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Spatial Stochastic Point Models for Reservoir Characterization

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Anne Randi Syversveen Spatial Stochastic Point Models for Reservoir Characterization Dr.Ing. Thesis Department of Mathematical Sciences Norwegian University of Science and Technology 1997 DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. Preface This thesis is submitted in partial fulfillment of the requirements for the degree “Doktor Ingenipr ” (Dr.Ing.). The work is financed by the Research Council of Norway under the Propetro program. My research has been carried out while I was a member of the statistics group at the Department of Mathematical Sciences, a period I have greatly enjoyed. I thank my supervisor Henning Omre for all his support and guidance. I am grateful to my coauthors, Havard Rue, Jesper Mgller and Rasmus Waagepetersen for the good collaboration we had. I thank Jesper M0ller for inviting me to Arhus in the autumn 1995 and for his hospitality during my stay. I also thank Hakon Tjelmeland for many enlightening discussions. At last, I thank Knut-Andreas for his encouragement, love and care, but also for technical support. Trondheim, November 1997 Anne Randi Syversveen Thesis outline The thesis consists of the following articles: I Conditioning of marked point processes within a Bayesian framework. (With Henning Omre) In Scandinavian Journal of Statistics, Volume 24, Number 3, 1997. II Marked point models for facies units conditioned on well data. (With Henning Omre) In Geostatistics Wollongong ’96, Volume 1. III An approximate fully Bayesian estimation of the parameters in a reservoir charac­ terization model. IV Bayesian object recognition with Baddeley ’s delta loss. (With Havard Rue) To appear in Advances in Applied Probability (SGSA) March 1998. V Log Gaussian Cox processes. (With Jesper M0ller and Rasmus Waagepetersen) To appear in Scandinavian Journal of Statistics. Appendix: A method for approximate fully Bayesian analysis of parameters. (With Havard Rue) In Communications in Statistics; Simulation and Computation, Volume 26, Issue 3, 1997. Papers I—III are on stochastic modeling of geology in petroleum reservoirs, and they are the main part of the thesis. It is natural to read paper I before paper II and III, although they can be read independently of each other. Paper IV is on object recognition in image analysis and in paper V a special type of spatial Cox processes is discussed. Co mmon for all papers is the use of spatial point processes, see the classical books Stoyan et al. (1995), Daley and Vere-Jones (1988) and Diggle (1983). We start by describing how stochastic modeling is used in reservoir characterization. After that, a short presentation of each paper is given. Background, on Reservoir Characterization In order to forecast future oil production from a reservoir, the flow properties must be modeled. Flow modeling is usually done by solving a set of partial differential equations numerically in a flow simulator, see for example Aziz and Settari (1979) and Lake (1989). Parameters in these equations are reservoir characteristics such as permeability, porosity and initial saturation. These characteristics must be known everywhere in the reservoir i in order to solve the equations, which in turn requires a description of the geology in the reservoir, for example the amount and distribution of different rock types or faults. However, there are very few observations of the geological properties of the reservoir. Observations can be collected from wells, but usually the number of wells is quite limited, due to high drilling costs. Other sources for information about reservoir characteristics are seismic data and production history, but the amount of observations from the reservoir is generally rather limited. On the other hand, extensive geological experience and knowledge are usually available. For example, geologists may have been studying outcrops which are believed to be analogs to the petroleum reservoir and they have knowledge about how geological processes develop. Because the information about geology is limited, a stochastic model is used to characterize the geologic properties of the reservoir. A realization from the stochastic model is used as input to the flow simulator, and oil production associated with the actual realization from the stochastic model is obtained. Repeating this several times, the uncertainty in oil production associated with the stochastic model is investigated. In order to utilize all available information about the reservoir, geological knowledge as well as observations should be included in the stochastic model of the geology. This is best achieved by a Bayesian framework, where expert knowledge guides the model parameter­ ization and choice of prior distribution. Observations from the reservoir under study are incorporated through likelihood functions. Realizations from the posterior distribution are used as input to the flow simulator. In this thesis, only well observations are considered. Examples on models where seismic data and data from production history are included are found in Hegstad and Omre (1997), Bide et al. (1996) and Tjelmeland and Omre (1997). The models described in the first two of these references describe reservoir characteristics such as permeability and porosity, while the last mentioned paper describes a model for spatial distribution of rocks in the reservoir. Modeling of rock distribution is also a theme for this thesis, as mentioned below. The spatial distribution of rock types in a petroleum reservoir can be described in two different ways, either as a mosaic phenomenon or as an event phenomenon; see Hjort and Omre (1994), Haldorsen and Damsleth (1990) and Ripley (1992). In the first case, different rock types are randomly packed without a background, for example varying facies types in a shallow marine environment. Markov random fields are often used to model this, see Tjelmeland and Besag (1996). In the second case, objects of one or more facies types are randomly distributed in a background of one rock type, for example shale units in a sand matrix. Marked point models may be used to model this phenomenon. The central geoscience reference on this field is Haldorsen and Lake (1984), who described a simple marked point model for shales in a background of sand. This thesis focuses on modeling of event phenomena by marked point models, and the model described is an extension of the model discussed by Haldorsen and Lake (1984). Mosaic phenomena are not discussed. Marked point models are also used to model fractures, or faults in the reservoir, see for example Munthe et al. (1994) and Wen and Sinding-Larsen (1997). Faults are usually n clustered, and marked point models with spatial attraction between points will be natural models. Summary The first three papers are on the reservoir characterization problem. In paper I a marked point model is defined for objects against a background in a two dimensional vertical cross section of the reservoir. The model extends the simple model defined by Haldorsen and Lake (1984) in several ways. The model handles conditioning on observations from more than one well for each object, it contains interaction between objects, and the objects have the correct length distribution when penetrated by wells. The Haldorsen model did not have any of these properties. Haldorsen and Lake (1984) modeled each object as a rectangle, while we model the objects as rectangles with Gaussian random fields superimposed on top and bottom. This makes it possible to condition on observations from more than one well. Our model has pairwise repulsion between objects and stochastic number of objects. Chessa (1995) pointed out that penetrated objects tend to be larger than non-penetrated, and the simulation algorithm takes care of this. The model is developed in a Bayesian setting. As mentioned before, few observations of the reservoir are available. Therefore it is natural to use Bayesian models to incorporate geological (prior) knowledge. Observations available are top and bottom of objects in a limited number of wells. Moreover, it is assumed as known whether observations from different wells come from the same object or not. The Metropolis-Hustings algorithm is used to simulate from the posterior distribution. Because the well observations are assumed to be exact, the proposal kernels must be chosen carefully. In image analysis, proposals are often drawn from the prior distribution. This will not work well here, since the acceptance rate will be zero because of the structure of the likelihood function. However, we are able to calculate marginal posteriors for the objects, and proposals are drawn from these distributions. The model and the simulation algorithm is demonstrated on an example with simulated data. The ideas regarding conditioning on well observations are implemented in the commercial software system STORM developed by Smevig Technologies a/s. Lia et al. (1996) give a description of the stochastic model used in STORM. They make a simple extension to three dimensions. Three types of objects are used, and one type is rectangular boxes with (two-dimensional) Gaussian fields on top and bottom. None of the object types are able to handle conditioning on all types of well configurations. For large objects and many wells, the conditioning might be impossible to fulfill. This is a possible area for further research. A solution might involve Gaussian fields added to the sides of the objects too, not only top and bottom. This can make objects flexible enough to fulfill conditioning. Paper II is written for a geostatistical audience, and presents almost the same model as paper I. However, one important extension is made: It is not longer assumed to be known iii whether observations from different wells come from the same object or not. Furthermore, a real data example with observations taken from an outcrop is also presented. Paper III is about parameter estimation in the model described in paper I. Two parame­ ters are estimated, one related to the intensity of objects and one related to the interaction between objects.
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