Doğal çatklaklı rezervarlara ait kuyu testi verilerinin doğrusal olmayan regrasyon yöntemleri ile analizi
Başlık çevirisi mevcut değil.
- Tez No: 55516
- Danışmanlar: DOÇ.DR. MUSTAFA ONUR
- Tez Türü: Doktora
- Konular: Petrol ve Doğal Gaz Mühendisliği, Petroleum and Natural Gas Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1996
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 228
Özet
ÖZET DO?AL ÇATLAKLI REZERVUARLARA AİT KUYU TESTİ VERİLERİNİN DO?RUSAL OLMAYAN REGRESYON YÖNTEMLERİ İLE ANALİZİ Bu çalışmanın amacı, doğal çatlaklı rezervuarlarda gerçekleştirilen kuyu basınç testlerinin doğrusal olmayan regresyon yöntemleri ile yorumlanarak bu rezervuarlara ait bilinmeyen model parametrelerinin tahmin edilmesidir. Çalışmanın odak noktasını pek çok alanda uygulamaları olan parametre optimizasyonu kavramı oluşturmaktadır. Bir kuyu testine ait veriler ile uygun bir matematiksel modelin nicel olarak en yakın çakışmasını sağlayan parametre değerlerinin tahmini durumunda, bu kavram bir regresyon uygulamasına dönüşmektedir. Söz konusu matematiksel modellerin, bu çalışmada dikkate alman doğal çatlaklı rezervuar modellerindeki gibi karmaşık olduğu ve parametrelerinin doğrusal bir fonksiyonu olmadığı durumlarda doğrusal olmayan regresyon yöntemleri uygulanmaktadır. Bilinen grafiksel yorumlama yöntemleri ile parametre tahmininin güç ve bazı durumlarda imkansız olabildiği bu tür problemlerin doğrusal olmayan regresyon yöntemleri ile bilgisayarlarda analizi, pratiklik ve hızlılık ile birlikte sonuçların geçerliliğinin sayısal anlamda sorgulanabilirlesin! de mümkün kılmıştır. Bu çalışmada özellikle doğal çatlaklı rezervuarlar temel alınacak şekilde, doğrusal olmayan regresyon yöntemleri detaylı bir şekilde araştırılmış, farklı yöntemlerin performansları ve söz konusu probleme uygulanmada gösterdikleri yararlılıkları tartışılmıştır. Bilinen grafiksel analiz yöntemlerinde olduğu gibi regresyon uygulamalarında da basınç verilerini temsil etmekte değişik model eğrileri dikkate alınmış, bilinen model eğrilerine ek olarak daha önce regresyonda denenmemiş yeni model eğrilerinin de performansları araştırılmıştır. Tez kapsamında, doğrusal olmayan regresyon ve sonuçların istatistiksel analizi için gerekli bilgisayar yazılımları geliştirilmiş ve gerek yapay gerekse gerçek saha verileri ile uygulamalar gerçekleştirilmiştir. Bunların sonucunda bilgisayarda doğrusal olmayan regresyon ile kuyu testi analizinin yararlarına ait genel yorumlara ek olarak, özelde çeşitli konular üzerinde ayrıntılı sonuçlar elde edilmiştir. Regresyonda alışılagelmiş tip eğrilerine alternatif olarak integral eğrileri ile basınç/türev oranı eğrilerinin kullanılması bunlara örnek olarak verilebilir. Test verilerinin gürültülü olduğu durumlarda integral eğrilerinin türev eğrilerine göre üstünlüğü ve diğer eğrilerin yakınsama problemleri olduğu durumlarda basınç/türev oranı eğrilerinin daha iyi performans gösterdiği, bu çalışmayla ortaya konmuştur. Model denklemlerinde etken kuyu yarıçapı kavramının kullanılmasının aşırı negatif zar faktörü değerlerinin neden olduğu sorunları önlediği gösterilmiştir. Bir başka önemli sonuç olarak basınç yükselim testlerinde üretim dönemini ihmal ederek analiz yapmanın neden olduğu hatalar ve bunun yerine bu testlere“değişken debili test”yaklaşımı ile regresyon uygulamasının sağladığı yararlar gösterilmiştir. XV
Özet (Çeviri)
SUMMARY ANALYSIS OF WELL TESTS FROM NATURALLY FRACTURED RESERVOIRS USING NONLINEAR REGRESSION The aim of this study is to investigate the applicability of parameter estimation using non-linear regression methods to analyze well-test data from naturally fractured reservoirs. The study is based upon the parameter optimization concept. Considering the problem of matching a well-test data and an appropriate analytical model and estimating the optimized model parameter values, this concept tums into a regression application. When these analytical models are too complex and are non-linear functions of the parameters to be estimated, like naturally fractured reservoir models studied here, non-linear regression analysis methods need to be applied. in conventional well-test analysis, various graphical representations like cartesian, semi-log ör log-log plots of well-test data are used to estimate some reservoir properties. For this purpose, either straight lines (semi-log analysis) ör pre-drawn model curves (type-curve analysis) are matched visually to these plots. Therefore, obviously the results of a conventional graphical analysis suffer from subjectivity and it is impossible to employ statistical analysis to evaluate the validity of results quantitatively. Besides, using öne of these graphical techniques, öne can estimate only a few parameters at önce. Parameter estimation using non-linear regression methods in well-test analysis have become increasingly popular for the last decade. The main advantages of this method över the conventional graphical techniques such as semi-log analysis and type-curve matching are that it allows for the estimation of a large number of unknown reservoir parameters, minimizes the subjectivity of the interpretation, and allows us to employ statistical analysis to evaluate the validity of results quantitatively. Typically, a nonlinear regression technique is based on finding the values of unknovvn reservoir model parameters vvhich will provide the closest fit between measured well-test data and appropriate interpretation model data. Regarding the number of unknovvn model parameters for naturally fractured reservoirs, parameter estimation using non-linear regression is extremely advantageous. in conventional graphical well test analysis, only a few parameters can be estimated through semi-log straight line techniques ör manual type curve matching at önce and that is stili problematic not only for these methods are subjective and have no repeatability, but they require a lot of effort to deal with many graphs ör determine the characteristic portions of data corresponding certain periods of pressure transient behavior as well. On the other hand, vvell test analysis using nonlinear regression lets us to estimate ali unknovvn model parameters at önce, without lack of repeatability xvior subjectivity problems, provided that; (i) an appropriate interpretation model is chosen, (ii) the interpretation model is not insensitive to these parameters, (iii) there is no strong correlation between these parameters to lead singularity problem, (iv) a portion of data which is controlled by a certain unknown parameter is not missing or hidden by the effect of another parameter (e.g., when the wellbore storage period lasts long, it may hide the transition period) and (v) the noise of well test data is normally distributed with zero mean. In general, this study is focused on naturally fractured reservoir systems. However, fixing the model parameters which represent the fracture system to their appropriate values, homogeneous reservoir systems can be evaluated using the same models as well. Both pseudo-steady state and transient state interporosity flow models with wellbore storage effect are included. Various idealized matrix shapes with and without interporosity skin effect are considered. Throughout this study, a fully penetrating vertical well in an infinitive naturally fractured reservoir was assumed. No outer boundary effect was considered except for the fault problem. Single fault and intersecting double faults with constant pressure or closed boundary effects are evaluated. Above models give the pressure transient behaviors of naturally fractured reservoirs producing at constant surface flow rate. However, using the superposition principle, these models can be used to obtain the pressure transient behavior of multirate problem as well. In this study, the performances of various regression techniques are investigated. One of them is the well known least squares (LS) technique and is based on the idea of minimizing the sum of squared residuals which can be represented as follows: n. np. 2 v-i 2.v-» / measured model \ 2>/ =2>/ -y, ) 1=1 1=1 Here, r, is the residual and is used to denote the difference between a measured data point and the corresponding model data point. Along with the conventional least squares technique, a number of least absolute values (LAV) approaches which are based on the idea of minimizing the sum of absolute residuals are studied. Similarly, sum of absolute residuals can be written as follows: measured model y, -y, »1=1 1=1 1=1 Results of this study verify that LAV type regression techniques are less sensitive to the existence of poorly measured data points, i.e. outliers, than LS technique and that is why they are also known as“Robust Regression Methods”. An outlier is a data point of more than 3 or 4 times of standard deviation from the mean. Outliers give much greater residual values than the rest of the data, therefore LS tecnique is strongly influenced by the existance XVIIof these points, since these residuals are squared and yield much greater values. However, LAV approach is not as easy to implement as LS approach, because the model gradients need to be calculated for the regression algorithm and by the definition of LAV approach it is not possible to achieve that directly. Instead, most LAV methods are based on minimizing an equivalent objective function which at the limit approaches to a weighed least squares technique and that makes the calculation of model gradients possible for LAV methods. Applying all the techniques mentioned above, as well as the pressure curve itself, various type curves derived from the pressure data, i.e. derivative and integral curves or different combinations of them are also studied. A comparison of these curves are made to discuss both reliability and efficiency of using them in non-linear regression analysis. One of these curves is presented in the Literature as pressure / pressure derivative ratio and is used for the first time in non-linear regression analysis during this study. This model curve is less sensitive to initial guess values of unknown parameters and therefore performs better in local minimum cases. In this study it is also shown that“pressure integral”and“pressure derivative integral”model curves prove useful in regression analysis of noisy pressure data cases where derivative curves fail to give reliable results. This advantage is obtained by the smoothing effect caused by integration of pressure data along time. In non-linear regression analysis, model gradients, i.e., derivatives of analytical model equations with respect to the unknown parameters need to be calculated repeatedly until a good match of measured and model data is obtained. For this purpose both numerical and analytical gradients are used to compare their performances. Finite differences approach is used to calculate numerical gradients. On the other hand, taking partial derivatives of the model equations with respect to the unknown parameters and rearranging, analytical gradient equations are written as compact and practical as possible. Later on, both approaches are implemented to compare their code-writing easiness and run time performances. The analytical gradient equations from well known fractured reservoir models are derived and presented for the first time during this study. In this study it is shown that the use of analytical gradients improves convergence scheme ande reduces the number of iterations needed, since they give more accurate gradient values. On the other hand, unless they can be written efficient and compact enough, computation of gradients take longer than those from the finite differences approach even if the number of iterations is smaller. Another point is the“sensitivity plots”from which the sensitivity of gradients can be seen. Sensitivity plots for both individual and grouped parameter sets are drawn to see where the gradients are sensitive to those parameters and where they are not. For a period where the sensitivity values of a certain parameter are close to zero, it can be concluded that the model is insensitive to the parameter within that period and therefore it is difficult to obtain good estimates from regression analysis for this parameter. XVIIIThe regression methods in this study are based on unconstrained parameter optimization techniques, i.e., during the iterations that unknown parameters are searched, they can have any value between -a> and +qo. Regarding the model equations that represent the pressure transient behavior of naturally fractured reservoirs and the physical limits (or constraints) for the unknown parameters (e.g. permeability or porosity are parameters which always have positive values) it is obvious that a regression procedure may be interrupted when an unrealistic value of a parameter that blows out the model is encountered. In this study, different constraining algorithms are implemented to keep the unknown parameters values within their physical limits. Each method is based on another approach completely;“Simple Bounding”method takes a fraction of the step back, when the value of a parameter exceeds the limits. This method is efficient and easy to employ.“Penalty Function”method is based on increasing the value of the sum of the squared deviations by simply adding a penalty function which increases its value drastically as the unknown parameter value gets closer to the limits.“Imaging Extention”method uses the multiple mirror images of the objective function to obtain equivalent constrained parameter values from unconstrained values. And the last method is“Line Search”which not only keeps the parameter values in their limits, and also searches the minimizing parameter value as compared to the previous iteration. This method may take longer computational time than all other methods mentioned above. In this study, while using different curves other than the pressure curve in non-linear regression analysis, numerical derivation and integration algorithms are used to calculate derivative, integral or combinations of them from the well test data. In practise, without the use of an efficient transform algorithm, these groups magnify the bad influence caused by the possible noise on well test data. For the synthetic well test examples even if no noise was introduced, numerical transform methods give derivative or integral type curves deviated from the ideal behavior since they can not help changing the characteristic trend of the actual data, and, therefore yielding a deformed type curve. Hence, to take the advantage of using derivative and integral type curves, efficient transform methods given in literature are implemented to generate them. One of the most significant advantages of well-test analysis using nonlineer regression is that a statistical analysis can be employed to evaluate the validity, or in other words, confidence of the results. In this study, for a comprehensive evaluation of the regression examples, various statistical analysis techniques like confidence limits, joint confidence regions and residual analysis are studied and implemented. Different types of residual plots aid the diagnosis of correlated parameters while confidence limits show the intervals that enclose the real parameter values, with a given percentage of probability. In case of correlated parameters, instead of individual confidence limits, joint confidence regions are used to check the validity of the results. These techniques are used to evaluate the results of a least squares (LS) type regression method. To check the validity of the results obtained from the least absolute values (LAV) type regression methods XIXwhich are based on minimizing an equivalent problem of weighed least squares, the same techniques are used. As an exception, a simplex LAV type regression method is followed by an LS regression which accepts its results as the initial guesses, and the final parameter estimates are analyzed using statistical techniques that belong to LS approach. In addition to these statistical methods, visual representations of the objective functions such as three-dimensional surface and contour plots are used to have an idea about the performance of regression algorithms and which parameters to be estimated from regression analysis. The problems mentioned above are investigated in detail through the synthetic well-test examples, each pointing at a specific topic, and example runs with field data from naturally fractured reservoirs are carried out to see the applicability and limitations of non-linear regression analysis. xx
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