Hasan Çavşak

Effective calculation of gravity effects of uniform triangle polyhedra

Uniform tetrahedra are commonly used elementary bodies for gravity calculations from which arbitrary polyhedra can be composed. A simple derivation of the gravity effect is presented for the apex P of the tetrahedron expanded from P to an arbitrarily oriented plane triangle. Integration of its potential effect in a rotated coordinate system applies vector algebra and renders the anomalous potential depending on the distance of P over the triangle plain and a function of the triangle coordinates. Partial differentiation by moving P infinitesimally in z-direction leads to two terms, a simple and a complex one; they can be understood as describing the same difference from two points of view: leaving P at the apex of the changed polyhedron or moving P off the unchanged polyhedron. Both views imply the same shape change and the sum over the polyhedron is thus numerically equal. Hence we need to calculate only the one of the terms of the differential which is simpler. The calculation of the gravity effect is numerically simplified and more stable. This has been tested for many models and is demonstrated by two examples.

A new method for determining lower density layer in prospection of hydrocarbon

Öz It is known that densities in formations are usually assumed to be constant for gravity model calculations. This also implies that formations are homogeneous and isotropic. However, the formations are usually heterogeneous and densities vary depending on heterogeneity. For this reason, densities should be taken into account as variables. Some scientists consider densities as variables in each formation in model calculations. In other words, density is defined as a function of the required parameters. In fact, functional change is regular. However, density is an irregular variable that depends on the change boundaries of seismic velocity. In this study, it is aimed to take density into account as a variable by using detected seismic velocity boundaries at which seismic velocity changes for each formation. In addition to main formations in model geometry in 3D inversion calculations, another formation was defined. This additional formation has been described by using a combination of all of the change boundaries of seismic velocity present in each formation in a specific order. The density calculated for the additional formation estimated the variation of density between the change boundaries of seismic velocity. This variation is added to the mass densities that are calculated for the description number of each zone. So, lower-density layer comprising oil can be determined by this method. The reliability of the results of the method depends on the reliability of seismic velocity boundaries. Moreover, the increasing number of seismic velocity boundaries leads to the increasing resolution of density variations. Bazı bilim adamları, 3 boyutlu gravite model hesaplamalarında, yoğunlukları her formasyon içinde değişken olarak ele alırlar. Yani yoğunluğu parametrelere bağlı bir fonksiyon olarak tanımlarlar. Bir yeraltı tabakası içindeki yoğunluk değişimi derinlikle orantılı olarak bulunur. Bu çalışmada, her formasyon içinde tespit edilen sismik hız sınırları kullanılarak, yoğunluğun değişken olarak göz önüne alınması amaçlanmıştır. Sismik hız sınırlarının izlediği yol, yoğunluk değişiminin bir göstergesidir. 3B ters çözüm hesaplarında model geometri içindeki ana formasyonlara ek olarak bir formasyon daha tanımlanmıştır. Bu ek formasyon tanımı, her formasyon içinde mevcut olan sismik hız sınırlarının tümü kesintisiz kullanılarak yapılmıştır. İşte bu ek formasyon için hesaplanan yoğunluk, sismik hız sınırları arasındaki yoğunluk değişim miktarı olarak kabul edilmiştir. Bu değişim, ana formasyonlar için hesaplanan yoğunluklara bir düzen içinde ilave edilerek, yoğunluğun derinlikle değişimi ayrıntılı olarak saptanmıştır. Bu çalışma, Adıyaman, Diyarbakır ve Gaziantep bölgesine ait sismik ve açılan kuyulara ait verilerin bir kısmının TPAO’dan alınmasıyla düşük hızlı yer altı modeli oluşturularak yapılmıştır. Bu çalışma sonunda sismik hız sınırlarının ekstra bir kütle olarak alınmasıyla yoğunluğun derinlikle nasıl değiştiği saptanmıştır. Böylece hidrokarbon içeren düşük yoğunluklu tabaka tespit edilmeye çalışılmıştır. Hidrokarbon aramalarında bu yöntem kullanılarak; daha az kuyu açılarak sonuca gidilebilir. Bu çalışmada, başlangıçta yoğunluklar sabit olarak dikkate alınmıştır. Fakat her tabaka içindeki yoğunluklar değişken olarak hesaplanmıştır.

Determining change of density with depth by using seismic velocity boundaries in 2D gravity inversion calculations

Densities in formations are usually assumed to be constant for gravity model calculations. This constancy implies that formations are homogeneous and isotropic. However, formations are usually heterogeneous, and densities vary depending on heterogeneity. For this reason, densities should be considered variables. Some scientists consider densities as variables in their model calculations of each formation. In other words, density is defined as a variable of the required parameters. In fact, variable change is regular, whereas density is an irregular variable that depends on the change boundaries of seismic velocity. This study aimed to take density into account as a variable by using determined seismic velocity boundaries at which the seismic velocity changes for each formation. The change boundaries of seismic velocity are an indication of the change of density in the formation. It is not possible for the respective description of this change to be variable for each point. In addition to defining the main formations in model geometry by using 2D inversion calculations, this study defined another formation, which is described with a combination of all of the change boundaries of seismic velocity that are present in each formation in a specific order. The reliability of the results of the method depends on the reliability of the seismic velocity boundaries. Moreover, an increasing number of seismic velocity boundaries lead to a higher resolution of density variations.

Erratum to: The Effects of the Earth’s Curvature on Gravity and Geoid Calculations

When this article was published online first, unfortunately there have been mistakes in the text and in the legend to Fig. 8. Opposite of Fig. 3, right column eighteenth line: ‘‘in the Cartesian -2,910.6858 km’’ should read ‘‘in the Cartesian -2,910.6838 km’’. Legend to Fig. 8, second line: ‘‘Spherical 4.496580E, 4.496580N’’ should read ‘‘Spherical 4,49658 E, 4,49658 N’’. Unfortunately, an error occurred in Fig. 4 and Fig. 9. The correct Fig. 4 and Fig. 9 are given on the next page.

Determining change of density with depth by using seismic velocity boundaries in 3D gravity inversion calculations

Densities in formations (layers) are usually assumed to be constant for gravity model calculations. This constancy implies that formations are homogeneous and isotropic. However, formations are usually heterogeneous, and densities vary depending on heterogeneity. For this reason, density should be considered a variable, and certain scientists do just this in their model calculations for each formation. In other words, density is defined as a function of the required parameters. In fact, functional change is regular, whereas density is an irregular variable that depends on the changing boundaries of seismic velocity. This study aims to look at density as a variable by using determined seismic velocity boundaries at which the seismic velocity changes for each formation. The change boundaries of seismic velocity are an indication of the change of density in the formation. It is not possible for the respective description of this change to be functional for each point. In addition to defining the main formations in model geometries using 3D inversion calculations, this study defines another formation, which is described by a combination of all of the change boundaries of seismic velocity that are present in each formation in specific order. The reliability of the results of the method depend on the reliability of the seismic velocity boundaries. In addition, an increasing number of seismic velocity boundaries lead to a higher resolution of density variations.

Comparing of the 2D–3D Gravity Calculations in Cartesian Coordinates and 3D in Cartesian-spherical Coordinates

In this study, various calculations comparisons were made to achieve the best results in gravity computation. In the first study, 2D and 3D gravity computation were compared by using a suitable synthetic model in cartesian coordinates. And also in the second study, 3D gravity calculations are compared by using a suitable synthetic model in spherical and cartesian coordinates. In the first study, accuracy of the 3D gravity calculation results were found by inversion in cartesian coordinates. And also in the second study, the 3D gravity calculation results were found to be true in the spherical coordinates instead of in cartesian coordinates. The two studies, forward and inversion solutions were made for these model geometries by intensity of adoption by using a special algorithm (Cavsak H. 1992). Thus, the three-dimensional gravity studies are necessary to do is tried to explain in spherical coordinates.