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 Hydraulic tomography is a method to identify the three-dimensional spatial distribution of hydraulic properties in the subsurface. We propose a tomographic approach providing the inversion of travel times of hydraulic or pneumatic tests conducted in a tomographic array. The inversion is based on the relation between the peak time of a recorded transient pressure curve and the diffusivity of the investigated system. The development of a transformation factor enables the inversion of further travel times besides the peak time of a transient curve. As the early travel times of the curve are mainly related to preferential flow features while the inversion based on late travel times are reflecting an integral behavior, it can be assumed that the different inversion results reflect the properties of the overall system. By comparing the different reconstructions the system interpretation therefore becomes more comprehensive and reliable. Furthermore, the similarity of the proposed hydraulic tomographic approach to seismic travel time tomography allows us to apply the inversion algorithms which are used for seismic tomography. We therefore apply the method of staggered grids, which enables to refine the grid resulting in a higher nominal resolution, to data from a set of interference tests arranged in a tomographic array. The tests were conducted in an unsaturated fractured sandstone cylinder in the laboratory. The three-dimensional reconstructions of the diffusivity distribution are found to be highly reliable and robust. In particular, the mapped fracture of the sandstone cylinder coincides with our reconstructed diffusivity distribution.
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 Hydraulic parameters estimated by conventional aquifer investigation methods like pumping or slug tests lead to integral information averaged over a large volume. This kind of information is insufficient to develop groundwater models requiring detailed information about the spatial distribution of the subsurface. To circumvent this problem tomographical methods are applied. The principle of tomography was developed for medical applications, e.g., to determine the spatial distribution of X-ray attenuation over cross sections of the head or body [Mersereau and Oppenheim, 1974; Scudder, 1978]. The application of geophysical methods in a tomographic array to create a three-dimensional image of the subsurface has been established since several years. It has to be noted, however, that geophysical methods like radar tomography [Davis and Annan, 1989; Hubbard et al., 1997], seismic tomography [Bois et al., 1972; Gelbke, 1988] or electrical tomography [Yorkey et al., 1987; Kohn and Vogelius, 1984] yield a geophysical parameter distribution which does not have to be in accordance with hydraulic properties of the subsurface. Consequently, it is necessary to perform a complicated and ambiguous parameter transformation [Dietrich et al., 1995, 1998, 1999].
 As opposed to geophysical methods, hydraulic tomography allows to directly determine hydraulic subsurface properties. The procedure of hydraulic tomography consists of a series of short-term tests conducted in a tomographical array. Up-to-date, several researchers have evaluated hydraulic tests by using drawdown as a function of time [Bohling, 1993; Gottlieb and Dietrich, 1995; Butler et al., 1999; Yeh and Liu, 2000]. Another approach is based upon the inversion of travel times of pressure changes. This approach follows the procedure of seismic ray tomography. In seismic ray tomography the influence of the velocity field on the travel time between a source and a receiver are described as a line integral relating the travel time to the velocity field. According to that, Vasco et al.  derived a line integral relating the arrival time of a “hydraulic signal” to the reciprocal value of diffusivity. The line integral relates the square root of the drawdown peak arrival time of a transient pressure curve obtained for a Dirac source at the origin directly to the square root of the reciprocal value of diffusivity. The derivation of the line integral is based on an asymptotic approach described in detail by Datta-Gupta et al. , Vasco et al. , Vasco and Datta-Gupta , Datta-Gupta and King , and Virieux et al. . Diffusivity is the quotient of hydraulic conductivity to storage and is a quantitative measure for the response of a formation on hydraulic stimulation.
 In this paper we propose a further development of the travel time line integral. This advanced approach relates any recorded travel time corresponding to the arrival of a certain percentage of the maximum signal (e.g., 1%, 10%, 20% of the maximum amplitude) to the peak time of a signal associated with a Dirac source. This relationship is derived for a Dirac source as well as for a Heaviside source. The advantage of simultaneously inverting several travel times is to exploit their different information content. As the signal propagation in a hydraulic tomographic experiment follows Fermat's principle, early arrivals characterizing the initial part of a signal, follow the fastest pathways between source and receiver. The fastest pathways are usually identical to preferential flow paths, and thus early travel times are dominated by preferential flow (if available). Whereas, late travel times, characterizing the final part of a signal, reflect integral behavior more or less, since the pressure difference between source and receiver stretches out over the complete system. Hence it can be assumed that the different inversion results provide improved knowledge about the properties of the overall system. Additionally, the comparison of the inversion results allows the identification of potential artifacts. For the inversion we have applied a least squares based inverse procedure which is established since several years in seismic tomography.
 Beyond this, the inversion results depend on the spatial position of the used grid because the hydraulic properties are averaged over one voxel. On account of this it could be that little inhomogeneities are not resolved because of the average effect within one voxel. Practically it is not possible to refine the grid as far as the averaging effects can be neglected because for each voxel, one model parameter has to be determined. Thus the number of performed short term tests is the upper limit for the model parameters we can safely invert [Menke, 1984]. To circumvent this problem without using any a priori information we use the method of staggered grids, which Vesnaver and Böhm  proposed for the inversion of seismic tomographic data. Especially, for hydraulic travel time tomography the method of staggered grids is of great interest, because the time exposure for the data acquisition is much higher in comparison to geophysical methods. By the method of staggered grids, the initial grid is displaced in several ways by a known factor. For each grid we get a slightly different image of the parameter distribution because inside each voxel we get a different averaged value. Finally, we have determined the arithmetic mean of the parameter distribution of all grids by staggering them. This leads to a refining of the grid and to a better resolved image without conducting additionally measurements or using a priori information.
 The introduced procedure has been applied to data from a set of pneumatic short term tests conducted in an unsaturated fractured sandstone cylinder. We have performed pneumatic instead of hydraulic short term tests in order to reduce the duration of the experiments. The experimental setup was designed to produce a streamline pattern covering the complete cylinder, whereby the source can be considered as a point source. The produced streamline patterns can be compared to the ray patterns of geophysical cross-hole tomography experiments. These recorded data make it possible to reconstruct a three-dimensional image of the diffusivity distribution of the cylinder and to verify the results by geological surface mapping. In this paper we use the terminology hydraulic instead of pneumatic tomography because the practical application of this method is the high resolution characterization of aquifers.
where tpeak is the travel time of the peak of a Dirac signal from the point x1 (source) to the observation point x2 (receiver) and D is the diffusivity (see Figure 1a for a Dirac impulse at the source). Equation (1) relates the travel time tpeak of the output signal to the diffusivity. For a Dirac source we develop a transformation factor relating each travel time corresponding to a certain percentage of the peak height to tpeak (see below). The arrows in Figure 1b indicate that the transformation factor has to be larger than 1 if the travel time is less than tpeak (part I) and if the travel times are larger than tpeak the transformation factor has to be less than 1 (part II).
 With regard to practicability, experiments were designed for a Heaviside source (Figure 2a). In order to avoid the differentiation of each recorded signal we derive a conversion factor providing a direct relationship between the travel times associated with a Dirac and a Heaviside source (Figure 2b).
2.1. Derivation of the Transformation Factor for a Dirac Source
 The diffusion equation in a homogeneous medium describes the evolution of head h in dependence of time t and space r
where S denotes the storage coefficient and k the hydraulic conductivity. Spherical coordinates are used here because each injection port represents a point source and the signal can be assumed to spread radially due to the opened measuring ports (see section 3 below). After the work by Häfner et al. , the solution of the diffusion equation in an infinite domain for a Dirac source is
where V is the product of the flow rate Q and the input time and hd (r, t) is the hydraulic head depending on space and time. The subscript d stands for a Dirac source. The peak time of a pressure pulse can be determined by means of the first derivative of equation (3)
The first derivative becomes zero when . Consequently, the peak time tpeak can be expressed in dependence of the hydraulic properties S and k and the distance r. The amplitude of the signal can be determined by inserting tpeak into equation (3):
In order to calculate the above mentioned transformation factor we introduce the head ratio αd via
The head ratio αd enables the comparison of the peak time with the travel times when the hydraulic signal exceeds or falls below some specified percentage of the maximum amplitude hpeak encountered at the receiver (Figure 3a). Using equations (3) and (5), the transformation factor for the corresponding travel time tα,d is given by
Equation (7) can be solved numerically for fα,d and is plotted in Figure 3b. The plot shows the quotient of which corresponds to the transformation factor fα,d in dependence of the head ratio αd. Two cases must be distinguished because each head ratio amplitude is associated with two travel times tα,d,1 and tα,d,2 (Figure 3a). As the travel time tα,d,1 is less than the peak travel time tpeak the transformation factor has to be greater than 1 and for tα,d,2 the transformation factor has to be less than 1 (Figure 3b). The general form of the line integral for a Dirac source can be written for tα,d as follows
Equation (8) demonstrates that it is possible to relate any recorded travel time tα,d with the diffusivity D by using the corresponding transformation factor fα,d.
2.2. Derivation of the Conversion Factor for a Heaviside Source
 For technical reasons a Heaviside source is easier to be put in practice, especially for the realization of an adequate signal strength needed for long distances. In order to avoid the differentiation of each recorded curve a conversion factor relating Heaviside and Dirac sources is developed. The solution of the homogeneous flow equation using a Heaviside source for an infinite domain is [Häfner et al., 1992]
where is equivalent to the maximum height hmax of the recorded signal. The subscript h stands for a Heaviside source. In accordance to the derivation of the transformation factor for a Dirac source we introduce a head ratio αh in order to convert the recorded travel times of a signal from a Heaviside source at the origin into the peak time of a signal from a Dirac source (Figure 4a). We have
with the conversion factor . Equation (11) is plotted in Figure 4b. By means of this graph we can determine the conversion factor fα,h for each travel time of a recorded signal with a Heaviside source at the origin. The conversion factor fα,h becomes 1 when the height of the signal is around 8.36% of hmax. This time is equivalent to the peak arrival time of a signal from a Dirac source. As expected, Figure 4b shows that all travel times being less than the peak time have to be multiplied by a conversion factor larger than 1. For larger travel times the conversion factor is less than 1. Therefore it is possible to convert all travel times of a hydraulic signal from a Heaviside source to the peak time of a Dirac signal. Furthermore, this conversion does not require differentiation or integration of the recorded signal, thus it is not necessary to smooth or fit disturbed data. The line integral for a Heaviside source can be written in accordance with equation (8) by using the deduced conversion factor fα, h as follows
3. Data Acquisition
 The presented methodology was applied to data from a set of cross-well pneumatic tests conducted in an unsaturated sandstone cylinder with a height of 34 cm and a diameter of 31 cm. For the recovery of the cylinder a Stubensandstein formation was chosen which is quarried in the southern part of Germany. This formation is part of a continental alluvial depositional system [Hornung and Aigner, 1999] and is mainly composed of arkose sandstone. The sample was situated in a bed load channel dominated facies. The sandstone cylinder was chosen out of a series of similar samples because of its simple structural composition exhibiting a single fracture is embedded in a more or less homogeneous matrix. Figure 5 illustrates the investigated unsaturated fractured sandstone cylinder prior to the preparation for the experiments. The potential position of the fracture was mapped and is illustrated in Figure 6. The porosity is approximately 8% and the fracture aperture ranges from 0.1 mm to 0.3 mm. In order to obtain fully controllable boundary conditions the cylinder was sealed with epoxy resin of approximately 5 mm thickness. To enable access to the sample 32 injection (source) and measuring (receiver) ports, respectively, attached in a regular grid onto the cylinder. The ports consist of circular metal plugs with 3 cm in diameter defining controlled input/output areas. In the center of each plastic plug a valve was attached allowing plugging in a tube. The setup was originally developed by McDermott  and it can be easily adapted to field problems by using multilevel packer systems [Butler et al., 1999].
 In order to receive a huge amount of experiments in a short time we have decided to conduct pneumatic instead of hydraulic tests. For the pneumatic experiments a fully automated multipurpose-measuring device with online data acquisition was developed [Leven, 2002]. The experimental setup is shown in Figure 7. At the injection port, which can be described as a point source, we have applied a Heaviside impulse. Thereby compressed air is injected into the unsaturated sandstone block and the flow rate is recorded at the measuring port. All other ports are opened allowing the release of the injected air. The flow rate is recorded until a stationary flow field is established between the injection and the measuring port. It takes about 20 seconds to reach steady state conditions. The pressure difference between the injection port and the measuring port is kept constant during the experiment by applying 0.5 bar at the injection port.
 In Figure 8, all 487 measured port-port-connections are indicated by white lines. Because of the experimental setup (infinitesimal distance between inside and outside of the cylinder) the gradient of the pressure is proportional to the absolute value of pressure. Consequently, not only the gradient of the pressure but also the pressure itself is proportional to the flow rate. For this reason it is possible to use the travel times of a pressure curve as well as the travel times of a flow rate curve for the inversion.
 With regard to practical applications the impact of two effects on travel times was investigated. First, it was found that compressibility can be neglected in our approach due to the modest pressure difference of 0.5 bar which is applied in the proposed setup. Transferring results from Jaritz , compressibility effects will alter travel times by less than 10% for the pressure difference applied. In addition, compressibility plays an even less important role in practical applications when experiments are conducted with water instead of gas.
 The second effect to be investigated is the relationship between pressure difference and hydraulic conductivity. As the latter often varies over several orders of magnitudes in natural systems, it is useful to adjust the pressure difference in order to generate a signal with adequate strength. The influence of the outflow signal strength was estimated by conducting several measurements with 0.3 and 0.5 bar pressure differences. The recorded curves were normalized to the maximum value of the flow rate (Figures 9a and 9b). The comparison of the normalized curves shows no evident differences among each other. For a better comparison of the curves we have plotted the travel times when 1%, 2%, 5%, 10%, 20%, 30% and 40% of the maximum is reached (Figure 9c). The plotted travel times can be fitted perfectly with a 45° line through the origin. This result shows the travel time to be independent of the signal strength. With decreasing pressure difference the part of the recorded curves reflecting the steady state condition becomes more diffuse (Figure 9a). This effect is due to the sensitivity of the measurement device, as the absolute value of the flow rate decreases with the pressure difference. For this reason we have decided to conduct the measurements with a pressure difference of 0.5 bar.
4. Discretization and Inversion
 The simplest linear inverse problem is an “Even-Determined Problem”. In even-determined problems there is just sufficient information to uniquely determine the model parameters. This unique solution has zero prediction error [Menke, 1989]. Thus the goal of the discretization is to adapt the number of cells or voxels to the number of measurements. In practice this can not be achieved because the streamline density depends on the spatial distribution of the sources and receivers and the parameter distribution itself. Consequently, the streamline density is varied from voxel to voxel. Thus we have to solve a “mixed-determined problem” where the imperfect angular coverage of the streamlines might lead to a mathematically nonunique solution [Menke, 1984; Jackson and Tweeton, 1994]. Resulting reconstructed parameter distributions might be characterized by smearing and ambiguity. In order to minimize these effects of the space dependent streamline density we have confined our investigation area of the central part of the cylindrical sample shown in Figure 8 (zmin = 6 cm, zmax = 27 cm) because the angular aperture covering the top and bottom is limited. The variations in streamline density are least in the central part. The central part was discretized by 500 voxels, 10 in x direction, 10 in y direction and 5 in z direction. We have chosen a higher number of voxels in x and y direction expecting larger lateral changes than vertical changes because of the position of the fracture being assumed to be nearly parallel to the axis of the cylinder.
 Furthermore, the results of tomographic investigation depend on the spatial position of the grid. The geological properties are averaged over one voxel. Thus it is possible that important properties are positioned on the border between voxels and so the anomaly is separated into two or more voxels. This can lead to an apparent dilution of the anomaly. In order to avoid such dilution effects by an unlucky choice of the grid position we apply the method of staggered grids proposed by Vesnaver and Böhm  for seismic tomography. This method is based on different viewpoints realized by shifting the grid. We have performed such a displacement of the initial grid three times in two directions (Figure 10). The displacements Δx and Δy are half of the voxel length in x- and y-direction, respectively. For each grid position we get a slightly different image of the cylinder because inside each voxel we average over a different volume. We have inverted each travel time four times and afterward we have determined the arithmetic average of all grids by staggering them. As a result, the final grid is composed of 2000 voxels, 20 in x-direction, 20 in y-direction and 5 in z-direction. This leads to a more complex image with a higher nominal resolution [Vesnaver and Böhm, 2000]. Another possibility to overcome the problem of averaging and dilution effects would have been to directly invert a more refined model but in this case therefore a larger number of measurements would have been required because the number of voxels for which inversion can be done reliably is limited by the number of performed measurements.
 The inversion of the defined line integrals was conducted with the commercial software package GeoTom 3D, which is based on the Bureau of Mines tomography program 3DTOM [Jackson and Tweeton, 1996]. Originally, the program was developed for seismic travel time tomography. Usually, seismic travel time tomography is based on the solution of the following line integral relating the travel time t, recorded between a source and a receiver, with the spatial velocity v distribution of an investigated area:
The comparison of equation (13) with equations (12) and (8) shows the similarity between both methods enabling to use the same application software. The program uses a least squares solution of a linear inverse problem. The algorithm is called SIRT (simultaneous iterative reconstruction technique) and belongs to the group of series expansion methods. These methods allow curved ray paths and streamline trajectories, respectively, through the target area and are therefore well suited for applications in seismic and hydraulic tomography. An explicit discussion of the series expansion methods and different ray bending algorithms would be beyond the scope of this paper. Interested readers are referred to the studies by Jackson and Tweeton , Um and Thurber , Dines and Lytle , Gilbert , and Lo and Inderwiesen . We like only mention, that the inversion procedure is based on a linearization of the proposed travel time integral. Thereby, the weight of each cell corresponds to the path length inside the cell. The path length is calculated for each iteration step based on the “diffusivity” distribution obtained in the previous step. In the case of the travel time based hydraulic tomographic the inversion procedure was used in order to choose a diffusivity distribution which minimizes the following functional , whereas tie is the estimated travel time, tim the measured travel time for ith measurement and n the number of measurements.
 In Figure 11 we show the overall residual for 10 iteration steps of the image discussed in section 5 (Figure 13). The residual decreases in a quasi exponential manner as iterations proceed and the values are nearly constant after 10 iteration steps, whereby the initial fit to the measured data is based on a uniform preliminary model. In this example the value of is 1396.26 s0.5. The comparison between and residual of the tenth iteration step (0.82 s0.5) indicates that ten iteration steps reduce the discrepancy to a very small value. A larger number of iteration steps lead to small fluctuations. On this account the inversion results proposed in section 5 are conducted with ten iteration steps. The inversion takes around 110 seconds on a 1 GHz Pentium CPU.
 In this section we present the inversion results showing the three-dimensional reconstruction of the diffusivity field of the investigated sandstone cylinder. The cylinder is characterized by a fracture ranging from top to bottom. The fracture is embedded in a more or less homogeneous matrix and it can be assumed that in this setting pronounced preferential flow may be observed. First, we discuss the results obtained by using the method of staggered grids. Next, we interpret the inversion results based on different travel times by comparing them among each other.
5.1. Staggered Grids
Figure 12 compares the inversion results for a staggered and four conventional horizontal grids at a selected slices (z = 11.3 cm in Figure 8). For staggering, the initial grid was displaced three times in x and y direction by half of the voxel length. The inversion was conducted using the travel times when 8.36% of the maximum flow rate was recorded at the receiver port. This travel time was chosen because it is equivalent to the peak time of a Dirac source (section 2.2). All inversion results show that significant variations in diffusivity are present whereby the largest diffusivity values coincide perfectly with the potential position of the fracture marked by a black line. The diffusivity distribution agrees with the well known fact that fractures usually have larger diffusivity values due to their higher conductivity and lower ability of storage. The comparison of the different slices in Figure 12 indicates that the method of staggered grids may lead to an improvement in resolution.
 The diffusivity pattern of the staggered grid is arranged systematically around the potential position of the fracture, while the diffusivity patterns of the conventional grids appear to be more random. The more systematic array of the staggered grid leads to a more precise, reliable and undisturbed image with reduced smearing effects. The random distribution of the yellow and red voxels of the conventional grids hamper the geological interpretation and reliability of the diffusivity distribution.
5.2. Inversion Results Based on Different Travel Times
 The advantage of inverting more than one value of the recorded signals is the possibility to verify and evaluate the reliability of the reconstructed parameter field by comparing them among each other. Thus it is possible to detect potential artifacts and to estimate the reliability of the anomalies. In order to illustrate the processing three inversion results are imaged (Figures 13, 14, and 15) based on travel times when 8.36%, 1% and 40% of the maximum flow rate was recorded. The travel times, when 1% and 40% of the maximum were recorded, are chosen to demonstrate the different information content of early and late travel times. The 1% mark equals the initial increase of the recorded signal which just can be determined reliably. Hence it can be assumed that the inversion based on the first arrivals are mainly dominated by preferential flow paths. It has to be noted, that usually under field conditions later travel times have to be chosen since the recorded signals are more influenced by noise.
 It was possible to image the different inversion results with the same scale although the travel times differ by more than one order of magnitude. This result points out the reliability of the developed method. All inversion results exhibit a high diffusivity feature crossing the complete sandstone cylinder. The spatial position and the shape of this zone agree with the location of the mapped fracture. In the slice taken at z = 0.218 m of Figure 13 an anomaly is imaged which could be interpreted as a small fracture connected with the main fracture under an angle of 45 degrees. This anomaly, however, cannot be recognized in Figure 14. Hence it is assumed that the described anomaly is an artifact or at least has to be interpreted carefully. In this way, the reliability of each anomaly can be verified.
 As the inversion results depend on the chosen travel time it can be assumed that each travel time is dominated by different components (matrix, fracture) of the investigated system. By the comparison of several inversion results the interpretation and verification becomes more comprehensive as the tomograms reflect the same investigated system but different details of it. The inversion using the travel time when 1% of the maximum flow rate is reached shows the best agreement between the high diffusivity zone and the mapped fracture of all inversions. This result agrees with our expectations that the first arrivals of a signal are dominated by preferential flow paths. In Figure 15 the influence of the matrix is much stronger in comparison to the other two inversions and consequently the high diffusivity zone features more smearing effects. By comparing the slice at z = 0.270 m of the three inversions it is remarkable that the high diffusivity zone is imaged much better in Figures 13 and 15. An explanation might be that the mapped fracture is partly filled and thus the later travel times are more capable to reconstruct this part of the fracture.
6. Summary and Conclusions
 The proposed travel time based hydraulic tomographic approach allows the inversion of several travel times of a pneumatic or hydraulic signal. A transformation factor derived in this study establishes the relationship of any travel time associated with a Dirac source with the corresponding peak time. In addition, a conversion factor was developed for a Heaviside source which is easier to be put in practice for technical reasons. The similarity of the theory between seismic and hydraulic travel time tomography allows us to use the same algorithms for inversion. Hence it is possible to use commercial software packages originally developed for seismic travel time tomography. The methodology was applied to data from a set of cross-well pressure tests (487 measurements) conducted in an unsaturated fractured sandstone cylinder. Three different inversions were performed in order to reconstruct the three-dimensional diffusivity distribution. Although the travel times used for the inversions differ by more than one order of magnitude it was possible to use the same scale for imaging the diffusivity tomograms. The interpretation of the three-dimensional diffusivity distribution was shown to become more comprehensive as the tomograms adapted from the inversions of different travel times reveal different details of the same investigated system. The inversions based on early travel times are mainly related to preferential flow features while the inversions based on late travel times reflect an integral behavior. Beyond this, we have applied the method of staggered grids. Thereby, we have shifted the grid three times and have staggered them in order to obtain more precise images reducing dilution and smearing effects. The images were found to be highly reliable and robust. It is planned to transfer the proposed approach to the field as it can be applied without any restrictions to real aquifers keeping in mind that pneumatic and hydraulic short-term tests are based on the same flow equations.
 The investigations were conducted with the financial support of the German Research Foundation to the “Hard Rock Aquifer Analogue: Experiments and Modeling” project under grant DI833/1.