[1] In this paper, we use a two-stage Markov chain Monte Carlo (MCMC) method for subsurface characterization that employs coarse-scale models. The purpose of the proposed method is to increase the acceptance rate of MCMC by using inexpensive coarse-scale runs based on single-phase upscaling. Numerical results demonstrate that our approach leads to a severalfold increase in the acceptance rate and provides a practical approach to uncertainty quantification during subsurface characterization.

Evapotranspiration (ET) of Amazon forests is a main driver of regional climate patterns and an important indicator of ecosystem functioning. Despite its importance, the seasonal variability of ET over Amazon forests, and its relationship with environmental drivers, is still poorly understood. In this study, we carry out a water balance approach to analyse seasonal patterns in ET and their relationships with water and energy drivers over five sub-basins across the Amazon basin. We used in-situ measurements of river discharge, and remotely sensed estimates of terrestrial water storage, rainfall, and solar radiation. 15We show that the characteristics of ET seasonality in all sub-basins differ in timing and magnitude. The highest mean annual ET was found in the northern Rio Negro basin (~1497 mm year -1 ) and the lowest values in the Solimões River basin (~986 mm year -1 ). For the first time in a basin-scale study, using observational data, we show that factors limiting ET vary across climatic gradients in the Amazon, confirming local-scale eddy covariance studies. Both annual mean and seasonality in ET are driven by a combination of energy and water availability, as neither rainfall nor radiation alone could explain patterns in ET. 20 In southern basins, despite seasonal rainfall deficits, deep root water uptake allows increasing rates of ET during the dry season, when radiation is usually higher than in the wet season. We demonstrate contrasting ET seasonality with satellite greenness across Amazon forests, with strong asynchronous relationships in ever-wet watersheds, and positive correlations observed in seasonally dry watersheds. Finally, we compared our results with estimates obtained by two ET models, and we conclude that neither of the two tested models could provide a consistent representation of ET seasonal patterns across the Amazon. 25 2012; Christoffersen et al., 2014;Hasler and Avissar, 2007;Restrepo-Coupe et al., 2016). ET is also an indicator of ecosystem 30 Earth Syst. Dynam. Discuss.,

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Summary The paper presents a novel method for rapid quantification of uncertainty in history matching reservoir models using a two-stage Markov Chain Monte Carlo (MCMC) method. Our approach is based on a combination of fast linearized approximation to the dynamic data and the MCMC algorithm. In the first stage, we use streamline-derived sensitivities to obtain an analytical approximation in a small neighborhood of the previously computed dynamic data. The sensitivities can be conveniently obtained using either a finite-difference or streamline simulator. The approximation of the dynamic data is then used to modify the instrumental proposal distribution during MCMC. In the second stage, those proposals that pass the first stage are assessed by running full flow simulations to assure rigorousness in sampling. The uncertainty analysis is carried out by analyzing multiple models sampled from the posterior distribution in the Bayesian formulation for history matching. We demonstrate that the two-stage approach increases the acceptance rate, and significantly reduces the computational cost compared to conventional MCMC sampling without sacrificing accuracy. Finally, both 2D synthetic and 3D field examples demonstrate the power and utility of the two-stage MCMC method for history matching and uncertainty analysis. Introduction Uncertainty exists inherently in dynamic reservoir modeling because of several factors, the primary ones being the modeling error, data noise, and the nonuniqueness of the inverse problems that causes several models to fit the dynamic data. Under a Bayesian framework, the uncertainty in the reservoir models can be evaluated by a posterior probability distribution, which is proportional to the product of a likelihood function and a prior probability distribution of the reservoir model. To quantify the uncertainty, it is necessary to generate a sequence of model realizations that are sampled appropriately from the posterior distribution. Rigorous sampling methods, such as Markov Chain Monte Carlo (MCMC) (Oliver et al. 1997; Robert and Casella 1999), provide the accurate sampling albeit at a high cost because of their high rejection rates and the need to run a full flow simulation for every proposed candidate. There is also additional cost associated with a burn-in time needed for the MCMC to assure that the starting state does not bias sampling. Approximate sampling methods, such as randomized maximum likelihood (RML) (Oliver et al. 1996; Kitanidis 1995), are commonly used to avoid the high cost associated with the MCMC methods. For linear problems (Gaussian posterior distributions), RML has an acceptance probability of unity; however, the assumptions made in RML may be too restrictive for nonlinear problems, which is typically the case for reservoir history matching. The main appeal of RML is its computational efficiency and ease of implementation within the framework of traditional automatic history matching via minimization. There are also some examples in the literature that the RML has favorable sampling properties for nonlinear problems (Liu et al. 2001), although it is likely to be problem-specific. There is a need for an efficient and rigorous approach to uncertainty quantification for general nonlinear problems related to history matching. We propose a two-stage MCMC approach for quantifying uncertainty in history matching geological models. Our proposed sampling approach is computationally efficient with a significantly higher acceptance rate compared to traditional MCMC algorithms. In the first stage, we compute the acceptance probability for a proposed change in reservoir parameters based on a fast linearized approximation to flow simulation in a small neighborhood of the previously computed dynamic data. In this stage, no reservoir simulations are needed to explore the model parameter space. In the second stage, those proposals that passed a selected criterion of the first stage are assessed by running full flow simulations to assure the rigorousness in sampling. Then, these samples are either rejected or accepted using the MCMC selection criterion. It can be shown that the modified Markov chain converges to a stationary state corresponding to the posterior distribution. Moreover, the two-stage approach increases the acceptance rate, and reduces the computational cost required for the MCMC sampling. To propose MCMC samples, we consider two instrumental probability distributions, the random walk sampler and the Langevin sampler (Robert and Casella 1999). Both 2D synthetic and 3D field examples demonstrate that the two-stage MCMC method is computationally more efficient than the conventional MCMC methods, but does not sacrifice their accuracy. The proposed method has been successfully used in conjunction with single-phase upscaling methods (Efendiev et al. 2005). All examples in the paper are based on the fine-scale geological models.

We describe the full automorphism group of the power (di)graph of a finite group. As an application, we solve a conjecture proposed by Doostabadi, Erfanian and Jafarzadeh in 2013.

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We have developed a new constrained optimization approach to the coarsening of 3D reservoir models for flow simulation. The optimization maximally preserves a statistical measure of the heterogeneity of a fine scale model. Constraints arise from the reservoir fluids, well locations, pay/non-pay juxtaposition, and large scale reservoir structure and stratigraphy. The approach has been validated for a number of oil and gas projects, where flow simulation through the coarsened model is shown to provide an excellent approximation to high resolution calculations performed in the original model. The optimal layer coarsening is related to the analyses of Li and Beckner (2000), Li, Cullick and Lake (1995), and Testerman (1962). It differs by utilizing a more accurate measure of reservoir heterogeneity and by being based on recursive sequential coarsening, instead of sequential refinement. Recursive coarsening is shown to be significantly faster than refinement: the cost of the calculation scales as (NX·NY·NZ) instead of (NX·NY·NZ)[2]. The more accurate measure of reservoir heterogeneity is very important; it provides a more conservative estimate of the optimal number of layers than the analysis of Li et.al.. The latter is shown to be too aggressive and does not preserve important aspects of the reservoir heterogeneity. Our approach also differs from the global methods of Stern (1999) and Durlofsky (1994). It does not require the calculation of a global pressure solution and it does not require the imposition of large scale flow fields, which may bias the analysis, Fincham (2004). Instead, global flow calculations are retained only to validate the reservoir coarsening. Our approach can generate highly unstructured, variable resolution, computational grids. The layering scheme for these grids follows from the statistical analysis of the reservoir heterogeneity. Locally variable resolution follows from the constraints (reservoir structure, faults, well locations, fluids, pay/non-pay juxtaposition). Our reservoir simulator has been modified to allow a fine scale model to be initialized and further coarsened at run time. This has many advantages in that it provides both simplified and powerful workflows, which allow engineers and geoscientists to work with identical shared models. Introduction The development of (coarsened) reservoir simulation models from high resolution geologic models remains an active field of research [1–7]. In the current study we will report upon our success in the use of coarsening algorithms to determine a ‘best’ reservoir simulation grid obtained by grouping fine scale geologic model cells into effective simulation cells. Our results differ from previous studies in that we have found a statistical analysis of the static properties of the model that appears to identify the best grid for dynamic reservoir simulation. Coarsening beyond the degree indicated by our analysis discards too much of the underlying heterogeneity. It will overly homogenize the properties on the simulation grid. Finer models will, of course, retain at least as much reservoir heterogeneity, but are more costly. Our analysis uses a statistical technique for layer grouping and a constrained approach for areal gridding. The resulting composite corner point grid (CCPG) has many of the advantages of unstructured PEBI grids in that they can follow major features of the geologic description [8]. Compared to PEBI grids, they have the advantage of exact alignment of the simulation cells with the geologic model, which will minimize property upscaling errors, and the practical advantage that they may be utilized without the development of new simulation pre and post processing applications. This approach also moves us closer to having an Earth Model shared between the reservoir engineer and the reservoir geologist: the 3D geologic model will provide the grid on which the high resolution initialization of the simulation model will be calculated.

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