Before understanding the difference between hydrologic and hydraulic modelling, it is essential to have an understanding of the differences between hydrology and hydraulics.

“Hydrology is the scientific study of the movement, distribution, and quality of water on Earth and other planets, including the hydrologic cycle, water resources and environmental watershed sustainability. A practitioner of hydrology is a hydrologist, working within the fields of earth or environmental science, physical geography, geologyor civil and environmental engineering. In numerous sub-braches of hydrology, Surface hydrology, Hydrometeorology, Drainage basin management and Isotope hydrology are very important. The overall objects of this branch of scientific study is managing agricultural productivity through designing irrigation and drainage system as well as providing fresh drinking water. Hydraulics is a topic in applied science andengineering dealing with the mechanical properties of liquids or fluids. “

Fluid mechanics provides the theoretical foundation for hydraulics, which focuses on the engineering uses of fluid properties. However,the water balance equation is probably the closet that hydrology comes to having a fundamental theory underlying it as a science, and hence almost all hydrological studies are based around it. In terms of this, hydrology and hydraulics are different in their theoretical foundation. Moreover, hydrology is concerned with the distribution of water on the earth surface and its movement over and beneath the surface, and through the atmosphere while hydraulics is more focused on the movement of water beneath and over the surface. In other words, hydrology is with an entire scope of hydrological cycle while hydraulics only with a part of it. In terms of methodology, hydrology is more descriptive whereas hydraulics is more numerate.

To some extent, both of them are applied science. Hydrology, as a applied science is to take a knowledge of why water on earth is not evenly distributed and is not evenly accessible, and try to lessen the impact of them. “Hydraulic, as a applied science,  whose topics range through some part of science and most of engineering modules, and cover concepts such as pipe flow, dam design, fluidics and fluid control circuitry,pumps,turbines, hydropower, computational fluid dynamics, flow measurement, river channelbehavior and erosion.”. In reality there are huge areas of overlap between the two and it is difficult to separate them, particularly when excessive terrain data and enhance computing power are available.

“Hydrologic models are simplified, conceptual representations of a part of thehydrologic cycle. They are primarily used for hydrologic prediction and for understanding hydrologic processes. Two major types of hydrologic models can be distinguished: stochastic and process-based models.”. “Hydraulic modeling is used to evaluate important elements of free surface fluid flow. Generally, hydraulic modeling can refer to both numeric modeling (in which a simulation is performed on a computer), or physical modeling (where the physical flow geometry is scaled in such a way that it can be modeled in the laboratory). Numeric models are usually two- or three-dimensional, whereas physical hydraulic models are always three-dimensional.”

The fundamentals under the hydrologic models is water balance (continuity equation) while the that under the hydraulic models is both continuity equation and momentum equation. Hydrologic models are more good at the description of runoff generation while hydraulic modes are more good at the description of runoff routing. “For example, a hydrologic study might tell you what the total volumetric flow for a river is based on certain conditions, but a hydraulics study might tell you the velocity, depth, flow regime and turbulence of that flow”.

All hydrological models have two basic components. The first deals with the conversion of rainfall into run-off and the second with routing of that run-off to the catchment outlet. Traditional rainfall-runoff modelling employs hydrological techniques for overland flow description based on the storage equation, with linear or non-linear storage function. The storage equation or continuity function is described as follows.                                       ds/dt=I(t)-Q(t)                                            (1)
The equation can be stated in words as “change in storage (S) of a hydrological system with time is equal to the difference in the iflow (I) and outflow (Q) from the system. There are two unknowns in the above equation, S and Q. To solve the equation, a storage function is used. Depending on the nature of the system being analysed, the storage function takes various forms. For example, in reservoir routing using the level pool method, the storage function is a nonlinear function of outflow only. TheMuskingum method for flow routing considers the storage function to be linearly related to inflow and outflow whereas in a linear reservoir model, the storage is a linear function of outflow.  However, these methods are limited in their approach as various catchment parameters are lumped together in a simplified manner.

With advancements in terrain data acquisition and enhanced computing powers, commercially available two-dimensional hydraulic models may be used to model the overland flow in the catchment. The benefits are two-fold, the hydraulic routing of flows is physically more accurate and the nature of two-dimensional modelling allows the development of a distributed catchment model where various landuses in the catchment are schematised more accurately. Surface flow or runoff is governed by the dynamic unsteady flow equations based on the laws of conservation of mass and momentum. Most distributed models use a simplified form of the fully dynamic equations of flow. The most widely used flow equations are the kinematic flow equations where the force of gravity is equated to the force of friction in the momentum equation. Kinematic representation for overland flow has been found to be adequate for sheet flow over the catchment surface but may or may not be suitable for routing flow through channels.  This type of modelling is limited to hydraulically steep parts of the catchment and is not suitable where backwater effects are pronounced. Dynamic wave modelling addresses these limitations. As the runoff process in the catchment is a spatially distributed phenomenon, a fully dynamic two-dimensional hydraulic model may provide a more realistic flow routing.

References：
(1) https://en.wikipedia.org/wiki/Hydraulics
(2) https://en.wikipedia.org/wiki/Hydrology
(3) http://www.whatisdifferencebetween.com/science/
(4) http://www.aldenlab.com/Hydraulic-Modeling
(5) Rehuman, el.al., 2003, hydrological Vs hudraulic routing Possibilities with Two-dimensional hydraulic modelling.
(6) John Gerrard, 2008, Fundamentals of Hydrology,Taylor & Francis Group.