SPH porous flow methods, which use explicit forces for internal fluid flow, we employ implicit non-pressure forces. These we solve as a linear system and strongly couple with fluid viscosity and solid elasticity. We capture the most common effects observed in porous flow, namely drag, buoyancy and capillary action due to adhesion. To achieve elastic behavior change based on local fluid saturation, such as bloating or softening, we propose an extension to the elasticity model. We demonstrate the efficacy of our model with various simulations that showcase the different aspects of porous flow behavior. To summarize, our system of strongly coupled non-pressure forces and enforced incompressibility across overlapping phases allows us to naturally model and stably simulate complex porous interactions.

Figures (15)

• Figure 1: Simulation of a sponge being soaked in water and subsequently squeezed. Our implicit incompressible porous flow solver is able to model the water leaving the sponge due to pressure forces that consider available pore space. These enable the two phases to overlap in a physically consistent manner, while the porous flow effects are simulated using momentum conserving coupling forces, including capillary action and drag.
• Figure 2: Particle volume definitions for both solid and fluid phase. A solid particle ${s}$ represents the porous object, such that its sampling volume ${{V}_{{s}}^{0}}$ includes void space based on the porosity $\phi$. The same solid mass ${{m}_{{s}}}$ without pores only covers the volume $(1 - \phi) {{V}_{{s}}^{0}}$. The volume $V_i^0$ of fluid particle ${i}$ on the other hand only represents the fluid phase, causing the particles to spread out when they overlap with the solid.
• Figure 3: Fluid seeping into a porous object, where an internal view of the particle distributions is achieved using cut planes. Shown is a closeup of the saturated states around the solid-fluid interface for different solid porosities. The porosity determines the free pore space inside the solid and therefore the distance between fluid particles.
• Figure 4: Porous block absorbing water after 20s with $C^{\text{cap},0} = 500 N \per m$ and two different capillary potential falloff parameters. The solid particles are colored based on saturation ranging from yellow (dry) to red (fully saturated). In (a), fully saturated solid particles no longer produce an adhesion force. A reduced falloff, as shown in (b), allows for more fluid to be pulled into the lower region of the block, leading to more total fluid absorption and a higher fully saturated region.
• Figure 5: Fluid (blue) interacts with a porous wall (gray, particles scaled down for internal visibility). Shown is the state $0.12 s$ after the water first comes into contact with the solid, using explicit or implicit coupling forces. The left two examples use a porous viscosity coefficient of $\mu^{\text{por}} = 10 Pa s$ while the right two use $\mu^{\text{por}} = 100 Pa s$. For a constant time step size of $1 m s$ explicit forces cannot sufficiently slow down fluid flow with the smaller coefficient and are unstable for larger one.