A Multiresolution Analysis of the Relationship Between Spatial Distribution of Reservoir Parameters and Time Distribution of Well-Test Data

Abstract

With recent advances in permanent downhole technology, well-test analysts now have to deal with enormous amounts of data. Using these data requires the development of efficient algorithms that are able to extract the relevant information from the data at minimal cost. We present a multiresolution wavelet approach to estimate the spatial distribution of reservoir parameters, by performing the nonlinear least-squares regression in the wavelet domains of both time and space. Wavelet transforms have the ability to reveal important events in time signals or spatial images. Thus, we transformed both the model space and the time-series pressure data into spatial wavelet and time wavelet domains, respectively, and used a thresholding to select a subset of wavelet coefficients from each of the transformed domains. These subsets were used subsequently in nonlinear regression to estimate the appropriate description of reservoir parameters. The appropriate subset is not only smaller; the problem is also reduced to the consideration of only the important components of the measured data and only the part of the reservoir description that depends on them.As a test of the approach, we first applied the model to well-test problems involving 1D (radially composite) reservoir systems. The inverse problem was solved to estimate the distributed permeability values by performing the nonlinear least-squares regression in the wavelet domains (time and space).Results obtained were compared with those obtained from the conventional nonlinear-regression approach, using all the pressure-time data and the full set of spatial reservoir parameters. The time/space wavelet approach proved to be efficient. By reducing the dimensions of the model and data spaces, the approach eliminates redundancy in the reservoir description and in the data set. Significantly, the approach reveals the true number of reservoir parameters that can be appropriately estimated from a given data set and also reveals which components of the full data set are active in constraining the reservoir model. Thus, the approach provides a good means to integrate different data properly while avoiding the inclusion of irrelevant data during nonlinear regression.

A black‐box automated approach to calibrate numerical simulations and optimize cover design: Application to a flow control layer constructed on an experimental waste rock pile

“…The numerical model discretization needs to be adapted to those constraints, which often results in a significant increase in the computational time. Parallelization (Tavakoli et al, 2013) and surrogate models (Awotunde & Horne, 2011) can reduce the calibration time and are included in many different hydrogeological calibration problems such as geostatistical inversion and geological inversion (Carrera et al, 2005;Zhou et al, 2014). Nowadays, the software PEST is often used for solving such groundwater flow inverse problems (Doherty, 2015(Doherty, , 2018.…”

An Improved Adjoint-Sensitivity Computation for Multiphase Flow Using Wavelets

“…The wp-wk approach presented in this article is a multiobjective analog of the wp dthr -wk approach presented in Awotunde and Horne (2011) and Awotunde (2010), except that the superscript dthr is dropped here for convenience. The wp-wk approach involves transformation and thresholding of both the data space (p) and the model space (k).…”

“…The choice is automatically determined by the nature of the sensitivity matrix. Observations (Awotunde and Horne 2011) have shown that as the iteration progresses, the number and indices of the wavelets selected gradually become constant (i.e., the composition of the model-space wavelets becomes steady). Once the wavelets are selected, a back transformation is applied to retrieve the permeability distribution.…”

“…In this work, we present a wavelet approach to adjoint-sensitivity computations that reduces the associated cost of obtaining sensitivities significantly. The method uses the data-space-dimension-reduction approach presented by Awotunde and Horne (2011) to reduce the size of the system of linear equations solved in the adjoint method.…”

Sensitivity-based upscaling for history matching of reservoir models

“…Thus, different methods of model space reduction have been proposed in the literature to reduce the number of model parameters to be estimated from the production history, thereby reducing the non-uniqueness associated with the inverse modeling. One such model reduction method is the wavelet multiscale inverse analysis (Lu and Horne 2000;Horne 2005, 2006a, b;Awotunde and Horne 2011a, b, 2012, 2013. The various methods of model space reduction often automatically produce some level of smoothening (homogenization) of the reservoir model parameters such as grid block permeabilities.…”