In the past 2 decades a new modern scaling technique has emerged from the
highly developed theory on the Lie group of transformations. This new method
has been applied by engineers to several problems in hydrology and
hydraulics, including but not limited to overland flow, groundwater dynamics,
sediment transport, and open channel hydraulics. This study attempts to
clarify the relationship this new technology has with the classical scaling
method based on dimensional analysis, non-dimensionalization, and the
Vaschy–Buckingham-

Scaling is an important tool that is used extensively in
engineering, mathematics, and physics. With scaling, conclusions about the
dynamics of a system can be based off of the dynamics of another system at a
more convenient scale. This is extremely important when it is necessary to
understand interactions of systems whose precise governing equations are
either unknown or too difficult to work with directly. Scaling can be used to
build models of phenomena for study in the laboratory and has important
implications in our understanding of very large- and very small-scale
phenomena. Examples of scaling in the sciences are numerous.

The goal of any scaling method is to predict information on one scale from
known information on another scale. This is done by scaling relevant
variables so that the dynamics of the system remain essentially unchanged.
The different scaling procedures each have a way of assigning relationships
between the variables being scaled. Three methods will be considered here.
The first technique is a classical scaling methodology based on dimensionless
groups of variables found through the Vaschy–Buckingham-

The first technique is based on dimensional analysis, and the scaling is
usually referred to as dimensional scaling. The scaling relationships are
determined based on the dimensions; e.g., dimensionless groups of variables
can be formed and related to ratios between forces, characteristic lengths
and times, fluxes, and other factors. There are numerous dimensionless groups
of variables that can be formed, and scaling of the system is based on
preserving the dimensionless groups. One of the most widely used tools in
dimensional scaling is the famous Vaschy–Buckingham-

The second scaling technique relies on the governing equations of the system.
In such methods, scaling transformations of the governing equations are
required to be invariant in order to preserve the dynamics of the system. An
invariant scaling method known as modified inspectional analysis is the basis
for physical models known as sandbox models. Modified inspectional analysis
is documented in

A third scaling method is a special case of a theoretical technique for symmetry analysis of differential equations. The basis for this general theory is the discovery that the set of invariant transformations has a special group structure, known as the Lie group of transformations. This has led investigators to develop an extensive theory completely characterizing all symmetry transformations that hold a system of equations invariant. The Lie scaling methodology uses this extensive mathematical background to provide an approach to obtain a physically based scaling transformation that depends on the system of equations modeling any dynamical process expressed as an initial-boundary value problem.

The Lie group of transformations is well documented in many mathematical
sources, and has been applied by several engineers over the last few decades.
The theory was originally put forth in

The focus of this study is on developing invariant scaling transformations,
but the Lie group method has implications outside of scaling. Indeed, the
discovery of any type of invariant symmetry transformation may have serious
consequences regarding the understanding of a system of equations. A theorem
due to

Application of the Lie group of invariant transformations has a second
important implication in scaling. The successful application of a scaling
transformation can be classified as one of two types of self similarity.
Type-1 self similarity corresponds to the usual result of a successful
dimensional scaling: when a set of dimensionless groups of parameters is
found and at least one member of the group can be shown through a limit to be
insignificant. This can be stated mathematically as

Type-2 self similarity is a subset of the case where the limit as

Analyses of type-1 self similarity in scaling problems are most common in
dimensional analysis and have utility since they can result in the reduction
of the number of arguments of the problem. Type-2 self similarity problems
have been seen in engineering and statistical mechanics and are promising for
solving difficult problems in hydrology and engineering. Examples can be seen
in

The Lie scaling method relies on theory developed to characterize all
symmetry transformations that leave a system of equations invariant. A
transformation

For a given system of equations

In general, finding the Lie groups of symmetries admitted by a system of
equations involves extending the operator above to include the independent
and dependent variables, as well as all derivates up to the differential
order of the system. The coefficients of the extended operator can be shown
to satisfy an overdetermined system of partial differential equations. In
some special cases the system of PDE (partial differential equation) for the
coefficients will be linear and homogeneous. Special cases include the wave
equation, nonlinear diffusion equations, and the advection–dispersion
equations; common equations for phenomena in hydrology and Earth science. The
solution to the overdetermined system of PDE allows the transformations
admitted by

The main object of study here is the sets of scaling transformations. The
general form of a scaling transformation is known:

As an example, consider a heterogeneous aquifer subject to a flux boundary
condition where the initial height of the saturated surface is

The Lie scaling transforms the entire system of equations: the governing
equations, as well as the boundary and initial conditions. The method begins
by gathering all relevant flow or medium variables and scaling them according
to the scaling parameter

The scaled flow equations produce conditions on the scaling exponents, the
external forcing, and the hydraulic conductivity. The scaling exponents must
satisfy the equations

In the event that any of the hydraulic or medium parameters are functions of
scaled quantities, those functions must also satisfy scaling relationships.
For example, if

In order to relate the Lie scaling technique to a more classical framework,
label the prototype system variables with subscript p and variables in the
model system with subscript m. Then

The saturated hydraulic conductivity depends on properties of the medium and
the fluid. It is often related to the permeability of the medium, as well as
the density and viscosity of the fluid:

The third equation in Eq. (

Once again, the quantity on the left is related to the Reynolds number with
velocity

The fourth equation in Eq. (

The characteristic velocity is the specific discharge over the saturated height. The Reynolds number can be seen on the left-hand side and the square of the Froude number on the right-hand side, and the characteristic lengths will be proportional to the horizontal dimensions of the aquifer. The ratios between the pore area and the aquifer area, and the saturated height to the horizontal extent, serve as relations between the dynamic quantities.

As may be seen above, the Lie scaling approach provides a grounded method for determining scaling conditions for invariance of a set of equations. Scale invariance conditions may be phrased in terms of such quantities as the Reynolds number, Froude number, and other useful non-dimensional properties. This is useful when considering design specifications of experiments and the nature of the forces that are preserved after transformation to the scaled system.

Equations (

The procedure above was structured to preserve two well-known non-dimensional
fluid groups as they are formed in the preceding analysis. Since

Other formulations may be of interest in terms of different characteristic velocities and lengths. There are other dimensionless groups that are used in fluid dynamics through porous media; one such number is the Péclet number, defined as the ratio of advective transfer rate to diffusive transfer rate. Investigation of the scaling implications in terms of the Péclet number could prove most interesting for subsurface problems.

Dimensional scaling is based on the idea that non-dimensional groups can be
formed based on the dimensions of the quantities involved in a physical
phenomenon. Quantities are usually spatial and temporal lengths, areas,
volumes, velocities, forces, resistances/conductances, densities, etc. If the
dominant forces and the quantities that the forces act on are known, then
dimensionless groups are formed using those forces and associated
characteristic properties. This idea is formalized with the famous
Buckingham

Dimensional analysis in fluid dynamics usually resolves to requiring
geometric, kinematic, and/or dynamic similarity. Geometric similarity
requires that all body/domain dimensions have the same linear scale ratio.
Kinematic similarity requires that the velocity scale ratios are identical.
Dynamic similarity requires that the force scale, or mass scale, ratio be the
same between the model and prototype.

As an example, consider flow in a confined aquifer. Under usual conditions
the flow velocity is quite low, so advective effects are reduced and the
viscous effects dominate the problem. In this case, there is no free surface
and the most important dimensionless quantity is usually taken to be the
Reynolds number, defined as the ratio of the inertial forces to the viscous
forces. Under the condition that the dynamics of the model be the same as the
dynamics of the prototype, the ratio of the Reynolds numbers should be the
same.

For example, in a hydrologic study modeling the fate of water infiltrated into a hillslope during a storm event, the rainfall rate at the surface can be measured, and the discharge through a seepage face can be measured. These two values give indications of the characteristic velocity of the water in the subsurface. The characteristic lengths will depend on the process. For the vertical infiltration through the hillslope, the characteristic length may be the depth from the surface to the water table. Once the water reaches the saturated zone, the nature of its movement changes from being primarily vertical flow to horizontal flow. For flow in the saturated zone the characteristic length will be related to the horizontal extent of the aquifer.

As a second example consider flow of water through an open channel. Flows
like these have relatively high velocities and large Reynolds numbers, and
the dimensionless quantity thought to be of greatest importance is known as
the Froude number. The Froude number is the ratio of the inertial forces to
the gravity forces. Scaling using this ratio proceeds by equating

The characteristic velocities are usually taken to be the average velocity of the flow in the channel and the length is usually taken to be the depth of the channel. For closed channels the characteristic length is the hydraulic diameter of the channel.

For the problem in Sect.

In typical problems, scaling is done either by

In the scaling problem for unconfined aquifer flow, the condition that the
Reynolds numbers between the model and prototype be the same is a condition
of the scaling scheme in Eq. (

A second established approach for scaling is called modified inspectional
analysis and is documented in Bear (

The scaling procedure is very similar to the Lie scaling technique.
Application of the modified inspectional analysis to the groundwater problem
in Sect.

Combining the six independent equations requires six variables to be chosen
arbitrarily. For example, if

The modified inspectional analysis is very similar in application to the Lie
scaling, producing similar results for the equations on the interior of the
domain. This method is difficult to apply to problems where the parameters

For a simple and straightforward example, consider a linear 1-D contaminant
transport problem; see

The concentration of the contaminant at location

Dimensional analysis can be applied very easily. The simplicity of the
problem does not warrant the full Buckingham-

The non-dimensional equation indicates that the Péclet number has a
profound effect on the dynamics of the system. For large values of

The Lie scaling approach begins with the direct scaling of all relevant
variables and parameters. Following the notation above the ratios

Invariance on the boundary requires that the functions

Comparing the Lie scaling and dimensional scaling, it is seen that the
results are the same for the equation on the interior of the flow domain. The
differences between the methodologies stem from the treatment of the boundary
by the Lie scaling. The conclusions from the boundary dictate that the
concentration must scale in the same way as the prescribed concentration
(either or both at the boundary or the initial concentration), and according
to Eq. (

The Lie scaling gives a complete picture of the requirements that must be satisfied in order for the full model to preserve the dynamics of the prototype boundary value problem. The other scaling methods focus on preserving a subset of dynamics, e.g., the dynamical similarity, the kinematic similarity, or invariance for the interior equations. The dimensional scaling methods do not consider parameter functions to be satisfying functional scaling relationships.

The three scaling techniques were described, applied, and compared. Each method will briefly be summarized below and comments made on each method's strengths and weaknesses.

The classical scaling method and most widely used technique is known as dimensional scaling. The basics of the method are an analysis through non-dimensionalization when equations describing the governing dynamics are known and application of the Vaschy–Buckingham theorem when governing equations are either not known or poorly understood.

The relative strengths of this method are its universal applicability, ease
of application, and identification of the important non-dimensional parameter
groups in a system and the effect they have on the solution process in the
system. It is extremely useful to be able to apply the Vaschy–Buckingham
theorem to analyze a problem about which very little is known. In cases where
the closed form equations are not available to model dynamics, very basic
information may be available on variables, parameters, and processes that
contribute to a phenomenon. Gathering the dimensions for these quantities is
enough to apply the

The weakness of the classical dimensional scaling lies in the limited detail gained through an analysis. Generally, dimensional analysis is only applied to the dynamics in the interior of the domain and not to the boundary phenomena. This results in the neglect of influences from the boundary, which can be substantial in many problems. Dimensional scaling also overlooks the need for parameter functions or processes depending on variables that are being scaled to satisfy self similarity conditions in order to preserve the dynamics of the system. This is somewhat a strength and a weakness. It is a strength in that it gives the modeler a larger amount of freedom in creating the model. It is a weakness because it fails to identify the conditions in which a model system is truly a scaled version of the prototype, i.e., when a scaling transformation is invariant.

The modified inspectional analysis is very similar to the Lie scaling method
both in development and in application. It is a somewhat intuitive version of
the Lie scaling technique. It operates by examining the governing equations,
applying a scaling transformation to the variables in the system, and
enforcing invariance of the scaling transformation. Similarly to the
dimensional scaling, this method does not explicitly consider that parameters
or secondary processes that are functions of the scaled state variables of
the system need to satisfy self similarity relationships. Rather, model
processes can be chosen to have different dynamical properties than the
prototype processes

The Lie scaling technique is a powerful method based on a well-developed
mathematical theory. In fact, the Lie scaling is an instance of a much larger
class of invariant transformations which act on a system. In application, it
involves development of the set of model variables by scaling all quantities
in the prototype system, performing the change in variables in the prototype
governing equations, and determining the precise scaling relationships by
enforcing invariance on both the governing equations and boundary conditions.
Note that the scaling transformation may also be found by applying the Lie
algorithm, outlined above and described in detail in

The strength of the Lie scaling technique is that it gives a complete picture
as to the set of conditions that a complete model must satisfy in order to
preserve the dynamics of the prototype system. The precise nature of the
dynamics to be preserved, e.g., kinematic similarity and dynamical
similarity, can be explored and specified along with the general invariance
of the transformation on the system of equations. It will not necessarily
always be the case that a non-trivial scaling is possible which preserves all
similarity relationships, but the simple fact that this can be unambiguously
determined for the set of all possible invariant scaling transformations is
itself fundamental. The Lie scaling approach requires that any known function
or process that depends on scaled variables satisfy a self similarity
relationship. Investigation of

The Lie scaling considers the complete model, meaning the interior and boundary conditions, enforcing invariance in scaling for all regions in the problem. This ensures that the effects of the boundary conditions will be accounted for in preservation of the dynamics. It also provides a link between the flow problems in domains adjacent to the problem domain. For example, for the investigation of a scaled model for a subsurface saturated zone receiving recharge from an unsaturated zone, either the forcing function in the Dupuit approximating equations or the boundary condition in the full 3-D nonlinear problem will require self similarity of the flow in the unsaturated zone. This is important for considering coupling dynamics between different systems. Examples include seawater intrusion into the groundwater system and the interaction of regional climate models with groundwater.

The Lie scaling method requires the governing equation for both the interior and the boundaries to be known, and the functional scaling properties for any variable dependent parameters to be known as well. Models based on the Lie scaling must include medium and flow parameters that have identical (scaled) structure to the prototype. This may introduce technical difficulties in producing precisely scaled physical models in the laboratory. The scaling procedure must be applied to problems individually. Any change in boundary conditions, initial conditions, flow, or medium parameters may significantly alter the scaling structure and existence of invariant transformations.

Lastly, the Lie scaling method is a special case of the more general method to classify the Lie group of transformations. The Lie group method systematically considers invariant changes of variables that make an equation integrable. While scaling is an important symmetry, other members may yield useful simplifications or insight into difficult problems and should be considered in future research.