文档介绍：Acta Math. Univ. Comenianae 1
Vol. LXX, 1(2001), pp. 1–13
Proceedings of Algoritmy 2000
ON THE SOLUTION OF SOME INVERSE PROBLEMS
IN POROUS MEDIA FLOW
D. CONSTALES and J. KACURˇ
Abstract. In this paper we discuss the numerical modellingof two inverse prob-
lems in porous media flow. In the first problem, we determine the soil parameters
that occur in the retention and hydraulic permeability curves of the Van Genuchten
ansatz for Richard’s equation. In the second problem we determine the hydraulic
permeability, longitudinal dispersivity and adsorption isotherms using the measure-
ments in a single injection-extraction well.
1. Introduction
The flow in unsaturated porous media is governed by Richard’s equation
(1) ∂tθ=div(k(Ψ) grad Φ)
where θ is a volumetric water content, Ψ is the matric potential (generated by the
capillary pressure) and Φ= ψ+ z is the totalhead including the gravitational
potential z. The hydraulic permeability we denote by k = k(ψ)andbyu we
θ−θr
s
denote the effective saturation, u = θs−θr where θ is the volumetric water content
at saturation and θr is the residualwater content. Van Genuchten [ 18] derived an
empiricalrelationshipbetween u and ψ and between k and ψ(constitutive laws)
in the form
1 1/2 1/m m 2
(2) u = ,k(ψ(u)) = ksu (1 −(1 − u ) ) ,
(1 + (αψ)n)m
where ks is the saturated hydraulic conductivity
ρwg
ks = k0
µ
with k0 the intrinsic permeability (depending on the structure of the porous media
only), ρw and µ are the density and dynamic viscosity of the water and g is the
gravitationalconstant. The values ks, α,0<m<1, n =1/(1 − m)andθr, θs
are soilparameters that have to be determined from additionalmeasurements in
infiltration. We assume that the porous medium is homogeneous. In this case
Received January 10, 2001.
2000 Mathematics Subject Classification. Primary 35R35, 49M15, 49N50.
Key words and phrases. Porous media equation, inverse problems, Richard’s equation.
The authors grate