Review of laboratory scale models of karst aquifers: approaches, similitude, and requirements

Abstract

This review focuses on investigations of groundwater flow and solute transport in karst aquifers through laboratory scale models (LSMs). In particular, LSMs have been used to generate new data under different hydraulic and contaminant transport conditions, testing of new approaches for site characterization, and providing new insights into flow and transport processes through complex karst aquifers. Due to the increasing need for LSMs to investigate a wide range of issues, associated with flow and solute migration karst aquifers this review attempts to classify, and introduce a framework for constructing a karst aquifer physical model that is more representative of field conditions. The LSMs are categorized into four groups: sand box, rock block, pipe/fracture network, and pipe-matrix coupling. These groups are compared and their advantages and disadvantages highlighted. The capabilities of such models have been extensively improved by new developments in experimental methods and measurement devices. Newer technologies such as 3-D printing, CT scanning, X-rays, NMR, novel geophysical techniques, and use of nano-materials allow for greater flexibilities in conducting experiments. In order for LSMs to be representative of karst aquifers, a few requirements are introduced: (1) the ability to simulate heterogeneous distributions of karst hydraulic parameters, (2) establish Darcian and non-Darcian flow regimes and exchange between the matrix and conduits, (3) placement of adequate sampling points and intervals, and (4) achieving some degree of geometric, kinematic, and dynamic similitude to represent field conditions.

Introduction

Carbonate aquifers with a total outcrop of 13.8% (Chen et al. 2017) provide freshwater for about 25% of the world’s population (Ford and Williams 2007). Unlike alluvial aquifers, which consist only of intergranular porosity, karst aquifers typically consist of different proportions of primary (rock matrix porosity), secondary (fractures and joints), and tertiary porosity (solution cavities and conduits) (Fig. 1). Mapping the heterogeneity and determination of hydraulic characteristics such as hydraulic conductivity (K), specific storage (S_s) and specific yield (S_y) of karst aquifers can be difficult due to the complexity of karst processes, resulting from a combination of several geochemical and mechanical agents. Moreover, groundwater can flow at very high velocities through enlarged conduits under a turbulent-flow regime and at slower, laminar flow regimes through the fracture-matrix system (Fig. 1). Accordingly, the assumption of Darcian flow is often not upheld for flow through conduits typically present in karstic aquifers. The interaction between conduits, fractures, and matrix is complex, requiring detailed scientific investigations (Martin and Dean, 2001; Reimann et al., 2011).

Fig. 1:

Schematic presentation of matrix (A), fractured-rock (B), and karst (C) aquifers (modified from Liedl et al. (2003).

Characterization of karst aquifers has been approached through a wide range of conventional tests and studies from regional to local scales. Although different types of data representing various components of karst aquifers can be obtained through conventional methods, quantitative methods to integrate data are needed to better map heterogeneity and to reduce uncertainty. Previous studies suggest that uncertainty reduction in mapping heterogeneity (Mohammadi and Illman 2019; Mohammadi, Raeisi, and Bakalowicz 2007; Geyer et al. 2013; Hartmann et al. 2014) is possible when complementary methods are employed for karst characterization. Detailed information on conduit geometry and hydrodynamic properties can lead to more accurate groundwater models, although this is difficult to obtain with existing approaches (Lazlo Kiraly 1998).

Groundwater modeling is a common approach for reconstructing the hydrogeological conditions of karst aquifers and predicting the response of the aquifer to hydraulic stresses and contaminants. In general, two types of models can be considered: mathematical and physical (e.g., laboratory). Mathematical modeling consists of analytical and numerical methods. Existing mathematical modeling techniques of karst aquifers have been reviewed by Kovács and Sauter, (2007), Ghasemizadeh et al., (2012), and Hartmann et al., (2014).

In porous media, laboratory scale models (LSMs) have been constructed to study processes occurring in groundwater systems (Prickett 1975; Silliman, Zheng, and Conwell 1998; Bear 1961; Cahill 1973; Kimbler 1970; Freeze and Banner 1970). Earlier studies have focused on basic groundwater flow behavior (Cahill 1973), variable density flow and mixing (Schincariol and Schwartz 1990), and solute dispersion behavior (Silliman 2001). Then, experiments were designed to study contaminant transport behavior such as DNAPL migration (Kamon et al. 2004) and contaminant plume migration (Y. F. Huang et al. 2006). Experiments have also been designed to test various groundwater flow modeling approaches (Illman et al. 2010; Berg and Illman 2011) and solute transport (Schincariol, Herderick, and Schwartz 1993), as well as to test the validity of new characterization methods such as hydraulic tomography (Illman, Liu, and Craig 2007).

LSMs have also been used to study groundwater flow and solute transport through karst aquifers. Studies have focused on investigations of exchanges between the conduit/fracture-matrix continuum (Faulkner et al. 2009; Shu et al. 2020; Lee J. Florea, Cunningham, and Altobelli 2009), testing of solute/contaminant transport (G. Li, Loper, and Kung 2008; Lee J Florea, Cunningham, and Altobelli 2008), testing of analytical and numerical modeling approaches (Hu et al. 2012; Loper 2013; Karay and Hajnal 2015), estimation of recharge (Wang et al. 2015), determination of hydrodynamic characteristics of aquifers such as K and hydrodynamic dispersion coefficient (Park, Lee, and Lee 2018; Cupola, Tanda, and Zanini 2015; Bouzaglou et al. 2018; Klammler, Nemer, and Hatfield 2014), simulation of saltwater intrusion (G. Li, Loper, and Kung 2008; F. lin Li et al. 2018; Oz et al. 2015), better understanding of tracer transport behavior (Field and Leij 2012; L. J. Florea and Wicks 2001) and more recently, evaluation of characterization methods (Zhao et al. 2017; K. Cunningham et al. 2008). Moreover, validation of numerical models and processes incorporated can be tested under controlled conditions. Other motivations for LSMs include obtaining insights into complex phenomena not fully described and data collection under controlled settings to test theoretical findings (Hughes 1993).

This review is the first attempt to highlight the improvements in laboratory-scale modeling of karst aquifers and to categorize various approaches. The objectives of this paper are: (1) to present an overview of LSMs in the literature with a primary focus on karst, and (2) to introduce some guidelines for laboratory scale modeling of karst aquifers.

Why laboratory models?

LSMs allow for greater insights into groundwater flow and solute transport because of their flexibility and advantages over field experiments. Some advantages of conducting experiments with LSMs include: 1) controlled forcing functions (initial and boundary conditions, and source/sink terms), 2) known subsurface heterogeneity and anisotropy, 3) ability to visualize experimental results, 4) opportunities to obtain high-resolution data, 5) time and cost savings compared to field experiments, 6) ability to directly observe processes for testing new theories and approaches, 7) ability to conduct multiple hydraulic and tracer tests, and 8) emplacing contaminants, typically not possible in the field due to regulatory requirements.

The flexibility of LSMs regarding the selection of the number of sampling points, allowing for differing hydraulic conditions, and having an accurate geometry of the studied media can be considered as an important advantage of these models in many cases. Generally, the number of boreholes and sampling points for obtaining hydraulic head and solute concentrations in the field is quite limited. Although the hydraulic condition of a field site in terms of hydraulic gradient, groundwater flow, forcing functions, and subsurface heterogeneity are not perfectly known LSMs allow for consideration of different scenarios with differing hydraulic conditions and/or differing combinations of recharge and discharge components as required. One could also study fluid flow behavior other than water such as air (Leven et al., 2004; McDermott et al., 2003) and nonaqueous phase liquids (Detwiler, Rajaram, and Glass 2001).

Simplification and scale effects are the main disadvantages of LSMs (Hughes 1993). Specifically, LSMs mainly suffer from the oversimplification of real conditions, difficulty in accurate 3D representation of heterogeneous pathways, as well as establishing realistic hydraulic conditions. In karst aquifers there are practical limitations to developing large scale flow paths through major faults or solution channels and caves in laboratory models. A major technical problem associated with any laboratory scale modeling effort is the problem of maintaining dynamic similitude.

Similitude in LSMs

When designing LSMs to study groundwater flow and solute transport through karst aquifers, maintaining similitude is extremely important. For scaling down or up a field-scale condition to a LSM, three types of similitudes need to be maintained: (1) geometric, (2) kinematic, and (3) dynamic, which specifically refer to similarities in form, motion, and forces, respectively. The ratio of corresponding variables (e.g., Z) of full scale and LSM may be quantified as a scale factor (R_Z) (Scheidegger 1963):

$${\text{R}}_{\text{Z}}=\frac{{\text{Z}}_{\text{fs}}}{{\text{Z}}_{\text{ls}}}$$ (1)

where Z is a variable related to form, motion, and force, such as length, velocity, and viscosity, respectively, and subscripts fs and ls refer to full and laboratory scales, respectively.

Geometric similitude requires implementing a scale factor for all length scales (e.g., length, area, and volume) to maintain the same shape, but different sizes between the LSMs and actual field conditions (Zohuri 2015; Heller 2011). Letting L_fs and L_ls to be the real lengths of full and laboratory scales, respectively, a length ratio of R_L as length scale factor can be calculated by (Chanson 1999):

$${\text{R}}_{\text{L}}=\frac{{\text{L}}_{\text{fs}}}{{\text{L}}_{\text{ls}}}$$ (2)

The length (or height), area, and volume of a LSM should be designed to be R_L, R_L², and R_L³, respectively.

Kinematic similitude is obtained if similarity between variables of fluid motion (e.g., velocity, acceleration, and discharge) are maintained. In fact, kinematic similitude depends on both similarity of time interval and length. The ratio of velocities (R_V) between the full-scale and LSMs (V_fl and V_ls, respectively) is computed by (Chanson 1999):

$${\text{R}}_{\text{V}}=\frac{{\text{V}}_{\text{fs}}}{{\text{V}}_{\text{ls}}}$$ (3)

and have to be maintained for other variables of fluid motion such as acceleration and discharge.

For the similitude of forces (e.g., inertia and viscosity), dynamic similarity should be maintained. The ratio of forces (R_F) should be the same between the full-scale (F_fs) and LSM (F_ls) as:

$${\text{R}}_{\text{F}}=\frac{{\text{F}}_{\text{fs}}}{{\text{F}}_{\text{ls}}}$$ (4)

In order to achieve dynamic similitude, geometric and kinematic similarities are requirements. Several dimensionless numbers (e.g., Froude and Reynolds) have been previously suggested for assessing true dynamic similitude. The Reynolds number has been mostly used as an indication of dynamic similitude (Blöschl and Sivapalan 1995; Heller 2011). For maintaining a similar Reynold number (referred to as the Reynold similitude) between the full field scale and LSM the following expression is applicable:

$${\left[\frac{\rho \text{VD}}{\text{μ}}\right]}_{\text{fs}}={\left[\frac{\rho \text{VD}}{\text{μ}}\right]}_{\text{ls}}$$ (5)

where ρ and μ are the density and dynamic viscosity of the fluid, respectively, and D is the length characteristics of the media. Rearrangement of Eq. (5) based on concept of scale factor in Eq. (1) provides:

$${\text{R}}_{\text{ρ}}{\text{R}}_{\text{V}}{\text{R}}_{\text{D}}={\text{R}}_{\text{μ}}$$ (6)

Assuming that the fluid used for the LSM is the same for the full-size condition in terms of temperature, R_ρ and R_μ are equal to unity. Moreover, R_D can be substituted by R_L, because they both have the same units of length. Therefore, the following ratio is required for achieving a Reynold similitude condition (Chanson 1999; Heller 2011):

$${\text{R}}_{\text{V}}=\frac{1}{{\text{R}}_{\text{L}}}$$ (7)

Keeping in mind the above aspects of similitude, LSMs designed to simulate field conditions should be evaluated by considering geometric, kinematic, and dynamic similarities. For example, if we make a LSM one thousand times smaller than the field scale problem under consideration (i.e., assuming R_L to be 1000), we must keep in mind that groundwater velocity of the LSM should be one thousand times larger than the field scale (i.e., R_V should be 0.001) to maintain Reynolds similitude. Ignoring one or all types of similitude can cause dissimilarity between field and laboratory conditions.

The mathematics of assuming laminar flow through matrix, conduit, and fracture as well as non-linear flow through conduit and defining several scaling factors is presented in Appendix S1. In order to establish kinematic as well as dynamic similarity, the coefficients C_D, C_H−P, C_D−W, and C_C (derived by assuming Darcy, Hagen-Poiseuille, Darcy-Weisbach equations, and cubic law, respectively, see Appendix S1) must be identical in both full and LSM conditions (Everdingen and Bhattacharyy 1963). These coefficients should not change with variation of scale and remain constant:

$${\left[C_D\text{or}{C}_{H-P}\text{or}{C}_{D-W}\text{or}{C}_C\right]}_{fs}={\left[C_D\text{or}{C}_{H-P}\text{or}{C}_{D-W}\text{or}{C}_C\right]}_{ls}$$ (8)

Given a scale factor of 1000 for length (i.e., R_l = 1000) for downscaling of a full-scale karst aquifer to a LSM, a suitable combination of variables (time, permeability, fluid density and viscosity, gravity and porosity) should be selected for LSMs such that Eq. (8) remains valid. For example, if the properties of porous media (permeability, porosity, and gravity) and fluid (fluid density and viscosity) remain unchanged between the full and LSMs, groundwater velocity is the only variable that differs between laboratory and field conditions (Appendix S1). Under this condition, groundwater will flow in the LSM one thousand times faster than real full-scale condition assuming Darcian flow. However, groundwater flows with a ratio of 31.5 and 0.001 in relation to full-scale condition assuming pipe flow under laminar and turbulent flow regimes, respectively (Appendix S1). Otherwise, if the time scale factor is set to 100, the scale factor for hydraulic conductivity and porosity must be held at 10. Sometimes the similarity characteristic (e.g., coefficients of C_D, C_H−P, C_D−W, and C_C) of LSMs do not conform to full-scale conditions and it is then necessary that the laboratory models be redesigned to find an optimum set of variables.

In order to examine the mathematics of similitude, the field characteristics of a karst aquifer from the Zagros Mountain Range, Iran was selected based on Mohammadi et al. (2018). Two alternative LSMs were designed assuming different similitude scales (Table 1). A combination of variables (time, permeability, fluid density and viscosity, gravity, and porosity) should be correctly selected for LSMs to maintain the similitude criteria of Eqs. S17 and S19. Assuming the fluid properties (fluid density and viscosity), porosity, friction factor, and gravity remain unchanged between the field and LSMs, we assigned different scale factors for length, time, matrix permeability, and conduit diameter to suggest the geometry, matrix hydraulic conductivity, and conduit diameter to setting up a LSM for karst.

Table 1:

The characteristics of two LSMs for a field karst aquifer (Mohammadi, Illman, and Karimi 2018) assuming different similitude scales.

Parameter	Field scale	LSM (1)	LSM (2)
Length (m)	40,000	40	8
Width (m)	6,000	6	1.2
Height (m)	350	0.35	0.084
Matrix K (m/s)	1 × 10−5	1 × 10−6	1.4 × 10−6
Diameter (m)	3	0.03	0.06

⁽¹⁾Scale factor for length, time, permeability of the matrix, and conduit diameter are 1000, 100, 10, and 100, respectively.

⁽²⁾Scale factor for length, time, permeability of the matrix, and conduit diameter are 5000, 700, 7, and 50, respectively.

LSM development

LSMs have been applied to study groundwater flow and solute transport for a wide range of flow conditions (e.g., steady and transient), porosity types (e.g., single and double), single phase (e.g., water, air, natural gas) and multiphase flow (e.g., LNAPLs, DNAPLs) (Table S1). In general, four categories of LSMs (including sandbox, rockblock, network of pipe/fracture, and coupling a pipe network within a sandbox) have been developed for simulating karst aquifer geometries and hydrodynamics. These four categories of LSMs are consistent with known conceptual modeling approaches for karst aquifers including equivalent porous media (EPM), dual continuum model (DCM), discrete fracture network (DFN), discrete conduit network (DCN), and coupled discrete-continuum (CDC) (Fig. 2; Teutsch and Sauter (1998) and Ghasemizadeh et al. (2012).

Fig. 2:

Schematic illustration of different approaches for modeling karst aquifers (Ghasemizadeh et al. 2012). See text for abbreviations.

Sandbox models

Sandbox models have a long history for studying groundwater flow and solute-transport processes in porous media. While sandbox models have its roots in simulating flow through porous media with an assumption of laminar flow, sandboxes with heterogeneous porous media (Table S2) could potentially be used to study karst aquifers when it can be treated as an EPM. In sandbox models, heterogeneity has been simulated as (1) gradually changing, from one side of the model to the other side (e.g., Huang et al., 1995; Silliman, 2001), and (2) discontinuous (e.g., Danquigny et al., 2004; Illman et al., 2007) (Fig. 3). Discontinuous heterogeneity patterns may be applicable to karst aquifers, as the large contrast in K between low- and high-K regions can mimic hydraulic and transport exchange processes between primary and secondary porosity areas. Moreover, sandbox models have been able to simulate some degree of fast flow through channel structures (i.e., high-K zone) (e.g., Danquigny et al., 2004).

Fig. 3:

Schematic illustration of different types of simulating heterogeneity used as sandbox models. The features are simplified after A: (Danquigny, Ackerer, and Carlier 2004), B: (Huang et al., 1995), and C: (Illman et al. 2010).

Despite the fact that different heterogeneous patterns can be created by sandbox models, they still result in substantial uncertainties when creating complex heterogeneous systems such as 3D networks of high-K zones, as well as heterogeneity at varying scales such as from lamination and microjoints to enlarged solution conduits, common in karst aquifers. Moreover, lack of fractures and conduits as well as the assumption of laminar flow are major disadvantages of application of sandbox models to karst aquifers.

Rockblock models

The experimental setup for these types of LSMs includes a block of rock, which allows for investigating fluid flow and solute transport through fractures and/or solution conduits (Fig. 4). These types of models differ from each other in terms of rock type, block size and shape, fluid flow, and measured parameters (Table S3). The experimental rockblock may be comprised of a fractured sandstone block (Brauchler et al. 2013; Brauchler, Liedl, and Dietrich 2003; Leven et al. 2004), a fractured dolostone block (Sharmeen et al. 2012), or a limestone block (DiFrenna, Price, and Savabi 2008; Cherubini, Giasi, and Pastore 2012; Develi and Babadagli 2015; Anaya et al. 2014). Typically, a rock sample is excavated from outcrops, but more recently, newer technologies such as CT-scans and digital optical imaging are being used to build different types of rockblock models by 3D printing (Otten et al. 2012; F. Hasiuk 2014; Ishutov et al. 2015; Lee J Florea, Cunningham, and Altobelli 2008; K. Cunningham et al. 2008). In order to capture major karst conduits, we probably need an impossibly large LSM. Rockblock models are applicable only in less-developed karst aquifers with simple and well-known fracture patterns and without large-scale major dissolution conduits. Generally, limitations regarding rock size for laboratory setup and unknown patterns of rockblock heterogeneity in terms of number, size, and connectivity of fractures and conduits are serious limitations of rockblock models. Since karst features develop at different scales (i.e., from small solution pores to underground caves) and contribute to groundwater flow and solute transport in different ways (i.e., storage or transmission), decisions for selecting the right dimension and size of rockblock models are accompanied by large uncertainties. Therefore, it is not straightforward to calculate the similitude criteria for these models. Challenges in designing LSMs utilizing rockblocks are: (1) to properly incorporate the topology, multiscale organization and connectivity of fractures and karst conduits, and (2) to design an appropriate sampling network and strategy in such a multiscale system.

Fig. 4:

Schematic features of different configurations used as rockblock models. The features are simplified after A: (Sharmeen et al. 2012; Brauchler et al. 2013), B: (Anaya et al. 2014), and C: (Leven et al. 2004; Cherubini, Giasi, and Pastore 2012).

Pipe/fracture network models

Pipe/fracture network models include a wide range of LSMs (Table S4) constructed by using an artificial fracture (F. J. Hasiuk, Florea, and Sukop 2016; K. J. Cunningham and Sukop 2011; F. Hasiuk, Ishutov, and Pacyga 2018; Qian et al. 2007; Tzelepis et al. 2015; Qian et al. 2011), a pipe (Wu, Hunkeler, and Goldscheide 2018; Anger and Alexander 2013), varying diameter tubes with channels (Field and Leij 2012), a set of fractures (Hull, Miller, and Clemo 1987), and a set of tubes (Karay and Hajnal 2015; Anger and Alexander 2013). These models all ignore the matrix, thus karst flow is simulated with a single or a series of interconnected fractures and/or pipes (Fig. 5).

Fig. 5:

Schematic features of different configurations used as pipe/fracture network models. The features are simplified after A: (Karay and Hajnal 2015); B: (Hull, Miller, and Clemo 1987); and C: (Qian et al. 2007, 2011; Tzelepis et al. 2015; Field and Leij 2012).

These models are restricted to constant geometrical patterns of fractures/conduits that are simplified versions of actual karst aquifers consisting of complex openings, spacing, orientations, and elongations. Despite the reliable application of fracture network models in testing theoretical issues in groundwater flow and mass transport and calibration of numerical models (e.g., Hull et al., 1987; Karay and Hajnal, 2015), simulation of real karst aquifers using these LSMs may oversimplify real fracture/conduit sets and could lead to difficulties in maintaining similitude criteria. Therefore, one should acknowledge these difficulties of pipe network models in simulating real karst aquifers, except where the role of the matrix can be neglected and a simple karst consisting of a series of conduits with fixed geometries is considered.

The ability to allow variations of flow regimes from laminar to turbulent through a wide range of pipe/fracture patterns from a single to a network of pipes/fracture is one major advantage of such LSMs. However, with current technology, the lack of ability to select very small (e.g., order of micron) and/or very large (e.g., order of meter) diameter pipes and fractures and no consideration for rock matrix, results in the lack of flow and transport exchange between matrix and pipe, which is the main drawback of these models. Moreover, the lack of sufficient information on fractures/conduit geometry and patterns in real aquifers limits these models when it comes to simulating actual aquifer conditions.

Pipe and matrix coupling models

The common feature of these LSMs summarized in Table S5 is the consideration of hydraulic and transport exchange between conduits and matrix. Conduit flow is simulated with a single pipe (Castro, 2017; Faulkner et al., 2009; Gallegos et al., 2013; Li et al., 2008; Xiao et al., 2018) or a series of interconnected pipes (Florea and Wicks, 2001; Mohammadi et al., 2019). The pipes are designed to be active inside (Castro, 2017; Florea and Wicks, 2001; Li et al., 2008; Mohammadi et al., 2019) or outside (Faulkner et al., 2009; Gallegos et al., 2013; Xiao et al., 2018) of the matrix (Fig. 6). The matrix consists of low-K material such as fine sand (Castro, 2017; Xiao et al., 2018), beads (Faulkner et al., 2009; Gallegos et al., 2013), ceramic clay (L. J. Florea and Wicks 2001), or bentonite mixed with sand (Mohammadi, Gharaat, and Field 2019).

Fig. 6:

Schematic features of different configurations used as pipe and matrix coupling models. The features are simplified after A: Mohammadi et al. (2019), B: Castro (2017); Li et al. (2008), and C: Faulkner et al. (2009); Gallegos et al. (2013).

Since pipe and matrix coupling models consider flow and solute transport through both the matrix and conduit and exchange between them, these models have a good potential for improved simulation of karst aquifers. In addition, adjusting experimental conditions to achieve laminar and turbulent flow is another advantage of these models. However, reduction of K in the matrix, clogging of slotted pipes, and the possibility of preferential flow paths developing at pipe boundaries are problems with these models.

Discussion: Requirements for valid laboratory scale karst models and future directions

Karst can range from diffuse to conduit flow, with larger conduits dominating groundwater flow and solute transport (L. Kiraly 2003). Accordingly, the applicability of LSMs depends on the karst system under investigation and such systems consisting of large caves with flowing streams may not be sufficiently represented in the laboratory. This review reveals that LSMs, if they are designed and operated under specific rules and conditions, could represent karst aquifers. The design conditions include geometric, kinematic and dynamic similitudes, which are achieved by varying experimental conditions such as hydraulic gradient, flow velocity, and contrast in hydraulic properties of conduits/matrix.

While LSMs are informative and useful in studying various processes in karst systems, maintaining similitude between a LSM and real karst aquifers is a significant challenge. For physical simulation of the characteristics of an aquifer at a laboratory scale, a reliable proportional dimension of length, width, and depth (i.e., geometric similitude) is necessary. In particular, coefficients C_D, C_H−P, C_D−W, and C_C (Eq. (8)) must be identical between the real full-scale aquifer and a LSM. Based on this rule, a valid LSM should have a set of porosity, K, time, and length that retains the same value of coefficients C_D, C_H−P, C_D−W, and C_C as the full-scale condition (Eq. 8). In addition, kinematic and dynamic similarities should be maintained by considering a fixed ratio between hydrodynamic parameters, such as groundwater velocity, discharge, and through the Reynolds number. In order to maintain a Reynolds similitude (Eq. 5), a valid LSM should hold an inverse relationship between length and velocity scale factors (Eq. (7)).

Once LSMs are deemed to be representative of field conditions, it could be utilized to conduct different experiments by varying fluid types, temperature, pressure head, hydraulic gradient, solute concentrations, tracer injection rates, etc. Experimental conditions could be varied to also consider laminar to turbulent flow and transport conditions.

Aside from design considerations, it is worth highlighting various novel technologies that could enhance future development of LSMs and corresponding experimental techniques. For example, the novelty and applicability of 3D printing for LSMs in geoscience, have been addressed for both groundwater research (Hasiuk et al., 2016) and petroleum industry applications (Ishutov et al. 2018). Specifically, a 3D printer was used to make different fracture sets (Suzuki et al. 2007; F. J. Hasiuk, Florea, and Sukop 2016), solutional karst porosity (Lee J Florea, Cunningham, and Altobelli 2008; K. Cunningham et al. 2008), sandstone porosity (Ishutov et al. 2015), and pore network (Otten et al. 2012) on printed rock-like specimens. These researchers used X-ray and CT-scan images to ensure that fracture sets are as accurate as possible. Elsewhere, 3D printing technology was used to investigate the effect of rock microstructure on permeability (Head and Vanorio 2016), underground stresses on the fracture structure (Ju et al. 2014), and rock deformation and failure under compressive and shearing stresses (Jiang et al. 2016).

Aside from 3D printing technology new materials, such as nano-materials, are being utilized in the design and setup of LSMs. Nano-materials have been tested in reduction of hydraulic conductivity of soil (Ng and Coo 2015), enhancement of mechanical properties of soils (Y. Huang and Wang 2016) and cement-based material (Kurapati Srinivas 2014), environmental remediation practices (Khin et al. 2012; Perreault, Fonseca De Faria, and Elimelech 2015), and to develop water-proof coatings and hydrophobic properties of traditional material (Kuo and Jeng 2014). Nano-materials that do not react with water could be applied in varying water temperatures. Therefore, stability of hydraulic and contaminant-transport conditions could be better controlled.

The applicability of new technologies, such as light transmission (Tidwell and Glass 1994), X-rays (Montemagno and Pyrak-Nolte 1999; Cnudde and Boone 2013), CT scans (Wildenschild et al. 2002; Vanghi, Iriarte, and Aranburu 2015; Krotkiewski et al. 2011; Dan et al. 2018), nuclear magnetic resonance (NMR) imaging (Lee J. Florea, Cunningham, and Altobelli 2009; Lee J Florea, Cunningham, and Altobelli 2008), and combinations of innovative logging tools and digital optical imaging (K. J. Cunningham et al. 2009; K. J. Cunningham and Sukop 2011; K. Cunningham et al. 2008) to image flow and solute behavior in LSMs, is also important for improved understanding of these processes. For example, the direction and magnitude of flow through LSMs could be visualized through NMR imaging of heavy water (D₂O) as a tracer (Lee J Florea, Cunningham, and Altobelli 2008). Moreover, salt tracers could be monitored with electrical resistivity tomography (ERT). In particular, ERT has been applied to investigate geological features of karstic unsaturated zones (Carrière et al. 2013) and underground buried caves (Martínez-Moreno et al. 2013; Yi, Kim, and Adepelumi 2013).

More recently, the use of nanoparticles (Torrese et al. 2014) and its monitoring with geophysical techniques have shown promise for mapping fluid flow and solute transport behavior in karst media. Such particles could act as moving sources that could be detected with ERT receiver arrays placed on and within karstic rock samples for LSMs.

Following the same idea, magnetic nanoparticles could be released into the groundwater flow field and their movement along the flow path would generate currents, which could be monitored with ERT monitoring arrays. This is a new approach that derives from magnetic particle imaging (MPI), which is an emerging non-invasive tomographic technique that directly detects superparamagnetic nanoparticle tracers in medical science (Vogel et al. 2019; Salamon et al. 2020). Currently, it is used in medical research to measure the 3D location and concentration of nanoparticles. Imaging does not use ionizing radiation and can produce a signal at any depth within the body. This technology has been recently advanced, but to date, has not been applied to hydrogeological and geophysical research, but could be a promising area of research.

Digital measurement of pressure head by transducers and tracer concentrations by chemical sensors or digital imaging may enhance spatial and temporal densities of available data for mapping heterogeneities and simulating LSM responses to different stresses. Although digital pressure transducers have been used extensively in the past to obtain measurements at a point, there are new techniques to measure pressure including high-pressure cell to measure chemo-thermo-mechanical characteristics of geomaterials (Abdul et al. 2020), novel orifice to measure wide range of discharge and pressure in pressurized pipe network (Rezazadeh, Bijankhan, and Mahdavi Mazdeh 2019), micro/nano pressure sensors for health monitoring and smart wearable devices (Chang et al. 2020; Song et al. 2020), and flexible sensors on surfaces (Fonov et al. 2005; Chang et al. 2020).

Assuming continuous progress in modern technologies that facilitate high spatial and temporal resolution of sampling and measurements, the future of LSMs for karst will be directed to challenge with issues such as: (1) improving capabilities of LSMs to more precisely simulate various phenomena in real karst aquifers, (2) capturing sharp variation of K-zones as local heterogeneity in karst aquifers by simultaneous using of synthetic materials as a coupled discrete-continuum approach, (3) attempting to identify, fully understand and formulation of the processes responsible for flow and solute transport in karst aquifers, and (4) reducing the gap between recent improvements of numerical and computational tools for modeling karst aquifers, current state of LSMs by generating new data consistent with assumptions, and the requirements of numerical models.

Supplementary Material

Table S1-S5

Table S1: General characteristics of the different categories of LSMs.

Table S2: Characteristics of representative sandbox models and their study purposes.

Table S3: Characteristics of rock block models.

Table S4: Characteristics of the pipe/fracture network models.

Table S5: Characteristics of the pipe and matrix coupling models.

Appendix S1

Appendix S1: mathematics of similitude.