Jeoistatistiksel, statik ve kararsız basınç testi verilerine koşullandırılmış heterojen geçirgenlik ve gözeneklilik sahalarının türetilmesi

Abstract

Rezervuar kayaç heterojenliğinin, rezervuar üretim performansını önemli ölçüde kontrol ettiği oldukça iyi bilinmektedir. Dolayısıyla, gözenekli ortamda akışı kontrol eden petrofizik özelliklerin (geçirgenlik, gözeneklilik gibi) rezeruvar içersinde nasıl bir dağılım (heterojenlik) gösterdiğinin belirlenmesi"ye tanjmlanması, rezervuar performans tahminlerinin doğru bir şekilde yapılmasında ön koşuldur^ Bu amaca ulaşmak için izlenecek en uygun yak4aşım,-ölçümlej::^a gözlemler yoluyla belirlenmiş eldeki tüm jeolojik, petrofizik^ jeofizik ve üretim verilerine doğrudan koşullanmış rezervuar kayaç özelliklerine ait dağılımların (genel anlamda, rezervuar tanımlamalanmn) belirlenmesine dayanır. Jeoistatistik; durağan (kuyu log, karot analiz, sismik, jeolojik, vb.) verilere doğrudan koşullandırilmiş^kayaç özellikleri dağılımlarının türetilmesinde kullanılan önemli bir araç olmakla beraber, dinamik (kuyu-basınç testi, kuyu-izleyici, üretim debisi, vb.) verilere doğrudan koşullanmış kayaç özellikleri dağılımlarının belirlenmesinde yetersiz kalmaktadır. Çalışmanın temel amacı, statik ve jeoistatistiksel verilerin yamsıra, eldeki kuyu basmç testi verilerini de koşullandırılmış rezervuar tanımlamalan türetmede kullanılabilen yöntemlerin araştırılması, genel ve etkin olanlarının okurlara sunulmasıdır. Bu amaç için, çalışmada üç aşamalı bir yaklaşım izlenmiştir. İlk aşamada, rezervuar heterojenliği ve jeoistatistik ile ilgili temel bilgiler ile eldeki statik ve/veya dinamik verilere koşullanmış rezervuar tanımlamalan türetmede kullanılabilen yöntemler ve bu yöntemlere ait bilgiler derlenmiştir (bkz. Bölüm 2). İkinci aşamada, heterojenliğin kuyu basmç testi verileri üzerinde nasıl bir etki yarattığına ait araştırmaları yapmak üzere bir ve iki boyutlu simulatörler geliştirilmiştir. Bu simülatörlerin geliştirilmesine ait teknik ayrıntılar, bu simülatörlerin doğrulanmasına yönelik kıyaslamalar Bölüm 3'de verilmektedir. Bölüm 4'de deterministik ve stokastik yöntemler kullanılarak oluşturulan heterojen geçirgenlik ve/veya gözeneklilik dağılımlı çeşitli rezervuar ortamlarındaki basmç testi verilerinin heterojenliğe karşı nasıl bir duyarlılık gösterdiği araştinlmaktadır. Bu bölümde, kuyu basmç testi verilerinin heterojenlikten oldukça karmaşık bir şekilde etkilendiği ortaya konularak, basmç testi verilerinden elde edilebilecek etken bir geçirgenlik (veya gözeneklilik) değerinin, özellikle heterojen geçirgenliğin (veya gözenekliliğin) bölgesel üişkilenme (korelasyon) gösterdiği durumlarda, basit bir harmonik, geometrik veya aritmetik bir ortalamayla temsilinin imkansız olduğu gösterilmiştir. Farklı heterojen geçirgenlik dağılımları göz önünde bulundurarak, sadece kuyu-basınç verilerinden hareketle tekil bir rezervuar geçirgenlik dağılımı elde edilebileceği düşüncesinin bir yanılgı olacağı da ayrıca gösterilmektedir. Bölüm 5'de, her türlü durağan (statik) ve dinamik veriyi koşullamada kullanılabilecek Bayes yaklaşımına dayanan ters problem metodolojisi tanıtılmış, bu metodolojinin eldeki statik, jeoistatistik ve kuyu basınç testi verilerine koşullandırılmış geçirgenlik ve gözeneklilik da&hmKS^b^rieMeTTçin tek- faz/bir boyutlu simulator aracılığıyla yâpflahüyguTamaları sunulmuştur

It has long been recognized that reservoir rock heterogeneities have strong influence on the performance of reservoirs during waterflooding and/or gas injection and EOR techniques. Therefore, a better description of reservoir rock property (such as permeability and porosity) fields is essential to make correct reservoir performance predictions. The appropriate way to reach this goal is to integrate all available data: geological, petrophysical, geophysical, and production data. However, how to effectively integrate these data is a challenge to the person or team working in the field of reservoir characterization requiring a multistage and cross-disciplinary work [1]. Recent developments in reservoir characterization methods have helped better describe reservoir rock property fields. Particularly, geostatistical methods viewing such reservoir properties as permeability and porosity as being regionalized random parameters with spatial correlation structures have provided many avenues to petroleum reservoir engineers in improving estimates of hydrocarbon reserves, helping to select appropriate locations for infill wells and in obtaining better predictions of primary, secondary and tertiary recovery processes [2]. It is important to realize that reservoir performance predictions require the values of reservoir parameters. A heterogeneous reservoir is in principle specified by an infinite number of parameters (say permeability and porosity) values. If it were practically feasible to obtain information at every point in reservoir, we would determine the values of all reservoir parameters and make performance predictions with great confidence, in practice, however, we have limited information and data available (from observations of some reservoir parameters and/or from reservoir performance measured at wells through some state variables such as pressure, production rates, etc.) to infer these parameters. Furthermore, a computational reservoir model (i.e., a numerical simulator, usually based on finite differences) for conducting performance predictions can only contain a finite number of parameter values. Mathematically, this means that the estimation of the values of reservoir (model) parameters for reservoir characterization purposes is inherently an undetermined problem (i.e., one having a non-unique solution) because of the large number of unknown parameters relative to the available data. Of course, it is always desirable to model the reservoir with as fine grid as possible. However, due to the limitations on today's computer technology, we can run reservoir simulators with 10,000 to 1,000,000 gridblocks for performance predictions [3]. Considering a single phase flow simulator with 10,000 gridblocks and reservoir porosity and permeability as unknown reservoir parameters, we see that running such a simulator requires 20,000 input values of permeability and porosity. Then, the question arises how we obtain so many values of reservoir parameters to make a prediction run. XDC It is clear that we have never enough information to determine reservoir parameters at all points in the reservoir, and that we have to estimate these parameters from limited information obtained from observations and data (either static or dynamic) obtained through measurements usually made where the wells are located. Furthermore, these available information and data normally will contain uncertainties due to measurement or observation errors. This indicates that we will have always uncertainty in the reservoir description that we attempt to generate from such available data. Consequently, we will have uncertainty in any reservoir performance prediction. Then, the ultimate goal of reservoir characterization is to asses the uncertainty in performance predictions. Having characterized the uncertainty in predicted performance, one can make reservoir management decisions that account for our lack of complete knowledge of the true reservoir. Although a reservoir intrinsically deterministic, a feasible approach to reservoir characterization should be based on a stochastic view point due to above mentioned problems [4]. We can better tackle with this problem by treating the parameters to be estimated as random variables and formulating the inverse problem in terms of probability calculus. Then, we can asses the uncertainty both in the model (i.e., a reservoir description) and in the performance prediction. By viewing reservoir property fields as random variables (actually as random variables with spatial correlation), geostatistics provides a tool to generate realizations of reservoir rock property fields from the so-called static data, which refers to data obtained from well logs, core analysis, seismic and geologic knowledge. (Here, we refer to a realization as a sample of reservoir parameters, which is representative of the true reservoir rock property field with a certain probability or occurrence, and we expect each realization to have a different probability). Usually, most of these data (except seismic data which is abundance at interwell distances) are available where the wells are located and abundance in vertical direction along the well. However, due to lack of closely spaced data in lateral directions, there is great uncertainty in realizations generated from geostatistical simulation methods such as kriging and co-kriging. Generally, realizations generated from static data cannot match the dynamic performance. To reduce the uncertainty in reservoir characterization and make reliable future performance predictions, it is required that one effectively incorporate dynamic data in addition to static data. Here, dynamic data, in general, represents data obtained under in-situ reservoir flow and/or transport conditions; e.g., data obtained from pressure transient testing, well tracer tests, production data, water cuts, gas-oil ratio, etc. As mentioned, geostatistical methods are usually well suited to generate realizations conditioned to static data and are difficult to apply when dynamic data are involved. The main difficulty in applying geostatistical techniques to condition both static and dynamic data is that static and dynamic data have different scale (or known as support volume) and resolution (errors) due to the fact that these data are obtained via different tools and under different conditions, and geostatistical methods cannot adequately address these issues as well as such issues as spatial resolution of estimates and the relative worth of different data. For example, permeability estimated from a core sample is representative of reservoir volume of 10"3 ft3, whereas permeability estimated from well test pressure data may represent a reservoir volume of 109 ft3 [5]. Furthermore, conditioning to dynamic data imposes also difficulties due to complex mathematical (non-linear) relationship between the reservoir description and dynamic data. For example, well test pressure versus time data is, in general, non-linear function of reservoir permeability and porosity. All these difficulties necessitate the development XX of effective methods to completely or systematically integrate dynamic data with static data. For the last several years, much work has been done to generate reservoir descriptions conditioned to both static and pressure transient data; see [5-13]. However, most attempts have not incorporated well test pressure data into reservoir description directly, but instead, have used the well test pressure data to estimate some parameters like permeability and porosity which are then used as constraints when generating reservoir descriptions; using prior information; see [5-10]. (Here, prior information refers to our prior knowledge of parameters derived from geostatistics and other sources of information, which is usually represented as prior means, variograms and cross- variograms for parameters of interest.) For example, Sağar et al. [5] and Holden et al. [7] used pressure data to compute an average permeability within a radius of investigation and then used this average permeability within a radius of investigation as constraint when generating a permeability field which matches the variogram. Using a numerical reservoir simulator within the context of Monte Carlo simulation, Alabert [8] has made an extensive study of how this average permeability and radius of investigation should be computed. Oliver [14] and Feitosa et al. [15] presented analytical equations for the support volume of permeability obtained from well test data using 1-D radial diffusion equation considering heterogeneous permeability fields. However, these formulas are limited to 1-D radial flow problems. A rigorous treatment of 2-D or 3-D flow problems with these analytical formulas is not an any easy task, if not impossible. As reported in Refs. 11-13, there are several disadvantages associated with these approaches: (i) any permeability averaging technique is only approximate, (ii) since usually simulated annealing (which is an optimization techniques for minimizing the objective function constructed using averaged permeability values and variogram) is used for generating reservoir descriptions conditioned to pressure data, it is computationally demanding and (iii) the descriptions obtained from simulated annealing as typically applied [5] may not represent proper sampling of the probability density function of the true reservoir parameters and thus this makes it very difficult to correctly asses the uncertainty in performance predictions; see Cunha et al. [13] for details. Recently, some authors [11,12,16] have presented more effective methods for generating reservoir descriptions conditioned to static, geostatistical and pressure transient data. Their methodology is based on the inverse problem theory using Bayesian parameter estimation. It is important to note that Gavalas et al. [17] were to first to apply this methodology to estimate heterogeneous permeability and porosity fields satisfying both prior information and production data within the context of automatic history matching for one dimensional single phase flow problems. Important aspects of this methodology are that: (i) in addition of static data and prior information represented by prior means and variograms, it allows one directly to incorporate pressure transient data (say measured pressure versus time data recorded during a drawdown (or a buildup) test at a single well or wells) to reservoir descriptions; (ii) it provides an estimation of the uncertainty in reservoir description; and (iii) it provides a recursive estimation algorithm in that as new data becomes available, one can easily update the reservoir descriptions. Furthermore, standard geostatistical methods such as simple kriging and co-kriging are intrinsically built in this methodology. It is also important to note that application of the methodology is not only limited by pressure transient data. It can be used to generate reservoir descriptions conditioned to other types of dynamic data. The standard XXI application of the methodology rests on the fact that prior information on the model (set of reservoir parameters to be estimated) satisfies a multi-normal distribution and measurement errors on static and dynamic data can be considered as normal random variables with zero mean and known variance. In this work, the overall objective is to investigate the use of effective methods for generating reservoir rock property fields of porosity and permeability conditioned to well test pressure data as well as static data and geostatistical data represented as prior means for permeability and porosity and variograms. The thesis is organized is as follows. In Chapter 2, we begin by presenting background material on the origin of reservoir heterogeneity, fundamentals of geostatisticts, and on the methods that can be used to generate reservoir descriptions conditioned to static and dynamic data. In Chapter 3, the development and implementation of 1-D (x) and 2-D (x-y) single-phase, multi-well, finite difference simulators, which are used to investigate the effect of heterogeneous permeability and/or porosity fields on pressure transient data, is presented. In Chapter 4, we investigate the effect of heterogeneous reservoir with both deterministic and/or random (both correlated and uncorrelated) variations in permeability and/or porosity on pressure transient data. Results in this chapter indicate that the effect of porosity and permeability heterogeneities on pressure transient tests are quite complex. It is shown that pressure transient tests act like filters to smooth the effect of heterogeneous permeability and porosity elements and, thus, heterogeneous nature of the reservoir even may not be detected from the pressure data, yet it still contains some information about the distribution of heterogeneities. However, an effective value of permeability (or porosity) computed by using a simple spatial geometric, harmonic or arithmetic average of gridblock permeabilities (or porosities) is not sufficient to represent pressure transient data obtained under a heterogeneous permeability (or porosity) field and, thus, not sufficient to describe the distribution of heterogeneous permeability (or porosity) field. Most importantly, it is shown that determination of permeability and porosity distributions from pressure transient data alone will not be unique. This necessitates the use of additional information and data from other sources to reduce non-uniqueness problem in reservoir descriptions to be generated from pressure transient data alone. In Chapter 5, we review the methodology based on Bayesian estimation as introduced in Refe. 11, 12, 16 and 17, which can be used to incorporate pressure transient data directly into reservoir descriptions. By presenting background material on the methodology, we show that the methodology provides an effective and general tool for generating reservoir descriptions honoring pressure transient data as well as static and geostatistical data. The methodology provides a means to asses the uncertainty in performance predictions and to evaluate the spatial resolution of parameters and the relative worth of different types of data. Applications of the methodology to 1-D single-phase flow problems considering synthetic pressure transient data generated from a 1-D multi-well simulator as well as synthetic hard and geostatistical data are also given in this chapter to demonstrate the utility and power of the methodology in reservoir characterization. In applications, both static and pressure tamsient data with or without measurement errors are used to generate the maximum a posteriori estimating using Bayesian approach. The applications presented in this chapter clearly indicate that the Bayesian parameter estimationis a powerful tool for incorporating pressure transient data into reservoir descriptions as well as incorporating geostatistical and static data types. Furthermore! it is shown that by sampling the maximum a posterior density function XXII conditioned to geostatistical, static and pressure data, one can properly evaluate the uncertainty in reservoir performance.