Hybrid Flow Modelling Approach: CAVE

To compute the flow field in the combined conduit/matrix system, the CAVE model (Clemens et al. 1996; Liedl et al. 2003) was used, which is a hybrid model coupling flow in a discrete conduit system to a 3-D porous continuum. CAVE models flow in the discrete conduit system with a pipe-flow modelling approach (Horlacher & Lüdecke 1992), which covers laminar and turbulent flow cases.

Flow in the porous continuum is simulated using the well-known MODFLOW code, a three-dimensional finite-difference formulation for solution of the partial differential equations describing laminar groundwater flow (Harbaugh &

McDonald 1996). CAVE was originally developed for modelling the genesis of

discrete karst aquifers. However, the simulation of groundwater flow in underground mines, discrete fractured systems as well as aquifers with intersecting boreholes, i.e., in systems with both, a highly conductive network with poorly interconnected conduits and a considerably less permeable, porous matrix can be treated in an analogous way to karstified carbonate aquifers with a discrete conduit system.

Fig. 2: Model components of CAVE, UMT3D and RUMT3D.

The three-dimensional continuum equation for flow in the porous matrix (ore material), including a further source/sink term γ that couples the continuum to the pipe-flow model, is described as follows:

m

PHT3D(= 3D MODFLOW/MT3DMS -based multicomponent transport model Prommeret al.2002)

MT3DMS(= modular 3D multi-species transport model developed by Zheng and Wang 1999)

RUMT3D(= 3D Reactive Underground Mine Transport Model)

merge Hybrid flow model

CAVE (= Carbonate Aquifer Void Evolution developed by Clemens et al.1996 or Liedlet al.2002)

Hybrid transport model Reactive transport model

UMT3D(= 3D

PHT3D(= 3D MODFLOW/MT3DMS -based multicomponent transport model Prommeret al.2002)

MT3DMS(= modular 3D multi-species transport model developed by Zheng and Wang 1999)

RUMT3D(= 3D Reactive Underground Mine Transport Model)

merge Hybrid flow model

CAVE (= Carbonate Aquifer Void Evolution developed by Clemens et al.1996 or Liedlet al.2002)

Hybrid transport model Reactive transport model

h_m (L): hydraulic head in the porous matrix

Km,xx, Km,yy, Km,zz (L T^-1): hydraulic conductivity along the co-ordinate axes in the porous matrix

S (L^-1): specific storage coefficient

W_m (T^-1): volumetric flux term per unit volume from a sink/source into the porous matrix, e.g., groundwater recharge

γ ^(T-1): volumetric rate of fluid transfer between the porous matrix and the conduit system per unit volume.

A conduit system is defined in the model as a pipe network consisting of cylindrical tubes. Conduit nodes are introduced between the connecting tubes to allow for exchange of flow between the different tubes from different faces of a cell and between a conduit node and the continuum (porous matrix) at different locations in the model domain. Only one conduit node can be placed in a porous matrix cell. There are 6 potential faces of a cell interfacing with a tube network node, i.e., top, bottom, front, back, left and right. A conduit tube may extend over one or more porous matrix cells depending on the respective geometry of the mineshafts, adits, boreholes or karst/fractured conduits and the locations of the sinks and sources, e.g., direct recharge and fixed heads. Conduit orientations can be freely designed, i.e., they do not necessarily need to be vertical or horizontal along the same continuum layer, so that the model can easily match the actual spatial co-ordinates of real mine networks, boreholes or karst/fractured conduits. It is assumed that the conduit system is fully saturated, an assumption that will hold for most flooded underground mines and saturated karst/fracture aquifers. Flow between the porous matrix and the conduit nodes is described by a linear relationship between the two systems (Barenblatt et al. 1960):

(

)

i

i

= h − h

Γ α

Γi (L³ T^-1): exchange flow rate between conduit node i and the porous matrix cell αi (L² T^-1): exchange coefficient between node i and the porous matrix

h_i (L): hydraulic head at conduit node i

h_i,m (L): hydraulic head in the porous matrix cell where conduit node i is located.

The magnitude of the exchange coefficient αi depends on the hydraulic conductivity of the porous matrix and geometrical factors, determined by the discretisation of the adjacent continuum cell.

Flow in each tube, i.e., flow from one to another conduit node can be determined by using Darcy-Weisbach equation:

g

λi, (-): friction factor of tube j connected to face f of conduit node i.

lj (L): length of tube j dj (L): diameter of tube j

g (L T^-2): earth’s gravitational acceleration

f

Substituting Hagen-Poiseuille equation which is valid for laminar flow

f

and equation 4 into equation 3, an expression for Q_i^f_,j describing laminar flow can be obtained:

For turbulent flow, the implicit Colebrook-White law is used:



Re (-): Reynolds number (=

By substituting the above equation and equation 4 into equation 3, an expression for Q_i^f_,j describing turbulent flow can be obtained (Horlacher & Lüdecke 1992):

j

Conservation of volume at any conduit node i can be determined by using Kirchhoff’s law that should evaluate to zero by summing up all in- and outflow terms at the respective node i:

, 0 addition to flow from the continuum and from the different connecting tubes to a conduit node, other sink/source terms can be applied such as direct recharge and a fixed head to the conduit node.

The sum of in- and outflow at any conduit node i can be computed by using conservation of volume at any node according to the Kirchhoff’s law but splitting up the flow terms after in and out, respectively, i.e.,

∑

The superscripts + and - differentiate in- and outflow terms from each other at conduit node i, respectively. Note that conservation of the volume can easily be

checked by separately summing up Q_iⁱⁿ and Q_i^out while the flow calculations and

dependent upon whether conduit node i is a source or a sink term to the conduit system.

Vi,m (L³): volume of the matrix cell where conduit node i is located (= xi,m·yi,m ·zi,m).

Clemens et al. (1996) or Liedl et al. (2003) provides a more detailed description of the pipe-flow model.