Geophysical flows impacting a flexible barrier: effects of solid-fluid interaction

Flexible barriers undergo large deformation to extend the impact duration, and thereby reduce the impact load of geophysical flows. The performance of flexible barriers remains a crucial challenge because there currently lacks a comprehensive criterion for estimating impact load. In this study, a series of centrifuge tests were carried out to investigate different geophysical flow types impacting an instrumented flexible barrier. The geophysical flows modelled include covered in this study include flood, hyperconcentrated flow, debris flow, and dry debris avalanche. Results reveal that the relationship between the Froude number, Fr, and the pressure coefficient α strongly depends on the formation of static deposits called dead zones which induce static loads and whether a run-up or pile-up impact mechanism develops. Test results demonstrate that flexible barriers can attenuate peak impact loads of flood, hyperconcentrated flow, and debris flow by up to 50% compared to rigid barriers. Furthermore, flexible barriers attenuate the impact load of dry debris avalanche by enabling the dry debris to reach an active failure state through large deformation. Examination of the state of static debris deposits behind the barriers indicates that hyperconcentrated and debris flows are strongly influenced by whether excessive pore water pressures regulate the depositional process of particles during the impact process. This results in significant particle rearrangement and similar state of static debris behind rigid barrier and the deformed full-retention flexible barrier, and thus the static loads on both barriers converge.     

泥石流冲击桥墩动力相互作用物理模型试验

在我国西部山区地震、地质活跃带,泥石流灾害对位于泥石流沟道、沟口等位置处的桥墩构成重大威胁。如何量化描述泥石流冲击桥墩的动力过程,是泥石流减灾领域拟要解决的一个重要科学问题。以泥石流灾害威胁成兰铁路沿线桥墩的工程背景为基础,依托大型泥石流模拟系统,进行多组室内大比例泥石流冲击桥墩物理模型试验。研究泥石流流速、流深以及流体特征参数与泥石流冲击压力的相关性。试验结果表明:冲击过程主要受到弗汝德数Fr和雷诺数Re两个无量纲数控制,稀性泥石流冲击压力主要控制参数为Fr,而对于黏性泥石流则同时有Fr和Re的影响;不论是对于峰值冲击力还是冲击功率谱,不同类型泥石流差别显著;在相同重度等条件下,稀性泥石流具有更大的冲击能量;此外,各种类型泥石流通过临界Fr线得到了本质上的区分。研究成果将为桥墩抗泥石流冲击结构设计提供技术支持及科学依据。     

The formation and development of debris flows in large watersheds after the 2008 Wenchuan Earthquake

The Wenchuan earthquake has caused abundance of loose materials supplies for debris flows. Many debris flows have occurred in watersheds in area beyond 20 km², presenting characteristics differing from those in small watersheds. The debris flows yearly frequency decreases exponentially, and the average debris flow magnitude increases linearly with watershed size. The rainfall thresholds for debris flows in large watersheds were expressed as I?=?14.7 D ^?0.79 (2 h?<?D?<?56 h), which is considerably higher than those in small watersheds as I?=?4.4 D ^?0.70 (2 h?<?D?<?37 h). A case study is conducted in Ergou, 39.4 km² in area, to illustrate the formation and development processes of debris flows in large watersheds. A debris flow develops in a large watershed only when the rainfall was high enough to trigger the wide-spread failures and erosions on slope and realize the confluence in the watershed. The debris flow was supplied by the widely distributed failures dominated by rill erosions (14 in 22 sources in this case). The intermittent supplying increased the size and duration of debris flow. While the landslide dam failures provided most amounts for debris flows (57 % of the total amount), and amplified the discharge suddenly. During these processes, the debris flow velocity and density increased as well. The similar processes were observed in other large watersheds, indicating this case is representative.     

Debris flow density determined by grain composition

Density is one of the most important parameters of debris flows. Because observing an active debris flow is very difficult, finding a method to estimate debris flow density is urgently needed for disaster mitigation engineering. This paper proposes an effective empirical equation in terms of grain size distribution (GSD) parameters based on observations in Jiangjia Gully, Yunnan Province, China. We found that the GSD follows P(D) = KD ^-μ exp(? D/Dc), with μ and Dc representing the fine and coarse grains, respectively. In particular, μ is associated with some characteristic porosity of soil in the natural state and increases with increased porosity. Dc characterizes the grain size range of the flow and increases with the grain concentration. Studies show that flow density is related to both parameters in power law. Here, we propose an empirical equation for estimating flow density: ρ = 1.26μ ^-0.132 + 0.049Dc^0.443, which provides not only an estimation of the density for a flow, but also describes the variation in density with the GSD of material composition; this provides important information related to the design of debris flow engineering structures.     

Simulation of interactions among multiple debris flows

Adjacent debris flows may interact in many ways: two or more concurrent debris flows may merge; one debris flow can run out over an existing debris flow fan. Such interactions may cause debris flow properties to change in the mixing process as well as more severe adverse effects than those caused by a single debris flow. This paper aims to investigate the interactions among channelized debris flows originated from adjacent catchments. Both concurrent and successive debris flows are considered. If several debris flows originate from different locations concurrently and merge, the volumetric sediment concentration (i.e., the ratio of the volume of solid material to the total volume of debris flow), C _v, is a good index to capture the mixing process of these debris flows. The change in C _v reflects where mixing occurs and the mixing degree. The debris flow properties (e.g., yield stress and dynamic viscosity) evolve in the mixing process and can be captured by the change in C _v. The debris flow with a larger volume dominates the mixing process, and the properties of the mixed debris flow are more similar to those of the larger debris flow. The inundated areas and runout distances of successive debris flows are smaller than those of concurrent debris flows of the same total volume due to the smaller scales of the individual events and blockage by the earlier debris flows. However, the deposit depth in the interacting part of the debris flow fans of successive debris flows can be much larger than that of concurrent debris flows, leading to more destructive cascading hazards (e.g., the formation of debris barrier lakes). The sequence of successive debris flows not only significantly influences the runout characteristics of the debris flows but also substantially affects the cascading hazards.     

Effect of roughness on water flow through a synthetic single rough fracture

A single fracture is usually idealized theoretically as smooth parallel plates, but the real fractures are rough-walled with points of contact. Though relative roughness is considered in quantifying the flow through a single rough fracture (SRF) previously, additional factors such as the distribution of rough elements and bending degree of streamlines should be considered in order to obtain more accurate results. Semiempirical friction factor (f) and discharge per unit width (q) equations are first deduced taking relative roughness, roughness elements distribution and streamline reattachment length into consideration. A horizontal SRF model was then set up and a series of experiments and simulations were performed. Main conclusions are drawn: Recirculation of streamlines arises in the rough element and the intensity of the recirculation increases with the angle from which the streamlines enter into the rough elements and Reynolds number (Re); streamlines are discontinuously distributed when asperity height is large and nonlinear flow occurs; the nonlinearity of the flow increases with the increasing the asperity height and Re; the critical value of related roughness used to judge whether the influence of roughness on water flow through a SRF can be ignored or not should be much lower than 0.033; the revised f and q equations under laminar flow through a SRF are proved to be better when calculating the f and q values.     

Dynamical evolution properties of debris flows controlled by different mesh-sized flexible net barriers

Because the flexible net barrier is a gradually developed open-type debris-flow counter-measure, there are still uncertainties in its design criterion. By using several small-scale experimental flume model tests, the dynamical evolution properties of debris flows controlled by large and small mesh-sized (equal to D₉₀ and D₅₀, respectively) flexible net barriers are studied, including the debris flow behaviors, segregation, and permeability of sediments, as well as the energy absorption rates and potential overtopping occurring when debris flows impact the small mesh-sized one. Experimental results reveal that (a) two sediment deposition patterns are observed depending on variations in debris flow textures and mesh sizes; (b) the aggregation against flexible net barriers is dominated by flow dynamics; (c) the segregation and permeable functions of the barrier are determined by the mesh size, concentration, and flow dynamics; and (d) the smaller mesh-sized flexible net barrier tends to be more efficient in restraining more turbulent debris flows and can absorb greater rate of kinematic energy, and finally, the great kinematic energy dissipation that occurs when secondary debris flows interact with the post-deposits in front of the small mesh-sized flexible net barrier is believed to cause the failure of overtopping phenomenon. The mesh size is concluded to be the decisive parameter that should be associated with debris flow textures to design the control functions of flexible net barriers.     

Design and performance of a novel multi-function debris flow mitigation system in Wenjia Gully,Sichuan

The post-earthquake debris flows in the Wenjia Gully led to the exposure of the shortcomings in the design of the original conventional debris flow mitigation system. A predicament for the Wenjia mitigation system is a large amount of loose material (est. 50 × 10⁶ m³) that has been deposited in the gully by the co-seismic landslide, providing abundant source material for debris flows under saturation. A novel design solution for the replacement mitigation system was proposed and constructed, and has exhibited excellent performance and resilience in subsequent debris flows. The design was governed by the three-phase philosophy of controlling water, sediment, and erosion. An Early Warning System (EWS) for debris flow that uses real-time field data was developed; it issues alerts based on the probabilistic and empirical correlations between rainfall and debris flows. This two-fold solution reduces energy of the debris flow by combining different mitigation measures while minimizing the impact through event forecasting and rapid public information sharing. Declines in the number and size of debris flows in the gully, with increased corresponding rainfall thresholds and mean rainfall intensity-duration (I-D) thresholds, indicate the high efficacy of the new mitigation system and a lowered debris flow susceptibility. This paper reports the design of the mitigation system and analyzes the characteristics of rainfall and debris flow events that occurred before and after implementation of the system; it evaluates the effectiveness of one of the most advanced debris flow mitigation systems in China.     

Prediction of local scour around complex piers using GEP and M5-Tree

Local scour around piers is one of the main causes of bridge failures. In this study, three robust techniques, artificial neural networks (ANNs), M5-Tree, and Gene Expression Programming (GEP), were employed for prediction of scour depth around complex piers. The clear water condition was chosen for all experimental tests. The results indicated that pier diameter (b _c) and foundation level (Y) are the main parameters for local scour. Furthermore, the minimum scour depth occurs in range of Y/b _c = 1.1~1.3. In next step, to evaluate the mentioned techniques, a wide range of dataset was collected from the present study and literature. The radial base function (RBF) with R ² = 0.945 and RMSE = 0.031 provides better prediction in comparison with conventional equations, M5-Tree (R ² = 0.883, RMSE = 0.292) and the GEP techniques (R ² = 0.811 and RMSE = 0.263). The equations developed by M5-Tree and GEP are more useful for practical purposes and can be easily employed to predict the depth of scour at complex piers.     

Analysis of the triggering conditions and erosion of a runoff-triggered debris flow in Miyun County,Beijing, China

The occurrence of debris flow from channel-bed failure is occasionally noted in small and steeply sloping watersheds where channelized water flow dominates debris flow initiation. On August 12, 2016, a debris flow from channel-bed failures occurred in the Caozhuangzi Watershed of the Longtan Basin, Miyun, Beijing. Rainfall records over 10-min intervals and field investigations including channel morphology measurements were used to study the triggering conditions and erosion process. The results indicated that the occurrence of this event lagged the peak 10-min rainfall interval and that the cumulative rainfall prior to the occurrence time played an important role in its formation. A mean 10-min rainfall intensity–duration expression in the form of I₁₀?=?5.0?×?D^?0.21, where I₁₀ denotes the mean 10-min rainfall intensity and D is the rainfall duration ranging from 10 to 60 h, was proposed. The debris flows have low proportions of grain size fractions <?0.1 mm and higher fractions of grains 0.1–2 mm in size, indicating that the flow had low viscosity and was coarse-grain dominated. Channel morphology analysis revealed that abrupt changes in topography in the study area, including a steep section, a concave stream bank area, and a partial concave stream section were eroded more extensively than other sites. The maximum sediment erosion volume and erosion depth were not proportional to the variation in stream gradient. Consideration of the degree of erosion in the channel at sites with abrupt morphology changes, the maximum sediment erosion volume, and the erosion depth and volume at the initial channel site and downstream region of forest area together showed that the prime factor controlling erosion was entrained sediment volume. This work, thus, provides a case study regarding the triggering conditions of runoff-triggered debris flows and the topographical changes by debris flow erosion.     

Stability and phase relations of nonstoichiometric clinopyroxenes in the join diopside-Ca-Eskola component at high pressures

The stability of nonstoichiometric clinopyroxenes in the Di-CaEsk join was experimentally studied, and phase diagrams were constructed for this join at pressures of 2.0 and 3.0 GPa. It was found that melting in the diopside part of the join occurs at anomalously low temperatures, and nonstoichiometric clinopyroxene coexists with a phase approaching diopside in composition. Phase relations along the Di-CaEsk join can be described and consistently interpreted only assuming that the diopside phase (α-diopside) is thermodynamically stable. The following phase volumes were observed along the solidus of the join at a pressure of 3.0 GPa: Cpx, αDi+Cpx, αDi+Cpx+Qtz, αDi+Cpx+Grt+Qtz, Cpx+Grt+Qtz, Cpx+Grt+Ky+Qtz, Grt+Ky+Qtz. Melting occurs via the eutectic reaction αDi+Cpx+Grt+Qtz=L at a temperature of about 1200°C in the diopside part of the system and via the eutectic reaction Cpx+Grt+Ky+Qtz=L at a temperature of 1400°C in the calcium-rich part of the system. At a pressure of 2.0 GPa, melting occurs at temperatures of 1200–1300°C via the eutectic reaction αDi+Cpx+ An + Qtz=L. The invariant equilibrium (L, An, Cpx, Grt, αDi, Qtz) lies within the pressure range 2.0–3.0 GPa. Nonstoichiometric clinopyroxenes form complex solid solutions, the compositions of which are not strictly confined to the Di-CaEsk join and depend on temperature, pressure, and phase association. Grossular garnets coexist with nonstoichiometric clinopyroxenes and α-diopside.     

Separate and Combined Effects of Estuarine Stress Gradients and Disturbance on Oyster Population Development on Restored Reefs

Disturbance combined with the effects of multiple stress gradients can produce biotic outcomes that are complex and perhaps not predictable based on knowledge of the individual stress variables. We analyzed oyster (Crassostrea virginica) colonization of novel substrate via structural equation modeling (SEM) to test cause-and-effect multivariate models posed a priori as hypotheses. We separately analyzed long-term data on water quality (WQ), canal flow, and rainfall to determine drivers of chlorophyll a for use in the oyster SEM. The best oyster SEM for adult (R ² = 0.74) and small <20-mm (R ² = 0.48) oyster abundances combined WQ stress gradients produced by normal canal flow with disturbance caused by extremely high flow. There was a ?0.26 direct negative effect of increasing salinity during normal canal flow on the small oyster size class possibly reflecting undocumented increases in marine predators and a negative total effect (negative indirect + direct effects) of the salinity gradient on adult oysters. Very low salinity occurring during extreme (disturbance) canal flows produced large negative direct and total effects on small oysters, but no significant total effect for adult oysters. Chlorophyll a (Chl-a) during normal canal flow had negative total effects on small oysters but positive total effects on adult oysters. The effect of max Chl-a on adult oysters was strongly negative during disturbance-level canal flow. Turbidity during normal canal flow had no effect on small or adult oysters. However, during disturbance flows, the maximum turbidity had strong negative effects. Stress and disturbance from freshwater releases impacted oyster recruitment and survival, affecting the colonization and growth of oysters.     

Assessment of dynamic impact force of debris flow in mountain torrent based on characteristics of debris flow

Landslides and debris flows that occur around residential areas are considered, globally, as significant disasters that cause damage to human life and property. With terrain slope defining the flow characteristics of debris flows, flow depth, flow velocity, and impact force vary by time and distance. In particular, when a structure is located in the flow path of debris flows, the flow characteristics of debris flows vary by terrain slope and direction angle. To simulate the flow characteristics of these debris flows, the simulation results obtained by FLO-2D were analyzed with six-stage conditions for the research area. In the analysis, the flow depth, flow velocity, and impact force were estimated on the basis of the outlet of the research area in the presence and absence of structure(s) at certain distances. With this, the variation of the impact force in accordance with the variation of the flow depth of the debris flows was highly similar to the simulation results obtained by FLO-2D, when the correction index (α) of the suggested dynamic impact force equation was 0.3–0.4. There were sections where the estimated value of the impact force was overestimated near the outlet, and it was judged that the fixed values of the terrain factors (width, roughness coefficient, slope, etc.) caused the impact force to be overestimated. However, the correlation analysis showed that the correlation index was above the normal ranges in the suggested dynamic impact force equation for debris flows with the application of the terrain factors.     

Location-based environmental factors contributing to rainfall-triggered debris flows in the Ba river catchment,northwest Viti Levu island,Fiji

Mass movements in tropical Pacific small island developing states (SIDS) can be devastating although studies are relatively few and contributing environmental factors are not often investigated in detail. On 25 January 2012, following 3 days of heavy monsoonal rainfall (c. 550 mm) during a La Niña episode, more than 150 debris flows were triggered in the western part of the Ba river catchment of northwest Viti Levu island, Fiji. Reconnaissance field survey and geographical information system (GIS) analyses using high-resolution satellite imagery were carried out to investigate factors that may have led to the occurrence of the debris flows in the catchment. We evaluated the correlation between the density of mass movements (number of mass movements/km²) and several continuous variables using data measured within the GIS. There was a weak but significant positive correlation between mass movement density and elevation (r = 0.38, p value < 0.01), cyclonic precipitation (r = 0.37, p value < 0.01) and stream density (r = 0.31, p value < 0.01). Ninety-three percent of the mass movements occur within a plantation of Pinus caribaea (Caribbean pine) on slopes oriented mainly to the northeast and east on (trade) windward slopes and may be significant factors for their development. Although forests generally have a stabilizing effect on slopes, the plantation at Ba was a mature stand on its second plantation cycle and is a species that has a shallow rooting system making it more susceptible to failure.     

Permeability of granular soil employing flexible wall permeameter

Understanding the changes in permeability of soil, when soil is subjected to high confining pressure and flow pressure, which may alter the textural and geomechanical characteristics of soil, is of great importance to many geo-engineering activities such as, construction of high-rise buildings near the coast or the water bodies, earthen dams, pavement subgrades, reservoir, and shallow repositories. It is now possible to evaluate the changes in permeability of soil samples under varying conditions of confining pressure and flow pressure using flexible wall permeameter (FWP). In the present study, investigation was carried out on a cylindrical sample of granular soil employing FWP under varied conditions of confining pressure (σ₃)—50–300 kPa, which can simulate the stress conditions equivalent to depth of about 20 m under the earth’s crust, and a flow pressure (f_p)—20–120 kPa, which is mainly present near the small earthen embankment dams, landfill liners, and slurry walls near the soft granular soil with high groundwater table. The obtained results indicate a linear relationship between hydraulic conductivity (k) with effective confining pressure (σ_eff.), k, decreasing linearly with an incremental change in σ_eff.. Further, k increases significantly with an increase in f_p corresponding to each σ_eff., and q increases significantly with increase in the f_p corresponding to each (σ₃). It was also observed that corresponding to the low f_p of 20 kPa, the reduction in k is nonlinear with σ_3. The percentage reduction in k is observed to be 9, 13, and 27% corresponding to σ₃ of 50–100, 100–200, and 200-300 kPa, respectively.     

Physicochemical model for the crystallization of rocks of the calc-alkaline series

The Al-rich region of the CaO-MgO-Al₂O₃-SiO₂ system was experimentally studied at pressures of 1.0–2.8 GPa and temperatures of 1300–1535°C. The slopes of the lines of the monovariant reactions An + Sp = Cpx + Cor + (Ga) and L = Cpx + Ga + Cor + Sp and the compositions of the phases involved in these reactions are determined. The results are utilized in the topological analysis of the aluminous region of the CaO-MgO-Al₂O₃-SiO₂ system. On this basis, the principal structure of the phase diagram is analyzed, and a phase diagram is constructed for the junction region of the quaternary system and the CaO-Al₂O₃-SiO₂ ternary system. A continuous series of the monovariant eutectic: L = Cpx + Opx + Fo + An, L = Cpx + Opx + An + Sp, L = Cpx + (Ga) + An + Sp, L = Cpx + Cor + (Ga) + An, L = An + Ga + Cpx + Ky and L = Ga + Cpx + Ky + Qz is examined within the pressure range from atmospheric to 3.0 GPa and higher. Analogous “telescoped” eutectic series are of fundamental character for interpreting the evolution of magmatic melts. A physicochemical model is suggested for the evolution of magmatic melts that produce rocks of the calc-alkaline series, with this model underlain by the fact that a change in the composition of magmatic melt at a pressure decrease should correspond to the minimum melting temperatures, i.e., to melts in the fundamental series of eutectic reactions. The comparison of our physicochemical model and rocks of the calc-alkaline series shows that the compositions of rocks of the calc-alkaline series are close to the compositions determined for the eutectic equilibria, and the mineralogical composition of xenoliths and megacrysts in volcanic and dike varieties of the rocks are similar to the subsolidus phases of the established fundamental eutectic system.     

Analytical expressions for three-phase generalized relative permeabilities in water- and oil-wet capillary tubes

We analyze three-phase flow of immiscible fluids taking place within an elementary capillary tube with circular cross-section under water- and oil-wet conditions. We account explicitly for momentum transfer between the moving phases, which leads to the phenomenon of viscous coupling, by imposing continuity of velocity and shear stress at fluid-fluid interfaces. The macroscopic flow model which describes the system at the Darcy scale includes three-phase effective relative permeabilities, K _{i j,r} , accounting for the flux of the ith phase due to the presence of the jth phase. These effective parameters strongly depend on phase saturations, fluid viscosities, and wettability of the solid matrix. In the considered flow setting, K _{i j,r} reduce to a set of nine scalar quantities, K _{i j,r} . Our results show that K _{i j,r} of the wetting phase is a function only of the fluid phase own saturation. Otherwise, K _{i j,r} of the non-wetting phase depends on the saturation of all fluids in the system and on oil and water viscosities. Viscous coupling effects (encapsulated in K _{i j,r} with i ≠ j) can be significantly relevant in both water- and oil-wet systems. Wettability conditions influence oil flow at a rate that increases linearly with viscosity ratio between oil and water phases.     

Fracture evolution and energy mechanism of deep-buried carbonaceous slate

In order to study the influence of confining pressure and water content on the mechanical properties, fracture evolution and energy damage mechanism of deep-buried carbonaceous slate, uniaxial and triaxial compression tests were carried out under natural and saturated states and acoustic emission monitored. The deep-buried carbonaceous slate samples were obtained at a depth of 1020 m from the Lanjiayan tunnel in Sichuan province, China, where the maximum in situ stress has been measured at 44.2 MPa. The results suggest that water has a significant softening effect on the strength and deformation characteristics of carbonaceous slate, but the effect decreases with an increase in the confining pressure. When both the confining pressure and water content are increased, the acoustic emission events and dissipated energy gradually increase at the pre-peak and post-peak stages. Thus, the AE evolution type seen in the natural state under low confining pressure usually presents as a main shock-type event, and it changes to a foreshock–main shock–after shock event when saturated and at high confining pressures. Based on the S-shaped energy evolution law, the damage evolution process of carbonaceous slate was analyzed. The damage stress thresholds σ _ea and σ _eb were obtained, which can be considered as the thresholds of the rock entering the energy-hardening and energy-softening stages. Finally, a new brittleness energy index BDE is proposed to describe the influence of confining pressure and water content on the damage mechanism of deep-buried carbonaceous slate.     

High-pressure thermo-elastic properties of beryl (Al<Subscript>4</Subscript>Be<Subscript>6</Subscript>Si<Subscript>12</Subscript>O<Subscript>36</Subscript>) from ab initio calculations,and observations about the source of thermal expansion

Ab initio calculations of thermo-elastic properties of beryl (Al₄Be₆Si₁₂O₃₆) have been carried out at the hybrid HF/DFT level by using the B3LYP and WC1LYP Hamiltonians. Static geometries and vibrational frequencies were calculated at different values of the unit cell volume to get static pressure and mode-γ Grüneisen’s parameters. Zero point and thermal pressures were calculated by following a standard statistical-thermodynamics approach, within the limit of the quasi-harmonic approximation, and added to the static pressure at each volume, to get the total pressure (P) as a function of both temperature (T) and cell volume (V). The resulting P(V, T) curves were fitted by appropriate EoS’, to get bulk modulus (K ₀) and its derivative (K′), at different temperatures. The calculation successfully reproduced the available experimental data concerning compressibility at room temperature (the WC1LYP Hamiltonian provided K ₀ and K′ values of 180.2 Gpa and 4.0, respectively) and the low values observed for the thermal expansion coefficient. A zone-centre soft mode \( P6/mcc \to P\bar{1} \) phase transition was predicted to occur at a pressure of about 14 GPa; the reduction of the frequency of the soft vibrational mode, as the pressure is increased, and the similar behaviour of the majority of the low-frequency modes, provided an explanation of the thermal behaviour of the crystal, which is consistent with the RUM model (Rigid Unit Model; Dove et al. in Miner Mag 59:629–639, 1995), where the negative contribution to thermal expansion is ascribed to a geometric effect connected to the tilting of rigid polyhedra in framework silicates.     

Higher resolution total velocity <Emphasis Type="Italic">Vt</Emphasis> and <Emphasis Type="Italic">Va</Emphasis> finite-volume formulations on cell-centred structured and unstructured grids

Novel cell-centred finite-volume formulations are presented for incompressible and immiscible two-phase flow with both gravity and capillary pressure effects on structured and unstructured grids. The Darcy-flux is approximated by a control-volume distributed multipoint flux approximation (CVD-MPFA) coupled with a higher resolution approximation for convective transport. The CVD-MPFA method is used for Darcy-flux approximation involving pressure, gravity, and capillary pressure flux operators. Two IMPES formulations for coupling the pressure equation with fluid transport are presented. The first is based on the classical total velocity Vt fractional flow (Buckley Leverett) formulation, and the second is based on a more recent Va formulation. The CVD-MPFA method is employed for both Vt and Va formulations. The advantages of both coupled formulations are contrasted. The methods are tested on a range of structured and unstructured quadrilateral and triangular grids. The tests show that the resulting methods are found to be comparable for a number of classical cases, including channel flow problems. However, when gravity is present, flow regimes are identified where the Va formulation becomes locally unstable, in contrast to the total velocity formulation. The test cases also show the advantages of the higher resolution method compared to standard first-order single-point upstream weighting.