Naftaly, A. Scher, S. Jiang and B.

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One-dimensional finite element method solution of a class of integro-differential equations: Application to non-Fickian transport in disordered media, Transport in Porous Media , , , doi Porta, A. Guadagnini, H. Yaron, B. Menahem, A. Transport of gadolinium- and arsenic-based pharmaceuticals in saturated soil under various redox conditions, Chemosphere , , , doi Raveh-Rubin, S.

Edery, I. Dror, H. Ciriello, V. Edery, A. Multimodel framework for characterization of transport in porous media, Water Resources Research , 51, , doi Dror, I. Interpretation and nonuniqueness of CTRW transition distributions: Insights from an alternative solute transport formulation, Advances in Water Resources , 74, , doi Kapetas, L. First-principles derivation of reactive transport modeling parameters for particle tracking and PDE approaches, Advances in Water Resources , 66, , doi Origins of anomalous transport in disordered media: Structural and dynamic controls, Water Resources Research , 50 , , doi Detection, fate and transport of estrogen family hormones in soil, Chemosphere , 95, , doi Frenk, S.

Ben-Moshe, I. Dror, B. Berkowitz and D. Minz Guadagnini, V. Di Federico, Y. Edery and B. Comparative analysis of formulations for conservative transport in porous media through sensitivity-based parameter calibration, Water Resources Research , 49, , doi Catalytic degradation of brominated flame retardants by copper oxide nanoparticles, Chemosphere , 93, ,doi Nowamooz, A. Radilla, M. Fourar and B. Non-Fickian transport in transparent replicas of rough-walled rock fractures, Transport in Porous Media , 98, , doi Kostinski, S.

Majumdar and B. Record-breaking statistics for random walks in the presence of measurement error and noise, Physical Review Letters , , , doi Bianchi Janetti, E. Dror, M. Riva, A. Guadagnini, X. Sanchez-Vila and B.

## Convective and Advective Heat Transfer in Geological Systems

Berkowitz, Y. Edery, H. Fickian and non-Fickian diffusion with bimolecular reactions, Physical Review E , 87, , doi Reactive transport in disordered media: Role of fluctuations in interpretation of laboratory experiments, Advances in Water Resources , 51, , doi Ben-Moshe, T. Frenk, I.

Dror, D. Minz and B. Effects of metal oxide nanoparticles on soil properties, Chemosphere , 90, , doi Gouet-Kaplan, M. Arye and B. Merom Jacov, A. Cortis and B. Sagee, O. Rubin, S. Fink, L. Enrofloxacin oxidative degradation facilitated by metal oxide nanoparticles, Chemosphere , 86, , doi Kostinski and B. Record setting during dispersive transport in porous media, Geophysical Research Letters , 38, L, doi Kuntz, B. Rubin, B. Berkowitz and K. Singha Dissolution and precipitation dynamics during dedolomitization, Water Resources Research , 47, W, doi Measurements of interactions between resident and infiltrating water in a lattice micromodel, Vadose Zone Journal , 10, , doi Bijeljic, B.

Rubin, H. Katz, G.

## Convective and Advective Heat Transfer in Geological Systems

Berkowitz, A. Guadagnini and M. Saaltink Arye, G. Fate and transport of carbamazepine in soil aquifer treatment SAT infiltration basin soils, Chemosphere , 82, , doi Scher, H. Willbrand and B. Transport equation evaluation of coupled continuous time random walks, Journal of Statistical Physics , , , doi Transport of metal oxide nanoparticles in saturated porous media, Chemosphere , 81, , doi Particle tracking model of bimolecular reactive transport in porous media, Water Resources Research , 46, W, doi Srinivasan, G.

Tartakovsky, M. Dentz, H. Viswanathan, B. Berkowitz and B. Robinson Random walk particle tracking simulations of non-Fickian transport in heterogeneous media, Journal of Computational Physics , , , doi Transport in disordered media with spatially nonuniform fields, Physical Review E , 81, , doi Anomalous transport in correlated velocity fields, Physical Review E , 81, , doi Contaminant geochemistry - a new perspective, Naturwissenschaften , 97, , doi Tartakovsky and B.

Simulation of the interplay between resident and infiltrating water in partially saturated porous media, Water Resources Research , 45, W, doi Nelkenbaum, E. Reductive dechlorination of atrazine catalyzed by metalloporphyrins, Chemosphere , 75, , doi Guadagnini, A. Sanchez-Vila, M. Saaltink, M. Bussini and B. Berkowitz , Application of a mixing-ratios based formulation to model mixing-driven dissolution experiments, Advances in Water Resources , 32, , doi Exploring the nature of non-Fickian transport in laboratory experiments, Advances in Water Resources , 32, , doi Cortis, I.

Dror and H. Laboratory experiments on dispersive transport across interfaces: The role of flow direction, Water Resources Research, 45, W, doi Modeling bimolecular reactions and transport in porous media, Geophysical Research Letters , 36, L, doi Oxidation of organic pollutants in aqueous solutions by nanosized copper oxide catalysts, Applied Catalysis B: Environmental , 85, , doi Dentz, M. Scher, D. Holder and B. Transport behavior of coupled continuous-time random walks, Physical Review E , 78, , doi Dror, and B.

Contaminant-induced irreversible changes in properties of the soil-vadose-aquifer zone: An overview, Chemosphere , 71, , doi Holder, D. Numerical study of diffusion on a random-mixed-bond lattice, Physical Review E , 77, , doi Emmanuel and H. Non-Fickian transport and multiple-rate mass transfer in porous media, Water Resources Research , 44, W, doi Yaron-Marcovich, D.

Behavior and stability of organic contaminant droplets in aqueous solutions, Chemosphere , 69, , doi Ovdat, H. Pore-scale imbibition experiments in dry and prewetted porous media, Advances in Water Resources , 30, , doi Emmanuel, S. Phase separation and convection in heterogeneous porous media: implications for seafloor hydrothermal systems, Journal of Geophysical Research , , B, doi Kapiluto, Y.

Yakir, P. Tans and B. Experimental and numerical studies of the 18 O exchange between CO 2 and water in the atmosphere-soil invasion flux, Geochimica Cosmica Acta , 71 11 , , doi Reductive hydrogenation of polycyclic aromatic hydrocarbons catalyzed by metalloporphyrins, Chemosphere , 68, , doi Effects of pore-size controlled solubility on reactive transport in heterogeneous rock, Geophysical Research Letters , 34, L, doi Continuous time random walks and heat transfer in porous media, Transport in Porous Media , 67, , doi An experimental analogue for convection and phase separation in hydrothermal systems, Journal of Geophysical Research - Solid Earth , , B, doi Cortis, M.

Dentz and H. Modeling non-Fickian transport in geological formations as a continuous time random walk, Reviews of Geophysics , 44, RG, doi Pore-scale study of drainage displacement under combined capillary and gravity effects in index-matched porous media, Water Resources Research , 42, W, doi Suppression and stimulation of seafloor hydrothermal convection by exothermic mineral hydration, Earth and Planetary Science Letters , , , doi Hornung, G.

Berkowitz and N. Barkai Morphogen gradient formation in a complex environment: An anomalous diffusion model, Physical Review E , 72, , doi Exact effective transport dynamics in a one-dimensional random environment, Physical Review E , 72, , doi Erratum: Physical Review E, 81, E , Amitay-Rosen, T. Magnetic resonance imaging and quantitative analysis of particle deposition in porous media, Environmental Science and Technology , 39, Cortis, A. Singurindy, O.

The role of fractures on coupled dissolution and precipitation patterns in carbonate rocks, Advances in Water Resources , 28, Mixing-induced precipitation and porosity evolution in porous media, Advances in Water Resources , 28, Baram and B. Use of nanosized catalysts for transformation of chloro-organic pollutants, Environmental Science and Technology , 39, Dedolomitization and flow in fractures, Geophysical Research Letters , 31 24 , L, doi Chen, H.

Quantitative characterization of pore-scale disorder effects on transport in "homogeneous" granular media, Physical Review E , 70, , doi Berkowitz and S. Gorelick Effects of air injection on flow through porous media: Observations and analyses of laboratory-scale processes, Water Resources Research , 40, W, doi Anomalous transport in ''classical'' soil and sand columns, Soil Science Society of America Journal , 68, Erratum: 69, Silliman and A. Dunn Impact of the capillary fringe on local flow, chemical migration, and microbiology, Vadose Zone Journal , 3, Berkowitz and R.

Lowell Carbonate dissolution and precipitation in coastal environments: Laboratory analysis and theoretical consideration, Water Resources Research , 40, W, doi Erratum: 41, W, doi Gallo, H. Numerical simulation of non-Fickian transport in geological formations with multiple-scale heterogeneities, Water Resources Research , 40, W, doi Diffusion in multicomponent systems: a free energy approach, Chemical Physics , , Cortis, H. Time behavior of solute transport in heterogeneous media: Transition from anomalous to normal transport, Advances in Water Resources , 27 2 , , doi Margolin, G.

Continuous time random walks revisited: First passage time and spatial distributions, Physica A , , Dentz and B. Continuous time random walk and multirate mass transfer modeling of sorption, Chemical Physics , 1 , Park, Y. Lee, G. Kosakowski and B. Transport behavior in three-dimensional fracture intersections, Water Resources Research , 39 8 , , doi Flow, dissolution, and precipitation in dolomite, Water Resources Research , 39 6 , , doi Levy, M.

Measurement and analysis of non-Fickian dispersion in heterogeneous porous media, Journal of Contaminant Hydrology , 64 , Transport behavior of a passive solute in continuous time random walks and multirate mass transfer, Water Resources Research , 39 5 , , doi Amitay, B.

Salt-pump mechanism for contaminant intrusion into coastal aquifers, Science , , Also: Reply to Comment, Science , , c, contains new material. Singurindy and R. Mixing-driven diagenesis and mineral deposition: CaCO 3 precipitation in salt water - fresh water mixing zones, Geophysical Research Letters , 30 5 , , doi Wang, Y.

## Profile Pages

Wang, A. Warshawsky and B. Evolution of hydraulic conductivity by precipitation and dissolution in carbonate rock, Water Resources Research , 39 1 , , doi Characterizing flow and transport in fractured geological media: A review, Advances in Water Resources, 25 , Margolin and B. Towards a unified framework for anomalous transport in heterogeneous media, Chemical Physics , , Klafter, R. Metzler, and H. Physical pictures of transport in heterogeneous media: Advection-dispersion, random-walk and fractional derivative formulations, Water Resources Research , 38 10 , , doi Thorenz, C.

Kosakowski, O. Kolditz and B. An experimental and numerical investigation of saltwater movement in coupled saturated-partially saturated systems, Water Resources Research , 38 6 , doi Vilensky, M. Berkowitz and A. Warshawsky In situ remediation of groundwater contaminated by heavy and transition metal ions by selective ion exchange methods, Environmental Science and Technology , 36 8 , Margolin, Metzler, R.

Klafter, and B. The dynamical foundation of fractal stream chemistry: The origin of extremely long retention times, Geophysical Research Letters, 29 5 , doi Silliman, S. Berkowitz, J. Simunek and M. Fluid flow and solute migration within the capillary fringe. Ground Water , 40 1 , Spatial behavior of anomalous transport, Physical Review E , 65, , , doi Dijk, P. Berkowitz and Y.

Yechieli Measurement and analysis of dissolution patterns in rock fractures, Water Resources Research , 38 2 , doi Davy and B. Advective transport in the percolation backbone in two dimensions, Physical Review E, 64, , Hansen A numerical study of the distribution of water in partially saturated porous rock, Transport in Porous Media , 45 2 , Lee and B. Transport and intersection mixing in random fracture networks with power law length distributions, Water Resources Research , 37 10 , Bonnet, E.

Bour, N. Odling, P. Davy, I. Main, P. While not being neither the most general nor the most efficient methods, Bayesian inversion based on the calculation of the Jacobian of a given forward model can be used to evaluate many quantities useful in this process. The calculation of the Jacobian, however, is computationally expensive and, if done by divided differences, prone to truncation error.

Here, automatic differentiation can be used to produce derivative code by source transformation of an existing forward model. We describe this process for a coupled fluid flow and heat transport finite difference code, which is used in a Bayesian inverse scheme to estimate thermal and hydraulic properties and boundary conditions form measured hydraulic potentials and temperatures.

The resulting derivative code was validated by comparison to simple analytical solutions and divided differences. Synthetic examples from different flow regimes demonstrate the use of the inverse scheme, and its behaviour in different configurations. Most of the methods applied will produce estimates of error associated with the parameters determined as a by-product of the inversion technique.

Inverse techniques usually make use of the definition of an objective function, whose value is minimized with respect to the parameters to be estimated. Most often, the observed data are sparse, in fact too sparse to determine the model alone. Therefore, this function will present a weighted measure of data fit combined with another term, which in some way incorporates our prior knowledge. Tarantola has given a comprehensive survey on the basic theory from a Bayesian viewpoint, while a more deterministic viewpoint is summarized by Parker Usually the latter are based on regularized least-squares techniques, which in turn make use of the differentiability of the objective function.

Depending on the special problem considered, deterministic methods that are used in this study can be much more efficient than their probabilistic counterparts. In deterministic methods, in the sense defined above, the differentiation of the forward modelling function g p of the vector of parameters p often cannot be evaluated easily: it may be very complex, including many, possibly non-linear, physical processes, modelled by finite numerical methods like finite difference FD schemes.

In this case, the evaluation of the parameter derivatives necessary for the non-linear optimization may be difficult. The calculation of sensitivities by divided differences is prone to additional truncation error, and may consume large amounts of computing time when the dimension of p is large. Here, the use of adjoint schemes see e. Automatic differentiation AD is an important technique for automatically generating programs to evaluate derivatives.

In AD, a computer program evaluating some function representing a forward problem is mechanically transformed into another computer program capable of evaluating the Jacobian or higher order derivatives of the given function Griewank ; Rall It does not only produce derivatives free of additional truncation error, but also allows for the flexible integration of new physical processes or constraints into a given code without increasing the human effort to produce the corresponding code for the computation of derivatives.

Thus this method is exceptionally adapted to the numerical needs in the geosciences, where complex geometry, advanced physics and coupling of processes are important. Surprisingly, AD techniques has not yet found its way to solid earth geosciences and hydrogeology. An extensive overview of AD publications may be found at the AD community portal. In most modelling studies of groundwater flow, data are sparse, and the geological knowledge of a given area may not extend to the thermophysical and hydrogeological properties of the subsurface.

Direct measurements of hydraulic head usually are confined to shallow boreholes, and tracer experiments may be infeasible for deep and low-permeability aquifers. Fortunately, as even small groundwater velocities may influence the temperature field, there is a good chance to constrain hydrological flow models by temperature measurements. Furthermore, temperature measurements are accurate and pose much less technical problems than densely spaced determinations of hydraulic head in deep boreholes. Thermal rock properties display much less variability if compared to the multidecade spread of hydraulic permeabilities.

Therefore, the integration of temperature into an hydrogeological inverse model is a promising approach. In this article we will describe the development and verification of an inverse modelling code from the corresponding forward model. In a follow-up article, the application to a real-world basin-scale problem will be demonstrated. The paper is organized as follows. The technique of AD is briefly sketched in Section 4. In the following sections, examples from the validation of the inverse code are described See 5 , and synthetic examples from different flow regimes are presented in Section 6.

Finally, Section 7 concludes the paper with a discussion and a summary. In its original form, SHEMAT is capable of solving the coupled transient equations for groundwater flow, heat transport, and the transport of reactive solutes at high temperatures. In this study, we confine ourselves to the simplified case of steady-state flow and heat transport. Also, the chemically reactive transport is not taken into account. Together with eq. To ease the adaptation to the highly variable conditions in the subsurface, the non-linearities resulting from fluid and rock properties were reformulated in a modular way.

The same approach was taken with respect to rock properties and mixing laws for the calculation of the effective properties of the two-phase system fluid-rock, giving the code a high degree of flexibility with respect to representation of physical properties and computational efficiency. It is well known that there exist several alternative formulations of the minimization problem considered here see Tarantola The results presented here, however, have all been obtained by the fundamental GN scheme shown in eq.

To guarantee positivity, and to improve its scaling, parameters can be partially replaced by their natural logarithm. This amounts to a scaling of the Jacobian J derived from AD to in eq. Though the choice of transformed parameters for practical inversion has proven to be advantageous in many cases, is has to be noted that this implies lognormal priors and posterior covariances. This assumption is probably justified for permeabilities, but is disputable for the other parameters mentioned. Physical parameters are assigned to the grid cells by means of an index array.

In order to keep the size of the inverse problem moderate, the parameters to be inverted for may be assigned to geological units with homogeneous properties, which may represent for instance layers, isolated bodies, or fault zones. The parameter vector p is generated concatenating the lists of properties for each unit. Using homogeneous units reduces the number of parameters to a manageable number, leaving the decision of necessary refinements to the user.

Presently, for reasons to be given in the following section, the number of parameters should be low, that is, much lower than the size of the underlying FD mesh.

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This can be achieved, for instance, by selecting a subset of optimized parameters for a large number of units, or by jointly estimating many parameters for a smaller number of units. The current inverse code gives the user considerable freedom to choose the appropriate set-up for a given problem. In this technique, a program is conceptually decomposed into a finite sequence of elementary operations such as the multiplication of two real values. This sequence may become very large for programs simulating phenomena from real-world applications. From a mathematical point of view, the whole sequence represents a composition of a potentially extremely large number of simple functions whose derivatives are known.

So, the derivatives of the complete sequence are given by accumulating the derivatives of the known elementary functions according to the chain rule of differential calculus. One of the main advantages of using AD is the absence of any truncation error. Neglecting rounding errors, the values produced by this technique are exact, whereas values computed by numerical differentiation using divided differences involve truncation error. From a computer science point of view, AD is similar to a compiler in that it transforms a given program into a new program.

However, unlike a compiler, AD changes the semantics of a given program by inserting additional statements for the computation of derivatives. The performance of programs generated by AD may, however, differ dramatically in terms of time and storage. In practical applications of AD, it is therefore important to choose an adequate technique for the given problem at hand. More details on AD are given in the books by Rall and Griewank Another source of information is the AD community portal.

When applying an AD tool to a given program, the user specifies a subset of its input and output variables as being independent and dependent , respectively. The program generated by AD then evaluates the derivatives of the dependent variables with respect to the independent variables. Intermediate variables occurring in the evaluation process are declared active and augmented with their corresponding derivative objects.

In particular, the derivative of a scalar function as 6 , that is, its gradient, can be calculated very cheaply Griewank This is important if we want to use the alternative optimization algorithms mentioned in Section 3 NLCG, LBFGS efficiently, as these only need the value of the objective function and its gradient.

This, however remains a task for further developments. When applying AD to a nontrivial code such as SHEMAT, the robustness of the AD tool based on source transformation is of crucial importance, because it has to be capable transforming large and complicated codes. Additionally, the F77 language standard is much simpler than F90, and the AD tools can usually manage such code better with fewer problems. To keep precision, all variables or constants should be explicitly converted to or defined as a uniform data type. The Basic Linear Algebra S ystem Lawson ; Dongarra , represents a standardized interface to the most important vector and matrix operations.

These interfaces are available in highly optimized versions on almost any hardware platform. F77 does not support dynamic memory allocation, so we need to transform all dynamic arrays into static ones. This can be done by setting up all dynamic arrays with a dummy size for each dimension. These derivative arrays should be allocated dynamically whenever the corresponding arrays in the original code are handled this way. Further, the initialization functions for the new memory allocation calls have to be adapted accordingly.

This does not pose any new problems, but is cumbersome to do. In this manner, the derivative function can quickly be updated, whenever changes in the forward code make this necessary. However, performance can be dramatically increased by taking into account knowledge on the forward code considered and the intended use of the derivative. In particular:. Of primary importance is the particular choice of the dependent and independent variables. For the computation of the Jacobian, this choice determines the dimensions of the derivative objects, and the corresponding memory requirements.

As the intended inverse scheme is unit oriented see Section 3 , access to inverse parameters is done indirectly associating parameters to mesh cells by means of an index array. These goals were achieved by transforming many existing intermediate arrays to function calls. This prevented the AD tool to declare them as active variables and associate corresponding derivative structures.

Also, indirect access to unit properties led to significant reduction of memory requirements. Additionally, the coupling between the eqs for h and T was also taken into account manually. This leads to a more efficient code, as near the fixed point of the non-linear solution, it is not required to execute the derivative code throughout the entire fixed point iteration Gilbert ; Christianson , When AD has produced an correct derivative code, it has to be interfaced with the inverse code developed. An appropriate choice may lead to considerable reduction of computer resources, especially because the flexibility of inverse configuration has to be maintained.

Also, we can reduce computation overhead in case only a linear combination of the Jacobian columns is needed. Though the approach described is not fully automatic, it offers great advantages. Apart from the lack of additional truncation error mentioned above, AD gives the opportunity to integrate revisions in the forward modelling code because of changes in the underlying physics or new discretization schemes. Once the necessary transformations described above have been made, and the required standards are maintained, the generation of derivative code can be done with only a small amount of additional work.

To verify the correctness of the AD generated code we used several analytical solutions to simplified problems, which represent only part of the model implemented in the code, assuming that the correct treatment of the parts leads to a correctly working full model. For the verification of this full model no analytical solutions are known, thus leaving us with the method of divided differences for comparison.

Though numerical less accurate, this method generally confirms the results obtained from AD. For the simple verification examples presented below, the non-linear dependencies of fluid and rock properties on T and P were generally neglected. Solid lines mark the values calculated form the analytical solution given in eqs 11 and Circles mark the results of an approximation of the derivatives using divided differences.

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Though their general trend follows the analytical and AD results, they display some scatter. To give a more realistic example, we calculated synthetic coupled 2-D models, schematically presenting the surroundings of salt diapir with cross-cutting tectonic faults. Model geometry and the association of units is shown in Fig. Unit 9 is associated with the nearly impermeable salt diapir, while units 10 and 11 represent the permeable fault zones. This geometry is the same in the all models presented, differing mainly in the driving forces and the values of permeability associated with the main units.

The sensitivities calculated by AD for both models are shown in Figs 5 and 6. The parameters used for the forward and inverse models are given in Table 2. The latter case is characterized by generally higher permeabilities in units 1—8, which leads to a stronger participation of these zones in the flow system.

They display, however, a similar temperature and flow field. Model geometry around a salt diapir with high thermal conductivity on the right. It is constructed from 11 units of different thermal and hydraulic properties. Units 1—8 represent sedimentary layers, while the salt is associated with unit 9. The cross-cutting faults, which in this model are highly permeable, are units 10 and Also shown is the assumed surface head driving the flow in the case of forced convection see Fig.

Free convection. Also shown are the Darcy velocities. Sensitivities of temperature left and hydraulic potential right with respect to isotropic permeability for zones 11 c and d and 7 e and f. These zones are marked in Fig. Unit 11 is the high-permeability fault zone in cross-cutting the model. Note that the sensitivities are calculated with respect to the natural logarithms of the parameters.

This simply amounts to a scaling of the original sensitivities by the parameter value. Forced convection. Sensitivities of temperature left and hydraulic potential right with respect to the natural logarithm of isotropic permeabilities. For explanation see Fig. Parameter assumed for the forward and inverse models. Though the single units themselves lack the permeability and vertical dimension to allow for Rayleigh convection, the permeable fault zones cause a short circuit leading to a large-scale guided convection through the high-permeability zones 10 and Sensitivities of temperature data and hydraulic head with respect to thermal conductivity and hydraulic permeability are given below.

The same quantities are shown for the topography-driven flow in Fig. Comparing these figures, a few observations can be made. Though the flow patterns and the corresponding perturbations of temperature are similar in both cases, the sensitivities are not. The areas with high sensitivities are distributed differently for both models. From this we may conclude, that the choice of data necessary for a successful inverse model depends strongly on the assumed model, for example, boundary conditions.

Though their values can in principle be obtained by the inversion, a conceptual model must be available. Moreover, the sensitivities display complicated, and sometimes contra-intuitive interrelations, as in Figs 5 c or 6 f. The sensitivities of the different data types sometimes complement each other, thus motivating hope that the joint inversion may produce better results than the ones based on head or temperature alone.

For the numerical experiments in the following we used the model set-up described in Table 2 and Fig. For each case, a set of eight models was created. In each of these models, a given number of boreholes was generated, where site and depth were chosen at random. In particular, depth varied normally with a mean of m and a standard deviation of m. The data were subsequently modified, adding zero-mean normal noise with a standard deviation or 0. This choice of errors is disputable: though temperature can be continuously measured at high accuracy, technical and geological noise may easily reach this value.

By geological noise we refer to small-scale parameter variations not accounted for in the model. The quasi-continuous head observations are somewhat idealized, because formation pressure is difficult to obtain in deep boreholes as assumed here. Often only integral values for special intervals or other derived quantities e.

Darcy velocities are known. These conditions could be considerably different in a shallower regime, where temperature differences may be much smaller, and heads are much easier to determine with high accuracy. For each of these models inversions were run to 50 iterations.

For some of the more problematic models, this turned out to be too few, but for nearly all of these convergence could be obtained after iterations.