B008 COMPARISON OF METHODS FOR DOWNSCALING OF COARSE SCALE PERMEABILITY ESTIMATES

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1 1 B8 COMPARISON OF METHODS FOR DOWNSCALING OF COARSE SCALE PERMEABILITY ESTIMATES Alv-Arne Grimstad 1 and Trond Mannseth 1,2 1 RF-Rogaland Research 2 Now with CIPR - Centre for Integrated Petroleum Research, University of Bergen Abstract Fine scale reservoir models with all available information integrated are important for predictions of future behavior. Traditionally, fine-scale realizations of the geostatistical model are adjusted in some way to integrate production data. Since these data do not support high resolution of the permeability representation, strong regularization of the history matching problem is needed. In this paper, we study how integration of information may be approached from the other direction. I.e., we study how information from well observations and a geostatistical model may be integrated with a coarse scale permeability estimate obtained through a multiscale estimation sequence (Adaptive multiscale estimation). There are several possible solutions to this problem, depending on which aspects of the geostatistical model it is important to match. We compare some methods with respect to how the resulting permeabilities match the available information. Introduction A major motivation for building reservoir models is to be able to make predictions about the future behavior of the reservoir under different scenarios. It is important to utilize as much information as possible when building the model. In addition, accurate simulation of some scenarios (e.g., investigation of placement of new wells) require a detailed numerical gridding of the model, at least in parts of the reservoir. Thus, we would like to have a fine scale reservoir model that is consistent with all available information. Several types of information about the permeability in the reservoir may be available. Geostatistical models try to quantify the spatial variability throughout the reservoir, but are based on assumptions about this variability. Well logs and core samples may be detailed, but informs only about the close vicinity of the wells. Production data, on the other hand, informs about the permeability in large areas between the wells, but only about large scale variations. Integrating all the information at the same scale of the reservoir model may not be advantageous, due to issues with computational cost and numerical instability. Some up- or downscaling of information will be necessary if we want to combine several information sources. A possible solution to the data integration task is the scale splitting method[3]; Adaptive multiscale estimation (AME)[2] followed by fine scale data integration. One aim of AME is to obtain a history matched estimate where the resolution of the permeability is determined by the information content in the available production data. After the history matching, information from other data types is integrated in a downscaling step. Several fine scale realizations may be generated from a single (or a few) history matched coarse scale model(s). This method is in contrast to history matching several fine scale realizations of the geostatistical model.

2 2 Since the time-consuming part of reservoir characterization is the conditioning to production history, the method with the fewest history matching runs should be the least expensive. However, the quality of the resulting permeability realizations should also be discussed. The match to production history and well observations is relatively easy to assess. Adherence to the parameters of the geostatistical model may also be quantified. But since the geostatistical model is only assumed, it may not be necessary to match all aspects of the model. It is not obvious which parts of the geostatistical model that are most important and need to be closely matched. The permeability is not uniquely determined by the available information, so the choice of which aspects of the geostatistical model to match will strongly influence the final permeability estimate(s). In this paper, we start with a discussion of some possible methods for downscaling a history matched (2D) coarse scale permeability estimate, i.e., integrating fine scale information such as well observations and geostatistical model. Some examples with application of the discussed methods are given, showing how the different ways of using the geostatistical model result in different types of permeability estimates. The values of various measurables of the resulting permeability estimates are calculated. These are the matches to the production history and well observations, and the histogram of the permeability field. Finally, we discuss and show how we may perform a final history matching step after the downscaling, using the permeability resolution obtained in AME, (i.e., using a limited number of parameters), but with smooth basis functions. Coarse scale history matched estimate We use Adaptive multiscale estimation (AME)[2] to obtain a coarse scale estimate of the permeability. AME solves a series of history matching problems with increasing resolution of the permeability representation. The permeability at each step is parameterized as k(x) =c T N ψ N(x), where ψ N is a vector of piecewise constant basis functions. The parameters c N are estimated by finding the minimum of the objective function J(c N )=(d m(c N )) T D 1 (d m(c N )), (1) where d is the measured production data, m is the corresponding data calculated with the reservoir simulator, and D is the covariance matrix of the measurement errors. Between the parameter estimation steps of AME, the parameterization is evaluated, investigating which refinement(s) would allow the largest improvement of the match to the data. This iterative procedure is terminated when the data is sufficiently well matched (data mismatch comparable to magnitude of measurement errors), or when no new improvement is achieved. In this way we obtain a coarse scale history matched permeability estimate. (More than one coarse scale estimate can be obtained with a modification of the algorithm.[]) Downscaling methods In this section we briefly discuss the methods we have investigated for downscaling of the coarse scale estimate. The references given for each method should be consulted for detailed explanations. Kriging Kriging (see, e.g., [1]) is a minimal-variance correction to an initial estimate, k, incorporating well observations (point measurements) of permeability in a manner consistent with a covariance model/function (the geostatistical model). The equations for the correction, k Kr,canbeformulated as: k Kr (x) =CA T (ACA T +Σ d ) 1 (d Ak (x)). (2)

3 3 In this formulation, Σ d is the error covariance matrix of the permeability measurements. C is the covariance matrix of the permeability field prior to incorporating the point measurements, and A is an observation operator, such that Ak are the estimated permeabilities corresponding to the observations d. To use Kriging for downscaling, we set k = k AME, and use equation (2) with C =(C 1 P + G) 1, where C P is the (prior) covariance matrix for the geostatistical model, G is the (fine scale) local basis information matrix from the AME estimation problem. This has the result that the perturbation k Kr only slightly (according to the uncertainty of the estimated permeability) alters the mean value of the permeability in the parameterization regions defined by AME. The Kriged estimate integrates production data and well observation information, and the permeability perturbation is consistent with the geostatistical model. However, since the coarse scale estimate, k AME, is piecewise constant, the downscaled estimate will only be piecewise smooth. Multiple realization of the permeability may be obtained by drawing realizations from a distribution with the updated covariance matrix after Kriging, and adding these to the Kriged permeability. The realizations obtained in this manner will, like the Kriged estimate, be piecewise smooth. Conditioning using Kriging The equation for the Kriging update may also be used in another way. If k is an unconditional realization from a distribution with covariance matrix C, thenk + k Kr,with k Kr given by equation (2), will also be a realization of the same distribution, but conditioned on d Ak = e, where e is a random vector with covariance matrix Σ d. (See, e.g., section 3.3 of [].) To use this in downscaling, we use as observations the estimated mean values in the parameterization regions determined in AME, and the available well observations. The observation operator, A, then consists of two parts: The first N AME rows extract the mean value of the permeability in the parameterization regions, while the N W last rows extract the permeability estimates in the grid blocks where the wells are located. Some consideration should go into the construction of the measurement error covariance matrix, Σ d. For the first part of the observation vector, we use the covariance matrix of the estimated region means. For the second part we use the covariance matrix of the permeability measurement errors. Since these are different types of measurements, the optimal scaling between the two are not obvious. However, for the cases in this paper, we have disregarded this issue. Regarding the cross covariance: Even if the mean permeability of a parameterization region and the permeability of a single block inside this region have a nonzero correlation, the observation error for these two values should not be correlated. Thus, we use [ ] ΣAME Σ d =, (3) Σ W where Σ AME is the estimation covariance matrix for the regional permeability means, and Σ W is the covariance matrix of the permeability measurement errors. This way, we can turn unconditional realizations of the prior geostatistical model into realizations conditioned on the coarse scale estimate and the well observations. Gradual deformation This was proposed as a method for conditioning realizations of a geostatistical model to production data (see, e.g., [4]). It consists of a series of perturbations of an initial realization. At each step, n k (k+1) = a 1 k (k) + a i k i, (4) i=2

4 4 where k (k) is the old estimate, k (k+1) is the new estimate, and {k i } are realizations of the geostatistical model. The coefficients {a i } that give the best match to the production data are sought. To ensure that the covariance properties of k (n) isthesameasthoseof{k i }, n i=1 a2 i = 1 is enforced. The method was further developed[7] by noting that the mean value is also conserved if n i=1 a i =1. To have non-trivial solutions for the coefficients {a i }, we need n>3. This development of the method allows one to use as {k i } realizations that are conditioned on well observations, and have k (n) also be conditioned on these observations. In this paper, we use this method with realizations based on the Kriged coarse scale permeability. This will give a better starting point for obtaining final realizations that match both types of data. Other methods Sequential simulation with Block Kriging[8] is another method for downscaling a coarse scale history matched estimate. Due to limited time, we could not examine this method closely. Final history matching step Since the fluid flow in the reservoir is influenced by fine scale permeability variations, the downscaling will most likely cause the match to the production history to deteriorate. A final history matching step may be needed to obtain a satisfactory match to all the data. However, since the downscaled permeability is strongly influenced by the history matched permeability, the required perturbations in a final history matching step will likely be small. This history matching step will need to use fine scale permeability representation, but there is no need for as many degrees of freedom as the number of grid blocks. The parameterization obtained in AME should be a good indication on the required resolution. The AME parameterization could be used as it is for the final perturbations, but if the geostatistical model prescribes smooth permeability variation, it could be better to use perturbations that are smooth. Both non-smooth and smooth perturbations will most likely be able to produce final estimates that match production data. The choice between the two approaches depends on how closely we want to follow the geostatistical model. In the following, we consider how to construct perturbations to the downscaled permeability that have a smoothness consistent with the geostatistical model, and a resolution consistent with the AME parameterization. To do this, we use equation (2) to define an expansion basis for the perturbations: Let the perturbations to the downscaled permeability be of the form k(p) = Bp, where B = CA T (ACA T +Σ d ) 1, () and minimize the objective function with respect to p. ThecolumnsofB are linear combinations of the columns of C, and the dimension of p is (N AME + N W ). Examples of the type of perturbations obtained in this way are shown in Figure in the examples section. Examples The examples presented in this section are based on a synthetic horizontal 2D case, where the reservoir is discretized into a simulation grid. Initially, the reservoir contains 1% oil, and it is produced using 3 water injection wells and production wells. The information available for reservoir characterization is: Production history data: Observed time series of pressure in injection and production wells, oil production rates and water production rates in production wells; 1 observations in each series, 27 observations in all. Well observations: The permeability at the position of the wells, 8 observations.

5 Geostatistical model: Assumed type of model (multigaussian with Gaussian covariance function), with assumed correlation function and mean value for the permeability. In addition, we assume known the porosity of the reservoir (uniform), and relative permeability and capillary pressure functions. Reference permeability and coarse scale history match The reference permeability is generated as a random realization of the geostatistical model. The logarithm of the permeability is assumed to follow a multigaussian distribution with a Gaussian correlation function. The mean value is log e (2) =.3, the variance is.3 and the range is 2 grid blocks. Synthetic production data is obtained by running the simulator with the reference permeability as input, and adding uncorrelated Gaussian noise to the simulator output. Figure 1: Left: Reference permeability. regions. The wells are shown as dots. Right: Coarse scale estimate obtained by AME with parameterization The reference permeability is shown in Figure 1, together with the coarse scale history matched permeability obtained with AME. The coarse scale estimate uses 17 parameters and obtains an objective function value of 27, which is close to the expected value, (N obs N par ). In the AME permeability we recognize many of the coarse scale features of the reference permeability. The match to the production history with the AME permeability is shown in Figure Figure 2: Production history from reference permeability and match obtained with the coarse scale estimate (AME). From left to right: Pressure in injection wells, pressure in production wells, water production rate and oil production rate. Downscaling step In this section we show the permeabilities obtained with the downscaling methods discussed above. Figure 7 should be consulted for measures of the match to production history, well observations and geostatistical model for each of the permeabilities. The permeability estimate obtained when we adjust the AME permeability by Kriging against the mismatch with observations in the well grid blocks is shown in the left plot of Figure 3. The right plot in the same figure shows the smooth change from the AME permeability. The resulting

6 permeability now contains small scale variation, but the discontinuities from AME still dominates Figure 3: Left: Downscaled estimate obtained with Kriging (Kr). Right: perturbation from coarse scale estimate. The first two plots in Figure 4 show realizations of the permeability obtained from the Kriged permeability in Figure 3. The small scale variability of these realizations is consistent with the geostatistical model, but the overall result is dominated by the discontinuities of the AME permeability. Still, the match to both production data and well observations is satisfactory. The last two plots of Figure 4 show realizations of the permeability obtained with Conditioning using Kriging. This method produce smooth permeabilities, as expected, but we note that a strong conditioning on the observations (regional mean and wells) may cause the variance of the permeability to increase somewhat. Figure 4: First two plots: Permeability realizations based on Kriged downscaled estimate (KrR). Last two plots: Downscaled estimates obtained with Conditioning using Kriging (CK) Two permeability realizations obtained when using Gradual Deformation on realizations based on the Kriged estimate (Figure 3) are shown in the first two plots of Figure. The discontinuities of the realizations based on Kriging are inherited. Figure : First two plots: Downscaled estimates obtained with the Gradual Deformation method (GD) on KrR realizations. Last two plots: Results after the CK permeabilities are subjected to a final HM step (CKHM).

7 7 Additional HM step The Gradual Deformation method is a way to perform a final history match on the downscaled estimates obtained with the Kriging method. This may be necessary since the match to production data has deteriorated as a result of the downscaling. Each minimization problem solved in the course of this method should be well posed due to the small number of parameters. However, because of the randomness in choice of perturbation direction, the progress is slow. The realizations from Conditioning using Kriging (last two plots of Figure 4) has also been subjected to a final history matching step. This history matching follows the method outlined in the section on Final history matching step. The resulting permeabilities are shown in last two plots of Figure. The change between k CK and k CKHM can be expressed as a linear combination of smooth basis functions, where each of the basis functions is a linear combination of the columns of the covariance matrix. A few of the basis functions are shown in Figure. Note that these functions have global support, but primarily change the value of only one observation of the A operator (e.g., the mean value of the permeability in one of the parameterization regions of AME). Due to the term Σ d, some change is allowed in the value of the other observations also Figure : Some of the basis functions for the final history matching step. The parameterization regions of AME are also shown. Discussion Figure 7 shows how the permeabilities downscaled with the different methods match the available information. In the left plot we note that the AME permeability (AME) gives a good match to the production history. As the permeability is Kriged (Kr), the match to well observations (middle plot) is improved at the expense of the match to production history. As expected, the realizations based on the Kriged permeability (KrR) show some variations in the match to both kinds of data AME Kr KrR GD CK CKHM AME Kr KrR GD CK CKHM 4 AME Kr KrR GD CK CKHM Figure 7: Left: Mean square normalized error in match to production history. Middle: Mean square normalized error in match to well observations. Right: Observed histograms. The permeabilities obtained with Gradual Deformation (GD) further improve both the history match and the match to well observations compared to the Kriging realizations.

8 8 The realizations obtained with Conditioning using Kriging (CK) exhibit a dramatic improvement in the match to production data through the final history matching step (CKHM). The match to well observations is also improved, but not as much. The right plot in Figure 7 shows the histograms of the permeabilities, with the prior model shown as a reference to the left. The results show that with a final history matching of the realizations obtained with Conditioning using Kriging, we obtain fine scale permeabilities that match both production data and well data, as well as the assumed histogram of the permeability distribution. The final history matching step is performed with a fine scale permeability, but with only few degrees of freedom, comparable to the resolution obtained in AME. Thus the computational work is reduced compared to minimizing with full grid block parameterization. Due to the relatively good starting point, the history match for each of the realizations converges with only a few iterations of the parameters. Summary The different areal support and associated length scale of available information make integration of the information into a permeability estimate a difficult task. Which aspects to match for each information type influences the integration method and the resulting estimates. The issue of what constitutes a good match (of the different types of information) deserves more attention. We have investigated some possibilities for downscaling a coarse scale history matched permeability estimate obtained with Adaptive multiscale estimation. The downscaling incorporates well observations and assumptions about the geostatistical model into the estimate. We have also shown how a final fine scale history matching step may be performed on the fine scale realizations using only few degrees of freedom, making good use of the permeability resolution obtained with Adaptive multiscale estimation. Acknowledgements The research presented in this paper has been financially supported by ENI SpA, Norsk Hydro ASA, and The Research Council of Norway. References 1. C. V. Deutsch and A. Journel. GSLIB Geostatistical Software Library and User s Guide. OxfordUni- versity Press, A.-A. Grimstad, T. Mannseth, G. Nævdal, and H. Urkedal. Adaptive multiscale permeability estimation. Computational Geosciences, 7(1):1 2, A.-A. Grimstad, T. Mannseth, J.-E. Nordtvedt, and G. Nævdal. Reservoir characterization through scale splitting. In Proceedings of ECMOR VII, Baveno, Italy, September M. Le Ravalec-Dupin and B. Nœtinger. Using the gradual deformation method for optimization. In Proceedings of ECMOR VII, Baveno, Italy, September 2.. L. K. Nielsen, S. Subbey, M. Christie, and T. Mannseth. Reservoir description using dynamic parameterisation selection with a combined stochastic and gradient search. Submitted, 24.. H. Rue and T. Follestad. GMRFLib: a C-library for fast and exact simulation of Gaussian Markov random fields. GMRFLib manual found at hrue/gmrflib/, X.-H. Wen, T. Tran, R. Behrens, and J. Gómez-Hernández. Production data integration in sand/shale reservoirs using sequential self-calibration and geomorphing: A comparison. In Proceedings of the 2 SPE ATCE, Dallas, Texas, 1 4 October 2. SPE S. Yoon, A. Malallah, A. Datta-Gupta, D. Vasco, and R. Behrens. A multiscale approach to production data integration using streamline models. In Proceedings of the 1999 SPE ATCE, Houston, Texas, Oct SPE 3.

Geostatistical History Matching coupled with Adaptive Stochastic Sampling: A zonation-based approach using Direct Sequential Simulation

Geostatistical History Matching coupled with Adaptive Stochastic Sampling: A zonation-based approach using Direct Sequential Simulation Eduardo Barrela* Instituto Superior Técnico, Av. Rovisco Pais 1,