Akhil Datta-Gupta

A semianalytic approach to tracer flow modeling in heterogeneous permeable media

A semianalytic approach for modeling tracer motion in heterogeneous permeable media is presented. The method is analytic along streamlines; the streamlines are derived from an underlying velocity field which is obtained numerically from a conventional fluid flow simulator. This generalizes the approach to any arbitrary configuration of wells and also to areally heterogeneous permeability fields. The semianalytic scheme is based on the observation that in a velocity field derived by finite difference, streamlines can be approximated by piecewise hyperbolic intervals. Along each of these intervals the evolution equation can be solved exactly. Thus, the approach is free from numerical time truncation error. Once tracer transit times to a producing well have been determined, the tracer response curve and the area swept by the tracer can be obtained from simple integral expressions. The transit time formalism allows for easy extension of the semianalytic approach to multiphase flow problems and the results are shown to be in excellent agreement with high resolution finite-difference simulations, but obtained at a fraction of the computation time. We have illustrated the semianalytic approach through application to tracer migration in homogeneous as well as heterogeneous quarter five spot patterns.

Asymptotic solutions for solute transport: A formalism for tracer tomography

An asymptotic approach to the solution of the transport equation, in the limit of rapid spatial and temporal variation, produces an extremely efficient formalism for the inversion of tracer data. The technique provides tracer concentration sensitivities to porosity, permeability, and pressure gradient variations in just a single simulation run. The calculated sensitivities compare well with those derived using a numerical perturbation method, at a fraction of the computational requirements. An application to a conservative tracer test at Hill Air Force Base in Utah indicates the efficiency and utility of the approach for characterizing three-dimensional variations in flow properties. On the basis of tracer concentration histories at 12 multilevel samplers and three extraction wells, some 44 tracer curves in all, significant small-scale variability in permeability is inferred. In general, the permeability is found to decrease as the lower boundary of the aquifer is approached. The permeability trends we find are consistent with tracer swept volume calculations based upon a moment analysis.

Resolution and uncertainty in hydrologic characterization

Hydrologists have applied inverse techniques to obtain estimates of subsurface permeability and porosity variations and their associated uncertainties. Although inverse methods are now well established in hydrology, important aspects of inverse theory, the analysis of resolution, and the trade-off between model parameter resolution and model parameter uncertainty have not been utilized. In this paper the concept of model parameter resolution is incorporated into the analysis of hydrological experiments. Model parameter resolution is a measure of the spatial averaging implicit in estimates of a distributed hydrological property such as permeability. There are two important uses of resolution and uncertainty estimates in hydrology. The first use is to plan a hydrologic testing program. Resolution matrices can be developed for proposed well tests in a variety of synthetic media. Then the effectiveness of the test design can be evaluated in terms of model parameter resolution and uncertainty. Secondly, when real data are available and used in an inversion determining the distribution of hydrologic parameters, estimates of model parameter resolution and uncertainty analysis can indicate the reliability of the solution. For synthetic tests in which the hydraulic conductivity varies and porosity does not, it is found that tracer data can provide better spatial resolution of subsurface hydraulic conductivity variations than transient pressure data. Pressure data are most sensitive to hydraulic conductivity variations immediately surrounding the well. Both pressure and tracer data better determine barriers to flow rather than channels to flow. The methodology is applied to a set of transient pressure data gathered at the Grimsel Rock Laboratory of the Swiss National Cooperative for the Storage of Radioactive Waste. In the fracture under study a low hydraulic conductivity region appears to partition the fracture plane into two distinct zones.

Optimal Transformations for Multiple Regression: Application to Permeability Estimation From Well Logs

Conventional multiple regression for permeability estimation from well logs requires a functional relationship to be presumed. Due to the inexact nature of the relationship between petrophysical variables, it is not always possible to identify the underlying functional form between dependent and independent variables in advance. When large variations in petrological properties are exhibited, parametric regression often fails or leads to unstable and erroneous results, especially for multivariate cases. In this paper we describe a nonparametric approach for estimating optimal transformations of petrophysical data to obtain the maximum correlation between observed variables. The approach does not require a priori assumptions of a functional form and the optimal transformations are derived solely based on the data set. An iterative procedure involving the alternating conditional expectation (ACE) forms the basis of our approach. The power of ACE is illustrated using synthetic as well as field examples. The results clearly demonstrate improved permeability estimation by ACE compared to conventional parametric regression methods.

Seismic imaging of reservoir flow properties: Time-lapse amplitude changes

Asymptotic methods provide an efficient means by which to infer reservoir flow properties, such as permeability, from time-lapse seismic data. A trajectory-based methodology, much like ray-based methods for medical and seismic imaging, is the basis for an iterative inversion of time-lapse amplitude changes. In this approach a single reservoir simulation is required for each iteration of the algorithm. A comparison between purely numerical and the trajectory-based sensitivities demonstrates their accuracy. An application to a set of synthetic amplitude changes indicates that they can recover large-scale reservoir permeability variations from time-lapse data. In an application of actual time-lapse amplitude changes from the Bay Marchand field in the Gulf of Mexico we are able to reduce the misfit by 81% in twelve iterations. The time-lapse observations indicate lower permeabilities are required in the central portion of the reservoir.

A Multiscale Approach to Production Data Integration Using Streamline Models

We propose a multiscale approach to data integration that accounts for the varying resolving power of different data types from the very outset. Starting with a very coarse description, we match the production response at the wells by recursively refining the reservoir grid. A multiphase streamline simulator is utilized for modeling fluid flow in the reservoir. The well data is then integrated using conventional geostatistics, for example sequential simulation methods. There are several advantages to our proposed approach. First, we explicitly account for the resolution of the production response by refining the grid only up to a level sufficient to match the data, avoiding over-parameterization and incorporation of artificial regularization constraints. Second, production data is integrated at a coarse-scale with fewer parameters, which makes the method significantly faster compared to direct fine-scale inversion of the production data. Third, decomposition of the inverse problem by scale greatly facilitates the convergence of iterative descent techniques to the global solution, particularly in the presence of multiple local minima. Finally, the streamline approach allows for parameter sensitivities to be computed analytically using a single simulation run and thus, further enhancing the computational speed. The proposed approach has been applied to synthetic as well as field examples. The synthetic examples illustrate the validity of the approach and also address several key issues such as convergence of the algorithm, computational efficiency, and advantages of the multiscale approach compared to conventional methods. The field example is from the Goldsmith San Andres Unit (GSAU) in West Texas and includes multiple patterns consisting of 11 injectors and 31 producers. Using well log data and water-cut history from producing wells, we characterize the permeability distribution, thus demonstrating the feasibility of the proposed approach for large-scale field applications.

EFFICIENT AND ROBUST RESERVOIR MODEL UPDATING USING ENSEMBLE KALMAN FILTER WITH SENSITIVITY-BASED COVARIANCE LOCALIZATION

Recently Ensemble Kalman Filtering (EnKF) has gained increasing attention for history matching and continuous reservoir model updating using data from permanent downhole sensors. It is a sequential Monte-Carlo approach that works with an ensemble of reservoir models. Specifically, the method utilizes cross-covariances between measurements and model parameters estimated from the ensemble. For practical field applications, the ensemble size needs to be kept small for computational efficiency. However, this leads to poor approximations of the cross-covariance matrix, resulting in loss of geologic realism. Specifically, the updated parameter field tends to become scattered with a loss of connectivities of extreme values such as high permeability channels and low permeability barriers, which are of special significance during reservoir characterization. We propose a novel approach to overcome this limitation of the EnKF through a ‘covariance localization’ method that utilizes sensitivities that quantify the influence of model parameters on the observed data. These sensitivities are used in the EnKF to modify the cross-covariance matrix in order to reduce unwanted influences of distant observation points on model parameter updates. In particular, streamline-based analytic sensitivities are easy to compute, require very little extra computational effort and can be obtained using either a finite difference or streamline-based flow simulator. We show that the effect of the covariance localization is to increase the effective ensemble size. But key to the success of the sensitivity-based covariance-localization is its close link to the underlying physics of flow compared to a simple distance-dependent covariance function as used in the past. This flow-relevant conditioning leads to an efficient and robust approach for history matching and continuous reservoir model updating, avoiding much of the problems in traditional EnKF associated with instabilities, parameter overshoots and loss of geologic continuity. We illustrate the power and utility of our approach using both synthetic and field applications.

Optimal Coarsening of 3D Reservoir Models for Flow Simulation

A new constrained optimization approach to the coarsening of 3D reservoir models for flow simulation has been developed. The optimization preserves a statistical measure of the heterogeneity of a fine-scale model. Constraints arise from the reservoir fluids, well locations, pay/nonpoy juxtaposition, and large-scale reservoir structure and stratigraphy. The approach has been validated for several oil and gas projects in that flow simulation through the coarsened model was shown to provide an excellent approximation to high-resolution calculations performed in the original model.

On the Sensitivity and Spatial Resolution of Transient Pressure and Tracer Data For Heterogeneity Characterization

This paper examines the sensitivities of interwell tracer and transient pressure response to spatial distribution of permeability heterogeneity. Based on the sensitivities, we describe a formalism to quantify the spatial resolution and averaging (smearing) associated with estimates of permeabilities derived through inversion of tracer and/or pressure data. The spatial resolution is a measure of the effectiveness of the data in estimating local-scale (grid block) permeabilities. The averaging kernels quantify the inherent averaging associated with our estimates due to limited data or sampling. By examining the spatial resolution and averaging kernels as a function of various data types, we can quantitatively evaluate the relative importance of tracer versus pressure data for heterogeneity characterization and the improvement in estimates obtained by combining the data types. We illustrate the concepts by application to a quarter five-spot geometry and also to an experimental tracer response from a well-characterized slab of Antolini sandstone. Tracer data is found to yield much better resolution compared to transient pressure response. Also, both transient pressure data and tracer data appear to better resolve barriers to flow rather than channels to flow.

Characterizing heterogeneous permeable media with spatial statistics and tracer data using sequential simulated annealing

Characterizing heterogeneous permeable media using flow and transport data typically requires solution of an inverse problem. Such inverse problems are intensive computationally and may involve iterative procedures requiring many forward simulations of the flow and transport problem. Previous attempts have been limited mostly to flow data such as pressure transient (interference) tests using multiple observation wells. This paper discusses an approach to generating stochastic permeability fields conditioned to geologic data in the form of a vertical variogram derived from cores and logs as well as fluid flow and transport data, such as tracer concentration history, by sequential application of simulated annealing (SA). Thus, the method incorporates elements of geostatistics within the framework of inverse modeling. For tracer-transport calculations, we have used a semianalytic transit-time algorithm which is fast, accurate, and free of numerical dispersion. For steady velocity fields, we introduce a “transit-time function” which demonstrates the relative importance of data from different sources. The approach is illustrated by application to a set of spatial permeability measurements and tracer data from an experiment in the Antolini Sandstone, an eolian outcrop from northern Arizona. The results clearly reveal the importance of tracer data in reproducing the correlated features (channels) of the permeability field and the scale effects of heterogeneity.