Modeling transient groundwater flow by coupling ensemble Kalman filtering and upscaling

The ensemble Kalman filter (EnKF) is coupled with upscaling to build an aquifer model at a coarser scale than the scale at which the conditioning data (conductivity and piezometric head) had been taken for the purpose of inverse modeling. Building an aquifer model at the support scale of observations is most often impractical since this would imply numerical models with many millions of cells. If, in addition, an uncertainty analysis is required involving some kind of Monte Carlo approach, the task becomes impossible. For this reason, a methodology has been developed that will use the conductivity data at the scale at which they were collected to build a model at a (much) coarser scale suitable for the inverse modeling of groundwater flow and mass transport. It proceeds as follows: (1) Generate an ensemble of realizations of conductivities conditioned to the conductivity data at the same scale at which conductivities were collected. (2) Upscale each realization onto a coarse discretization; on these coarse realizations, conductivities will become tensorial in nature with arbitrary orientations of their principal components. (3) Apply the EnKF to the ensemble of coarse conductivity upscaled realizations in order to condition the realizations to the measured piezometric head data. The proposed approach addresses the problem of how to deal with tensorial parameters, at a coarse scale, in ensemble Kalman filtering while maintaining the conditioning to the fine-scale hydraulic conductivity measurements. We demonstrate our approach in the framework of a synthetic worth-of-data exercise, in which the relevance of conditioning to conductivities, piezometric heads, or both is analyzed.The authors acknowledge Wolfgang Nowak and three anonymous reviewers for their comments on the previous versions of the manuscript, which helped substantially to improve it. 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Prediction of petro-physical properties for carbonates

This thesis is concerned with the inversion of lattice pore-network model parameters of carbonate rocks using only the capillary pressure, and then the use of the inverted parameters to predict the water-flooding relative permeabilities of the carbonate rocks. Background: There has been a tendency to claim that pore-network modelling using three-dimensional micro-computed tomography or 3D mathematically created images can predict imbibition relative permeabilities for wettabilities other than strongly water/oil-wetting. This is based on the flexibility for matching data, which is a weakness of pore-network modelling. The method proposed in this thesis is important because a large percentage of the porosity in carbonates is microporosity. Conclusions: We applied stochastic inversion of lattice pore-network model parameters using Hamiltonian Dynamics (Hamiltonian Monte Carlo) to three carbonate rock samples and we predicted water-flooding relative permeabilities with good accuracy by using as constraint only routinely obtained data, such as mercury intrusion capillary pressure (MICP) and oil/water capillary pressure. We found that there is a strong correlation between the amount of microporosity and the volume exponent parameter. This suggests that when microporosity is ignored, the volume exponent will tend to be systematically strongly underestimated. HMC found large variability in wettability that causes mid-sized pores to be invaded at the same level of pressure as larger pores. The coexistence of these events reduces the tendency for preferential flow through large pores, resulting in more uniform flow at the pore scale compared with the case in which flow is restricted only to large pores. Mid-sized pores have an important effect on the connectivity because they could have higher contact angles than larger pores. Therefore, they do not spontaneously imbibe and shield larger pores, improving water-flooding displacement. The wettability of micropores could better explain the gentle curvature of the imbibition water relative permeability compared with the generally assumed mixed-wet large wettability model. The importance of the maximum and minimum observed capillary pressure is directly connected to accounting for the full pore-size distribution. Thus, the common assumption that microporosity can be ignored is unsatisfactory. The ranges of advancing contact angles obtained from the HMC inversion were wider than the ranges of apparent advancing contact angles obtained with analytical determinations in previous studies, and in one case our results were contradictory to the analytical determination. It follows that variability in advancing and receding contact angles is not reflected in the apparent contact angle data outside porous media. Apparent contact angle data outside porous media cannot completely characterise the wettability in porenetwork models because the data does not capture the contact angle variability in porous media. The existence of wetting films depends on the maximum capillary pressure during drainage, and thus wettability alteration during ageing. Our results suggest that matching both connate water at the maximum drainage capillary pressure before ageing and matching residual oil at the minimum imbibition capillary pressure leads to better estimation of the advancing and receding variability in the contact angles

Upscaling and Inverse Modeling of Groundwater Flow and Mass Transport in Heterogeneous Aquifers

Dividimos el trabajo en tres bloques: En el primer bloque, se han revisado las técnicas de escalado que utilizan una media simple, el método laplaciano simple, el laplaciano con piel y el escalado con mallado no uniforme y se han evaluado en un ejercicio tridimensional de escalado de la conductividad hidráulica. El campo usado como referencia es una realización condicional a escala fina de la conductividad hidráulica del experimento de macrodispersión realizado en la base de la fuerza aérea estadounidense de Columbus en Misuri (MADE en su acrónimo inglés). El objetivo de esta sección es doble, primero, comparar la efectividad de diferentes técnicas de escalado para producir modelos capaces de reproducir el comportamiento observado del movimiento del penacho de tritio, y segundo, demostrar y analizar las condiciones bajo las cuales el escalado puede proporcionar un modelo a una escala gruesa en el que el flujo y el transporte puedan predecirse con al ecuación de advección-dispersión en condiciones aparentemente no fickianas. En otros casos, se observa que la discrepancia en la predicción del transporte entre las dos escalas persiste, y la ecuación de advección-dispersión no es suficiente para explicar el transporte en la escala gruesa. Por esta razón, se ha desarrollado una metodología para el escalado del transporte en formaciones muy heterogéneas en tres dimensiones. El método propuesto se basa en un escalado de la conductividad hidráulica por el método laplaciano con piel y centrado en los interbloques, seguido de un escalado de los parámetros de transporte que requiere la inclusión de un proceso de transporte con transferencia de masa multitasa para compensar la pérdida de heterogeneidad inherente al cambio de escala. El método propuesto no sólo reproduce el flujo y el transporte en la escala gruesa, sino que reproduce también la incertidumbre asociada con las predicciones según puede observarse analizando la variabilidad del conjunto de curvas de llegada.Li ., L. (2011). Upscaling and Inverse Modeling of Groundwater Flow and Mass Transport in Heterogeneous Aquifers [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/12268Palanci

History Matching of 3D Reservoir Models with Complex Non-Gaussian Distributions of the Model Parameters

Although the first application of the ensemble Kalman filter (EnKF) as a technique for sequential assimilation of noisy measurements was to a numerical weather prediction problem, remarkable research progress has been made in adapting this technique for application to data assimilation problems in reservoir engineering. Since its first application to a fairly simple parameter estimation problem in petroleum engineering less than a decade ago, the ensemble Kalman filter has been applied to fairly complex sequential model calibration problems in reservoir engineering with remarkable success. The reason for the rapid increase in the application of EnKF to data assimilation problems in reservoir engineering is partly due to the ease of implementing this technique; it is unnecessary to determine sensitivities from adjoint equations and the correlations between the model parameters and predicted data are estimated from the ensemble. Also the information from previously assimilated data are stored within the ensemble of conditional models such that it is unnecessary to repeat the history matching on previously assimilated data whenever new data are available for assimilation.Despite the encouraging performance of EnKF applied to sequential model calibration problems in reservoir engineering, the formulation of the ensemble Kalman filter is based on some critical assumptions (linear forward model and Gaussian model priors) that are generally not valid for reservoir engineering problems. The EnKF performance is optimal if at each data assimilation timestep, the prior state vector is linearly related to the predicted data and the distribution of the prior state vector is multivariate Gaussian. The state vector in reservoir engineering applications consists of the static model variables and the dynamic model variables, i.e., gridblock pressures and saturations for a black oil model. Some potential issues with EnKF applied to data assimilation problems in reservoir engineering include the following: non-Gaussian dynamic and/or static model parameters in the state vector, nonlinear model--data relationship, etc. Usually, the presence of severe nonlinearities is resolved by iteration, which also resolves the problem of non-Gaussian dynamic variables. However, for history matching problems with complex non-Gaussian model parameters, such as facies variables, vertical flow barriers, multimodal model variables, etc, the nonlinear effects are exacerbated and the performance of the EnKF in adjusting the models to obtain predictions that match production data is significantly degraded.In this dissertation, I focus on history matching problems with non-Gaussian model parameters for which the standard EnKF will perform very poorly without modifications. I identify some of the difficulties in the application of EnKF to the problem of updating facies models to match both production measurements and facies observations at the well locations and propose some modifications at the update step that improves the overall performance of the ensemble Kalman filter. I also introduce the concept of using pseudo-model variables for jointly updating the discrete facies variables and the multimodal rock properties in a way that is consistent with the EnKF updating scheme. The problem of updating reservoir models with vertical flow barriers using EnKF is also addressed in this dissertation. Estimation of vertical flow barriers is of practical importance in reservoir simulation studies as these flow barriers influence recovery mechanisms, gravity drainage processes and the selection of optimal well performance parameters. I outline three very efficient and fairly general methods for parameterizing the vertical transmissibility barriers between reservoir zones so that zonal communication can be shut off if production data indicate that there should be no communication. The EnKF will generally perform very poorly in generating conditional samples of the reservoir models if the posterior PDF is multimodal. I introduce a two-stage ensemble Kalman filter technique for application to history matching problems with multiple modes. I demonstrate the advantage of two-stage EnKF technique on a fairly complex low-order reservoir model with non-Gaussian model parameters and show that it converges to a better history match solution than the standard EnKF

Characterisation of the transmissivity field of a fractured and karstic aquifer, Southern France

International audienceGeological and hydrological data collected at the Terrieu experimental site north of Montpellier, in a confined carbonate aquifer indicates that both fracture clusters and a major bedding plane form the main flow paths of this highly heterogeneous karst aquifer. However, characterising the geometry and spatial location of the main flow channels and estimating their flow properties remain difficult. These challenges can be addressed by solving an inverse problem using the available hydraulic head data recorded during a set of interference pumping tests.We first constructed a 2D equivalent porous medium model to represent the test site domain and then employed regular zoning parameterisation, on which the inverse modelling was performed. Because we aim to resolve the fine-scale characteristics of the transmissivity field, the problem undertaken is essentially a large-scale inverse model, i.e. the dimension of the unknown parameters is high. In order to deal with the high computational demands in such a large-scale inverse problem, a gradient-based, non-linear algorithm (SNOPT) was used to estimate the transmissivity field on the experimental site scale through the inversion of steady-state, hydraulic head measurements recorded at 22 boreholes during 8 sequential cross-hole pumping tests. We used the data from outcrops, borehole fracture measurements and interpretations of inter-well connectivities from interference test responses as initial models to trigger the inversion. Constraints for hydraulic conductivities, based on analytical interpretations of pumping tests, were also added to the inversion models. In addition, the efficiency of the adopted inverse algorithm enables us to increase dramatically the number of unknown parameters to investigate the influence of elementary discretisation on the reconstruction of the transmissivity fields in both synthetic and field studies.By following the above approach, transmissivity fields that produce similar hydrodynamic behaviours to the real head measurements were obtained. The inverted transmissivity fields show complex, spatial heterogeneities with highly conductive channels embedded in a low transmissivity matrix region. The spatial trend of the main flow channels is in a good agreement with that of the main fracture sets mapped on outcrops in the vicinity of the Terrieu site suggesting that the hydraulic anisotropy is consistent with the structural anisotropy. These results from the inverse modelling enable the main flow paths to be located and their hydrodynamic properties to be estimated