Fluid Mass Balance Research Articles

Numerical analysis of geomechanical behavior of fractures and faults in a deformable porous medium

In this study, geomechanical behavior of fractures and faults in rocks as a saturated deformable discrete fracture-porous medium has been evaluated using coupled fluid flow-geomechanics numerical modeling. The purpose of this paper is to observe and evaluate the effects of fractures and faults on the pore pressures during fluid flow through reservoirs. This issue involves solving the equations that have been derived from the Biot consolidation theory such as fluid mass balance equation, Darcy law and momentum balance equations. Govern coupled equations were solved using the standard Galerkin finite element method for continuous porous medium. Elements called "zero-thickness elements” were also used to discretize the fault as a discontinuous part of the porous medium. In compared with the previous and similar methods, the method introduced in this paper, made modifications in either the choice of the element and the method of solving the governing equations. The main advantage of this paper is providing clear precise formulations of the double node zero-thickness element in hydro-mechanical modeling of fractures and faults. Verification of the proposed process and models presented in this paper were done by providing three index problems which their analytical and numerical solutions are available. The results of our model provide a good agreement to these reference solutions which indicates the accuracy of the method presented in this paper.

CFD investigations on the behavior of a solar dryer used for medicinal herbs dehydration under climatic conditions of Constantine, Algeria

Drying is an essential process, which requires significant energy consumption. It is applied in the conservation or transformation of medicinal plants. These biological products are much sensitive to temperature, during the dehydration and they may lose their active components during the process. Therefore, they deserve special care. This work examines the behavior of Myrtus Comminus L during the drying process. The main reason Myrtus Comminus L has been selected is that previous drying studies have never been carried out on this type of plant. Drying curves (kinetics) and sorption isotherms for the Myrtus Comminus L are determined experimentally, in an assisted indirect solar dryer and in a laboratory, respectively.Then an indirect solar dryer located in Constantine, in Algeria and adapted for Myrtus Comminus L, is analyzed using a computational fluid dynamic. Fluid mechanics, mass and energy balance equations are implemented and solved using COMSOL software, considering climatic data for Constantine, the drying kinetics and sorption isotherms. The parametric study focused on the effects of the gap distance between the glass and the absorber, the length of the collector, in addition to the air velocity inside the dryer. Drying air inside the longer collector and at low velocities accumulates more energy, and the best gap distance is 5 cm. The computational fluid dynamics analysis of the whole drying process, combined with experimental work gives a better understanding of the air distribution inside the dryer and leads to avoid the non-uniformity and enhances the design.

Coupled poro‐mechanical tangent formulation applied to sedimentary basin modeling

AbstractSedimentary basin modeling involves large space domains and time periods. Reproducing the compaction processes of the sediment material requires the constitutive model to be formulated in the framework of large irreversible strains, taking both geometric and physical nonlinearities into account. This work presents a tangent formulation for the coupled poro‐mechanical system of equations resulting from the weak form of the momentum and fluid mass balance equations. The proposed workflow is integrated in a thermo‐poro‐mechanical finite element basin simulator. At material level, purely mechanical and chemo‐mechanical deformations are respectively addressed by means of plastic and viscoplastic components in the macroscopic state equations. The accuracy and efficiency of the proposed tangent formulation are assessed through the simulation of a basin geological scenario involving gravitational compaction and tectonic shortening. Both drained and undrained behaviors of the basin rocks are simulated. The procedure has significantly improved the convergence rate and reduced the computational cost with comparison to the original formulation based on the standard poro‐elastic coefficients. In order to illustrate the potential of the basin simulator to deal with real case applications, a well model from the Neuquén basin in Argentina is also presented. The results are validated against available numerical and field data for porosity, pore‐pressure and temperature.

An Open-Source Code for Fluid Flow Simulations in Unconventional Fractured Reservoirs

In this article, an open-source code for the simulation of fluid flow, including adsorption, transport, and indirect hydromechanical coupling in unconventional fractured reservoirs is described. The code leverages cutting-edge numerical modeling capabilities like automatic differentiation, stochastic fracture modeling, multicontinuum modeling, and discrete fracture models. In the fluid mass balance equation, specific physical mechanisms, unique to organic-rich source rocks, are included, like an adsorption isotherm, a dynamic permeability-correction function, and an Embedded Discrete Fracture Model (EDFM) with fracture-to-well connectivity. The code is validated against an industrial simulator and applied for a study of the performance of the Barnett shale reservoir, where adsorption, gas slippage, diffusion, indirect hydromechanical coupling, and propped fractures are considered. It is the first open-source code available to facilitate the modeling and production optimization of fractured shale-gas reservoirs. The modular design also facilitates rapid prototyping and demonstration of new models. This article also contains a quantitative analysis of the accuracy and limitations of EDFM for gas production simulation in unconventional fractured reservoirs.

Algebraically stabilized Lagrange multiplier method for frictional contact mechanics with hydraulically active fractures

Accurate numerical simulation of coupled fracture/fault deformation and fluid flow is crucial to the performance and safety assessment of many subsurface systems. In this work, we consider the discretization and enforcement of contact conditions at such surfaces. The bulk rock deformation is simulated using low-order continuous finite elements, while frictional contact conditions are imposed by means of a Lagrange multiplier method. We employ a cell-centered finite-volume scheme to solve the fracture fluid mass balance equation. From a modeling perspective, a convenient choice is to use a single grid for both mechanical and flow processes, with piecewise-constant interpolation of Lagrange multipliers, i.e., contact tractions and fluid pressure. Unfortunately, this combination of displacement and multiplier variables is not uniformly inf–sup stable, and therefore requires a stabilization technique. Starting from a macroelement analysis, we develop two algebraic stabilization approaches and compare them in terms of robustness and convergence rate. The proposed approaches are validated against challenging analytical two- and three-dimensional benchmarks to demonstrate accuracy and robustness. These benchmarks include both pure contact mechanics problems and well as problems with tightly-coupled fracture flow.

Performance analysis of desalination system working on humidification-dehumidification coupled with solar assisted air heater and wind tower: Closed and open water cycle

The world which is hot and arid strives for fresh water and thermal comfort, thus there are two basic requirements. In this work, performance of wind tower in combination with solar air heater assisted humidification-dehumidification desalination system working with closed and open air water cycle is analysed numerically for cooling effect and fresh water production. The behaviour of system has been studied by different air flow rate in tower, temperature drop due to energy stored in wall and clay conduits, temperature drop due to evaporative cooling, humidity of moist air available for cooling, amount of potable water produced in dehumidifier and its variation with design and operating variables.A mathematical model based on fluid flow, heat transfer, energy and mass balance in components of system has been developed. Study reveals that comfort condition is achieved mainly by evaporative cooling in wind tower. Optimized column height is found as 8–9 m suitable to design the wind tower with 65% decrease in air velocity, 31% decrease in air temperatures and change in relative humidity up to 90%, however 42% reduction in water requirement in humidifier is also achieved at 5 m height with a cooling effect of 3.5 kW. Air mass flow rate of 0.032–0.035 kg/sec in solar air heater found suitable with maximum productivity of 5 and 4.2 kg/day for closed and open water cycle respectively. Water mass flow rate of 0.035–0.038 kg/sec in dehumidifier has been found suitable for 200% increase in productivity.

Coupling of meshfree peridynamics with the Finite Volume Method for poroelastic problems

Peridynamics is a non-local theory of continuum mechanics that has been developed primarily for understanding material failure due to different mechanisms, including fluid-driven crack propagation during hydraulic fracturing of subsurface reservoirs. Because of its non-local nature, this theory is computationally expensive. To improve its performance, a scheme was recently proposed for coupling Peridynamics (PD) with less expensive Finite Element Method (FEM) for static equilibrium problems. This scheme has been adapted in the current paper to couple a PD-based poroelastic model with the Finite Volume Method (FVM) for simulating problems in porous media. Coupling is implemented by dividing the computational domain into two subdomains, one of which is discretized and solved with PD and the other with FVM. The formulation is developed for porous flow involving fluid mass balance and is extended for poroelastic problems to include rock momentum balance. The coupled model is verified against the analytical solutions to classical problems. No spurious behavior is observed near the PD-FVM interface region. Improvements in computational performance over the pure PD model are demonstrated. Moreover, due to differences in the sparsity patterns and the magnitudes of PD and FVM transmissibility/Jacobian terms, it is shown that appending the PD equations after all the FV equations in the global matrix has additional computational benefits.

A semi-continuum model of saturation overshoot in one dimensional unsaturated porous media flow

A semi-continuum model for fluid flow in saturated-unsaturated porous medium in one spatial dimension is presented. The model is based on well-established physics, measurable parameters and material characteristics. The porous material is characterized by porosity, intrinsic permeability, main wetting and draining branches of the retention curve, and the saturation dependence of the relative permeability. The fluid is characterized by its density and dynamic viscosity. The only physics involved is the mass balance of fluid in porous media together with the Darcy-Buckingham Law for fluid flow in unsaturated porous media. The model is a cellular automaton based on the Macro Modified Invasion Percolation concept of dividing the porous medium into blocks which are not infinitesimal and are assumed to retain the characteristics of a porous medium. The cellular automaton repeats three successive rules: saturation update in each block, pressure update in each block, and flux update between neighboring blocks. The model tracks the evolution of the relative saturation, the fluid capillary pressure, and the fluid flux. The model is shown to reproduce qualitatively and quantitatively all features of one dimensional saturation overshoot behavior reported in the literature.

An implicit joint-continuum model for the hydro-mechanical analysis of fractured rock masses

Fractures control the hydro-mechanical behavior of rock masses. Explicit numerical analyses require detailed information on fracture properties, spacing, and orientation. This paper advances an implicit joint-continuum model for the coupled hydro-mechanical analysis of regularly spaced-persistent fractured rock masses. The stiffness tensor combines the compliance of each fracture set and the intact rock; similarly, the permeability tensor adds the fluid transport through fractures and the matrix. The fully coupled hydro-mechanical analysis incorporates the rock mass stiffness and permeability tensors, and satisfies force equilibrium and macroscopic fluid mass balance. We implement the implicit joint-continuum model within a finite element framework and verify the numerical simulator against closed-form solutions for simple boundary conditions. The application of the code to the hydraulic stimulation of a fractured rock mass shows the effect of stress anisotropy and fracture orientation on the development of open-mode discontinuities (i.e., hydraulic fracture) and hydro-shearing. The implicit joint-continuum model can be readily extended to more complex coupled processes, including thermal and chemical phenomena.

Extension and evaluation of a macroscopic model for syngas-fueled chemical looping combustion

Syngas-fueled Chemical Looping Combustion (syngas-CLC) which can be integrated with ex-situ gasification of coal has advantages over the direct use of coal in CLC: (i) no requirement of carbon stripper, (ii) no interaction of oxygen carrier with coal ash, (iii) no loss of oxygen carrier with the draining stream of ash. Few works on simulation of syngas-CLC were performed, although experimental investigations were extensively carried out. In this work, a macroscopic fuel reactor model is extended to a lab-scale 500 Wth reactor. The model based on fluid dynamics, mass balance and reduction kinetics is solved with MATLAB® codes and validated against experiments. Influences of various operation parameters are evaluated to study the flexibility of this model. It is shown that the model can give satisfactory predictions for fuel reactor of a syngas-CLC system, independent on the operation conditions. Variations of syngas composition, temperature, solids circulation and oxygen carrier inventory show different effects on flue gas composition and combustion efficiency. After thorough simulation, a region for a combustion efficiency of ηC = 99.9% is proposed, with which the optimized conditions for the 500 Wth reactor are established. An oxygen carrier inventory as low as 50 kg/MWth can assure the complete syngas combustion.

Large deformation poromechanics with local mass conservation: An enriched Galerkin finite element framework

SummaryNumerical modeling of large deformations in fluid‐infiltrated porous media must accurately describe not only geometrically nonlinear kinematics but also fluid flow in heterogeneously deforming pore structure. Accurate simulation of fluid flow in heterogeneous porous media often requires a numerical method that features the local (elementwise) conservation property. Here, we introduce a new finite element framework for locally mass conservative solution of coupled poromechanical problems at large strains. At the core of our approach is the enriched Galerkin discretization of the fluid mass balance equation, whereby elementwise constant functions are augmented to the standard continuous Galerkin discretization. The resulting numerical method provides local mass conservation by construction with a usually affordable cost added to the continuous Galerkin counterpart. Two equivalent formulations are developed using total Lagrangian and updated Lagrangian approaches. The local mass conservation property of the proposed method is verified through numerical examples involving saturated and unsaturated flow in porous media at finite strains. The numerical examples also demonstrate that local mass conservation can be a critical element of accurate simulation of both fluid flow and large deformation in porous media.

A stabilized element-based finite volume method for poroelastic problems

The coupled equations of Biot's poroelasticity, consisting of stress equilibrium and fluid mass balance in deforming porous media, are numerically solved. The governing partial differential equations are discretized by an Element-based Finite Volume Method (EbFVM), which can be used in three dimensional unstructured grids composed of elements of different types. One of the difficulties for solving these equations is the numerical pressure instability that can arise when undrained conditions take place. In this paper, a stabilization technique is developed to overcome this problem by employing an interpolation function for displacements that considers also the pressure gradient effect. The interpolation function is obtained by the so-called Physical Influence Scheme (PIS), typically employed for solving incompressible fluid flows governed by the Navier–Stokes equations. Classical problems with analytical solutions, as well as three-dimensional realistic cases are addressed. The results reveal that the proposed stabilization technique is able to eliminate the spurious pressure instabilities arising under undrained conditions at a low computational cost.

Numerical simulation of solid deformation driven by creeping flow using an immersed finite element method

An immersed finite element method for solid–fluid interaction is presented with application focus on highly deformable elastic bodies in a Stokes flow environment. The method is based on a global balance equation which combines the solid and fluid momentum balances, the fluid mass balance and, in weak form, the interface conditions. By means of an Updated Lagrangian description for finite elasticity, only one analysis mesh is used, where the solid particles are backtracked in order to preserve the deformation history. The method results in a full coupling of the solid-fluid system which is solved by an exact Newton method. The location of the material interface is captured by a signed distance function and updated according to the computed displacement increments and the help of an explicit surface parameterisation; no body-fitted volume meshes are needed. Special emphasis is placed on the accurate integration of finite elements traversed by the interface and the related numerical stability of the shape function basis. A number of applications for compressible Neo-Hookean solids subject to creeping flow are presented, motivated by microfluidic experimentation in mechanobiology.

Numerical modelling of waterhammer pressure pulse propagation in sand reservoirs

Abstract This paper presents a numerical model with a new approach for analyzing the propagation of pressure waves in porous media and investigates the dynamic response of sand in relation to the attributes of pore pressure pulses. There are various instances in which dynamic phenomena can have a significant impact on porous media in a reservoir. One notable example is the possible influence of waterhammer pressure pulsing on sand fluidization around injection wells in oil reservoirs following a hard wellbore shut-in, which can result in massive sand production. In some extreme cases, this phenomenon can even result in the loss of the wellbore. Nevertheless, the pore pressure wave propagation in porous media has often been neglected in modelling likely due to mathematical complexity. The proposed model solves the momentum balance of fluid and solid coupled with the fluid mass balance equation in the prediction of dynamic fluid flow and mechanical deformation in porous media. The model is a two-dimensional, elasto-plastic, axisymmetric, single-phase and sequentially coupled model. The numerical model was validated against experimental data for a step wave in a shock tube and good agreement between model calculations and measured data has been obtained. Two distinct waves have been observed as a result of a shock pore pressure wave. The first one is an undrained wave where fluid and solid travel at the same speed. The other one is a wave which is often damped far from the source due to the friction between fluid and solid as they no longer travel together. It is found that tortuosity plays an important role on the amplitude of the waves. The results were then compared to the predictions by Biot's theory for waves through porous media. Biot's theory is shown to be inaccurate in predicting the transient dynamic behaviour, but it is sufficient in capturing the overall trends. Finally, the model is used to predict waterhammer response in near wellbore reservoir.

Groundwater balance estimation in karst by using simple conceptual rainfall–runoff model

The objective of this work is the study of Opacac karst spring which geographically lies in Dalmatia (Croatia). Numerous studies have been carried out in karst aiming the investigation of groundwater regime. The karst spring hydrograph can reflect the groundwater regime and consequently the analysis is based on them. A simple conceptual rainfall–runoff model is used for the estimation of groundwater balance components including the influences of time-invariant catchment boundaries (Jukic and Denic-Jukic 2009). The proposed parameter estimation procedure merges the soil-moisture balance and the groundwater balance approaches to obtain the complete groundwater budget. The effective rainfall is calculated by using mathematical model based on soil-moisture balance equations, i.e. Palmer’s fluid mass balance method. The parameters of model of effective rainfall are determined by using simple conceptual rainfall–runoff model consisting of two linear reservoirs representing the fast and slow flow component of the recession. The weight coefficient between the fast and slow component is determined by using base flow index analysis of hydrograph. Recession coefficient of the slow flow component and the weight coefficient are determined from hydrograph analysis. Available data from nearby meteorological station includes daily average discharge, the amount of precipitation, the average temperature and the humidity from 1995 to 2010. The average catchment area is also estimated with the average yearly runoff deficit using Turc’s method and compared with the values obtained from the application of the rainfall–runoff model. Nash–Sutcliffe model efficiency, root mean square error and Pearson’s correlation coefficient for simulated hydrograph are applied to assess the predictive power of model. Calculated groundwater balance shows that the Opacac spring aquifer contains a significant storage capacity. The application of series of linear reservoirs is a classical and common technique, but the proposed simple approach enables the estimation of the components of groundwater balance in karst areas.

Mine-Water Flow between Contiguous Flooded Underground Coal Mines with Hydraulically Compromised Barriers

Groundwater flow entering closed contiguous underground coal mines may be strongly influenced by leakage across inter-mine barriers. This study examines a complex of multiple closed and flooded mines that developed into a nearly steady-state groundwater flow system within 10 to 50 years after closure. Field water-level observations, mine geometry, barrier hydraulic conductivity, recharge rates, and late-stage storage gains were parameterized to match known pumping rates and develop a fluid mass balance. Vertical infiltration (recharge and leakage) estimates were developed using a depth-dependent model based on the assumption that most vertical infiltration is focused in areas with <75 m of overburden. A MODFLOW simulation of the nearly steady-state flow conditions was calibrated to hydraulic heads in observation wells and to known pumping rates by varying barrier hydraulic conductivity. The calibrated model suggests significant head-driven leakage between adjacent mines, both horizontally through coal barriers and vertically through inter-burden into a shallower mine in an overlying seam. Calibrated barrier hydraulic conductivities were significantly greater than literature values for other mines at similar depths in the region. This suggests that some barriers may be hydraulically compromised by un-mapped entries, horizontal boreholes, or similar features that act as drains between mines. These model results suggest that post-mining inter-annual equilibrium conditions are amenable to quantitative description using mine maps, sparse observation-well data, accurately estimated pumping rates, and depth-dependent vertical infiltration estimates. Results are applicable to planning for post-flooding water-control schemes, although hydraulic testing may be required to verify model results.

Equilibrium modeling of ion adsorption based on Poisson–Boltzmann equation

Abstract A phenomenological model representing ion adsorption equilibrium is proposed. It is an extension of the model used to represent surface adsorption, and it is intended to interpret bulk adsorption data. The fundamental equations of the model were established for fluid and solid phases mass balance and for the spatial distribution of ions across the sample. The electric field distribution in the system, modeled on Poisson's equation, and the chemical potential behavior were determined as a function of fluid phase equilibrium concentration. The concentration of ions as a function of electric potential is given by Boltzmann distribution. The proposed model was successfully tested to represent equilibrium adsorption of metallic ions in zeolite, showing results similar to conventional isotherms.

Simulations of ecosystem hydrological processes using a unified multi-scale model

This paper presents a unified multi-scale model (UMSM) that we developed to simulate hydrological processes in an ecosystem containing both surface water and groundwater. The UMSM approach modifies the Navier–Stokes equation by adding a Darcy force term to formulate a single set of equations to describe fluid momentum and uses a generalized equation to describe fluid mass balance. The advantage of the approach is that the single set of the equations can describe hydrological processes in both surface water and groundwater where different models are traditionally required to simulate fluid flow. This feature of the UMSM significantly facilitates modelling of hydrological processes in ecosystems, especially at locations where soil/sediment may be frequently inundated and drained in response to precipitation, regional hydrological and climate changes. In this paper, the UMSM was benchmarked using WASH123D, a model commonly used for simulating coupled surface water and groundwater flow. Disney Wilderness Preserve (DWP) site at the Kissimmee, Florida, where active field monitoring and measurements are ongoing to understand hydrological and biogeochemical processes, was then used as an example to illustrate the UMSM modelling approach. The simulations results demonstrated that the DWP site is subject to the frequent changes in soil saturation, the geometry and volume of surface water bodies, and groundwater and surface water exchange. All the hydrological phenomena in surface water and groundwater components including inundation and draining, river bank flow, groundwater table change, soil saturation, hydrological interactions between groundwater and surface water, and the migration of surface water and groundwater interfaces can be simultaneously simulated using the UMSM. Overall, the UMSM offers a cross-scale approach that is particularly suitable to simulate coupled surface and ground water flow in ecosystems with strong surface water and groundwater interactions.

A numerical study of dynamic capillary pressure effect for supercritical carbon dioxide‐water flow in porous domain

Numerical simulations for core‐scale capillary pressure (Pc)‐saturation (S) relationships have been conducted for a supercritical carbon dioxide‐water system at temperatures between 35°C and 65°C at a domain pressure of 15 MPa as typically expected during geological sequestration of CO2. As the Pc‐S relationships depend on both S and time derivative of saturation ( ) yielding what is known as the “dynamic capillary pressure effect” or simply “dynamic effect,” this work specifically attempts to determine the significance of these effects for supercritical carbon dioxide‐water flow in terms of a coefficient, namely dynamic coefficient (τ). The coefficient establishes the speed at which capillary equilibrium for supercritical CO2 (scCO2)‐water flow is reached. The simulations in this work involved the solution of the extended version of Darcy's law which represents the momentum balance for individual fluid phases in the system, the continuity equation for fluid mass balance, as well as additional correlations for determining the capillary pressure as a function of saturation, and the physical properties of the fluids as a function of temperature. The simulations were carried out for three‐dimensional cylindrical porous domains measuring 10 cm in diameter and 12 cm in height. τ was determined by measuring the slope of a best‐fit straight line plotted between (1) the differences in dynamic and equilibrium capillary pressures against (2) the time derivative of saturation (dS/dt), both at the same saturation value. The results show rising trends for τ as the saturation values reduce, with noticeable impacts of temperature at 50% saturation of aqueous phase. This means that the time to attain capillary equilibrium for the CO2‐water system increases as the saturation decreases. From a practical point of view, it implies that the time to reach capillary equilibrium during geological sequestration of CO2 is an important factor and should be accounted for while simulating the flow processes, for example, to determine the CO2 storage capacity of a geological aquifer. In this task, one would require both the fundamental understanding of the dynamic capillary pressure effects for scCO2‐water flow as well as τ values. These issues are addressed in this article. © 2014 American Institute of Chemical Engineers AIChE J 60: 4266–4278, 2014

Comprehensive Theory of the Deans’ Switch As a Variable Flow Splitter: Fluid Mechanics, Mass Balance, and System Behavior

The Deans' switch is an effluent switching device based on controlling flows of carrier gas instead of mechanical valves in the analytical flow path. This technique offers high inertness and a wear-free operation. Recently new monolithic microfluidic devices have become available. In these devices the whole flow system is integrated into a small metal device with low thermal mass and leak-tight connections. In contrast to a mechanical valve-based system, a flow-controlled system is more difficult to calculate. Usually the Deans' switch is used to switch one inlet to one of two outlets, by means of two auxiliary flows. However, the Deans' switch can also be used to deliver the GC effluent with a specific split ratio to both outlets. The calculation of the split ratio of the inlet flow to the two outlets is challenging because of the asymmetries of the flow resistances. This is especially the case, if one of the outlets is a vacuum device, such as a mass spectrometer, and the other an atmospheric detector, e.g. a flame ionization detector (FID) or an olfactory (sniffing) port. The capillary flows in gas chromatography are calculated with the Hagen-Poiseuille equation of the laminar, isothermal and compressible flow in circular tubes. The flow resistances in the new microfluidic devices have to be calculated with the corresponding equation for rectangular cross-section microchannels. The Hagen-Poiseuille equation underestimates the flow to a vacuum outlet. A corrected equation originating from the theory of rarefied flows is presented. The calculation of pressures and flows of a Deans' switch based chromatographic system is done by the solution of mass balances. A specific challenge is the consideration of the antidiffusion resistor between the two auxiliary gas lines of the Deans' switch. A full solution for the calculation of the Deans' switch including this restrictor is presented. Results from validation measurements are in good accordance with the developed theories. A spreadsheet-based flow calculator is part of the Supporting Information.