Geostatistics application in spatial analysis of geomechanical properties

Abstract—Geomechanics applications play an important role in both drilling and production of oil and gas field. There are many important properties such as Unconfined compression strength (UCS), Poison ratio (PR), Internal Friction Coefficient (IF) and Porosity (PHIE) need to be estimated properly. To estimate these properties, there are many methods that can be used but geostatistics has more advantages. This research presents geomecanical propertiesfor two offset wells according to experiment relations existing. Then, variogramand spatial continuity will be analyzed. The OrdinaryKriging (OK) methods will be used to interpolatethe properties in the cross section between two offset wells and then for a planned well. The predicted properties were compared with the actual measured data to find the linear correlation coefficient. Most of these values arenearly 1. As a result, the quality of the modelbuilt could be practically accurate and reliable to predict geomechanical properties for planned wells used in wellbore stability, sanding studies.

However, similar to other conventional geomechanics studies, the results shown estimated geomechanical properties with depth-stretched method that is equivalent to correlation in petrophysics study. The other study and papers currently are still used the same workflow with applying correlation and choosing the closest well for estimation geomechanical properties [8,11]. In oil price downturn situation, it is more difficult to drill new exploration wells and challenge to drill successfully. Furthermore, the geologic pattern has become more complex and extremely risky. In addition, the budget for core test also reduced and limited. Because of insufficiency of information required, right access to a method capable to determine properly geomechanics information on the existing information is highly interested. This study will utilized the concepts of variogram, krigingand spatial analysisto predict geomechanics properties with high accuracy.

Variogram and Covariance
Variogram is a mathematical function, basic tool to quantify correlation of spatial variables [1], defined as: Where: There are three standard variogram models: Spherical, Exponential and Gaussian. In practice, we need to replace empirical variogram with a most matched variogram model.
Covariance measures similar variation of 2 random variables, defined as: And obeyed the following relationship:

2.3
Kriging: Kriging is a geostatistical technique for optimally interpolating values at unsampled locations. Kriging employs variogram model, so it is a weighted method with respect to both distance and trend of data. It generates Best Linear Unbiased Estimation (BLUE) at each location.
Simple Kriging (SK): The simplest kriging and rarely applied in reality. Global mean is assumed known and constant in the study area, which is not really actual [2]. The value at an unsampled location can be estimated by: λi are calculated from minimum variance condition, as below simplified covariance matrix: Where:  Z * (u0): Estimated value at location u0.  Z(ui): Nearby sample value at location ui.  n: Total number of samples selected in the study area.  λi: Weights assigned to each sample  λ0 = a constant.  m: Global mean value in area .  C(ui, uj): Covariance value between points located at ui and uj.  C(ui, u0): Covariance between sampled location ui and unsampled location u0 Ordinary Kriging (OK): Assuming that there are many local means and calculated from nearby values [2]. This also assumes true global mean is unknown so it is "ordinarily" used more than SK. The estimation is written as: By forcing λ0 to be zero, the necessity of mean m is eliminated which constitutes Eq.(6) by Eq. (9) λi are calculated from minimum variance condition, as below simplified covariance matrix: Where μ is Lagrange parameter. CoKriging: Cokriging is used to estimate one variable value with co-variable. Two common examples are the estimation of permeability using porosity data and the estimation of porosity data using seismic data [1]. Estimation equation is: Where:  CZ and CY: covariance for the Z and Y variables, respectively.  CC: Cross-variance between 2 variables. μZ and μY: Lagrange parameters.

GEOMECHANICAL MODEL
The geomechanical properties were calculated from correlations based on well-logging data. Then, core and experiments data were used to calibrate.

Unconfined compressive strength:
Unconfined compressive strength is defined as the maximum axial compressive strength that a right-cylindrical sample can withstand under unconfined conditions.There are many correlations to determine UCS based on seismic and well logging data. This study used MsNally's correlation which can be applied for sand reservoir

Prediction workflow
Step 1: Building geomechanical models  Getting the inputpetrophysical data (velocity and density)  Build geomechanical models for along the offset wells by using the above empirical correlations.  Validate rock properties estimated with core samples.
Step 2: Building variogram models  Calculating experimental variogram for the cross section based on the data between 2 offset wells,  Choosing the standard variogram models.  Cross-validating to find the best-fit variogram model for each property.
Step 3: Predicting geomechanical model Using chosen variogram models to interpolatethe 2D geomechanical model between 2 offset wells and then for the planned well using Ordianary Kriging (OK).

4.2
Geomechanics model and results Petrophysics offset data from field A of Cuu Long basin, Vietnam is used in this research to build geomechanics model along the well path. The study presented the fundamental ideas of geostatistics for interpolating the geomechanics detail of the area with the boundaries from 2 offset well: XX-2P and XX-3P (Figure 1). The distance between two offset wells is about 2.6 km. Estimated points are compared with the actual points based on regression value and coefficient of correlation.
Building geomechanical models The input data from XX-2P and XX-3P included: velocity (DTC, DTS) and density log (Rho). Because DTC data just only for the section (3470m -4247m), so we must combine with the velocity log(Vel) from seismic(Vp) to have a full DTC log. Due to a more limited data of DTS, we ought to calculate the DTC-DTS regression to build the full DTS log. Similarly, density log cannot be measured for the whole wellbore. So that, we might use Garner's correlation to build density log based on velocity log for unmeasured well sections: Building variogram models: For each property, the unique experimental variograms for both of two wells were calculated, then checked with the standar variogram models. Overall, these variogram had a very good correlation coefficient (r2). The above variogram model of UCS (Figure 4) is shown to be the best-fit with Gaussian model with r2= 99.7%  The variogram model of PR (Figure 6) is shown to be the best-fit with Gaussian model with r2=94%.
The variogram model of IFC (Figure 7) is shown to be the best-fit with Gaussian model with r2=99.2%. The above variogram model of PHIE (Figure 8) is shown to be the best-fit as Exponential model with r2=98.1%.

Cross-validation for best-fit variogram model
Each of these variogram models was then used for cross-validated to evaluate the accuracy variogram model before applying Kriging techniques. To obtain the best-fit variogram, we may eliminate or edit some outliner points which may be due to invalid measurements. As shown below figures, all of correlation factors are higher than 95%.
Cross-validated correlation factor (r2) of PR equals to 97.4% Cross-validated correlation factor (r2) of IFC equals to 99.6%. Predicting geomechanical model The best-fit variogram models are used to predict geomechanical properties for well XX-4P.

Cross-checking the predicted values
To validate the model used for prediction, crosscheck is used for verifying all geomechanical properties. This process is used variogram model choosen to re-estimate all measured data.
Cross-checking UCS between actual and estimated values with r2= 77.73%.
Cross-checking PR between actual and estimated values with r2= 58.5%.
Cross-checking IFC between actual and estimated values with r2= 94.19% Cross-checking PHIE between actual and estimated values with r2=88.13%.

CONCLUSION
In this study, geomechanical model has been studied using empirical correlations to calculate geomechanical properties of between two wells XX-2P and XX-3P from petrophysical data. Then geostatistics was applied to predict 2D geomechanical model between two offset wells and then the planned well XX-4P, including properties: UCS, PR, IFC and PHIE. The best-fit variogram have been choose and validated as following tables.
Clearly, these correlation factors (r2) for variogram models are similar to each other. The best-fit variogram had the highest r2 (in bold). The calculation results were summarized in table 3.
In table 3, the variogram models for UCS, PR and IFC were Gaussian. It means that there was a high correlation over short range and these were continuous phenomena. Where as, the PHIE variogram model was Exponential, which means this had a short scale variability.
As also in Table 3, these variogram had a good correlation coefficient (r2). These greatly good r2 (>95%) of all variogram models and crossvalidations showed that these chosen variogram models are greatly accurate.
In Table 4, the correlation coefficient (r2) of all properties are fairly high, roughly 80%. However, Poisson's ratio (PR) with r2=58.5% may be due to lack of solid actual data for calibration.

DISCUSSION
There were just two wells used in calculating variograms which are applied isotropic analysis. If more wells were used, anisotropic variograms would have been calculated and compared with isotropic ones to select the best-fit variogram, resulting more accurate models.
The log-derived rock strengths should be calibrated by more rock test data to initiate better accuracy. So that, the predicted model will minimize the uncertainties of consequent geomechanics application, particularly in well bore stability.