Determination of DynamicElastic Properties of Anah Formation, Near Rawa city / Western Iraq Using Ultrasonic Technique

The aim of the current study is to determine the elastic properties of carbonate rocks using ultrasonic method. Forty rock samples of Anah formation were collected at different depths from four wells drilled at the study area . The relationship between wave velocities and elastic properties of rocks was defined. Regression analyses to define these relations were applied. The results indicate that the elastic properties of the rocks show a linear relationship with both Pand S-wave velocities. The best relationship was obtained between both Young's modulus and Shear modulus with Vs in the determination of the coefficient ( R 2 ), with values of 0.91 and 0.94, respectively. Bulk modulus and Lame’s constant were better correlated with Vp than with Vs in the determination of R 2 ,with values of 0.92 and 0.83, respectively. Poisson’s ratio showed a good correlation using the ratio of Vp/Vs in the determination of R 2 , with a value of 0.81. The main output of this study shows that the ultrasonic method is a useful tool for the prediction of the elastic dynamic properties of sample rocks and that it can be used as an economical , simple and nondestructive method, especially for engineering purposes.


Introduction
Physical and mechanical properties of rocks are of a remarkable interest in many fields, including materials science, petrophysics, geophysics and engineering geology[1]. In rock mechanics, ultrasonic wave velocity tests are becoming very popular due to their non-destructive nature, high precision, easy application, and cost effectiveness [2].Three types of sonic velocity methods are available, namely the ultrasonic, low frequency sonic wave, and frequency resonant techniques [3,4]. Among these methods, the ultrasonic technique is the most convenient to be used in rock mechanics. The advantages of this method is its capability to yield both seismic velocities (Vp and Vs) as well as ultrasonic elastic constants that are very useful in characterizing the dynamic properties of rock, such as Young's modulus, shear modulus, bulk modulus and Poisson's ratio . This method is simple, fast and can be used in both field and laboratory [5,6].Many researchers used ultrasonic methods to characterize different geomaterials, and concluded that these methods are among the best ways to investigate the elastic properties of such materials [7,[8][9][10][11].Several correlations between elastic modulus and physical properties of rocks with ultrasonic velocities were established [2,3,[12][13][14][15][16][17]). The results indicate that the seismic velocity is closely related with rock properties .The aim of the current study is to use an ultrasonic method for the prediction of dynamic elastic properties of carbonate rocks of Anah Formation in a cost-effective manner by establishing correlations between elastic wave velocities and rock properties.

Location of the Study Area
The study area is located close to the both banks of the Euphrates River, between latitudes 34º 25 ′ 0"-34º 30′ 0" N and longitudes 41º 42′ 0"-41º 45′ 0" E. The area represents a proposed site for building a new dam near Rawa city northwest of Al-Anbar governorate, western Iraq ( Figure-1).

Geology of the Study Area
According to the four wells located at the study area, Lithostratigraphy can be described from the oldest to the youngest as follows [18,19] ;Baba Formation (U.Oligocene)comprised of hard dolomitic limestone of white color with a thickness varying between 99 and149 m. The formation that overlies Baba is Anah Formation which has a thickness in the typical section ranging between 7 and 99 m. and composed of chalky limestone ,coral reef and breccias. The Euphrates Formation (L. Miocene) is mainly composed of basal conglomerate, dolomitic limestone chalky limestone, and marly limestone, with a thickness ranging between 28 and 75 m. The Fatha Formation (M. Miocene) is composed of marl, limestone, gypsum, bituminous gypsum, and Claystone. Quaternary deposits comprised of river terrace(sandy gravel) sediments of the Pleistocene are also present. Pleistocene-Holocene deposits are represented by slope sediments, floodplain sediments, valley and depression fill sediments, and Sabkha's sediments, including salt crust mixed with silt and clay.

Samples collection and preparation
Forty rock samples of Anah formation were collected from four wells drilled at the study area.The vertical interval distances between the collected samples were irregular . The thickness of Anah formation and the number of the collected samples in each well are illustrated in table 1. Particular attention was dedicated to the collection procedure to avoid sampling in altered layers. All samples were calibrated to be in a cylindrical shape with an approximate length of around 10-16cm and approximate diameter of 5 cm. End surfaces of the samples were polished to a sufficiently smooth plane to provide good coupling between the transducer face and the rock surface.

Density measurements
The density of each core sample was measured after removing moisture. The density of dry samples was obtained from the following formula: (kg/m 3 ) = The density of samples at the study area shows relatively wide range of dry density due to the effect of compaction and micro fractures. The value of dry density varied from 2029kg/m 3 to 2839kg/cm 3 with an average value of 2506kg/m 3 .

Velocity measurements
The velocities of compressional and shear waves were measured on core samples using a New Sonic Viewer (model-5217A) available in Baghdad University;The ultrasonic impulses are transmitted to the core sample and then the transit time (T) is recorded. The core sample was positioned between the transmitter and receiver and the transducers were pressed to the ends of the sample until a stable transit time was recorded. In order to ensure good contact between the surface of sample and transducers , special grease was used in this study. Compressional and shear wave velocity values(V p and V s ) were calculated as follows: ( 2)

Dynamic elastic moduli
The elastic properties of rocks can be described by many types of elastic moduli termed Young's modulus, bulk modulus, Poisson's ratio, shear modulus, and Lame's constant [20]. The knowledge of Vp and Vs and densities ( ) of the substance allow calculating values of elastic moduli, according to the formulas below: .
where E is Young's modulus, K is the bulk modulus, is the Poisson's ratio, G is the shear modulus,λ is the Lame's constant , is the density, is P-wave velocity, and is the S-wave velocity.

Results and discussion
The main properties obtained from the velocity measurement for the core samples collected from wells 1,2,3 and 4are presented in Table 2. The elastic properties of core samples showed difference values. The young modules (E) values ranged from 1.44 E+10 to 8.94E+10,the bulk modulus (K) ranged from 1.35E+10to 8.86 E+10, the poisons ratio (σ) ranged from 0.109to 0.42 , the shear modulus (G) ranged from 0.54E+10to 3.43E+10,and the Lame's constant (λ) ranged from 0.53E+10 to 7.73E+10 Pascal .The high difference between these two values reflects structural and textural variations that occurred in rocks of Anah formation in the investigated area.In order to describe the relation between ultrasonic wave velocity and elastic modulus of the tested core samples , a regression analysis was achieved .The equations of the best fit line and the regression coefficients (R 2 ) were established for each relation.The present section aims to present and discuss correlations that enable the estimation of rocks properties from ultrasonic velocities..

Correlation between Vpand Vs
The correlation between Vp and Vswere established.A positive linear relationship with a determination coefficientof R 2 =0.66 is shown in Figure-2.The equation for these relations is given below: Vs = 0.3888Vp +563.08 where both Vp and Vs are in m/s.

Correlation between ρ and Vp and Vs
The correlation between (ρ) and (VP-and VS) for rock samples is plotted in Figures-(4,  5), respectively. According to the plots, Vp and Vs increased linearly with increasing (ρ). The empirical relations between ( ρ) and Vp and Vs are given below:

Correlation between (E) and (Vp and Vs)
Young's modulus measures the resistance of a material to elastic deformation under load.Compacted cemented rocks have a higherYoung's modulus than the porous ones.This means that the greater value of Young's modulus, the larger stress that is required to accomplish the deformation. Figures-5 and 6 show a linear relationship with high regression coefficient values (R 2 =0.76 and 0.91)between E and (P-and S) wave velocities ,respectively . The empirical relations between E and Vp and Vs are given below: -

Correlation between (K) and (Vp and Vs)
Bulk modulus is another elastic constant that has been investigated in this study. This property is to examine the resistance of rocks to compression.A high value of K can be obviously noted in the compacted cemented rocks than porous rocks .The relations between K and (P-and S-wave) velocities are plotted in Figures-7 and8, respectively. The high regression coefficient (R 2 = 0.92) reveals a strong linear positive correlation between the K and Vp , while the low regression coefficient (R 2 =0.43) reveals weak linear positive correlation between the K and Vs. The following equation defines these relationships:

Correlation between (G) and (Vp and Vs)
The shear modulus or modulus of rigidity is defined as the ratio of shear stress and shear strain. This property is to examine how stiff a rock is to shearing strain with no change in the volume [20]. Rigid compacted rocks have higher shear modulus than soft rocks .This implies that the greater value of shear modulus , the larger stress is required to produce deformation. Figure-10 shows a linear relationship between Vp and G, with a regression coefficient of 0.69 .There is a similar relationship between Vs and G ( Figure-11) but with a larger determination coefficient (R 2 =0.94). The following equation defines this relationship: . G=6E+06Vp -1E+10 G=1E+07 Vs -2E+10

Correlation between (λ) and (Vp and Vs)
The correlation between (λ) and (Vp and Vs) for rock samples are plotted . Figure-12 shows a linear relationship with high regression coefficient (R 2 =0.94) betweenVp and λ ,which implies that the λ The dispersion of data is illustrated in Figure-13. It shows aweak relationship with a low regression coefficient (R 2 =0.27) between Vs and λ ,which indicates that the value of Vs cannot be used to estimate λ with an acceptable accuracy. The empirical relations between (λ) and Vp is given below:λ =2E+07Vs-2E+10

CONCLUSIONS
The current study aimed to determine some of the elastic properties of carbonate rocks by establishing correlations between rock properties and ultrasonic wave velocities. The results indicate that the dynamic elastic properties of carbonate rock showed linear relationships with both Pand S-wave velocities, but with different values of determination coefficient (R 2 ). Some parameters such us density, bulk modulus and lame's constant are strongly related with the P-wave velocity compared with the S-wave velocity, where the determination coefficient between P-wave velocity and these parameters is higher than 0.67. Other parameters such us Young's modulus and Shear modulus are strongly related with S-wave velocity compared with P-wave velocity, where the determination coefficient between S-wave velocity and these parameters higher than 0.91, while Poisson's ratio is strongly related with velocity ratio in the determination of R 2 , with a value of 0.81. Empirical equations obtained in the current study revealed that the elastic properties of carbonat rocks at the study area can be estimated by determining ultrasonic wave velocities.