In this paper, a 2-D depth averaged flow model has been developed taking into account the effects of the aperture variation on the natural rock joint. First, we confirm the model verification using the 1-D sinusoidal model. Then, we carry out the estimation of flow behavior in a single rock joint under direct shear processes using our developed 2-D flow model. The model is verified by comparing the calculated results to 1-D analytical solution of an idealized sinusoidal surface joint roughness under the simple hydraulic conditions. Through this comparison, it is confirmed that there exists good agreement between numerical and analytical solution. And, an increase of amplitude-to-wavelength ratio (increase of surface roughness) caused a proportionate decrease of discharge can be confirmed.
Flow in fractures was initially estimated using the conceptually parallel plate model [1]. In parallel plate model an individual fracture is represented by two infinite smooth parallel plates separated by constant distance (aperture) between them. The flow is assumed to be laminar with a parabolic velocity profile across the aperture. This led to the well-know cubic law [2] relating fluid flux to aperture as follows:
[Equation available in full paper]
where Q is the volumetric flow rate, W is the width of the fracture perpendicular to flow, D is the aperture size, µ is the fluid viscosity and p is the fluid driving pressure. The important implication of the cubic law is that the fluid is characterized by the separation distance (aperture) although the velocity varies across the distance. Real fractures, however, have rough (irregular) surface walls. Therefore, variable apertures in addition, locations exist where the two surfaces of the fracture may come into contact, thereby creating zero aperture size. In considering the effect of aperture variation, a simplified form of Navier-Stokes equations the Reynolds equation has always been used as follows [3 ~ 6]:
[Equation available in full paper]
where p is the pressure and D(x,y) is the aperture size at coordinates x, y. Theoretically speaking, solving the Navier-Stokes equations under complicated fracture surfaces will provide details on pressure and flow velocity distributions in fractures and avoid restrictions involved in using the cubic law and Reynolds equation and thus estimate the flow in fractures more correctly. The solution of Navier-Stokes equations, however, is by no means computationally straight forward.
This paper is concerned with developing the depth averaged flow model for estimating flows in single fractured joint of the laboratory experiments. The authors were motivated to perform this research basing on the fact that the herein derived model includes inertia term, viscous term and fracture surface variation components which could not be incorporated in the previous models (i.e. cubic law and Reynolds equation). We believe this approach avoids the restrictions involved in the using the Reynolds equation and is computationally tractable.
