Conceptual models represent hydrologic systems using
physical concepts. The conceptual model is used as the starting point for defining the important model components. The relationships between model components are then specified using
algebraic equations,
ordinary or
partial differential equations, or
integral equations. The model is then solved using
analytical or
numerical procedures. Conceptual models are commonly used to represent the important components (e.g.,
features, events, and processes) that relate hydrologic inputs to outputs. These components describe the important functions of the
system of interest, and are often constructed using entities (stores of water) and relationships between these entitites (flows or fluxes between stores). The conceptual model is coupled with scenarios to describe specific events (either input or outcome scenarios). For example, a watershed model could be represented using
tributaries as boxes with arrows pointing toward a box that represents the main river. The conceptual model would then specify the important watershed features (e.g., land use, land cover, soils,
subsoils, geology, wetlands, lakes), atmospheric exchanges (e.g., precipitation, evapotranspiration), human uses (e.g., agricultural, municipal, industrial, navigation, thermo- and hydro-electric power generation), flow processes (e.g., overland, interflow, baseflow, channel flow), transport processes (e.g., sediments, nutrients, pathogens), and events (e.g., low-, flood-, and mean-flow conditions). Model scope and complexity is dependent on modeling objectives, with greater detail required if human or environmental systems are subject to greater risk.
Systems modeling can be used for building conceptual models that are then populated using mathematical relationships.
Example 1 The
linear-reservoir model (or Nash model) is widely used for rainfall-runoff analysis. The model uses a cascade of linear reservoirs along with a constant first-order storage coefficient,
K, to predict the outflow from each reservoir (which is then used as the input to the next in the series). The model combines continuity and storage-discharge equations, which yields an ordinary differential equation that describes outflow from each reservoir. The continuity equation for tank models is: :{dS(t) \over dt} = i(t) - q(t) which indicates that the change in storage over time is the difference between inflows and outflows. The storage-discharge relationship is: :q(t) = S(t)/K where
K is a constant that indicates how quickly the reservoir drains; a smaller value indicates more rapid outflow. Combining these two equation yields :K {dq \over dt} = i - q and has the solution: : q= i(1-e^{-t/k})
Example 2 Instead of using a series of linear reservoirs, also the model of a
non-linear reservoir can be used. In such a model the constant
K in the above equation, that may also be called
reaction factor, needs to be replaced by another symbol, say
α (Alpha), to indicate the dependence of this factor on storage (S) and discharge (q). In the left figure the relation is quadratic: :
α = 0.0123
q2 + 0.138
q - 0.112
Governing equations Governing equations are used to mathematically define the behavior of the system. Algebraic equations are likely often used for simple systems, while ordinary and partial differential equations are often used for problems that change in space in time. Examples of governing equations include:
Manning's equation is an algebraic equation that predicts stream velocity as a function of channel roughness, the hydraulic radius, and the channel slope: :v = {k \over n} R^{2/3} S^{1/2}
Darcy's law describes steady, one-dimensional groundwater flow using the hydraulic conductivity and the hydraulic gradient: :\vec q = -K \nabla h
Groundwater flow equation describes time-varying, multidimensional groundwater flow using the aquifer transmissivity and storativity: :T \nabla^2 h = S {\partial h \over \partial t}
Advection-Dispersion equation describes solute movement in steady, one-dimensional flow using the solute dispersion coefficient and the groundwater velocity: :D {\partial^2 C \over\partial x^2} - v {\partial C \over\partial x} = {\partial C \over\partial t}
Poiseuille's law describes laminar, steady, one-dimensional fluid flow using the
shear stress: : {\partial P \over \partial x} = - \mu {\partial \tau \over \partial y}
Cauchy's integral is an integral method for solving boundary value problems: :f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\, dz
Solution algorithms Analytic methods Exact solutions for algebraic, differential, and integral equations can often be found using specified boundary conditions and simplifying assumptions.
Laplace and
Fourier transform methods are widely used to find analytic solutions to differential and integral equations.
Numeric methods Many real-world mathematical models are too complex to meet the simplifying assumptions required for an analytic solution. In these cases, the modeler develops a numerical solution that approximates the exact solution. Solution techniques include the
finite-difference and
finite-element methods, among many others. Specialized software may also be used to solve sets of equations using a graphical user interface and complex code, such that the solutions are obtained relatively rapidly and the program may be operated by a layperson or an end user without a deep knowledge of the system. There are model software packages for hundreds of hydrologic purposes, such as surface water flow, nutrient transport and fate, and groundwater flow. Commonly used numerical models include
SWAT,
MODFLOW,
FEFLOW, PORFLOW,
MIKE SHE, CREST, and
WEAP.
Model calibration and evaluation is used to determine the ability of the calibrated model to meet the needs of the modeler. A commonly used measure of hydrologic model fit is the
Nash-Sutcliffe efficiency coefficient. ==See also==