Prediction of Oil Production With Confidence Intervals

We present a prediction methodology for reservoir oil production rates which assesses uncertainty and yields confidence intervals associated with its prediction. The methodology combines new developments in the traditional areas of upscaling and history matching with a new theory for numerical solution errors and with Bayesian inference. We present recent results of coworkers and ourselves. Introduction A remarkable development in upscaling 2 allows reduction in computational work by factors of more than 10,000 compared to simulations using detailed geological models, while preserving good fidelity to the oil cut curves generated from solutions of the highly detailed geologies. In common engineering practice, the detailed geology models are too expensive for routine simulation. This is especially the case if an ensemble of realizations of the reservoir is to be explored. The ensemble allows consideration of distinct geological scenarios, an issue of greater importance in many cases than errors associated with upscaling of detailed geology to obtain a coarse grid solution. Upscaling allows rapid solutions and is a key to good history matching. We formulate history matching probabilistically to allow quantitative estimates of prediction uncertainty . A probability model is constructed for numerical solution errors. It links the history match to prediction with confidence intervals. The error analysis establishes the accuracy of fit to be demanded by the history match. It defines a Bayesian posterior probability for the unknown geology. Thus history matching defines a revised ensemble of geologies, with revised probabilities or weights. Prediction is based on the forward solution, averaged with these weights. Confidence intervals are also defined by the probability weights for the ensemble together with error probabilities for the forward solution. Results of the prediction methodology will be described, based on simulated geologies and simulated reservoir flow production rates. Efficient scaleup allows a sizable number of geologies to be considered. The Bayesian framework incorporates prior knowledge (for example from geostatistics or seismic data) into the prediction. We show that a history match to past production rates improves prediction significantly. The plan of this paper is to pick one fine grid reservoir from an ensemble and regard its solution as a stand in for production data. Other reservoirs in the ensemble are evaluated on the basis of the quality of their match to this data. They are upscaled, simulated on a coarse grid, and the upscaled solution is compared to production history from the data. Probability of mismatch between the coarse grid solution and the data weights each realization in a balanced manner according to (a) its prior probability and (b) the quality of its match to data. We thus define a posterior probability on the ensemble, which is used for prediction. Uncertainty in the prediction has two sources: uncertainty in the geology, or history match, as discussed above, and uncertainty in the forward simulation, also conducted on coarse grids. The total uncertainty receives contributions from these two sources, and its analysis leads to confidence intervals for prediction. The intended application of this prediction methodology is to guide reservoir development choices. For this purpose, simulation of an ensemble of reservoir scenarios is important to explore unknown geological possiblities. Statistical methods are important to assess the ensemble of outcomes. The methods are intended for use by reservoir managers and engineers. For this purpose, the methods will need to be augmented by inclusion of factors omitted from the present study. The significance of our methods is their ability to predict the risk, or uncertainty associated with production rate forecasts, and not just the production rates themselves. The latter feature of this method, which is not standard, is very useful for evaluation of decision alternatives. Stochastic History Matching Problem Formulation. Stochastic history matching is based on an ensemble of geological realizations. To simplify this study, we fix the geologic model aside from the SPE 66350 Prediction of Oil Production With Confidence Intervals James Glimm, SPE, SUNY at Stony Brook and Brookhaven National Laboratory; Shuling Hou, SPE, Los Alamos National Laboratory; Yoon-ha Lee, SUNY at Stony Brook; David Sharp, SPE, Los Alamos National Laboratory; and Kenny Ye, SUNY at Stony Brook. JAMES GLIMM SPE 66350 2 permeability field, which is taken to be a random variable simple form of the Darcy and Buckley-Leverett equations 0 = ∇ − = p K v λ ; 0 = ∇v , ...................(1) ( ) 0 = ⋅ ∇ + ∂ ∂ s f v t s , ...............................(2) where λ is a relative mobility, K the absolute permeability, v velocity, p pressure, s the water saturation and f the fractional flow flux. We consider these equations in a two dimensional (reservoir cross section) geometry, 1 0 ≤ ≤ x , 1 0 ≤ ≤ z in dimensionless units. Assume no flow across the boundaries 1 , 0 = z and a constant pressure drop across the boundaries 1 , 0 = x . The absolute permeability K is spatially variable, with an assumed log normal distribution. We characterize the covariance ( ) K ln by correlation lengths 50 / 1 = z l and = lx ( ) 0 1 8 0 6 0 4 0 2 0 . , . , . , . , . . Thus, ( ) K ln is actually a Gaussian mixture, and is not Gaussian. This distribution for K is called the prior distribution. Each realization is a specific choice of K . We consider an ensemble defined by 500 realizations of K , 100 for each of the five correlation lengths, selected according to the above Gaussian distribution. Each K is specified on a 100 x 100 grid (the fine grid). K and the fractional flow functions f are then upscaled to grids at the levels 5 x 5, 10 x 10, and 20 x 20. Each of the coarse grid upscaled reservoirs is also solved, in all cases for up to 1.4 pore volumes of injected fluid (1.4 PVI). We select one of the geologies, 0 i K , as representing the exact but unknown reservoir. We observe the oil cut 0 i f generated by the fine grid solution for times 0 0 t t ≤ ≤ (PVI). This data represents past, historical data, and using it, we seek to predict production for 1 0 t t t ≤ ≤ = 1.4 PVI, i.e. into the future. The solution is (a) history matching, to select a revised ensemble of geologies, which reflect agreement with history data, and (b) forward simulation, averaged over the revised (posterior) ensemble, to predict the future production. The Bayesian Framework. In the Bayesian framework, the prediction problem is solved by assigning a probability, or likelihood to any degree of mismatch between the coarse grid oil cut ( ) t c j and the observed history ( ) { } 0 0 , 0 t t t f O i ≤ ≤ = , where ( ) t fi0 is the oil cut for the reservoir 0 i K computed on the fine grid (and the fine grid is conceptually considered to be exact). The probability or likelihood of the observation given the geology K is denoted ( ) K O p | . A mismatch could arise due to measurement errors, or as we consider here, due to use of a coarse grid in a simulation analysis. According to Bayes’ theorem, the posterior probability for the geology defined by the permeability realization K is ∫ = dK K p K O p K p K O p O K p ) ( ) ( ) ( ) ( ) ( , .........................(3) where ( ) K p is the prior probability for the realization K. The prior probability is defined, for example, by methods of geostatistics 6, 7, 8, 9, , and in the present context it is defined by the above mixture of Gaussians with specified correlation lengths. In the absence of errors, there would be no mismatch, and we could accept geology j K as a history match only if ( ) ( ) t f t c i j 0 ≡ . This is of course unrealistic, as errors do occur. Since ( ) j K O p | assumes 0 i j K K = is exact, the mismatch is assumed to be due to an error in determining j c . We write j j j c f e − = as the error. Measurement errors also contribute to the mismatch likelihood, but for simplicity we concentrate on scale up and numerical solution errors only. Thus, ( ) j K O p | is the probability of