Governing equations

The developed approach is to transform the groundwater flow equation first into the frequency domain by means of Fourier transformation. Based upon an asymptotic solution for the flow in the frequency domain, the flow equation can thus be trans-formed into the eikonal equation. Through inverse Fourier transformation, the flow equation will be transformed back into the time domain and will be combined with the solution of the eikonal equation. Through these derivations a relationship between the diffusivity and the travel time of a hydraulic signal can then be established with the following line integral (Vasco et al., 2000; Kulkarni et al., 2001):

2.1 Hydraulic travel time inversion METHODOLOGY OF INVERSION the observation point x2 (receiver) and D is the diffusivity. Based on this line integral, the travel time of a hydraulic signal is directly related to the reciprocal value of diffu-sivity. In the following a brief introduction of the derivation of the proposed proce-dures from Vasco et al. (2000) and Kulkarni et al. (2001) is given.

In a heterogeneous medium, the time (t) and space (x) dependent head h

 

x,t is described by Bear (1972) and de Marsily (1986) with the following equation:

   

where K(x) denotes the hydraulic conductivity and S(x) denotes the storage coef-ficient. This equation can be transformed into the frequency domain by means of the Fourier transformation

Thus the diffusion equation in a heterogenous medium in the frequency domain, which describes the evolution of head H with dependence on frequency  and space x is

The equation 2.5 can be written as

     

, ,

   

, 0

2     

 H x x H x  i x H x  ,

Eq. 2.6

by defining

2.1 Hydraulic travel time inversion METHODOLOGY OF INVERSION

where (x) is the inverse of the diffusivity.

An asymptotic solution of Equation 2.6 is given by Virieux et al. (1994) and Fatemi et al. (1995), by which the medium is assumed to have a smooth variation in conduc-tivity k

 

x :

This kind of asymptotic expansion is the one that has been used in the ray theory except that the i term for wave propagation has been replaced by  i for dif-fusive transport. This replacement can later help simplify equation 2.6 into the eikonal equation. In this expansion

 

x represents the phase of a propagating wave. It corre-sponds to the geometry of a propagating front and has a dimension of square root of time.  is the frequency of the wave and ^An

 

^x are real functions that relate to the amplitude of the wave.

Asymptotic expansions have been widely used in the electromagnetic and wave propagation. The expansion with inverse power of  has the initial terms of the se-ries which represent rapidly varying (large , high frequency) components of the so-lution and the successive terms are associated with lower frequency behaviour (Vasco and Datta-Gupta, 1999). Although this expansion has a sum of an infinite number of functions ^An

 

^x , the propagation of a sharp front is described only by the initial terms of the sum, which can be related to important physical quantities.

In order to obtain expressions for these quantities the sum (Equation 2.9) is sub-stituted into Equation 2.6. This substitution leads to an expression with an infinite number of terms and each term contains i to some order. The equations with terms of i and



^i



² retain one’s attention because solving them will give the propagating phase 

 

x and the amplitude A₀

 

x used for the zero-order term of the solution. These are identical to both the eikonal and the transport equations used for

2.1 Hydraulic travel time inversion METHODOLOGY OF INVERSION

ray tracing of seismic waves and permit us to compute 

 

x and A₀

 

x with any stan-dard ray-tracing program (Virieux et al., 1994; Lambare, 1992). In the case of this study the terms of order



^i



²are considered, leading to the following equation

     

 ₀  

   

₀ 0

 x x A x i x A x

i    .

Eq. 2.10

Assuming that A0(x) and  are unequal to zero, Equation 2.10 can be expressed as follows

     

  0

 x  x x .

Eq. 2.11

Equation 2.11 is known as the eikonal equation, which describes many types of propagation processes e.g. wave propagation. In this case it relates the function 

 

x to the flow properties as contained in (x). A physical interpretation of 

 

x is ob-tained if the zeroth-order term in expansion of Equation 2.9 is considered

 

x, A0

 

x e^{ i (x)} H   ^{ } .

Eq. 2.12

Taking the inverse Fourier transformation with respect to , Virieuxet al.(1994) transformed Equation 2.12 back to the time domain under the assumption of a Dirac source at the origin

The first derivative of Equation 2.13 at a fixed position x

 

^   



^

   ^{   }

^ _ time is a function of 

 

x , the inverse of the diffusivity of the medium. The spatially varying quantity A₀

 

x will generally ensure that the amplitude of the peak drawdown observed at various positions will differ.

2.1 Hydraulic travel time inversion METHODOLOGY OF INVERSION

Now, it is possible to derive a line integral by defining a trajectory s

 

t between a source (x1) starting at t = 0 and a receiver (x2) using a curvilinear coordinate system.

In these coordinates 

 

x only varies with (s) and  is tangent to the s coordinate curve. Thus from Equation 2.11, it is possible to define the following

 

s

where t_peak is the travel time of the pressure from the source to the receiver and D is the diffusivity. Note that this equation is a result of a high frequency assumption.

Since a smooth continuous medium is necessary for the asymptotic approach of Virieux et al. (1994), it is assumed that the permeability and porosity vary smoothly with respect to the spatial wavelength of the propagation (Vireux et al., 1994; Vasco et al., 2000). The resulting limits of this inversion technique based on the asymptotic approach are also discussed in Chapters 3.2 and 4.4.

Im Dokument Hydraulic Tomography: A New Approach Coupling Hydraulic Travel Time, Attenuation and Steady Shape Inversions for High-Spatial Resolution Aquifer Characterization (Seite 25-29)