A Modular, Model, Library Framework (DebrisLib) for Non-Newtonian Geophysical Flows

Abstract

Non-Newtonian mud and debris flows include a wide range of physical processes depending on the setting, concentration, and soil properties. Numerical modelers have developed a variety of non-Newtonian algorithms to simulate this range of physical processes. However, the assumptions and limitations in any given model or software package can be difficult to replicate. This diversity in the physical processes and algorithmic approach to non-Newtonian numerical modeling makes a modular computation library approach advantageous. A computational library consolidates the algorithms for each process. This work presents a flexible numerical library framework (DebrisLib) that has a diverse range of software implemented to simulate geophysical flows using steady flow, kinematic wave, diffusion wave, and shallow-water models with finite difference, finite element, and finite volume computational schemes. DebrisLib includes a variety of non-Newtonian closures that predict a range of geophysical flow conditions and modular code designed to operate with any Newtonian parent-code architecture. This paper presents the DebriLib algorithms and framework and laboratory validation simulation. The simulations demonstrate the utility of the algorithms and the value of the library architecture by calling it from different modeling frameworks developed by the US Army Corps of Engineers (USACE). We present results with the one-dimensional (1D) and two-dimensional (2D) Hydrologic Engineering Center River Analysis System (HEC-RAS) and the 2D Adaptive Hydraulics (AdH) numerical models, each calling the same library.

1. Introduction

As extreme flood events [1], atmospheric rivers, and wildfires [2,3] become more prevalent, secondary hazards from mud and debris flows will be more common. These flows develop over the denuded, hydrophobic soils that wildfires leave behind. In addition, Lahars and mine tailing dams also threaten communities and infrastructure around the world [4]. The high sediment concentration exacerbates damages from these flows, which have been documented around the world [5,6,7,8,9,10]. These events all diverge from classic Newtonian shallow-water flow assumptions commonly applied in flood risk management studies.

Mud, debris, and ash flows are unsteady, gravity-driven events that involve complex mixtures of sediment, water, and entrained material (i.e., organics, woody debris, unconsolidated substrate). These non-Newtonian flows have high sediment concentrations–including cohesive sediment that increases the viscosity of the fluid phase and can carry large boulders, trees, and uprooted infrastructure (e.g., upstream structures or bridge debris). These flows are commonly modeled using shallow-water equations with either (1) single-phase, (2) two-phase, or (3) mixture theory using non-Newtonian closure approaches and approximations [11,12,13]. However, these events can behave very differently depending on the prevalence and type of solids and the physical setting that determines the regime of solid interactions (e.g., increased fluid viscosity, particle collisions, geotechnical friction). The grain-size distribution, sediment concentration, and flow-stress state (as a function of slope) determine the geophysical flow conditions.

In addition to the process complexity of non-Newtonian hazards, modelers face algorithmic diversity challenges. The processes (e.g., hyper concentration, mudflow, and debris flow) do not even have consistent definitions across the literature, and each has multiple modeling approaches. Researchers have developed a wide range of algorithms to help apply these rheological and geotechnical modeling libraries to different classifications of geophysical flows. The methods that simulate these non-Newtonian flood events are commonly grouped into four main categories:

(1)

Linear [13] and non-linear viscoplastic models [11,12]. These approaches use rheological modes as heuristics for cohesive and/or viscous stresses in the fluid and are commonly used to describe the rheology of laminar mud flows.

(2)

Dispersive-turbulent stress models [14]. This approach adds a turbulent stress term to the dispersive model to describe the inter-particle mechanics within a clay, silt, and organic matrix. This model is commonly applied to sediment mixtures containing cohesive sediment in quantities greater than 10% by volume.

(3)

Dispersive fluid models [15,16,17]. The dispersive models use Bagnold’s theory to describe the non-linear fluid stress that develops from particle-to-particle interaction between granular clastic (i.e., silt, sand, or gravel) particles where the fluid fluctuations keep the particles suspended. This model is commonly applied to sediment–water mixtures containing mostly sand and gravel with lower quantities of fine sediment (≤5–10% by volume).

(4)

Coulomb-based frictional models [10,18,19]. Friction-dominated models such as Coulomb or Vollemy [10] take a more geotechnical approach to simulate grain flows that approach library-supported conditions where the internal mixture stresses are dominated by inter-granular friction. These approaches are often applied to heterogeneous, poorly sorted (well-graded) clastic debris flows approaching land-slide classifications.

Non-Newtonian flows include several regimes (or flow states). The names and criteria for these different flow states vary across the literature [20,21], but most conceptual models classify geophysical flows by the solid concentration of the fluid and the grain size (mostly the cohesive/cohesionless division) of the sediment [21,22,23]. In general, as concentration increases and grain size coarsens, non-Newtonian fluid–sediment mixture passes through five classifications: (1) Newtonian Flow (clear water or alluvial sediment transport), (2) hyper-concentrated flow, (3) mudflow, (4) debris flow and (5) clastic flow. While these flow types are continuous and overlapping, the taxonomy and conceptual model in Figure 1 simplify the non-Newtonian flow continuum as a function of concentration and grain size distribution. Figure 1 also includes a rheological approach often associated with each process.

Researchers have developed a range of algorithms and numerical models to simulate these different processes for hazard assessment, flood risk evaluation, and mitigation design [14,19,24]. These physics-based numerical models simulate advection with constitutive laws of fluid mechanics in one and two dimensions. Fread (1988) and Savage and Hutter (1989) [24,25] provided the foundation for predicting non-Newtonian flow using Saint-Venant-based shallow-water equations, which have been commonly applied to rapid mass movements over irregular geometry [14,26,27,28,29,30,31]. Since then, many of the major hydrodynamic models (e.g., FLOW-2D, FLOW-3D, RiverFlow2D, TELEMAC, and TUFLOW) have non-Newtonian algorithms. Several specialized and academic non-Newtonian models are available (FLDWAV, RAMMS, DCLAW, DAN3D, TITAN2D) [32,33,34,35,36,37,38,39].

Emergency management practitioners are becoming more aware of the importance of non-Newtonian modeling in these high-concentration flow applications. However, the diversity of non-Newtonian flows (floods, mudflows, debris flows, etc.) and multiple algorithms available to simulate each process led the study team to develop a formal, modular library that consolidates these algorithms and provide them to multiple hydrodynamic codes to make non-Newtonian implementation more transparent, repeatable, and robust. DebrisLib is a library of validated non-Newtonian algorithms that provides a flexible, modular, numerical modeling framework for single-phase geophysical flows that developers can call from any one-dimensional or two-dimensional, shallow-water-based hydraulic or hydrologic model. Consolidating multiple non-Newtonian closures associated with the continuum of geophysical flow processes in a modular library and sharing it between hydrodynamic codes has three main advantages:

• The library consolidates diverse literature, making a suite of algorithms available to each linked model, avoiding duplication of effort.
• The library makes non-Newtonian modeling more transparent by standardizing algorithm implementation.
• The library leverages the validation and verification activities of multiple development communities, accelerating debugging and the quality of the code.

This paper documents the numerical library development, enhancements, and linkage architecture necessary for predicting large non-Newtonian flood events across different numerical models, specifically the USACE’s one and two-dimensional numerical models.

DebrisLib was developed to simulate the continuum of extreme flood responses along these concentration and grain size gradients, based on rheology and Coulomb theory, to operate independently of the shallow-water parent codes. The non-Newtonian library assembles closures that different shallow-water codes can leverage and can select the appropriate flow condition and algorithm with non-dimensional threshold parameters. The library also estimates a non-Newtonian drag coefficient [40,41], addresses hindered settling [42,43,44], computes relative buoyancy terms and accounts for increased viscosities and mass density [45,46]. The diversity of the processes and sub-diversity of algorithms for each process makes a shared library approach particularly useful for non-Newtonian model development.

2. Methods

2.1. Shallow-Water Numerical Models

The shallow-water flow equations solve the continuity and momentum equations simultaneously to compute water stage and velocity. The frictional forces between the fluid and the solid boundary are the primary resisting forces in the standard Newtonian clear-water hydraulic equations. In one dimension, the conservation of mass and momentum are the Saint-Venant equations:

$${\displaystyle \frac{\partial Q}{\partial x}}+{\displaystyle \frac{\partial A}{\partial t}}-q=0$$

(1)

$${\displaystyle \frac{\partial Q}{\partial x}}+{\displaystyle \frac{\partial Q\overline{u}}{\partial x}}+gA\left({\displaystyle \frac{\partial \eta }{\partial x}}+S_f+S_{MD}\right)=0$$

(2)

where

• Q = volumetric flow discharge (m³/s)
• x = downstream distance in a channel (m)
• t = time (s)
• A = cross-sectional area of channel (m²)
• q = lateral inflow or outflow (m²/s)
• $\eta$ = water surface elevation (m)
• $S_f$ = Newtonian friction slope (m/m).
• $S_{MD}$ = Mud and debris friction slope (m/m)
• g = gravitational acceleration (m/s²)
• $\overline{u}$ = depth-averaged velocity (m/s)

In two dimensions, the shallow-water equations vertically integrate the mass and momentum equations under the assumptions of incompressible flow and hydrostatic pressure. Assuming negligible free surface shear and pressure variations at the free surface, the two-dimensional shallow-water equations are:

$${\displaystyle \frac{\partial Q}{\partial t}}+{\displaystyle \frac{\partial F_x}{\partial x}}+{\displaystyle \frac{\partial F_y}{\partial y}}+H=0,$$

(3)

$$Q=\left\{\begin{array}{c}h\\ uh\\ vh\end{array}\right\}$$

(4)

$$F_x=\left\{\begin{array}{c}uh\\ u^{2}h+\frac{1}{2}g{h}^2-h{\displaystyle \frac{{\sigma }_{xx}}{{\rho }_m}}\\ uvh-h{\displaystyle \frac{{\sigma }_{yx}}{{\rho }_m}}\end{array}\right\}$$

(5)

$$F_y=\left\{\begin{array}{c}vh\\ uvh-h{\displaystyle \frac{{\sigma }_{xy}}{\rho }}\\ vh+\frac{1}{2}g{h}^2-h{\displaystyle \frac{{\sigma }_{yy}}{\rho }}\end{array}\right\}$$

(6)

$$H=\left\{\begin{array}{c}0\\ gh{\displaystyle \frac{\partial z_b}{\partial x}}+gh{\displaystyle \frac{n^{2}u\sqrt{u^2+v^2}}{h^{\frac{4}{3}}}}+gh{S}_{MDx}\\ gh{\displaystyle \frac{\partial z_b}{\partial y}}+gh{\displaystyle \frac{n^{2}v\sqrt{u^2+v^2}}{h^{\frac{4}{3}}}}+{ghS}_{MDy}\end{array}\right\}$$

(7)

where

• ${\rho }_m$ = mixture density (kg/m³)
• $u$ = flow velocity in the x direction (m/s)
• $v$ = flow velocity in the y direction (m/s)
• $h$ = flow depth (m)
• ${\sigma }_{ij}$ = Reynolds stresses due to turbulence (Pa)
• $S_{MDx}$ = Mud and debris slope in the x direction (m/m)
• $S_{MDy}$ = Mud and debris slope in the x direction (m/m)
• $\overline{u}$ = depth-averaged velocity (m/s)

where the first subscript indicates the direction and the second indicates the face on which the stress acts.

• $g$ = gravitational acceleration (m/s²)
• $z_b$ = bottom elevation (m)
• $n$ = Manning’s roughness coefficient

The Reynolds stresses are determined using the Boussinesq approach to the gradient in the mean current:

$${\sigma }_{xx}=2{\rho }_m\nu {\displaystyle \frac{\partial u}{\partial x}}$$

(8)

$${\sigma }_{yy}=2{\rho }_m\nu {\displaystyle \frac{\partial v}{\partial y}}$$

(9)

$${\sigma }_{xy}={\sigma }_{yx}={\rho }_m\nu \left({\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial v}{\partial x}}\right)$$

(10)

The depth-averaged shallow water equation model in HEC-RAS solves volume and momentum conservation equations and includes temporal and spatial accelerations as well as horizontal mixing, while the diffusive wave equation model ignores these processes but is, therefore, simpler and more computationally efficient [47]. The 2D volume conservation of the water–solid mixture is given by:

$${\displaystyle \frac{\partial \eta }{\partial t}}+{\displaystyle \frac{\partial \left(hu\right)}{\partial x}}+{\displaystyle \frac{\partial \left(hv\right)}{\partial y}}=q$$

(11)

where

• $\eta$ = the flow surface elevation (m)
• $t$ = time (s)
• $h$ = water depth (m)
• $V$ = velocity vector (m/s)
• $q$ = a source or sink term (m³/s)

to account for external and internal fluxes.

The depth-averaged momentum conservation equations may be written as [48]

$${\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}=-g{\mathrm{c}\mathrm{o}\mathrm{s}}^2\phi {\displaystyle \frac{\partial \eta }{\partial x}}+{\displaystyle \frac{1}{h}}{\displaystyle \frac{\partial }{\partial x}}\left(v_{t}h{\displaystyle \frac{\partial u}{\partial x}}\right)+{\displaystyle \frac{1}{h}}{\displaystyle \frac{\partial }{\partial y}}\left(v_{t}h{\displaystyle \frac{\partial u}{\partial y}}\right)-{\displaystyle \frac{{\tau }_x}{{\rho }_{m}R}}{\displaystyle \frac{\mathrm{c}\mathrm{o}\mathrm{s}\psi }{\mathrm{c}\mathrm{o}\mathrm{s}\phi }},$$

(12)

$${\displaystyle \frac{\partial v}{\partial t}}+u{\displaystyle \frac{\partial v}{\partial x}}+v{\displaystyle \frac{\partial v}{\partial y}}=-g{\mathrm{c}\mathrm{o}\mathrm{s}}^2\phi {\displaystyle \frac{\partial \eta }{\partial y}}+{\displaystyle \frac{1}{h}}{\displaystyle \frac{\partial }{\partial x}}\left(v_{t}h{\displaystyle \frac{\partial v}{\partial x}}\right)+{\displaystyle \frac{1}{h}}{\displaystyle \frac{\partial }{\partial y}}\left(v_{t}h{\displaystyle \frac{\partial v}{\partial y}}\right)-{\displaystyle \frac{{\tau }_y}{{\rho }_{m}R}}{\displaystyle \frac{\mathrm{c}\mathrm{o}\mathrm{s}\psi }{\mathrm{c}\mathrm{o}\mathrm{s}\phi }}$$

where

• $g$ = gravitational acceleration (m/s²)
• $v_t$ = a turbulent eddy viscosity (Pa)
• $\tau =({\tau }_x,{\tau }_y)$ is the total basal stress (Pa), which is a function of the friction slopes, including $S_{MD}$
• ${\rho }_m$ = the water–solid mixture density (kg/m³)
• $R$ = the hydraulic radius (m)
• $\phi$ = the water surface slope (m/m)
• $\psi$ = the inclination angle of the current velocity direction (°)

In the above equations, the second term on the right-hand-side ${\displaystyle \frac{1}{h}}{\displaystyle \frac{\partial }{\partial x}}\left(v_{t}h{\displaystyle \frac{\partial v}{\partial x}}\right)$ represents the horizontal mixing due to turbulence and also, in the case of a debris flow, horizontal mixing due to particle collisions. Utilizing the conservative form of the mixing terms is essential for accurate momentum conservation [49]. The bottom friction coefficient is computed utilizing Manning’s roughness coefficient as $\tau ={\tau }_{turbulent}+{\tau }_{MD}$, where ${\tau }_{turbulent}$ is the turbulent stress and ${\tau }_{MD}$ is the mud and debris stress, which includes all non-Newtonian stresses. The turbulence bottom shear stress is computed as a function of Manning’s roughness coefficient. The mud and debris stress are described in detail in the section discussing the rheology models. When the non-Newtonian stress is equal to zero, and the cosine functions (slope corrections) are ignored, the above 2D shallow-water equations reduce to the clear-water equations utilized in HEC-RAS.

2.2. Non-Newtonian Shallow-Water Closure

Mud and debris flows generate additional resisting forces. Increasing the solid content increases the viscosity of non-Newtonian flows, generating internal resisting forces within the fluid. At higher concentrations, particularly with coarse particles, particle collision and friction dissipate more energy. Most of the theoretical and numerical modifications to Newtonian flows integrate new internal fluid forces in the momentum equation. The depth-averaged equations can be adapted for non-Newtonian simulations by adding an additional loss slope term (S_MD) to the classic friction slope term (S_f) in the conservation of momentum equation (Equations (1) and (7)).

DebrisLib computes this mud and debris slope that the hydrodynamic models add to the momentum equation by computing internal shear stresses that the different non-Newtonian processes generate based on rheology or Coulomb models (see Figure 1). The library then converts the internal, rheological shear (τ_MD) from the hypothesized stress–strain characteristics into a fluid loss mud and debris slope (S_MD):

$$S_{MD}={\displaystyle \frac{{\tau }_{MD}}{{\rho }_{m}gR}}$$

(13)

where

• ${\rho }_m$ = the sediment–fluid mixture density (kg/m³)
• $R$ = the hydraulic radius (m)
• ${\tau }_{MD}$ = the total mud-and-debris shear stress (Pa)
• g = gravitational acceleration (m/s²)

The O’Brien et al. (1993) [14] quadratic rheological model combines the four stress components of non-Newtonian sediment mixtures: (1) cohesion between particles, (2) internal friction between fluid and sediment particles, (3) turbulence, and (4) inertial impact between particles. The quadratic model separates the stress–strain relationships into these four additive components such that the shear stress is:

$${\tau }_{MD}={\tau }_{yield}+{\tau }_{viscous}+{\tau }_{turbulent}+{\tau }_{dispersive},$$

(14)

where

• ${\tau }_{MD}$ = the total mud-and-debris shear stress (Pa)
• ${\tau }_{yield}$ = yield stress (Pa)
• ${\tau }_{viscous}$ = viscous shear stress (Pa)
• ${\tau }_{turbulent}$ = turbulent shear stress (Pa) (similar to Manning’s roughness in O’Brien et al., 1993) [14]
• ${\tau }_{dispersive}$ = dispersive shear stress (Pa)

O’Brien et al. (1993) [14] define these terms, yielding a quadratic model based on the strain ($d{v}_x/dz$):

$${\tau }_{MD}={\tau }_y+{\mu }_m\left({\displaystyle \frac{d{v}_x}{dz}}\right)+{\rho }_m{l_m^2\left({\displaystyle \frac{d{v}_x}{dz}}\right)}^2+c_{Bd}{\rho }_s{\left({\left({\displaystyle \frac{C_{\mathrm{*}}}{C_v}}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}-1\right)}^{-2}{d}_s^2{\left({\displaystyle \frac{d{v}_x}{dz}}\right)}^2$$

(15)

where,

• $d{v}_x/dz$ = the shear rate (1/s) computed as a function of depth-averaged velocity and flow depth
• ${\mu }_m$ = mixture dynamic viscosity (Pa s)
• ${\rho }_m$ = sediment mixture mass density (kg/m³)
• $l_m$ = mixing length (m)
• $c_{Bd}$ = Bagnold impact empirical coefficient ($c_{Bd}\cong 0.01)$
• ${\rho }_s$= sediment particle density (kg/m³)
• $C_{*}$= maximum volumetric sediment concentration (-)
• $C_v$ = volumetric sediment concentration (-)
• $d_s$ = sediment grain size (mm)

Takahashi [50] experimentally bounded the Bagnold impact coefficient $\left(c_{Bd}\right)$ between 0.35 and 0.5, which is significantly larger than the value recommended by Bagnold [16,17]. Iverson [10] defined the strain (or shear rate) $\left(3\overline{u}/h\right)$ is based on a vertical integration of a parabolic velocity profile or $\left(2\overline{u}/h\right)$ for linear velocity profile conditions, where,

• $\overline{u}$ = depth-averaged velocity (m/s)
• $h$ = flow depth (m).

Therefore, the quadratic model requires two new terms: the mixture density and the Prandtl mixing length (l_m). The equation for the Prandtl mixing length is defined as

$$l_m=kz,$$

(16)

where

• $k$ = the Von Karmen coefficient ($\cong$0.41)
• $z$ = the proportional distance from the boundary (bed).

This quadratic model combines linear and non-linear rheological modes to compute internal shear. Rheological models do not perform as well as the mixtures become more clastic (i.e., high concentrations of coarse particles). DebrisLib simulates clastic debris flows with a Coulomb approximation based on the Johnson and Rodine [51] Coulomb viscous model [29]. This approach replaces the Bingham yield strength (${\tau }_y$) a geotechnical Coulomb yield stress defined as,

$${\tau }_{yc}={\rho }_{m}gh\cos \alpha \tan \phi ,$$

(17)

where,

• ${\tau }_{yc}$ = Coulomb yield stress (Pa)
• $\alpha$ = bed slope angle (°)
• $\phi$ = Coulomb friction angle (°) with typically ranges between 30° and 40° [10,52]
• g = gravitational acceleration (m/s²)
• h = depth (m)
• ${\rho }_m$ = sediment mixture mass density (kg/m³)

The Coulomb friction angle is a function of the individual grain friction angle and the packing geometry of the particles along the failure plane.

2.3. Numerical Model Discretization

Equations (1)–(15) can be discretized in time and space. Temporal discretization of the equations can use explicit or implicit solution schemes. Explicit schemes rely on the solutions from the previous time steps to obtain the new time step solution, and implicit schemes rely on the previous and present solutions to obtain the new time step solution. Numerical techniques for spatial discretization include the finite difference method (FDM), finite volume method (FVM), finite element method (FEM), and meshless methods. The models in this study used FDM, FEM, and FVM for the spatial discretization. We used the USACE 1, 2D HEC-RAS, and 2D AdH models to test the developed techniques and library architecture.

HEC-RAS 2D solves the depth-averaged shallow water equations (SWEs) and the diffusion wave equation (DWE), which ignores all the terms except the friction and pressure gradient terms. The DWE is solved using an implicit FVM. The SWEs are solved with a combination of finite difference and finite volume methods and a semi-implicit time-stepping scheme. Water volume conservation is ensured by a finite volume discretization of the continuity equation. The momentum equation is discretized semi-implicitly using the finite-difference method. HEC-RAS uses a subgrid modeling approach that describes the high-resolution subgrid terrain using hydraulic property tables, allowing for larger computational cells and time steps while still maintaining accuracy. The subgrid approach leads to a mildly nonlinear system of equations, which is solved using a Newton-type iteration algorithm [23,49,53].

The AdH suite is a collection of solvers for the unsaturated Richard’s equations, Reynolds Averaged Navier Stokes Equations (RANS), full momentum shallow water equations (FMSWE), and the DWE. The FMSWE and the DWE can be meshed in space using the Cartesian or the spherical coordinate system. AdH uses the FEM method with implicit time stepping to solve the equations of motion and conservation of mass [54,55,56,57].

2.4. Verification and Validation Datasets

Several flume experiments were selected for model verification and validation to represent the continuum of non-Newtonian flow behavior under both steady and unsteady conditions. These included the Jeyepalan [58,59] and Hungr [18] unsteady dam break analytical solutions, Haldenwang [60] steady-state flume experiments, and the large-scale US Geological Survey (USGS) flume experiments of high concentration debris flow conditions from Iverson et al. [61,62]. The 1D analytical solutions of Jeyapalan [59] and Hungr [18] reproduce an idealized 30 m high tailings dam subjected to instant liquefaction failure. The primary verification variables included failure runout distance, velocity, depth, and cessation of flow. Haldenwang et al. [63] conducted flume experiments using kaolinite, bentonite, and solutions of carboxymethyl cellulose polymer with various concentrations ranging from 1% to 10% by volume. The experiments include a 10 m long and 300 mm wide flume which were used in this study. The slope varied, ranging from 1° to 5°. The final experiments utilized were the USGS debris-flow flume at H.J. Andrews Experimental Forest, Oregon, United States [61,62]. The USGS conducted large-scale debris flow experiments between 1994 and 2004. The USGS flume experiments consisted of rapid releases of saturated, nonuniformly sized sediment mixtures [62]. Experiments were conducted in a 95 m long, 2 m wide flume with a maximum slope of approximately 31° for both fixed and mobile-bed conditions for a wide range of sediment gradations and concentrations. The headgate is positioned at 12.5 m downslope. This steep slope transitions to a reach that follows a catenary curve descending 2.2 m vertically before reaching an outlet to a 107.5 m runout surface with a gentle slope of 2.4°. Rough concrete tiles were positioned between 6 and 79 m from the release gate. The height of debris behind the headgate was 1.9 m. The simulations presented here were conducted for fixed-bed experiments using debris mixtures of about 56% gravel, 37% sand, and 7% mud particles [62].

The HEC-RAS grid for the USGS flume experiment had a constant grid resolution of 0.15 and 0.2 in the downslope and transverse directions, respectively, with a total of approximately 13,350 cells. Input conditions and calibration data for Hungr [18], Haldenwang et al. [63], and Iverson et al. [62] experiments are provided in

Table 1. Input parameters for the modeled experiments.

Variables	Hungr (1995) [18]	Haldenwang et al. (2006) [63]	Iverson et al. (2010) [62]
Impoundment Height (m)	30	-	1.9
Volumetric Concentration (%)	55	10	61.2
Mixture Density (kg/m3)	2500	1165	2010
τy (Pa)	1500	21.3	-
ds (mm)	0.025	0.02–0.04	0.0625–5.0
O’Brien Power Coefficient	0.75	0.05	-
O’Brien Yield Coefficient	6	9	-
O’Brien Viscosity Coefficient	9.1	8	-
Hershel Bulkley Coef (k) (Pa sn)	-	0.524	-
Hershel Bulkley Power (n)	-	0.468	-
Yield Strength (Pa)	1496.4	23.84	-
Dynamic Viscosity (Pa s)	101.2	0.00316
Domain Length (m)	2000	10	107.5
Domain Width (m)	-	0.015	2

.

3. Results

The model library and the linkage architecture were evaluated by simulating three non-Newtonian verification datasets by calling the library from two different hydrodynamic codes.

3.1. Jeyapalan (1983) [58] Validation

The 2D HEC-RAS and AdH (Bingham) results for the Jeyapalan [59] and Hungr [18] dam breach simulations are included in Figure 2 (left and right, respectively). These plots represent conditions of dynamic unsteady conditions following an impoundment failure. The initial condition profile represents the 30 m high pre-dam break conditions, and the final profile is the analytical solution for the runout profile when the flow ceases. This analytical dam removal computes “run out length,” which is the distance required for this high-energy mass flow to come to rest (i.e., the shear stress drops below the yield strength). Both numerical simulations compute run out lengths very similar to Hungr’s [18] result and have similar final depth profiles. Both HEC-RAS and AdH sufficiently replicated the impoundment failure to include unsteady runout, flow depth, and velocity. These results were consistent with other publications based on this analytical solution (e.g., [29]). Simulations were conducted using Bingham, Herschel–Bulkley, and O’Brien’s quadratic model with similar flow behavior and results. The Bingham results are presented in Figure 2 to compare to the results presented in Hungr [18].

3.2. Haldenwang et al. (2006) [63] Validation

The results from the Haldenwang et al. [63] simulations are included in Figure 3. The AdH results use the same 2D, finite element, unsteady flow scheme used in the other simulations (with a constant flow boundary condition) to simulate stages associated with multiple experimental flows with four different rheological assumptions. The HEC-RAS simulation, however, used the 1D, unsteady-flow FD solver, adding a third model framework to the DebrisLib validation. The results are plotted for sixteen flows and depths from Haldenwang et al. [63] 5-degree slope for the 10 m flume measured during a sequence of stepped, steady-flow conditions for 10 percent kaolinite. Model results of the Haldenwang flume experiments are provided in Figure 3. In both cases, the non-Newtonian simulations compute the fluid stage better than clear water flow. The Bingham results fit these data well in both models for flows greater than 7.5 liters/second (L/s). The O’Brien quadratic method is presented for comparison, but because this method adds additional non-linear terms to account for turbulent and dispersive processes (that the Bingham model lumps into the yield and viscosity), it requires lower yield and viscosity parameters [47]. The differences between HEC-RAS and AdH results are likely due to a range of factors, including how each model accounts for initial conditions, model discretization, grid size, and time steps. All rheology closure models for both models predict depth for flows smaller than approximately 7.5 L/s. This systematic low-flow under prediction could be the result of several factors, including the strain rate approximation of ${\displaystyle \raisebox{1ex}{$3u$}\!\left/ \!\raisebox{-1ex}{$h$}\right.}$ approximation, change in velocity profile, potential sediment deposition, or a combination. Subsequent research is investigating this divergence. All models predict transitional and turbulent conditions better than laminar. The turbulent laminar condition is represented with the vertical red line near 5 L/s in Figure 3 which corresponds to the low-flow divergence in both models. Perez [64] also compared shallow-water results to the Haldenwang dataset, with similar findings.

3.3. Iverson et al. (2010) [62] Validation

The DebrisLib connection to AdH and HEC-RAS was also evaluated with a mesoscale experiment with a mixture of sand, gravel, and mud (silts and clays) and a section of bumpy concrete tiles. The comparison of AdH and HEC-RAS water levels compared to the measurements from one of the USGS experiments is presented in Figure 4. Both AdH and HEC-RAS water surface time series compared reasonably well with pressure measurements. Differences in AdH and HEC-RAS results are due to differences in the model formulation, numerical assumptions, and solution methods. This demonstrates that the optimal rheological parameters will likely be different for different models. Both models tend to perform better at the upstream locations, specifically at x = 2 m and x = 32 m. Model results continue to deviate from flume conditions slightly for x = 66 and x = 90. AdH retained the pressure peak farther downstream and matched downstream magnitudes better because the adaptive mesh adjusted the resolution as the front progressed. The models still perform reasonably well considering this limitation.

4. Discussion

DebrisLib, the non-Newtonian model library, includes a wide range of rheological algorithms to simulate mud and debris flows across multiple numerical-model platforms. The process diversity and the multiple models in the non-Newtonian literature make a modular, computation, library-based framework advantageous by consolidating the algorithms for each process. With a few common variables such as depth, velocity, mixture density, and slope, the model library can be linked with most existing shallow-water Newtonian numerical models, regardless of dimension, discretization scheme, dimensionality, or solution method to predict non-Newtonian flows. DebrisLib is under development and is available in the latest releases of HEC-RAS [47] and ADH and could be called by other models. Simulating a range of flow conditions across multiple non-Newtonian transport models will increase access to non-Newtonian modeling capabilities and reduce uncertainty.

The flexibility will allow broader access to the non-Newtonian modeling capabilities for practitioners, managers, and researchers to improve their understanding of the model limitations and operational modeling for post-wildfire emergency management and flood risk management. During the development of the model library, the team identified some key disadvantages and advantages associated with library-based frameworks in research and development. An overview of the advantages and challenges associated with the development of a flexible library-based model framework is provided in

Table 2. Advantages and challenges associated with the development of Library-based frameworks.

Advantages	Disadvantages/Challenges
Standardization and transparency Leverages contributions from multiple developers on a combined product (e.g., reduces duplication and redundant effort) Improve debugging and development Encourages scientific communication and collaboration	Requires continuous communication Research and development can take longer Requires team commitment

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The library approach requires more communication and collaboration because code modifications cannot be unilateral. This is a limitation. A library approach adds effort and limits the flexibility of each contributor. Because of this constant coordination, research and development takes longer. The authors have contributed to other code libraries in the past that quickly diverged and just became customized appendages (often vestigial appendages) to each parent code. A shared library requires a long-term commitment from all collaborators in order to accrue the benefits; however, the contributors to this work have found the benefits far outweigh the costs.

Standardizing the computational approaches and closure between model frameworks is a substantial advantage. Most models include undocumented constitutive theory or algorithmic customizations that make results difficult to compare, reproduce, and interpret. Standardizing these algorithms makes those assumptions shared and transparent. The library architecture also leverages contributions from multiple researchers and teams, minimizing duplication. The savings from avoiding duplication more than offset the costs of the library approach in this initiative. The library becomes a framework that encourages scientific communication between research teams by requiring explicit negotiations when addressing challenges and setbacks during development. Finally, one of the biggest advantages of a shared library is the QA/QC (Quality Assurance/Quality Control) benefits of the library approach. Each software development team conducted independent validation and verification studies (including those documented above) to evaluate the reliability and robustness of the DebrisLib algorithms. Each team found different bugs and issues with the code. Each developer tested code they did not write, automatically building code-checking into the development process, increasing the rate of bug discovery and rectification.

5. Conclusions

Production level flood risk management models will be pressed into service to assess mud and debris impacts more often as mine tailing dams age and wildfire potential increases. Post-wildfire flooding and mine tailing dam failure are among the most common applications of these methods [53], but risk management for other large-scale geophysical hazards such as lahars and ice floes will benefit from this development. This work consolidates the state-of-the-practice in a modular non-Newtonian library implemented in two widely used flood risk models (HEC-RAS and AdH). The study conducted parallel verification and validation studies of the same non-Newtonian libraries through different software packages, increasing the reliability of the individual software packages for these applications, and the reliability of the shared code. This modular library improves and enhances prediction capabilities to assist with planning, management, and mitigation in a range of environments using practical science-based approaches and integrated numerical approaches.