MATHEMATICAL MODEL

MODELO MATEMÁTICO

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A mathematical model uses mathematical language to describe a system. The process of developing a mathematical model is termed mathematical modelling (also modeling). Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines, but also in the social sciences (such as economics, psychology, sociology and political science); physicists, engineers, computer scientists, and economists use mathematical models most extensively.


Eykhoff (1974) defined a mathematical model as 'a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in usable form'.[1]
Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures.


Examples of mathematical models


-Population Growth. A simple (though approximate) model of population growth is the Malthusian growth model. A slightly more realistic and largely used population growth model is the logistic function, and its extensions.
-Model of a particle in a potential-field. In this model we consider a particle as being a point of mass m which describes a trajectory in space which is modeled by a function x : R → R3 giving its coordinates in space as a function of time. The potential field is given by a function V : R3 → R and the trajectory is a solution of the differential equation

*Note:this model assumes the particle is a point mass, which is certainly known to be false in many cases in which we use this model; for example, as a model of planetary motion.
-Model of rational behavior for a consumer. In this model we assume a consumer faces a choice of n commodities labeled 1,2,...,n each with a market price p1, p2,..., pn. The consumer is assumed to have a cardinal utility function U (cardinal in the sense that it assigns numerical values to utilities), depending on the amounts of commodities x1, x2,..., xn consumed. The model further assumes that the consumer has a budget M which is used to purchase a vector x1, x2,..., xn in such a way as to maximize U(x1, x2,..., xn). The problem of rational behavior in this model then becomes an optimization problem, that is:

subject to:


Diagram showing definitions and directions for Darcy's law.
The total discharge, Q (units of volume per time, e.g., ft³/s or m³/s) is equal to the product of the permeability (κ units of area, e.g. m²) of the medium, the cross-sectional area (A) to flow, and the pressure drop (Pb − Pa), all divided by the dynamic viscosity μ (in SI units e.g. kg/(m·s) or Pa·s), and the length L the pressure drop is taking place over. The negative sign is needed because fluids flow from high pressure to low pressure. So if the change in pressure is negative (in the x-direction) then the flow will be positive (in the x-direction). Dividing both sides of the equation by the area and using more general notation leads to
where q is the filtration velocity or Darcy flux (discharge per unit area, with units of length per time, m/s) and DP is the pressure gradient vector. This value of the filtration velocity (Darcy flux), is not the velocity which the water traveling through the pores is experiencing[2].
The pore (interstitial) velocity (v) is related to the Darcy flux (q) by the porosity (φ). The flux is divided by porosity to account for the fact that only a fraction of the total formation volume is available for flow. The pore velocity would be the velocity a conservative tracer would experience if carried by the fluid through the formation.
In 3D
In three dimensions, gravity must be accounted for, as the flow is not affected by the vertical pressure drop caused by gravity when assuming hydrostatic conditions. The solution is to subtract the gravitational pressure drop from the existing pressure drop in order to express the resulting flow,
where the flux Q is now a vector quantity, K is a tensor of permeability, D is the gradient operator in 3D, g is the acceleration due to gravity, ez is the unit vector in the vertical direction, pointing downwards and ρ is the density.
Effects of anisotropy in three dimensions are addressed using a symmetric second-order tensor of permeability:
where the magnitudes of permeability in the x, y, and z component directions are specified. Since this a symmetric matrix, there are at most six unique values. If the permeability is isotropic (equal magnitude in all directions), then the diagonal values are equal, while all other components are 0. The permeability tensor can be interpreted through an evaluation of the relative magnitudes of each component. For example, rock with highly permeable vertical fractures aligned in the x-direction will have higher values for Kxx than other component values.
Assumptions
Darcy's law is a simple mathematical statement which neatly summarizes several familiar properties that groundwater flowing in aquifers exhibits, including:
if there is no pressure gradient over a distance, no flow occurs (this of course, is the hydrostatic condition),
if there is a pressure gradient, flow will occur from high pressure towards low pressure (opposite the direction of increasing gradient—hence the negative sign in Darcy's law),
the greater the pressure gradient (through the same formation material), the greater the discharge rate, and
the discharge rate of fluid will often be different — through different formation materials (or even through the same material, in a different direction) — even if the same pressure gradient exists in both cases.
A graphical illustration of the use of the steady-state groundwater flow equation (based on Darcy's law and the conservation of mass) is in the construction of flownets, to quantify the amount of groundwater flowing under a dam.
Darcy's law is only valid for slow, viscous flow; fortunately, most groundwater flow cases fall in this category. Typically any flow with a Reynolds number (based on a pore size length scale) less than one is clearly laminar, and it would be valid to apply Darcy's law. Experimental tests have shown that flow regimes with values of Reynolds number up to 10 may still be Darcian. Reynolds number (a dimensionless parameter) for porous media flow is typically expressed aswhere ρ is the density of the fluid (units of mass per volume), v is the specific discharge (not the pore velocity — with units of length per time), d30 is a representative grain diameter for the porous medium (often taken as the 30% passing size from a grain size analysis using sieves), and μ is the dynamic viscosity of the fluid.
Derivation
Assuming stationary, creeping, incompressible flow, the Navier-Stokes equation simplify to the Stokes equation
where μ is the viscosity, ui is the velocity in the i direction, gi is the gravity component in the i direction and p is the pressure. Assuming the viscous resisting force is proportional to the velocity, and opposite in direction, we may write
where φ is the porosity.This gives the velocitywhich gives Darcy's law
***********Additional forms of Darcy's law*************
Time derivative of flux
For very short time scales or high frequency oscillations, a time derivative of flux may be added to Darcy's law, which results in valid solutions at very small times (in heat transfer, this is called the modified form of Fourier's law),where τ is a very small time constant which causes this equation to reduce to the normal form of Darcy's law at "normal" times (> nanoseconds). The main reason for doing this is that the regular groundwater flow equation (diffusion equation) leads to singularities at constant head boundaries at very small times. This form is more mathematically rigorous, but leads to a hyperbolic groundwater flow equation, which is more difficult to solve and is only useful at very small times, typically out of the realm of practical use.
Brinkman term
Another extension to the traditional form of Darcy's law is the Brinkman term, which is used to account for transitional flow between boundaries (introduced by Brinkman in 1947),

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DARCY'S LAW

In fluid dynamics and hydrology, Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments (published 1856)[1] on the flow of water through beds of sand. It also forms the scientific basis of fluid permeability used in the earth sciences.

Darcy's law is a simple proportional relationship between the instantaneous discharge rate through a porous medium, the viscosity of the fluid and the pressure drop over a given distance.

where β is an effective viscosity term. This correction term accounts for flow through medium where the grains of the media are porous themselves, but is difficult to use, and is typically neglected.

Multiphase flow

For multiphase flow, an approximation is to use Darcy's law for each phase, with permeability replaced by phase permeability, which is the permeability of the rock multiplied with relative permeability. This approximation is valid if the interfaces between the fluids remain static, which is not true in general, but it is still a reasonable model under steady-state conditions.

Assuming that the flow of a phase in the presence of another phase can be viewed as single phase flow through a reduced pore network, we can add the subscript i for each phase to Darcy's law above written for Darcy flux, and obtain for each phase in multiphase flow

where κi is the phase permeability for phase i. From this we also define relative permeability κri for phase i as
κri = κi / κ
where κ is the permeability for the porous medium, as in Darcy's law.
Dupuit-Forchheimer equation for non-Darcy flow
For a sufficiently high flow velocity, the flow is nonlinear, and Dupuit and Forchheimer have proposed to generalize the flow equation to

In membrane operations
In pressure-driven membrane operations, Darcy's law is often used in the form,
where,
J is the volumetric flux (m.s − 1),
ΔP is the hydraulic pressure difference between the feed and permeate sides of the membrane (Pa),
ΔΠ is the osmotic pressure difference between the feed and permeate sides of the membrane (Pa),
μ is the dynamic viscosity (Pa.s),
Rf is the fouling resistance (m − 1),
and Rm is the membrane resistance (m − 1).

This model has been used in general equilibrium theory, particularly to show existence and Pareto efficiency of economic equilibria. However, the fact that this particular formulation assigns numerical values to levels of satisfaction is the source of criticism (and even ridicule). However, it is not an essential ingredient of the theory and again this is an idealization.
Neighbour-sensing model explains the mushroom formation from the initially chaotic fungal network.
Modelling requires selecting and identifying relevant aspects of a situation in the real world.

Background
Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations.
A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. The values of the variables can be practically anything; real or integer numbers, boolean values or strings, for example. The variables represent some properties of the system, for example, measured system outputs often in the form of signals, timing data, counters, and event occurrence (yes/no). The actual model is the set of functions that describe the relations between the different variables.
Building blocks
There are six basic groups of variables[citation needed]: decision variables, input variables, state variables, exogenous variables, random variables, and output variables. Since there can be many variables of each type, the variables are generally represented by vectors.
Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables).
Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases.

Classifying mathematical models
Many mathematical models can be classified in some of the following ways:

1.Linear vs. nonlinear: Mathematical models are usually composed by variables, which are abstractions of quantities of interest in the described systems, and operators that act on these variables, which can be algebraic operators, functions, differential operators, etc. If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise.
The question of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model.
Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.
2.Deterministic vs. probabilistic (stochastic): A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables. Therefore, deterministic models perform the same way for a given set of initial conditions. Conversely, in a stochastic model, randomness is present, and variable states are not described by unique values, but rather by probability distributions.
3.Static vs. dynamic: A static model does not account for the element of time, while a dynamic model does. Dynamic models typically are represented with difference equations or differential equations.
4.Lumped vs. distributed parameters: If the model is heterogeneous (varying state within the system) the parameters are distributed. If the model is homogeneous (consistent state throughout the entire system), then the parameters are lumped. Distributed parameters are typically represented with partial differential equations.