Evaluation of a first‐order water transfer term for variably saturated dual‐porosity flow models

Abstract

Variably saturated water flow in a dual-porosity medium may be described using two separate flow equations which are coupled by means of a sink source term F•,, to account for the transfer of water between the macropore (or fracture) and soil (or rock) matrix pore systems. In this study we propose a first-order rate expression for Fw which assumes that water transfer is proportional to the difference in pressure head between the two pore systems. A general expression for the transfer coefficient aw was derived using Laplace transforms of the linearized horizontal flow equation. The value of aw could be related to the size and shape of the matrix blocks (or soil aggregates) and to the hydraulic conductivity K a of the matrix at the fracture/matrix interface. The transfer term Fw was evaluated by comparing simulation results with those obtained with equivalent one-and two-dimensional singleporosity flow models. Accurate results were obtained when K a was evaluated using a simple arithmetic average of the interface conductivities associated with the fracture and matrix pressure heads. Results improved when an empirical scaling coefficient yw was included in aw. A single value of 0.4 for 3% was found to be applicable, irrespective of the hydraulic properties or the initial pressure head of the simulated system. INTRODUCTIONDual-or double-porosity models, initially introduced to simulate single-phase flow in fissured groundwater reservoirs [e.g., Barenblatt et al., 1960], assume that a porous medium consists of two separate but connected continua. Of these, one continuum is associated with a system or network of fractures, fissures, macropores, or interaggregate pores, while the other continuum involves the porous matrix blocks or soil aggregates. Hence dual-porosity models usually involve two flow equations which are coupled by means of a sink/source term to account for water transfer between the pore systems.The dual-porosity concept has been popularly used to describe the preferential movement of water and solutes at the macroscopic scale, a phenomenon that is widely believed to occur in most natural (undisturbed) media [e.g., van Genuchten et al., 1990; Gish and Shirmohammadi, 1991; Wang, 1991]. A large number of double-porosity type models have been proposed to predict water flow in fractured reservoirs [Barenblatt et al., 1960; Warren and Root, !963; Duguid and Lee, 1977; Moench, 1984] or solute transport during both steady state flow [Coats and Smith, 1964; van Genuchten and Wierenga, 1976] as well as transient groundwater flow [Bibby, 198!]. Recently, attempts have been made to extend the concept to transient water flow and solute transport in variably saturated fractured rock formations and structured soils [Dykhuizen, 1987; Dudley et al., 1988; Peters and Klavetter, 1988; Jarvis et al., 1991; Chen and Wagenet, 1992a, b; Gerke and van Genuchten, 1993]. One of the most critical components of dual-porosity models is the source/sink term describing the exchange of water between the fracture and the matrix pore s...

Numerical evaluation of a second‐order water transfer term for variably saturated dual‐permeability models

“…On average, a scaling factor g w of 0.4 was deemed to give best results. However, best fit scaling factors differed by up to 5 times for different initial pressure head values between À30 and À3000 cm and varied by 20% or less between medium and fine textured soils [Gerke and van Genuchten, 1993a, Tables 2 and 3]. Even for the best fit scaling factors, the relative cumulative horizontal infiltration was initially underestimated and later overestimated, revealing the inherent limitations of the firstorder approach.…”

“…Geometry was represented by a factor b in equations (7) and (8). Values for b in equation (7) could be derived, for example, by comparing the solution of the first-order transfer term, assuming a certain geometry and a step-function boundary condition, with a Laplace transform of the linearized horizontal water flow equation (e.g., b equals 3 for slab-type matrix blocks) [Gerke and van Genuchten, 1993a]. By analogy, the same b values were used in equation (8).…”

“…First, it is theoretically valid only for quasi-steady water transfer [Zimmerman et al, 1993] and therefore underestimates highly transient transfer flux for early times [e.g., Zimmerman et al, 1993;Gerke and van Genuchten, 1993a]. Second, in a lumped formulation such as equation (1), K m needs to be evaluated as some blockaveraged, effective hydraulic conductivity function of the pressure head in the matrix and fractures that best represents the conductivity of the actual pressure head profile at all times.…”

“…Second, in a lumped formulation such as equation (1), K m needs to be evaluated as some blockaveraged, effective hydraulic conductivity function of the pressure head in the matrix and fractures that best represents the conductivity of the actual pressure head profile at all times. Gerke and van Genuchten [1993a] evaluated the conductivity in equation (1) as an arithmetic average function of pressure heads in both matrix and fractures and scaled the first-order transfer term to match the exact solution at 50% of the cumulative horizontal infiltration. Moreover, they assumed an effective hydraulic conductivity at the fracturematrix interface, K a [L T À1 ], affected by fracture coatings [Gerke and van Genuchten, 1993a], to obtain following modified first-order transfer term:…”

Monte Carlo simulation of radionuclide migration in fractured rock for the performance assessment of radioactive waste repositories

“…These equations appeal to our physical intuition as they represent the balance between the production and the destruction processes for both the solutons and the fracturons populations at ' z in dt . The forward and backward rates in equations (1.a,b) can be determined by analogy with the equivalent dual-permeability advection-dispersion equations [16] [17], whereas the exchange rates can be identified on the basis of detailed physical analyses [16][31] [32] (see Appendix A for more details).…”