# 7 Equations of Groundwater Flow

In almost every field of science and engineering the techniques of analysis are based on an understanding of the physical processes, and in most cases, it is possible to describe these processes mathematically. Hydrogeology is no exception. Mathematical relationships describing groundwater flow are the foundation used to develop quantitative equations and models for specific groundwater systems and sites. In the previous 6 Sections of this book, principles and definitions have been presented which will be used to develop groundwater flow equations. These include the concepts of the representative elementary volume (REV), Darcy’s Law, head, gradient, hydraulic conductivity, transmissivity and storativity. These concepts are used in conjunction with the equation that describes conservation of mass (water) (e.g., mass inflow = mass outflow + change in mass storage) to develop universal mathematical equations that describe groundwater flow under various conditions.

The basic law governing groundwater flow in a porous medium is Darcy’s Law which describes the relationships between discharge, Q, or flux, q, and the hydraulic conductivity, gradient (head per distance) and cross-sectional area. When Darcy’s Law is combined with an equation of continuity that describes the conservation of fluid (water) mass, then the universal equations governing groundwater flow are formulated. Though the mathematics, which use partial differential equations, may appear daunting, it is straight forward. The equation development presented here starts with groundwater flow in one dimension and then introduces the same equations in three dimensions.

To apply these general equations to a particular groundwater problem, a problem domain, they are constrained to specific conditions by assigning site boundary conditions and, in some cases, initial conditions, that represent a particular groundwater setting. Analytical solutions and computer simulation are tools of the groundwater industry that incorporate the governing equations presented here making the equations accessible to groundwater professionals. These tools are used to quantify groundwater behavior in a wide variety of hydrogeologic settings under varied conditions. Some examples of the application of these tools to solve groundwater problems are presented in Section 7.5. It is important to understand the underlying assumptions and simplifications used to formulate methods used to solve groundwater problems. The goal of this section is to present the governing equations and explain those assumptions and simplifications.