Agustín Medina

Geostatistical inversion of coupled problems: dealing with computational burden and different types of data

Practical application of geostatistical inversion to coupled problems is hampered by a number of difficulties. In this paper, we address two of them: first, the computational cost of sensitivity (Jacobian) matrices and, second, the evaluation of the relative weights of different types of data. Regarding the first, we revise the adjoint state equations to propose a form whose cost is independent of the number of unknown parameters and only grows with the number of observation wells. Regarding the second, we derive expressions for the relative weights of different types of data. These expressions are based on minimizing the expected likelihood, rather than the likelihood itself. The efficiency of both improvements is tested on a synthetic example. The example analyzes a wide range of groundwater flow and solute transport conditions. Yet, the expected likelihood consistently yields the optimal weights. The proposed form of the adjoint state equations leads to one order of magnitude reduction in CPU time with respect to the conventional sensitivity equations.

Inverse modelling of seawater intrusion in the Llobregat delta deep aquifer

Abstract A flow and mass transport mathematical model has been calibrated to simulate piezometric head and chloride evolution in the Llobregat delta confined aquifer. A twenty-year period (1965–1985) is simulated with monthly time steps. Automatic estimation of flow and transport parameters are obtained by means of a code that simulates flow and mass transport of constant density fluids. In this case, density differences owing to groundwater salinity changes by marine intrusion can be neglected because the aquifer thickness is relatively small and because both the horizontal density gradient and the aquifer dip are small. Calibration of the flow problem leads to an excellent fit between measured and computed heads, which, however, can be obtained with different transmissivity patterns. When switching to the transport problem, best results in terms of concentration fits are obtained by associating the seawater intrusion plumes with high transmissivity zones that can be attributed to palaeochannels. In short, transient head and concentration data have been used jointly for automatic calibration of a regional model, which has proven advantageous both because it facilitates model selection and because it has led to a model with an improved conceptual basis.

Computational techniques for optimization of problems involving non-linear transient simulations

Many optimization problems in engineering require coupling a mathematical programming process to a numerical simulation. When the latter is non-linear, the resulting computer time may become unaffordably large because three sequential procedures are nested: the outer loop is associated to the optimization process, the middle one corresponds to the time marching scheme and the innermost loop is required for solving iteratively the non-linear system of equations at each time step. We propose four techniques for reducing CPU time. First, derive the initial values of state variables at each time (innermost loop) from those computed at the previous optimization iteration (outermost loop). Second, select time increment on the basis of those used for the previous optimization iteration. Third, define convergence criteria for the simulation problem on the basis of the optimization process, so that they are only as stringent as really needed. Finally, computations associated to the optimization are shown to be greatly reduced by adopting Newton–Raphson, or a variant, for solving the simulation problem. The effectiveness of these techniques is illustrated through application to three examples involving automatic calibration of non-linear groundwater flow problems. The total number of iterations is reduced by a factor ranging between 1·7 and 4·6. Copyright

Crystalline particle packings on constant mean curvature (Delaunay) surfaces

We investigate the structure of crystalline particle arrays on constant mean curvature (CMC) surfaces of revolution. Such curved crystals have been realized physically by creating charge-stabilized colloidal arrays on liquid capillary bridges. CMC surfaces of revolution, classified by Delaunay in 1841, include the 2-sphere, the cylinder, the vanishing mean curvature catenoid (a minimal surface), and the richer and less investigated unduloid and nodoid. We determine numerically candidate ground-state configurations for 1000 pointlike particles interacting with a pairwise-repulsive 1/r(3) potential, with distance r measured in three-dimensional Euclidean space R(3). We mimic stretching of capillary bridges by determining the equilibrium configurations of particles arrayed on a sequence of Delaunay surfaces obtained by increasing or decreasing the height at constant volume starting from a given initial surface, either a fat cylinder or a square cylinder. In this case, the stretching process takes one through a complicated sequence of Delaunay surfaces, each with different geometrical parameters, including the aspect ratio, mean curvature, and maximal Gaussian curvature. Unduloids, catenoids, and nodoids all appear in this process. Defect motifs in the ground state evolve from dislocations at the boundary to dislocations in the interior to pleats and scars in the interior and then isolated sevenfold disclinations in the interior as the capillary bridge narrows at the waist (equator) and the maximal (negative) Gaussian curvature grows. We also check theoretical predictions that the isolated disclinations are present in the ground state when the surface contains a geodesic disk with integrated Gaussian curvature exceeding -π/3. Finally, we explore minimal energy configurations on sets of slices of a given Delaunay surface, and we obtain configurations and defect motifs consistent with those seen in stretching.

Land surface temperature as an indicator of the unsaturated zone thickness: A remote sensing approach in the Atacama Desert

Land surface temperature (LST) seems to be related to the temperature of shallow aquifers and the unsaturated zone thickness (∆Zuz). That relationship is valid when the study area fulfils certain characteristics: a) there should be no downward moisture fluxes in an unsaturated zone, b) the soil composition in terms of both, the different horizon materials and their corresponding thermal and hydraulic properties, must be as homogeneous and isotropic as possible, c) flat and regular topography, and d) steady state groundwater temperature with a spatially homogeneous temperature distribution. A night time Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) image and temperature field measurements are used to test the validity of the relationship between LST and ∆Zuz at the Pampa del Tamarugal, which is located in the Atacama Desert (Chile) and meets the above required conditions. The results indicate that there is a relation between the land surface temperature and the unsaturated zone thickness in the study area. Moreover, the field measurements of soil temperature indicate that shallow aquifers dampen both the daily and the seasonal amplitude of the temperature oscillation generated by the local climate conditions. Despite empirically observing the relationship between the LST and ∆Zuz in the study zone, such a relationship cannot be applied to directly estimate ∆Zuz using temperatures from nighttime thermal satellite images. To this end, it is necessary to consider the soil thermal properties, the soil surface roughness and the unseen water and moisture fluxes (e.g., capillarity and evaporation) that typically occur in the subsurface.

Interpretation of Field Tests in Low Permeability Fractured Media. Recent Experiences

We have been treating fractured media as the result of embedding conductive 2D fractures in a 3D continuum medium. Automatic calibration has normally succeeded in producing models capable of predicting independent data sets. However, in recent times, we have faced a set of tests performed in very low conductive, highly heterogeneous granite where using exclusively the standard procedure has not produced good results. Difficulties include the following: (1) complex fracture geometry; (2) spurious effects caused by ill-shaped tetrahedra; (3) weak responses to pumping; (4) superposition of natural head data (controlled by boundary conditions) with drawdowns caused by pumping; (5) coupling these two flow conditions to produce the flow field needed for the tracer test is extremely sensitive to small errors. Finally, these difficulties are aggravated by the large computer times required by a fully 3D medium. Suggestions for overcoming these problems are outlined.

A Natural Parameterization of the Roulettes of the Conics Generating the Delaunay Surfaces

We derive parametrizations of the Delaunay constant mean curvature surfaces of revolution that follow directly from parametrizations of the conics that generate these surfaces via the corresponding roulette. This uniform treatment exploits the natural geometry of the conic (parabolic, elliptic or hyperbolic) and leads to simple expressions for the mean and Gaussian curvatures of the surfaces as well as the construction of new surfaces.

Inverse Geostatistical Modeling of Pumping and Tracer Tests Within a Shear-Zone in Granite

A numerical model of the hydraulic conditions within a heterogeneous shear-zone has been built using a geostatistical inverse approach for the analysis of static heads and pumping tests. Results indicate the existence of channels. The estimated transmissivity field serves as a basis for tracer test evaluation. The conservative tracer uranine seems to diffuse only insignificantly into the immobile zones of the shear-zone.

Combining point and regular lattice data in geostatistical interpolation

Abstract This work studies how to include both point and areal measurements when estimating gaussian fields by kriging. To achieve this objective, three geostatistical approaches are considered for the areal distributed data: (a) regionalized measurements that are geographically referenced by their centroid as if they were point measurements, (b) regionalized measurements that are explicitly accounted by formally computing all the needed covariances (i.e. area-to-area, area-to-point and point-to-point covariances, respectively) and (c) regionalized measurements that are used as an external drift variable. Results indicate that the measurement error corresponding to the areal data plays a key role to decide when the spatial support of the areal measurements is relevant. For small measurement errors, it is necessary to explicitly consider the spatial support of the areal measurements to avoid large estimation variances. For large measurement errors, the difference between defining areal measurements by using their actual spatial support and defining areal measurements by referencing them by their centroids (i.e. gravity centre) is small. In this situation, it is possible to use the areal measurements as an external drift instead of merging both types of information (i.e. point and areal data) as measurements for kriging. In this case, the cross-validation analysis shows a larger coefficient of determination, similar average kriging variance and smaller mean square error than the obtained in the case of merging point and areal measurements for kriging.