# ZOOMQ3D

ZOOMQ3D is a numerical finite-difference model, which simulates groundwater flow in aquifers. The program is used by hydrogeologists to investigate groundwater resources and to make predictions about possible future changes in their quantity and quality. The code is written in C++, an object-oriented programming language and can compile and run on Windows and Unix operating systems.

## Groundwater flow equation

ZOOMQ3D applies a quasi-three-dimensional finite-difference approximation to the general three-dimensional governing partial differential groundwater flow equation:

${\displaystyle {\frac {\partial }{\partial x}}\left[K_{xx}{\frac {\partial \phi }{\partial x}}\right]+{\frac {\partial }{\partial y}}\left[K_{yy}{\frac {\partial \phi }{\partial y}}\right]+{\frac {\partial }{\partial z}}\left[K_{zz}{\frac {\partial \phi }{\partial z}}\right]=S_{S}{\frac {\partial \phi }{\partial t}}-q}$

where:

• ${\displaystyle \phi (x,y,z,t)}$ is the potentiometric head at a point ${\displaystyle (x,y,z)}$ and time ${\displaystyle (t)}$ (L)
• ${\displaystyle K_{xx}}$, ${\displaystyle K_{yy}}$ and ${\displaystyle K_{zz}}$ are the values of hydraulic conductivity along the x, y, and z coordinate axes (LT−1)
• ${\displaystyle q}$ is a volumetric flux per unit volume representing sources and/or sinks of water, where negative values are abstractions, and positive values are injections (T−1) and,
• ${\displaystyle S_{S}}$ is the specific storage of the porous material (L−1)

This equation is derived by considering a flow balance for an infinitesimally small volume element located anywhere within a body of saturated aquifer. A number of assumptions underlie this equation. First, the fluid is assumed to be of constant density; this allows the flow balance to be a consequence of mass conservation within the element. Next, the Cartesian coordinate system is aligned with the principal axes of the hydraulic conductivity tensor; this avoids the need for cross derivatives.

A model, based on the above equation, incorporating appropriate boundary and initial conditions, would be truly three-dimensional. ZOOMQ3D takes a simplifying approach to the solution of the three-dimensional equation by recognising that in many aquifers it is possible to identify a layered structure. If the layers are aligned parallel to the horizontal coordinate axes, then the three-dimensional equation can be integrated vertically across the layer to produce an equation which describes the flow within a layer and its interactions with adjacent layers. Such an equation is:

${\displaystyle {\frac {\partial }{\partial x}}\left[T_{xx}{\frac {\partial h}{\partial x}}\right]+{\frac {\partial }{\partial y}}\left[T_{yy}{\frac {\partial h}{\partial y}}\right]=S_{c}{\frac {\partial h}{\partial t}}-q-L_{\mathrm {above} }+L_{\mathrm {below} }}$

where:

• ${\displaystyle h}$ is the potentiometric head within a layer (L)
• ${\displaystyle t,}$ is time (T)
• ${\displaystyle T_{xx}}$ and ${\displaystyle T_{yy}}$ are the values of transmissivity along the x and y coordinate axes (L2T−1)
• ${\displaystyle q}$ is a volumetric flux per unit plan area representing sources and/or sinks of water, where negative values are abstractions, and positive values are injections (LT−1)
• ${\displaystyle S}$ is the storage coefficient of the porous material (L0) and,
• ${\displaystyle L}$above and ${\displaystyle L}$below are leakage rates from layers above and below (LT−1)

## Model features

Feature Description
Multiple layers ZOOMQ3D can incorporate multiple layers of finite difference nodes. The elevation of these layers can vary across the model and the base elevation of one layer can be higher than the top of the layer below it. The separation of model layers simplifies the representation of groundwater systems that contain aquifers separated aquitards. This because the flow through low permeability layers, which is assumed to be vertical, is represented by the vertical leakage term connecting two finite difference nodes within the upper and lower aquifer.
Local grid refinement ZOOMQ3D incorporates a mesh refinement procedure which aids the solution of problems related to scale. The density of finite difference nodes can be increased by adding successively finer rectangular grids in discrete areas of the model domain. The mesh can be refined in separate areas and grids can be refined multiple times in the same location in order to zoom into a specific model feature, for example an abstraction borehole or a river reach.
Confined - unconfined conditions Both confined and unconfined aquifers can be modelled. At confined finite difference nodes transmissivity and storage are independent of groundwater head. At unconfined nodes transmissivity is a function of saturated thickness and the storage term incorporates specific yield. In the top model layer finite different nodes can be defined as being confined, unconfined or convertible. Convertible nodes switch between unconfined and confined behaviour when the groundwater head rises above its top elevation. In each of the lower model layers, all the nodes must be specified as being either confined or convertible.

Finite difference nodes dewater as the groundwater head drops below their base. In this case the node is removed from the matrix of finite difference equations.

Heterogeneity and anisotropy Models can be heterogeneous and anisotropic. Different hydraulic parameter values can be specified at each finite difference node and hydraulic conductivity may be different in the x and y-directions. It is assumed that the Cartesian co-ordinate system is aligned with the principal axes of the hydraulic conductivity tensor.
Moving boundaries Model nodes can de-water and re-wet. Nodes are made inactive when the groundwater level falls below their base and vice versa. The re-wetting of model nodes depends on the groundwater head in adjacent finite difference nodes.
Variable hydraulic conductivity with depth (VKD) Vertical variations in hydraulic conductivity with depth can be specified within model layers or across model layers by defining VKD profiles. The transmissivity at a node is calculated by integrating the hydraulic conductivity over the vertical saturated thickness of the node.
Recharge Recharge can vary spatially and temporally. Recharge is always applied to the upper-most active node.
Abstraction wells Pumped boreholes can be placed at any node within the model domain. Abstraction rates can vary temporally and wells can both abstract water from the aquifer and inject water into it.
Rivers Dendritic rivers basins are simulated using a series of interconnected river reaches. The hydraulic parameters characterising a reach can vary along the river as can the degree of connection with the aquifer. The transfer of water between the aquifer and rivers is simulated as is the accretion of baseflow along each river branch. Discharges to the river can be specified in any reach, for example to represent a sewage treatment works, and the discharge rate can vary over time. Both fully penterating and perched rivers can be simulated.
Head-dependent leakage nodes In addition to rivers, a second head-dependent leakage mechanism is included in ZOOMQ3D. The flow through leakage nodes is proportional to the difference between its elevation and the groundwater head at the finite difference node to which it is connected. Flow can occur in either direction i.e. into or out of the aquifer. Leakage nodes can be used to model spring flows, lakes or estuaries, for example.
Springs This model feature has been developed to simulated spring flows specifically. The flow out of a spring depends on the transmissivity of the surrounding finite difference nodes. Spring flows are represented by an ‘abstraction’ which removes water from the aquifer at the location of the spring until the water table falls below the level of the ground surface.
Time discretisation Simulation time is divided into time-steps, stress periods and blocks. The length of a time-step is equivalent to the length of time between which successive solutions are calculated for the model’s state variables. A stress period represents a period of time during which all model stresses remain constant e.g. recharge, groundwater abstraction or discharge to rivers. Stress periods are divided into one or more time-steps. A block is composed of one or more stress periods. The rationale for the use of blocks is predominantly related to the simplification of the organisation of time-variant data, for example, groundwater abstraction or recharge rates, within input files. The number of stress periods in each block is the same for all blocks within a simulation.

## History of development

The groundwater flow model ZOOMQ3D is one of the codes in the ZOOM family of numerical groundwater models which also consists of the advective transport particle tracking code ZOOPT and the distributed recharge model ZOODRM. Each of these models has been developed using object-oriented techniques, a programming approach commonly applied in commercial software development but only relatively recently adopted in numerical modelling for scientific analysis.

ZOOMQ3D and ZOOPT have been developed through a tri-partite collaboration between the School of Civil Engineering of the University of Birmingham, UK, the British Geological Survey and the Environment Agency of England and Wales. The distributed recharge model ZOODRM has been developed by the British Geological Survey. All models are maintained by the British Geological Survey.

## Link to the geological modelling package GSI3D

A link exists between ZOOMQ3D and the geogical modelling software GSI3D. The link facilitates the transfer of the structure of a geological model into a ZOOMQ3D groundwater model. After a GSI3D geological model is constructed it can be used to create a hydrogeological model of an aquifer. This is achieved by assigning hydrogeological parameters, such as hydraulic conductivity, to the geological units. Once this has been completed the resulting hydrogeological model can then be converted to a layered ZOOMQ3D groundwater model.

## References

• Jackson CR. (2001). The development and validation of the object-oriented quasi-three-dimensional regional groundwater flow model ZOOMQ3D. British Geological Survey Internal Report IR/01/144.
• Jackson CR & Spink AEF. (2004). User’s manual for the groundwater flow model ZOOMQ3D. British Geological Survey Internal Report IR/04/140.
• Spink AEF, Hughes AG, Jackson CR & Mansour MM. (2006). Object-Oriented Design in Groundwater Modeling. Proceedings of MODFLOW 2006 conference, Golden, Colorado, US. May 2006.