Finite element modeling approach to assess the stability of debris and rock slopes: a case study from the Indian Himalayas

Landslides and slope failures are recurrent phenomena in the Indian Himalayas. The study area comprises the hill slopes along a road stretch of 1.5 km at a distance of 9 km from Pipalkoti on Chamoli–Badrinath highway (NH-58) in the Garhwal Himalayas, India. Based on the field survey, contour map, and the hillshade, the study area has been divided into different zones. Three different zones/slopes in this study area including one potential debris slide, one stable debris slope, and one potential rock slide have been undertaken for investigation and modeling. Field mapping, data collection related to slope features and soil/rock sample collection, and discontinuity mapping for all the slopes have been carried out in field. Soil samples have been tested in the laboratory to determine the physico-mechanical properties. These properties along with some material properties from the literature have been used as input parameters for the numerical simulation. To investigate the failure process in the debris/rock slides as well as stable debris slope, the slopes were modeled as a continuum using 2D finite element plain strain approach. Shear strength reduction analysis was performed to determine the critical strength reduction factor. The computed deformations and the stress distributions, along the failure surface, have been compared with the field observations and found to be in good agreement. The analysis results indicated rock/debris slide slopes to be highly unstable. The debris slide modeling depicted failures both above and below road levels as observed in field. The rock slide modeling could depict the exact pattern of failure involving 3 sets of discontinuities simultaneously as observed in real-field scenario which is a major limitation in case of limit equilibrium analysis. The field-observed stable slope comes to be stable through FE analysis also. Based on these analyses, landslide hazard assessment of the study area could be done.     

Undrained creep failure of a drainage canal slope stabilized with deep cement mixing columns

This paper presents an analysis of the slope failure of a Suvarnabhumi drainage canal during construction. The Suvarnabhumi drainage canal project includes a large drainage canal with a road on both sides. The width of the bottom of the drainage canal is 48.0 m, the depth of the drainage canal is 3.0 m, and the length of the drainage canal is 10.5 km. Because the project was constructed on very soft Bangkok clay, deep cement mixing (DCM) columns were employed to increase the stability of the excavated canal. The failure of the drainage canal slope occurred 25 days after the end of excavation. The field monitoring data show that lateral movement of the canal slope continuously increased with time, which caused failure due to the instability of the canal slope. The time-dependent deformation and undrained creep behavior of very soft clay was suspected to be the cause of the canal failure. A laboratory investigation of undrained creep behavior and a finite element analysis (FEA) using the soft soil creep (SSC) model were performed to confirm the causes of the canal failure. The results indicate that very soft clay specimens that are subjected to deviator creep stress levels of 70 and 100 % of the peak strength failed by creep rupture within 60 days and 8 min, respectively. The factor of safety for the canal slope, which was obtained from the FEA, shows significant reduction from the initial value of 1.710 to 1.045 within 24 days after the end of excavation due to the effect of undrained creep. This paper also describes a solution method that is applied to a new section of the canal. Field monitoring and an FEA of the new trial section were performed to prove the effectiveness of the solution method.     

2D viscoplastic finite element modelling of slow landslides: the Portalet case study (Spain)

This paper proposes a hydro-geomechanical finite element model to reproduce the kinematic behaviour of large slow landslides. The interaction between solid skeleton and pore fluids is modelled with a time dependent u–p _w formulation and a groundwater model that takes into account recorded daily rainfall intensity. A viscoplastic constitutive model based on Perzyna’s theory is applied to reproduce soil viscous behaviour and the delayed creep deformation. The proposed model is applied to Portalet landslide (Central Spanish Pyrenees). This is an active paleo-landslide that has been reactivated by the construction of a parking area at the toe of the slope. The stability analysis reveals that, after the constructive solutions were undertaken, the slope is in a limit equilibrium situation. Nevertheless, time-dependent analysis reproduces the nearly constant strain rate (secondary creep) and the acceleration/deceleration of the moving mass due to hydrological changes. Overall, the model reproduces a 2-m displacement in the past 8  years that coincides with in situ monitoring data. The proposed model is useful for short- and mid-term predictions of secondary creep. However, long-time predictions remain uncertain, stability depends strongly on the position of the water table depth and new failures during tertiary creep due to soil temporal microstructural degradation are difficult to calibrate.     

Evolution Mechanism and Rainfall Warning Criteria for Maijianwo Slope in Henan Province,China

In this study, two different research methods are applied to investigate the evolution mechanism and rainfall warning criteria for Maijianwo slope located in Henan Province, China. On the one hand, an indoor-model test is performed under artificial rainfall and based on similarity theory. A set of monitoring system is utilized to track the moisture content, deformation and cracks of critical points of the model during the test. On the other hand, the numerical simulation is carried out to provide an insight into the variation of unstable zone and factor of safety for the landslide with the increasing cumulative rainfall. Results indicate that the evolution process of Maijianwo slope is composed of three stages of initiating, accelerating and failure respectively, and stability of slope decreases gradually as cumulative rainfall increases. Based on the evolution mechanism of retrogressive landslide verified by both model test and numerical simulation, cracking time of critical positions on the slope prior to each stage were set as the initiating time of each stage and the cumulative rainfall associated with each initiating time (E₁ = 75 mm, E₂ = 180 mm) has been defined as the warning criteria for the Maijianwo slope. As the cumulative rainfall in Maijianwo slope reaches 75 and 180 mm, the landslide orange and red warning codes are issued, respectively. Otherwise, the slope is in a safe condition when the cumulative rainfall is less than 75 mm.     

Geological Characteristics and Stability Evaluation of Wanjia Middle School Slope in Wenchuan Earthquake Area

To investigate the formation mechanism and the stability of Wanjia middle school slope in Wenchuan Earthquake Area, the macroscopic geological characteristics and the failure process of the landslide are researched by engineering geology analysis method, limit equilibrium method, and finit element method. The results show that after the Wenchuan Earthquake, retaining walls, houses and other infrastructure on the foot of Wanjia middle school slope were severely destroyed, 10 cm wide tension fracture appeared at the trailing edge of the slope. Wanjia middle school slope is a type of medium-sized soil landslide. The area of the deformation body is about 19,314 m², the total volume of the deformation body is about 23 × 10⁴ m³. There may be two potential sliding surfaces in the unstable slope: shallow and deep landslide. The analysis results of the limit equilibrium method and the finite element method show that: under dead weight, dead weight + rainstorm, dead weight + earthquake conditions, the plastic zone occurs mainly at the middle part or the trailing edge of the slope, and it doesn’t fully cut through the deep landslide body, so the deep landslide is stable. However, under rainstorm or earthquake conditions, the plastic zone almost completely cut thorough the shallow landslide body, it shows that the shallow landslide is in the understable–basic stable state. It is found that the results of finite element method is concordant with the results of the limit equilibrium method (F _s = 1.06–1.29, the shallow landslide is in the basic stable–stable state). The calculation results show that shallow landslides are likely to occur in Wanjia middle school slope during a rainstorm or an earthquake, so monitoring and control of the slope should be strengthened. The shallow landslide should be managed by some measures, such as anti slide pile retaining structures and drainage works, and the dangerous rock bodies on the slope surface should be cleaned up.     

Nonstationary flow surface theory for modeling the viscoplastic behaviors of soils

This paper presents a three-dimensional elastic viscoplastic model that can describe the time-dependent behaviors of soft clays. The constitutive model is formulated based on the nonstationary flow surface theory and incorporates new developments, including (i) an improved definition of the nonstationary flow surface that is capable of capturing the stress–strain behaviors under different loading paths, (ii) a unique stress–strain—viscoplastic-strain-rate equation that is able to explicitly describe the nonstationary flow surface, and (iii) a final stable state concept that identifies the final equilibrium state at the end of creep and stress relaxation, which is also used to simplify the loading criteria. The consistency condition is validated for the proposed model, and the viscoplastic multipliers are calculated by solving the consistency equations. The model performance is investigated and validated via simulation of both oedometer and triaxial tests. The numerical results demonstrate that the proposed model is able to reproduce the main viscoplastic behaviors of soils, including creep, undrained creep rupture, stress relaxation, rate effect and accumulated effect.     

Prediction of the water waves generated by a potential semisubmerged landslide in La Yesca reservoir,Mexico

This paper presents a study of the effects of a potential landslide in La Yesca Reservoir, Jalisco-Nayarit, Mexico. The main purpose of the paper is to predict the maximum wave amplitude, wave run-up, and dam overtopping. The landslide is formed by an unstable slope of more than 24 Mm³ that is partially submerged for the range of the reservoir operation levels. The dynamics of the sliding mass were obtained in detail considering that it moves over a pair of failure surfaces with the potential rupture of a third surface. The paper presents results of a physical model of the reservoir based on Froude similitude (scale 1:200). Impulse waves are produced with a solid wedge shape slide as it moves on rails. The movement was calibrated to reproduce the dynamics of the landslide. Also, numerical modelling of the event was performed with a 2D implicit model that solves the two-dimensional shallow water equations. In this case, the impulse waves were generated at each time increment with the variation of the ground elevation (obtained from the dynamics of the landslide) for the mesh points where the landslide passes. The results of both studies are similar.     

Three-dimensional numerical modeling of land subsidence in Shanghai,China

Shanghai, in China, has experienced two periods of rapid land subsidence mainly caused by groundwater exploitation related to economic and population growth. The first period occurred during 1956–1965 and was characterized by an average land subsidence rate of 83 mm/yr, and the second period occurred during 1990–1998 with an average subsidence rate of 16 mm/yr. Owing to the establishment of monitoring networks for groundwater levels and land subsidence, a valuable dataset has been collected since the 1960s and used to develop regional land subsidence models applied to manage groundwater resources and mitigate land subsidence. The previous geomechanical modeling approaches to simulate land subsidence were based on one-dimensional (1D) vertical stress and deformation. In this study, a numerical model of land subsidence is developed to simulate explicitly coupled three-dimensional (3D) groundwater flow and 3D aquifer-system displacements in downtown Shanghai from 30 December 1979 to 30 December 1995. The model is calibrated using piezometric, geodetic-leveling, and borehole extensometer measurements made during the 16-year simulation period. The 3D model satisfactorily reproduces the measured piezometric and deformation observations. For the first time, the capability exists to provide some preliminary estimations on the horizontal displacement field associated with the well-known land subsidence in Shanghai and for which no measurements are available. The simulated horizontal displacements peak at 11 mm, i.e. less than 10 % of the simulated maximum land subsidence, and seems too small to seriously damage infrastructure such as the subways (metro lines) in the center area of Shanghai.     

Assessing hydro-morphological changes in Mediterranean stream using curvilinear grid modeling approach - climate change impacts

The objective of this work was the estimation of time-space hydraulic (water depth, flow velocity) and morphological (sediment transport and bank erosion) characteristics in the downstream part of a Mediterranean stream under current and future climatic conditions. The two-dimensional hydraulic model MIKE 21C was used, which has been developed specifically to simulate 2D flow and morphological changes in rivers. The model is based on an orthogonal curvilinear grid and comprises two parts: (a) the hydrodynamic part and (b) the morphological changes part. The curvilinear grid and the bathymetry file were generated using a very high-resolution DEM (1 m × 1 m). Time series discharge data from a hydrometric station introduced in the hydrodynamic part of the model. Regarding the morphological part of the model, field measurements of suspended sediment concentration and of bank erosion were used. The model was calibrated and verified using field data that were collected during high and low flow discharges. Model simulation was in good agreement with field observations as indicated by a variety of statistical measures. Next, for predicting the riverbank change, future meteorological data and river flow data for the next 10 years (2017–2027) were employed. These data series were created according to a lower and a higher emission climate change scenario. Based on the results, an increase in rainfall intensity may cause significant changes in river banks after 10 years (more than 5 m of soil loss in river meanders). Using the obtained simulation results, extreme hydrological events such as floods transporting large sediment loads and changes in river morphology can be monitored. The proposed methodology was applied to the downstream part of the Koiliaris River Basin in Crete, Greece.     

Monitoring and early warning of the 2012 Preonzo catastrophic rockslope failure

In this paper, we describe the investigations and actions taken to reduce risk and prevent casualties from a catastrophic 210,000 m³ rockslope failure, which occurred near the village of Preonzo in the Swiss Alps on May 15, 2012. We describe the geological predisposition and displacement history before and during the accelerated creep stage as well as the development and operation of an efficient early warning system. The failure of May 15, 2012, occurred from a large and retrogressive instability in gneisses and amphibolites with a total volume of about 350,000 m³, which formed an alpine meadow 1250 m above the valley floor. About 140,000 m³ of unstable rock mass remained in place and might collapse partially or completely in the future. The instability showed clearly visible signs of movements along a tension crack since 1989 and accelerated creep with significant hydromechanical forcing since about 2006. Because the active rockslide at Preonzo threatened a large industrial facility and important transport routes located directly at the toe of the slope, an early warning system was installed in 2010. The thresholds for prealarm, general public alarm, and evacuation were derived from crack meter and total station monitoring data covering a period of about 10 years, supplemented with information from past failure events with similar predisposition. These thresholds were successfully applied to evacuate the industrial facility and to close important roads a few days before the catastrophic slope failure of May 15, 2012. The rock slope failure occurred in two events, exposing a compound rupture plane dipping 42° and generating deposits in the midslope portion with a travel angle of 39°. Three hours after the second rockslide, the fresh deposits became reactivated in a devastating debris avalanche that reached the foot of the slope but did not destroy any infrastructure. The final run-out distance of this combined rock collapse–debris avalanche corresponded to the predictions made in the year 2004.     

Limit Equilibrium Analyses for Internal Design of Geosynthetic Reinforced Slopes: Influence of Potential Failure Surface and Strength Distribution

The paper presents results from a computer code, based on limit equilibrium analyses, able to quantify earth pressure coefficients for the internal design of geosynthetic reinforced soil structures and identify the potential failure surfaces. Failure mechanisms assuming bilinear or logarithmic spiral failure surfaces are considered. The influence of the potential failure surface and geosynthetic strength distribution on the earth pressure coefficient is analysed. Required reinforcement tensile strengths calculated by the developed program are compared with values published in the literature. To further evaluate the capabilities of limit equilibrium analyses, the numerical modelling of a geosynthetic reinforced steep slope, designed at ultimate limit state conditions (FS = 1), is also presented. Good agreement was achieved between the potential failure surfaces predicted by limit equilibrium analyses and those obtained with numerical modelling.     

Probabilistic landslide hazard assessment using Copula modeling technique

A new probabilistic methodology for landslide hazard assessment in regional scale using Copula modeling technique is presented. The current probabilistic landslide hazard analyses are performed under the assumption that landslide hazard elements, such as magnitude, frequency, and location, are independent. In this paper, a general approach is proposed to consider the possible dependence among hazard elements. Part of the Seattle, WA area was selected to evaluate the competence of the presented method. A total of 357 slope failure events and their corresponding topography and geology data were included in the study to develop and test the model. Based on the results, the mean success rates of the presented model in predicting landslide occurrence are 90 % in hazardous area and 12 % in safe locations on average, while these success rates are 63 and 44 % when these hazard elements were treated as mutually independent.     

Dynamic analysis and numerical modeling of the 2015 catastrophic landslide of the construction waste landfill at Guangming,Shenzhen, China

Since lots of underground and slope excavation works were conducted during the urbanization process, an increasing number of sites in ravines around a city have been used to stockpile a large amount of excavated soils. This brings a huge challenge for researchers and managers in the risk evaluation and mitigation of potential dangers of these man-made construction waste landfills. This paper describes a recently large landslide of the construction waste landfill, which occurred at a site of Guangming new district in Shenzhen, China, on December 20, 2015. This catastrophic landslide caused the death of 69 persons and 8 persons are still missing. In this paper, this landslide was numerically simulated and analyzed. In spite of neither high-intensity rainfall nor antecedent rainfall, a slope of this landfill with a relative height of 111 m sided and caused about 2.34 million cubic meters of the soils to travel over a gentle terrain more than 1.2 km. This means that the landslide mobility index (H/L = 0.092) is much lower than a general designed value and the values in most other cases. A depth-integrated continuum method and a MacCormack-TVD finite difference algorithm are adopted, in this paper, to numerically simulate the dynamic process of this large landslide. It is found that a Coulomb friction model with consideration of the pore water pressure effects can well reproduce the main characteristics of the dynamic process of this landslide. Sensitivity analysis has demonstrated that the high pore water pressure in the soils plays a significant role in its mobility and is a key factor to the severity of this landslide.     

How does the water–rock interaction of marly rocks affect its mechanical properties in the Three Gorges reservoir area,China?

Marly rock is a sensitive rock group of landslides in the Three Gorges reservoir area, China. It is composed predominantly of carbonate and clay minerals, water–rock interaction (WRI) of which could activate landslides in the reservoir area. To study the mechanism by which WRI affects the mechanical properties of marly rock, samples were collected from two boreholes (depth 301.78 and 307.14 m) and slope surface. Then, laboratory tests were designed to study the quantitative relationship between mineral contents and mechanical properties of intact rock, to analyze both change process of mineral composition and microstructure under short-term and long-term WRI. Finally, the change in mechanical properties and its effect on slope stability are suggested. This study indicates that the uniaxial compressive strength and Poisson ratio can be estimated by linear regression equations: (1) σ _c = 8.959 × (C/Q) ? 0.744 × CM + 58.516; (2) μ = 0.014 × (C/Q) ? 0.001 × CM + 0.234. The chemical reactions of WRI mainly included dissolution and ion exchange. On the slope surface, dissolution mainly acts on calcite, illite, dolomite, feldspar and other minerals dissolved in water. Underground, both chemical reactions of dissolution and ion exchange easily approach equilibrium with long-term seepage. Small-size minerals and micropores damage the stable microstructure of marly rock. These changes of minerals and microstructure can trigger shallow slope failure and develop deep creep deformation along some crash zones in the reservoir shoreline.     

Bioclogging in porous media under continuous-flow condition

Bioclogging extensively exists in porous media, such as permeable reactive barrier, constructed wetland, reverse osmosis, and biofilter systems. Microorganisms overproduce and affect the efficiency of sewage treatment. In this paper, variations in biochemical and hydraulic parameters during the clogging process were obtained using various column experiments. The hydraulic conductivity first decreased sharply to 18.32 % of the original value at the 12th day and decreased to 2.71 % at the end of the experiment, a reduction of more than an order of magnitude. The hydrodynamic dispersion had the highest increase at 7.13 times the initial value and ultimately decreased to 29 %. The porosity decreased to 47.24 % of the initial value, and the total bacterial count in the inlet of the column increased from 3.4 × 10⁶ to 8.8 × 10⁸ cells/mL. Based on the biochemical and hydraulic parameter variation, the clogging process can be divided into four stages: (1) severe clogging occurs, and aerobic microorganisms reproduce rapidly in the inlet; (2) clogging exists in the entire column, and hydrodynamic dispersion increases sharply as aerobic and anaerobic microorganisms reproduce; (3) anaerobic microorganisms reproduce rapidly and produce more gas, and hydrodynamic dispersion decreases quickly; (4) aerobic and anaerobic microorganisms multiply continuously, and hydrodynamic dispersion, hydraulic conductivity, and porosity decrease steadily. Bioclogging then transforms into a steady stage.     

Stability of embankments constructed from soil mixed with stone dust in quarry reclamation

This study examines the stability of an embankment used in stone quarry reclamation built of local natural soil mixed with stone dust. Stone dust is a fine-grained and low-strength material, whereas soil is coarser with higher strength. Test results show that the shear strength and maximum dry unit weight of the mixed material increase with a decrease in the stone dust. After reviewing various domestic and international standards for slope stability, this study suggests a three-grade slope stability rating system (“unstable,” “requires attention” and “stable”) for embankments built of soil mixed with stone dust. A series of slope stability analysis aims to define stable heights, slope angles and mix ratios for such embankments. Safe slope gradients should be 1:1.8 or higher for 10 m high embankments. Embankments 15 m in height should use a mix ratio less than 50% and a slope gradient of 1:1.8 or higher. The safe dimensioning and design guideline values determined by this study should be useful for the construction of new, as well as the stability evaluation of existing embankments that have been made from mixed soils containing stone dust.     

3D modeling of stratified and irregularly jointed rock slope and its progressive failure

Little has been published on the three-dimensional (3D) simulation of the progressive failure of rock slopes, possibly because the process of failure involves a complex, nonlinear evolution from initiation, through propagation and crack. In addition, rock is typically anisotropic, which makes it difficult to identify and describe the slope constituents and failure processes accurately. Despite such difficulties, further study of the fracture process is just as important as analyzing stress fields in 3D rock slope failures. In this paper, the 3D realistic failure process analysis code using finite element programming, and an extended version of numerical centrifugal method, is used to simulate slopes failure with different dip angles. The numerical centrifugal analysis results in this paper are found that the critical failure surface develops along the weak structural surface when the slope dip angle β is below 30°; conversely, the failure surface is formed along the toe of circular sliding when β is above 30°. In addition, it is also found that whether or not including the irregularity of joint into modeling to analyze the 3D slope stability problem will lead to a significant difference in factors of safety, it can reach 8.41 % at the same slope angle. Furthermore, the acoustic emission analyzing reveals deformed location characters of rock slope during the failure processes. With such capabilities, the approach contributes significantly to the in-depth study of the mechanisms of rock slope instability process.     

Time evolution and spatial accumulation of progressive failure for Xinhua slope in the Dagangshan reservoir,Southwest China

This paper presents a preliminary study of time evolution and spatial accumulation of progressive failure for ancient landslide deposits in Xinhua slope. According to the geological response after impoundment, the Xinhua slope has shown the spatial accumulation of deformation, such as ground cracks in the rear edge, toe collapse, local shallow slides in intense rainfall, and progressive creep displacement. Approximately 2 years of monitoring was performed for the Xinhua slope with the assistance of the global navigation satellite system (GNSS), unmanned aerial vehicles (UAVs), and field investigations. The deformation process of a reservoir landslide is considered to be a comprehensive and complicated combination of geological influence from various adverse factors. Field investigations and monitoring indicate that the major serious influence after completion of dam construction comes from the initial large-scale impoundment, the fluctuation of water level, and the existence of a flood season. The creep/slip deformation of slope deposits is a result of integration with adverse hydraulic conditions, e.g., strong rainfall, intense currents and transient seepage flow inside the slope deposits, and activation by water level fluctuation, which can be verified from the twofold evident deformation in the flood season. For the reservoir with daily regulation ability, the occurrence of evident deformations in July highlights that the regulation plan for water level in the flood season is important for controlling the deformation of slope deposits, where the fluctuation of the water level is no more than 10 m in the operation period.     

Runoff and sediment yield modeling by means of WEPP in the Bautzen dam catchment,Germany

Soil erosion by water is one of the most widespread forms of soil degradation in Europe. There are many undesirable consequences of soil erosion due to water such as loss of water storage capacity in reservoirs and transfer of pollutants from farmland to water bodies. The objectives of this study were to calibrate and validate the Water Erosion Prediction Project watershed model (WEPP 2012.8) in the Bautzen dam catchment area with monthly and daily single events for runoff and sediment yield. This is to our knowledge the first study using WEPP in Germany. The catchment (310 km²) was subdivided into small sub-catchments with an area of <260 ha as recommended in WEPP. A sensitivity analysis revealed that the runoff is highly sensitive to the effective hydraulic conductivity in Bautzen, whereas the sediment yield is highly sensitive to rill erodibility, critical shear stress, and to the effective hydraulic conductivity as well. All these parameters were initially calculated using WEPP’s built-in equations and parameters, which, however, produced very poor results for both runoff and sediment yield. Therefore, the model was calibrated for 2 years (2005–2007) and validated for another 2 years (2008–2009) against monthly measurements, in addition to 14 daily single events from the calibration period and 2010. The monthly results were compared with the monthly measurements on the basis of a continuous simulation. Results of calibration and validation periods show a satisfactory performance of WEPP with a determination coefficient R ² above 0.6 and Nash–Sutcliffe efficiency coefficients above 0.50 for runoff and sediment yield. Thus, the model could be used to simulate runoff and sediment yield, and used in scenario studies in the Bautzen dam catchment area.     

Influence of source thickness on steady-state plume length

The impact of source thickness on steady-state plume length is studied using modifications of the analytical expressions provided in Liedl et al. (2005, 2011) for 2D and 3D scenarios. For comparison, 2D and 3D numerical experiments were performed, and the following three important conclusions were obtained: first, the modified expressions overestimate the plume length only up to a factor of 2 when the source thickness (M _s ) is at least 50 % of the aquifer depth (M). Second, overestimates do not exceed plume length by a factor of 10 (2D scenario) or 5 (3D scenario) for 25 % < M _s /M < 50 %. Third, numerical techniques are recommended for M _s  < 25 %. In addition, it was observed that the degradation from the top dominates for M _s /M > 50 %. As far as the numerical experiments are concerned, it is important to note that the employed finite element approach was applied to the transformed transport equation provided in both Liedl et al. works. This transformation, which can also be applied to more complex scenarios than those studied here, eliminates reaction terms from the model equations and therefore largely facilitates numerical computations.