#### Transcript Introduction to Groundwater Modelling

Introduction to Groundwater Modelling C. P. Kumar Scientist ‘F’ National Institute of Hydrology Roorkee – 247667 (Uttaranchal) India Email: [email protected] Webpage: http://www.angelfire.com/nh/cpkumar/ Presentation Outline Groundwater in Hydrologic Cycle Why Groundwater Modelling is needed? Mathematical Models Modelling Protocol Model Design Calibration and Validation Groundwater Flow Models Groundwater Modelling Resources Groundwater in Hydrologic Cycle Types of Terrestrial Water Surface Water Soil Moisture Ground water Pores Full of Combination of Air and Water Unsaturated Zone / Zone of Aeration / Vadose (Soil Water) Zone of Saturation (Ground water) Pores Full Completely with Water Groundwater Important source of clean water More abundant than SW Baseflow Linked to SW systems Sustains flows in streams Groundwater Concerns? pollution groundwater mining subsidence Problems with groundwater Groundwater overdraft / mining / subsidence Waterlogging Seawater intrusion Groundwater pollution Why Groundwater Modelling is needed? Groundwater • An important component of water resource systems. • Extracted from aquifers through pumping wells and supplied for domestic use, industry and agriculture. • With increased withdrawal of groundwater, the quality of groundwater has been continuously deteriorating. • Water can be injected into aquifers for storage and/or quality control purposes. Management of a groundwater system, means making such decisions as: • The total volume that may be withdrawn annually from the aquifer. • The location of pumping and artificial recharge wells, and their rates. • Decisions related to groundwater quality. Groundwater contamination by: Hazardous industrial wastes Leachate from landfills Agricultural activities such as the use of fertilizers and pesticides MANAGEMENT means making decisions to achieve goals without violating specified constraints. Good management requires information on the response of the managed system to the proposed activities. This information enables the decision-maker, to compare alternative actions and to ensure that constraints are not violated. Any planning of mitigation or control measures, once contamination has been detected in the saturated or unsaturated zones, requires the prediction of the path and the fate of the contaminants, in response to the planned activities. Any monitoring or observation network must be based on the anticipated behavior of the system. A tool is needed that will provide this information. The tool for understanding the system and its behavior and for predicting this response is the model. Usually, the model takes the form of a set of mathematical equations, involving one or more partial differential equations. We refer to such model as a mathematical model. The preferred method of solution of the mathematical model of a given problem is the analytical solution. The advantage of the analytical solution is that the same solution can be applied to various numerical values of model coefficients and parameters. Unfortunately, for most practical problems, because of the heterogeneity of the considered domain, the irregular shape of its boundaries, and the non-analytic form of the various functions, solving the mathematical models analytically is not possible. Instead, we transform the mathematical model into a numerical one, solving it by means of computer programs. Prior to determining the management scheme for any aquifer: We should have a CALIBRATED MODEL of the aquifer, especially, we should know the aquifer’s natural replenishment (from precipitation and through aquifer boundaries). The model will provide the response of the aquifer (water levels, concentrations, etc.) to the implementation of any management alternative. We should have a POLICY that dictates management objectives and constraints. Obviously, we also need information about the water demand (quantity and quality, current and future), interaction with other parts of the water resources system, economic information, sources of pollution, effect of changes on the environment---springs, rivers,... GROUND WATER MODELING WHY MODEL? •To make predictions about a ground-water system’s response to a stress •To understand the system •To design field studies •Use as a thinking tool Use of Groundwater models • • Can be used for three general purposes: To predict or forecast expected artificial or natural changes in the system. Predictive is more applied to deterministic models since it carries higher degree of certainty, while forecasting is used with probabilistic (stochastic) models. Use of Groundwater models • • To describe the system in order to analyse various assumptions To generate a hypothetical system that will be used to study principles of groundwater flow associated with various general or specific problems. ALL GROUND-WATER HYDROLOGY WORK IS MODELING A Model is a representation of a system. Modeling begins when one formulates a concept of a hydrologic system, continues with application of, for example, Darcy's Law to the problem, and may culminate in a complex numerical simulation. Ground Water Flow Modelling A Powerful Tool for furthering our understanding of hydrogeological systems Importance of understanding ground water flow models Construct accurate representations of hydrogeological systems Understand the interrelationships between elements of systems Efficiently develop a sound mathematical representation Make reasonable assumptions and simplifications Understand the limitations of the mathematical representation Understand the limitations of the interpretation of the results Introduction to Ground Water Flow Modelling Predicting heads (and flows) and Approximating parameters h(x,y,z,t)? Solutions to the flow equations Most ground water flow models are solutions of some form of the ground water flow equation The partial differential equation needs to be solved to calculate head as a function of position and time, i.e., h=f(x,y,z,t) “e.g., unidirectional, steady-state flow within a confined aquifer Darcy’s Law dh q dx K x q K ho x Integrated h h0 dh q K 0 x 0 dx h h0 qx K x h(x) x x h( x ) h0 qx K Groundwater Modeling The only effective way to test effects of groundwater management strategies Takes time, money to make model Conceptual model Steady state model Transient model The model is only as good as its calibration Processes we might want to model • Groundwater flow calculate both heads and flow • Solute transport – requires information on flow (velocities) calculate concentrations MODELING PROCESS ALL IMPORTANT MECHANISMS & PROCESSES MUST BE INCLUDED IN THE MODEL, OR RESULTS WILL BE INVALID. TYPES OF MODELS CONCEPTUAL MODEL QUALITATIVE DESCRIPTION OF SYSTEM "a cartoon of the system in your mind" MATHEMATICAL MODEL MATHEMATICAL DESCRIPTION OF SYSTEM SIMPLE - ANALYTICAL (provides a continuous solution over the model domain) COMPLEX - NUMERICAL (provides a discrete solution - i.e. values are calculated at only a few points) ANALOG MODEL e.g. ELECTRICAL CURRENT FLOW through a circuit board with resistors to represent hydraulic conductivity and capacitors to represent storage coefficient PHYSICAL MODEL e.g. SAND TANK which poses scaling problems Mathematical Models Mathematical model: simulates ground-water flow and/or solute fate and transport indirectly by means of a set of governing equations thought to represent the physical processes that occur in the system. (Anderson and Woessner, 1992) Components of a Mathematical Model • Governing Equation (Darcy’s law + water balance equation) with head (h) as the dependent variable • Boundary Conditions • Initial conditions (for transient problems) Derivation of the Governing Equation R x y Q q z x y 1. Consider flux (q) through REV 2. OUT – IN = - Storage 3. Combine with: q = -K grad h Law of Mass Balance + Darcy’s Law = Governing Equation for Groundwater Flow --------------------------------------------------------------div q = - Ss (h t) q = - K grad h (Law of Mass Balance) (Darcy’s Law) div (K grad h) = Ss (h t) (Ss = S / z) General governing equation for steady-state, heterogeneous, anisotropic conditions, without a source/sink term h h h ( Kx ) ( Ky ) ( Kz ) 0 x x y y z z with a source/sink term h h h ( Kx ) ( Ky ) ( Kz ) R * x x y y z z General governing equation for transient, heterogeneous, and anisotropic conditions h h h h ( Kx ) ( Ky ) ( K z ) Ss R* x x y y z z t Specific Storage Ss = V / (x y z h) h h b Unconfined aquifer Specific yield S=V/Ah S = Ss b Confined aquifer Storativity Figures taken from Hornberger et al. (1998) General 3D equation h h h h ( Kx ) ( Ky ) ( K z ) Ss R* x x y y z z t 2D confined: 2D unconfined: h h h (Tx ) (Ty ) S R x x y y t h h h ( hKx ) ( hKy ) S R x x y y t Storage coefficient (S) is either storativity or specific yield. S = Ss b & T = K b Types of Solutions of Mathematical Models • Analytical Solutions: h= f(x,y,z,t) (example: Theis equation) • Numerical Solutions Finite difference methods Finite element methods • Analytic Element Methods (AEM) Limitations of Analytical Models The flexibility of analytical modeling is limited due to simplifying assumptions: Homogeneity, Isotropy, simple geometry, simple initial conditions… Geology is inherently complex: Heterogeneous, anisotropic, complex geometry, complex conditions… This complexity calls for a more powerful solution to the flow equation Numerical modeling Numerical Methods All numerical methods involve representing the flow domain by a limited number of discrete points called nodes. A set of equations are then derived to relate the nodal values of the dependent variable such that they satisfy the governing PDE, either approximately or exactly. • Numerical Solutions Discrete solution of head at selected nodal points. Involves numerical solution of a set of algebraic equations. Finite difference models (e.g., MODFLOW) Finite element models (e.g., SUTRA) Finite Difference Modelling 3-D Finite Difference Models Requires vertical discretization (or layering) of model K1 K2 K3 K4 Finite difference models may be solved using: • a computer program (e.g., a FORTRAN program) • a spreadsheet (e.g., EXCEL) Finite Elements: basis functions, variational principle, Galerkin’s method, weighted residuals • Nodes plus elements; elements defined by nodes • Properties (K, S) assigned to elements • Nodes located on flux boundaries • Able to simulate point sources/sinks at nodes • Flexibility in grid design: elements shaped to boundaries elements fitted to capture detail • Easier to accommodate anisotropy that occurs at an angle to the coordinate axis Hybrid Analytic Element Method (AEM) Involves superposition of analytic solutions. Heads are calculated in continuous space using a computer to do the mathematics involved in superposition. The AE Method was introduced by Otto Strack. A general purpose code, GFLOW, was developed by Strack’s student Henk Haitjema, who also wrote a textbook on the AE Method: Analytic Element Modeling of Groundwater Flow, Academic Press, 1995. Currently the method is limited to steady-state, two-dimensional, horizontal flow. Modelling Protocol What is a “model”? Any “device” that represents approximation to field system Physical Models Mathematical Models Analytical Numerical Modelling Protocol Establish the Purpose of the Model Develop Conceptual Model of the System Select Governing Equations and Computer Code Model Design Calibration Calibration Sensitivity Analysis Model Verification Prediction Predictive Sensitivity Analysis Presentation of Modeling Design and Results Post Audit Model Redesign Purpose - What questions do you want the model to answer? Prediction; System Interpretation; Generic Modeling What do you want to learn from the model? Is a modeling exercise the best way to answer the question? Historical data? Can an analytical model provide the answer? System Interpretation: Inverse Modeling: Sensitivity Analysis Generic: Used in a hypothetical sense, not necessarily for a real site Model “Overkill”? Is the vast labor of characterizing the system, combined with the vast labor of analyzing it, disproportionate to the benefits that follow? ETHICS There may be a cheaper, more effective approach Warn of limitations Conceptual Model “Everything should be made as simple as possible, but not simpler.” Albert Einstein Pictorial representation of the groundwater flow system Will set the dimensions of the model and the design of the grid “Parsimony”….conceptual model has been simplified as much as possible yet retains enough complexity so that it adequately reproduces system behavior. Select Computer Code Select Computer Model Code Verification Comparison to Analytical Solutions; Other Numerical Models Model Design Design of Grid, selecting time steps, boundary and initial conditions, parameter data set Steady/Unsteady..1, 2, or 3-D; …Heterogeneous/Isotropic…..Instantaneous/Continuous Calibration Show that Model can reproduce fieldmeasured heads and flow (concentrations if contaminant transport) Results in parameter data set that best represents field-measured conditions. Calibration Sensitivity Analysis Uncertainty in Input Conditions Determine Effect of Uncertainty on Calibrated Model Model Verification Use Model to Reproduce a Second Set of Field Data Prediction Desired Set of Conditions Sensitivity Analysis Effect of uncertainty in parameter values and future stresses on the predicted solution Presentation of Modelling Design and Results Effective Communication of Modeling Effort Graphs, Tables, Text etc. Postaudit New field data collected to determine if prediction was correct Site-specific data needed to validate model for specific site application Model Redesign Include new insights into system behavior NUMERICAL MODELING DISCRETIZE Write equations of GW Flow between each node Darcy's Law Conservation of Mass Define Material Properties Boundary Conditions Initial Conditions Stresses At each node either H or Q is known, the other is unknown n equations & n unknowns solve simultaneously with matrix algebra Result H at each known Q node Q at each known H node Calibrate Steady State Transient Validate Sensitivity Predictions Similar Process for Transport Modeling only Concentration and Flux is unknown NUMERICAL MODELING Model Design MODELs NEED Geometry Material Properties (K, S, T, Φe, R, etc.) Boundary Conditions (Head, Flux, Concentration etc.) Stress - changing boundary condition Model Design • • • • • • • • • Conceptual Model Selection of Computer Code Model Geometry Grid Boundary array Model Parameters Boundary Conditions Initial Conditions Stresses Concept Development • Developing a conceptual model is the initial and most important part of every modelling effort. It requires thorough understanding of hydrogeology, hydrology and dynamics of groundwater flow. Conceptual Model A descriptive representation of a groundwater system that incorporates an interpretation of the geological & hydrological conditions. Generally includes information about the water budget. May include information on water chemistry. Selection of Computer Code • Which method will be used depends largely on the type of problem and the knowledge of the model design. • Flow, solute, heat, density dependent etc. • 1D, 2D, 3D Model Geometry • Model geometry defines the size and the shape of the model. It consists of model boundaries, both external and internal, and model grid. Boundaries • Physical boundaries are well defined geologic and hydrologic features that permanently influence the pattern of groundwater flow (faults, geologic units, contact with surface water etc.) Boundaries • Hydraulic boundaries are derived from the groundwater flow net and therefore “artificial” boundaries set by the model designer. They can be no flow boundaries represented by chosen stream lines, or boundaries with known hydraulic head represented by equipotential lines. HYDRAULIC BOUNDARIES A streamline (flowline) is also a hydraulic boundary because by definition, flow is ALWAYS parallel to a streamflow. It can also be said that flow NEVER crosses a streamline; therefore it is similar to an IMPERMEABLE (no flow) boundary BUT Stress can change the flow pattern and shift the position of streamlines; therefore care must be taken when using a streamline as the outer boundary of a model. TYPES OF MODEL BOUNDARY NO-FLOW BOUNDARY Neither HEAD nor FLUX is Specified. Can represent a Physical boundary or a flow Line (Groundwater Divide) SPECIFIED HEAD OR CONSTANT HEAD BOUNDARY h = constant q is determined by the model. And may be +ve or –ve according to the hydraulic gradient developed TYPES OF MODEL BOUNDARY (cont’d) SPECIFIED FLUX BOUNDARY q = constant h is determined by the model (The common method of simulation is to use one injection well for each boundary cell) HEAD DEPENDANT BOUNDARY hb = constant q = c (hb – hm) and c = f (K,L) and is called CONDUCTANCE hm is determined by the model and its interaction with hb Boundary Types Specified Head/Concentration: a special case of constant head (ABC, EFG) Constant Head /Concentration: could replace (ABC, EFG) Specified Flux: could be recharge across (CD) No Flow (Streamline): a special case of specified flux (HI) Head Dependent Flux: could replace (ABC, EFG) Free Surface: water-table, phreatic surface (CD) Seepage Face: pressure = atmospheric at ground surface (DE) Boundary conditions in Modflow • Constant head boundary • Head dependent flux – River Package – Drain Package – General-head Boundary Package • Known Flux – – – – Recharge Evapotranspiration Wells Stream • No Flow boundaries Initial Conditions • Values of the hydraulic head for each active and constant-head cell in the model. They must be higher than the elevation of the cell bottom. • For transient simulation, heads to resemble closely actual heads (realistic). • For steady state, only hydraulic heads in constant head-cell must be realistic. Model Parameters • Time • Space (layer top and bottom) • Hydrogeologic characteristics (hydraulic conductivity, transmissivity, storage parameters and effective porosity) Time • Time parameters are specified when modelling transient (time dependent) conditions. They include time unit, length and number of time steps. • Length of stress periods is not relevant for steady state simulations Grid • In Finite Difference model, the grid is formed by two sets of parallel lines that are orthogonal. The blocks formed by these lines are called cells. In the centre of each cell is the node – the point at which the model calculates hydraulic head. This type of grid is called block-centered grid. Grid • Grid mesh can be uniform or custom, a uniform grid is better choice when – Evenly distributed aquifer characteristics data – The entire flow field is equally important – Number of cells and size is not an issue Grid • Grid mesh can be custom when – There is less or no data for certain areas – There is specific interest in one or more smaller areas • Grid orientation is not an issue in isotropic aquifers. When the aquifer is anisotropic, the model coordinate axes must be aligned with the main axes of the hydraulic conductivity. • Regular vs irregular grid spacing Irregular spacing may be used to obtain detailed head distributions in selected areas of the grid. Finite difference equations that use irregular grid spacing have a higher associated error than FD equations that use regular grid spacing. Considerations in selecting the size of the grid spacing Variability of aquifer characteristics (K,T,S) Variability of hydraulic parameters (R, Q) Curvature of the water table Vertical change in head Desired detail around sources and sinks (e.g., rivers) MODEL GRIDS Grids It is generally agreed that from a practical point-of-view the differences between grid types are minor and unimportant. USGS MODFLOW employs a body-centred grid. Boundary array (cell type) • Three types of cells – Inactive cells through which no flow into or out of the cells occurs during the entire time of simulation. – Active, or variable-head cells are free to vary in time. – Constant-head cell, model boundaries with known constant head. Hydraulic conductivity and transmissivity • Hydraulic conductivity is the most critical and sensitive modelling parameter. • Realistic values of storage coefficient and transmissivity, preferably from pumping test, should be used. Effective porosity • Required to calculate velocity, used mainly in solute transport models Calibration and Validation Calibration parameters are uncertain parameters whose values are adjusted during model calibration. Identify calibration parameters and their reasonable ranges. Typical calibration parameters include hydraulic conductivity and recharge rate. In a real-world problem, we need to establish model specific calibration criteria and define targets including associated error. Calibration Targets associated error calibration value 0.80 m 20.24 m Target with smaller associated error. Target with relatively large associated error. Targets used in Model Calibration • Head measured in an observation well is known as a target. • The simulated head at the node representing the observation well is compared with the measured head. • During model calibration, parameter values are adjusted until the simulated head matches the observed value. • Model calibration solves the inverse problem. Calibration to Fluxes When recharge rate (R) is a calibration parameter, calibrating to fluxes can help in estimating K and/or R. In this example, flux information helps calibrate K. q = KI H1 H2 In this example, discharge information helps calibrate R. Calibration - Remarks • Calibrations are non-unique. • A good calibration does not ensure that the model will make good predictions. • You can never have enough field data. • Modelers need to maintain a healthy skepticism about their results. • Need for an uncertainty analysis to accompany calibration results and predictions. Uncertainty in the Calibration Involves uncertainty in: Targets Parameter values Conceptual model including boundary conditions, zonation, geometry etc. Ways to analyze uncertainty in the calibration Sensitivity analysis is used as an uncertainty analysis after calibration. Use an inverse model (automated calibration) to quantify uncertainties and optimize the calibration. Uncertainty in the Prediction Reflects uncertainty in the calibration. Involves uncertainty in how parameter values (e.g., recharge) will vary in the future. Ways to quantify uncertainty in the prediction Sensitivity analysis Stochastic simulation How do we “validate” a model so that we have confidence that it will make accurate predictions? Modeling Chronology 1960’s Flow models are great! 1970’s Contaminant transport models are great! 1975 What about uncertainty of flow models? 1980s Contaminant transport models don’t work. (because of failure to account for heterogeneity) 1990s Are models reliable? “The objective of model validation is to determine how well the mathematical representation of the processes describes the actual system behavior in terms of the degree of correlation between model calculations and actual measured data”. How to build confidence in a model Calibration (history matching) “Verification” requires an independent set of field data Post-Audit: requires waiting for prediction to occur Models as interactive management tools KEEPING AN OPEN MIND Consider all dimensions of the problem before coming to a conclusion. Considering all the stresses that might be imposed and all the possible processes that might be involved in a situation before reaching a conclusion. KEEPING AN OPEN MIND is spending 95% of your TIME DETERMINING WHAT YOU THINK IS HAPPENING and only 5% of your TIME DEFENDING YOUR OPINION. AVOID the common human TRAP of REVERSING THOSE PERCENTAGES. Groundwater Flow Models Groundwater Flow Models • The most widely used numerical groundwater flow model is MODFLOW which is a three-dimensional model, originally developed by the U.S. Geological Survey. • It uses finite difference scheme for saturated zone. • The advantages of MODFLOW include numerous facilities for data preparation, easy exchange of data in standard form, extended worldwide experience, continuous development, availability of source code, and relatively low price. • However, surface runoff and unsaturated flow are not included, hence in case of transient problems, MODFLOW can not be applied if the flux at the groundwater table depends on the calculated head and the function is not known in advance. MODFLOW USGS code Finite Difference Model • MODFLOW 88 • MODFLOW 96 • MODFLOW 2000 MODFLOW (Three-Dimensional Finite-Difference Ground-Water Flow Model) • When properly applied, MODFLOW is the recognized standard model. • Ground-water flow within the aquifer is simulated in MODFLOW using a block-centered finite-difference approach. • Layers can be simulated as confined, unconfined, or a combination of both. • Flows from external stresses such as flow to wells, areal recharge, evapotranspiration, flow to drains, and flow through riverbeds can also be simulated. MT3D (A Modular 3D Solute Transport Model) • MT3D is a comprehensive three-dimensional numerical model for simulating solute transport in complex hydrogeologic settings. • MT3D is linked with the USGS groundwater flow simulator, MODFLOW, and is designed specifically to handle advectively-dominated transport problems without the need to construct refined models specifically for solute transport. FEFLOW (Finite Element Subsurface Flow System) FEFLOW is a finite-element package for simulating 3D and 2D fluid density-coupled flow, contaminant mass (salinity) and heat transport in the subsurface. HST3D (3-D Heat and Solute Transport Model) The Heat and Solute Transport Model HST3D simulates ground-water flow and associated heat and solute transport in three dimensions. SEAWAT (Three-Dimensional Variable-Density Ground-Water Flow) • The SEAWAT program was developed to simulate threedimensional, variable- density, transient ground-water flow in porous media. • The source code for SEAWAT was developed by combining MODFLOW and MT3D into a single program that solves the coupled flow and solute-transport equations. SUTRA (2-D Saturated/Unsaturated Transport Model) • SUTRA is a 2D groundwater saturated-unsaturated transport model, a complete saltwater intrusion and energy transport model. • SUTRA employs a two-dimensional hybrid finite-element and integrated finite-difference method to approximate the governing equations that describe the two interdependent processes. • A 3-D version of SUTRA has also been released. SWIM (Soil water infiltration and movement model) • SWIMv1 is a software package for simulating water infiltration and movement in soils. • SWIMv2 is a mechanistically-based model designed to address soil water and solute balance issues. • The model deals with a one-dimensional vertical soil profile which may be vertically inhomogeneous but is assumed to be horizontally uniform. • It can be used to simulate runoff, infiltration, redistribution, solute transport and redistribution of solutes, plant uptake and transpiration, evaporation, deep drainage and leaching. VISUAL HELP (Modeling Environment for Evaluating and Optimizing Landfill Designs) • Visual HELP is an advanced hydrological modeling environment available for designing landfills, predicting leachate mounding and evaluating potential leachate contamination. Visual MODFLOW (Integrated Modeling Environment for MODFLOW and MT3D) • Visual MODFLOW provides professional 3D groundwater flow and contaminant transport modeling using MODFLOW and MT3D. Groundwater Modelling Resources Groundwater Modeling Resources Kumar Links to Hydrology Resources http://www.angelfire.com/nh/cpkumar/hydrology.html USGS Water Resources Software Page water.usgs.gov/software Richard B. Winston’s Home Page www.mindspring.com/~rbwinston/rbwinsto.htm Geotech & Geoenviron Software Directory www.ggsd.com International Ground Water Modeling Center www.mines.edu/igwmc Ground Water Modelling Discussion Group An email discussion group related to ground water modelling and analysis. This group is a forum for the communication of all aspects of ground water modelling including technical discussions; announcement of new public domain and commercial softwares; calls for abstracts and papers; conference and workshop announcements; and summaries of research results, recent publications, and case studies. Group home page Post message Subscribe Unsubscribe List owner : http://groups.yahoo.com/group/gwmodel/ : [email protected] : [email protected] : [email protected] : [email protected] Visual MODFLOW Users Group Visual MODFLOW is a proven standard for professional 3D groundwater flow and contaminant transport modeling using MODFLOW-2000, MODPATH, MT3DMS AND RT3D. Visual MODFLOW seamlessly combines the standard Visual MODFLOW package with Win PEST and the Visual MODFLOW 3D-Explorer to give a complete and powerful graphical modeling environment. This group aims to provide a forum for exchange of ideas and experiences regarding use and application of Visual MODFLOW software. Group home page Post message Subscribe Unsubscribe List owner : http://in.groups.yahoo.com/group/visual-modflow/ : [email protected] : [email protected] : [email protected] : [email protected] THANKS HAPPY MODELLING