Hongxue Han, Shunde Yin

Civil and Environmental Engineering, University of Waterloo, ON, N2L 3G1, Canada

Keywords: Adjusted borehole size In-situ stress Caliper log Borehole deformation Artificial neural network Genetic algorithm Probabilistic analysis

ABSTRACT This paper proposes an integrated method of analytical calculation, artificial intelligence, and probabilistic analysis to cost-effectively determine geomechanical properties and in-situ stresses from borehole deformation via caliper logs.It's also demonstrated in this paper that the actual borehole size can not be simply taken as the bit size by default, and adjusted borehole size has to be used to find the reasonable borehole deformation.In the proposed method, an artificial neural network (ANN) is applied to map the relationship among in-situ stress, adjusted borehole size, geomechanical properties, and borehole displacements.The genetic algorithm (GA) searches for the set of unknown stresses and geomechanical properties that match the objective borehole deformation function.Probabilistic analysis is conducted after ANN-GA modeling to estimate the most possible ranges of the parameters.The hybrid method has been demonstrated by a field case study to estimate the adjusted borehole size, Young's modulus, and the two horizontal in-situ stresses using borehole deformation information reported from four-arm caliper logs of a vertical borehole in Liard Basin in Canada.

1.Introduction

In petroleum engineering, it is critical to determine in-situ stresses and geomechanical properties in order to ensure successful drilling, quality completion, and reservoir containment analysis.Mechanical properties of rocks, such as Young's modulus and Poisson's ratio, are normally measured from core samples in lab tests.These properties can also be calculated dynamically from sonic and density logs.However, such calculated properties require calibrations by the lab static measurements.These processes are expensive and time-consuming.Three principal in-situ stresses are assumed for the convenience of analysis: one vertical and two orthogonal horizontal stresses.The magnitude of vertical normal stress is assumed to be equal to the weight of overlying rock and can be calculated from the integration of bulk density logs.The magnitude of smallest principal in-situ stress can be measured by leak off test (LOT), hydraulic fracturing test [1,2], or small scale diagnostic fracturing test.These methods are commonly used nowadays and considered most reliable.For a vertical drilled borehole in a normal faulting stress regime [3], the measured smallest principal in-situ stress is the minimum horizontal in-situ stress.The maximum horizontal in- situ stress magnitude is often determined by using Kirsch equation [4] from borehole breakouts, minimum horizontal stress, and geomechanical properties such as cohesion, friction angle, UCS (unconfined compressive strength), etc.[5,6].Other means of in-situ stress determination have been investigated in the past several tens of years.For example, Ervin and Bell used breakdown pressure or leak off pressure from formation leak off test to calculate the maximum horizontal stress [7].Cornet and Valette developed a method based on normal stress measurements and fast flow rate reopening tests to calculate in-situ stress [8].

Geomechanics laboratory tests and field in-situ stresses measurements in the actual petroleum engineering practices require a substantial overhead and a prolonged standby time.However, the results might only be available for limited formations in only few wells in an oilfield.Furthermore, properties achieved from lab test are not in-situ, where mismatched confining stresses must be applied to mimic the in- situ conditions underground.Therefore, many attempts have been made to develop techniques to determine geomechanical properties or in-situ stress in an economical and prompt manner while maintaining the measurement in-situ.Some researchers [9]; [10] tried to establish empirical relations between rock mechanical properties and sonic and density logs or rock physical properties.In 1980s, Aadnoy developed a method to determine the orientation and magnitudes of horizontal in- situ stresses based on the formation breakdown pressure of a circular borehole of arbitrary trajectory [11].In recent years, artificial intelligence method and genetic algorithm based methodologies have been practiced in many fields including energy industry [12,13,14].In petroleum industry, artificial intelligence method and genetic algorithm have been used to map the relationship between in-situ stress and the displacements of a borehole wall and to calculate the in-situ stress [15,16-18].

However, in the abovementioned methods, two major challenges exist in the determination of in-situ stresses and rock mechanics properties from borehole deformation data.First, multiple solutions appear when the number of factors that affect borehole deformation increases to certain level; second, the calculation using bit size as the original borehole size gives controversial results when compared to field observations, adjusted borehole size needed to be considered.To address these challenges and to facilitate the field application of the method, we developed a hybrid soft-computing based methodology to estimate both in-situ stress and geomechanical properties from borehole deformation, considering borehole size as an unknown.The salient features of the proposed methodology are:

(1) Integration of analytical calculation, artificial intelligence model, and probabilistic analysis enables estimating the ranges of rock mechanical properties and the in-situ stresses from borehole deformation information.The methodology is cost effective.

(2) Adjusted original borehole size is considered as an unknown parameter to obtain reasonable borehole deformation in the field.Thus, dilemma of using bit size as the default original borehole size is solved.

In the following sections of this paper, the mathematical basis of the proposed approach is introduced; sensitivity of geomechanics parameters to the borehole deformation was analyzed; combined workflow of artificial neural network modeling, genetic algorithm method, and probabilistic analysis is illustrated; and finally, a case study in Liard Basin is presented, which demonstrates the feasibility of estimating geomechanical properties and in-situ stress from borehole deformation.

2.Mathematics of borehole displacement

When a borehole is drilled, there will be several scenarios about the borehole shape (depending on the mud pressure, geomechanical properties, and the in-situ stresses), such as breakouts, drilling induced fractures, elliptical borehole, circular hole, or the combination of two or more of these events.If the borehole rock has not yielded under two different stresses that are orthogonal to the borehole cross section, an elliptical borehole will be formed as shown in Fig.1, with a vertical borehole considered as an example.

According to the elasticity theory, when a circular hole is drilled, the displacements around borehole can be written as:

where:ris original borehole radius;Ris the distance from borehole center;μrris the displacement of the borehole atRdistance from the borehole center;Eis Young's modulus; υ is Poisson's ratio;is vertical stress;is maximum horizontal stress;is minimum horizontal stress;Pmis borehole mud pressure;θis the angle from maximum horizontal stress direction.

WhenR=r, the displacement will occur on the wellbore wall.If we consider point A in Fig.1, the location along the maximum horizontal stress direction on the wellbore wall, the displacement can be determined by the following equation:

On the borehole wall along the minimum horizontal stress direction, point B in Fig.1, the displacement can be determined by equation (2):

The diameters of longer axis (C13) and shorter axis (C24) of the four-arm caliper measurements, which are corresponding to the deformation of the borehole wall at point A and point B respectively, can be determined by equations (4) and (5):

Therefore, the above equations (4) and (5) describe the relationship among the caliper measurements (longer axisC13, shorter axisC24), original borehole size, geomechanical properties, and the in-situ stresses.

Fig.1.Schematics of borehole deformation.

Theoretically, for a vertical well, if Young's modulus, Poisson's ratio, vertical stress, original borehole size, borehole pressure, and two horizontal stresses are known, equation (4) and equation (5) can be applied to calculate the longer and shorter diameters of the elliptical borehole.In actual cases of drilled borehole, the longer and shorter diameters can be measured by four-arms-caliper tools or borehole televiewers.Vertical stress can be calculated from overburden weight using density logs.Borehole pressure can be estimated form mud weight.Young's modulus, Poisson's ratio, and two horizontal stresses are usually unknowns.Although bit size is known in any drilling process, the borehole size is not supposed to be equal to it, due to the factors such as lithology, as well as erosion, damage and whirling of the bit.

3.Original borehole size consideration

3.1.Necessity for adjusted borehole size

Theoretically, if vertical stress, maximum horizontal stress, minimum horizontal stress, borehole mud pressurePm, original borehole radiusr, as well as rock mechanical properties Young's modulusEand Poisson's ratio υ, are known, the displacement of the borehole at borehole wallμrrcan be calculated by using equations (4) and (5).To see if this explains the field observation, we took a case study on a vertical well A-006-C/094-O-08 in the Liard basin in Western Canada [19].The well has four-arms caliper data, which enables us to compare the theoretical and measured deformation of the borehole.Geomechanical parameters were picked from Liard basin stress analysis report [19].Fig.2 shows the four-arms caliper log of the well A-006-C/ 094-O-08.We chose a section between around 4550 and 4670 feet that with no breakouts observed.The section can be identified using PFAS (Planning and Field Application Software) of ITC a.s.of Tonsberg, Norway [19].Zoom-in view of a section from 4620 feet to 4650 feet is shown in Fig.3, which is corresponding to top of Fort Simpson shale formation.The measured longer axisC13 is around 8.7 inches, the measured shorter axisC24 is around 8.5 inches.The bit size used in drilling this section is 8.5 inches.It seems that the borehole has enlarged in one direction (C13) which indicates minimum horizontal stress direction.In-situ stresses, borehole pressure while drilling, and the Poisson's ratio are taken from the Liard basin stress analysis [19].There is no Young's modulusEreported in Bell's report.We referenced several papers about the possible Young's modulus value in gas shale rock at various depth [20,21,22,23,24].The Young's modulus of shale can vary from 1 to 3 GPa [20,23,24] to as high as 20 GPa [21].Therefore, a number of Young' modulus values were considered for the calculations.The input parameters and the calculated theoretical borehole deformation in terms of displacement (μrrAandμrrB), borehole diameter (C13 andC24), and the ratio (C24/C13) are listed in Table 1.

Fig.2.Four arms caliper log of well A-006-C/094-O-08.

Fig.3.Measured four-arm caliper log of well A-006-C/094-O-08 at the top of Fort Simpson shale formation.

In Table 1, borehole diameters of 8.5 inches are used as the original borehole size; other parameters of vertical stress, maximum horizontal stressH, minimum horizontal stress, borehole mud pressurePm, and Poisson's ratio υ, are taken from Liard basin stress analysis report [19].While the rock mechanical property of Young's modulusEwere assigned arrange of values from 500 Mpa to 20 Gpa.By using equations (4) and (5), theoretical borehole displacement (μrrAandμrrB), deformed borehole diameter (C13 andC24), and the axis ratio of elliptical borehole (C24/C13) were calculated.It can be seen that the calculated theoreticalC24 (7.23-8.47inches) andC13 (8.27-8.49 inches) are both smaller than the measuredC24 (8.5 inches) andC13 (8.7 inches); furthermore, the calculated theoretical shorter and longer axesC24 andC13 are even larger than the bit size.The calculation results indicate borehole shrinkage, while observation indicates borehole expansion, based on the assumption that the borehole size is equal to the bit size.This requires the reassessment of the actual borehole size.Although the bit size is known in any drilling process and borehole was seemingly enlarged if we considered the original borehole as the default the bit size, the actual original hole size should at least be equal to the bit size or be larger than the bit size due to the factors such as lithology, as well as erosion, damage and whirling of the bit; otherwise it won't be pulled out easily.In the case example of well A-006-C/094-O-08 in the Liard Basin, where there is a measured longer axis is 8.7 inches, the original borehole size should be at least 8.7 inches considering the theoretically calculated borehole shrinkage.Therefore, there must be an adjustment of the borehole size from the bit size before it can be used in calculation.

3.2.Determination of adjusted borehole size and Young's modulus when in- situ stress is known

Based on equations (4) and (5), the ratio between the two axes, c = C24/C13, can be determined:

This ratio can be obtained via the two axes of the elliptical borehole from the four-arm caliper log.Therefore, Young's modulus can be calculated from equation (6).In the case study example of the deformation section as shown in Fig.3 in well A-006-C/094-O-08 in Liard Basin in Canada, the average shorter diameter is 8.5 inches, the average longer diameter is 8.7 inches.The ratio of the shorter and longer diametercis 0.977011.Using the known parameters of in-situ stresses, borehole pressure, and Poisson's ratio in Table 1, the calculated Young's modulus is 2.686 GPa.With Young's modulus obtained, by either Equation (4) or (5), the adjusted original hole size value can be calculated, which is 8.744 inches in this case.The ratio of hole size over bit size is 1.029.

Table 1 Input parameters and calculated borehole deformation results for well A-006-C/094-O-08.

Fig.4.Measured four arms caliper log of well A-045-E/094-O-10 at the top of Scatter formation in Liard Basin Canada.

We repeated this calculation for another well in the Liard Basin, Well A-045-E/094-O-10.Similarly, by using four arms caliper log we identified non-breakout section as shown in Fig.4, which is around the top of Scatter formation at depth around 2425 feet (～739 m).The section was drilled with bit size of 7.75 inches, and from the caliper log, the average shorter diameter C24 is 7.78 inches and the average longer diameter C13 is 8.03 inches.The ratio c of shorter diameter and longer diameter is 0.968867.

For this case, the calculated Young's modulus is 1.031 GPa, and the calculated adjusted borehole size is 8.087 inches.Therefore, the ratio of adjusted hole size over bit size is 1.043 in this case.

The previous two case studies in the Liard Basin showed that the reasonable original borehole sizes in both cases are larger than the bit sizes, and the ratios of the adjusted borehole sizes over the bit sizes are around 1.029 to 1.043.It can be seen the ratio varies because of the variation of lithology and the drilling operations.

3.3.Sensitivity of borehole deformation to geomechanical properties

In order to find out how sensitive the borehole deformation is to the variation of these parameters, sensitivity of borehole deformation to the variation of each of these rock mechanical properties was analyzed by using the case of Well A-006-C/094-O-08 as an example.The sensitivity of borehole deformation to the variation of Poisson's ratio and Young's modulus are listed in Table 2 and Table 3.

In Table 2, an entire range of Poisson's ratio (0-0.5) was used; while all the other parameters: borehole diameters, vertical stress, maximum horizontal stress, minimum horizontal stress, borehole mud pressurePm, and Young's modulusEwere kept constant are taken from Liard basin stress analysis report [19].While the rock mechanical property of as shown if the other parameters are constant.The calculated the borehole deformation is up to 0.0909 inches (0.0455 on each side).The difference of the axis ratio is only 0.0056, which is less than 0.6%.When considering the Poisson's ratio of around 0.15 to 0.35 for most common rock types, the differences of deformation and variation of axis ratio will be even smaller, which are less than 0.02 inches and 0.0023 respectively.

Unlike Poisson's ratio, Young's modulus has a larger range of variations.For shale, it can be as low as several hundred of MPa to severaltens of GPa [20,21,22,23,24].It can be observed from Table 3 that borehole deformation is more sensitive to the variation of Young's modulus than Poisson's ratio.In various oilfields, the Young's modulus value can vary from several hundred MPa to several tens of GPa, which may result in as high as 18% borehole deformation differences.Therefore Young's modulus is a relatively more sensitive parameter.

Table 2 Sensitivity of borehole deformation to Poisson's ratio.

Table 3 Sensitivity of borehole deformation to Young's modulus.

Sensitivity of borehole deformation to the maximum and minimum horizontal in-situ stresses are listed in Table 4 and Table 5 respectively.

It can be seen that for a reasonable range of maximum horizontal stresses, the differences among the borehole deformation can reach up to 41%, and for a reasonable range of minimum horizontal stresses, the differences among the borehole deformation can reach up to 20%.

4.Estimation of in-situ stresses and rock mechanical properties using artificial intelligence method and probabilistic analysis

4.1.Artificial intelligence metho d

Theoretically, equation (4) and equation (5) can be applied to calculate the longer and shorter diameters of the elliptical vertical borehole, should Young's modulus, Poisson's ratio, vertical stress, original borehole size, borehole pressure, and two horizontal stresses be known, but not the other way around.There will be too many unknowns to solve the only two equations.The relationship will be nonlinear, and the solution will not be unique.Therefore, Artificial Intelligence (AI) method is considered to map the relationship and to find the optimum results.AI methods such as Genetic programming and automated neural network search for modeling systems and rules.The advantage of these methods is that it has ability to build the explicit models for complex systems and adapt to non-linear equations based on only the data.This model can then be used offline or can be integrated in system for real- time monitoring of system in fields of medical, energy, and engineering [25-27,12].Anther extensively used AI method in recent years is the Artificial Neural Network (ANN), which can be integrated with Genetic Algorithm (GA) in practices in many fields including energy and petroleum industry [15,12,13,14,16-18].ANN model has the following advantages: ANN model is nonlinear model that is easy to use and understand; ANN model is non-parametric model while most of statistical methods are parametric model that need higher background of statistic; ANN with Back propagation (BP) learning algorithm is widely used in solving various classification and forecasting problems.Therefore, in our practice to estimate the unknowns of these in-situ stress and geomechanical properties with the limited known parameters from caliper logs, we firstly applied ANN model to map the relationship between the geomechanical parameters and the borehole deformations.However, ANN is black box learning approach, cannot interpret relationship between input and output and cannot deal with uncertainties.To overcome this, we integrated ANN with geneticalgorithm (GA), and probabilistic analysis.The purpose of such methodology is to get an estimate of these parameters with acceptable tolerance of error.

Table 4 Sensitivity of borehole deformation to maximum horizontal in-situ stress.

Table 5 Sensitivity of borehole deformation to minimum horizontal in-situ stress.

Fig.5.Schematic diagram of multilayer perception ANN model.

An ANN model was originally developed by McCullock and Pitts based on algorithms and mathematics [28].Now ANN models have evolved and have been considered modeling tools in finding patterns between the inputs and outputs based on the characterized relationship.An ANN model usually consists of an input layer, one or more hidden layers, and an output layer [29].A schematic diagram of multilayer perception model is shown in Fig.5.In the case study described in the following section, two hidden layers are used.A number of sets of borehole deformation, calculated from a range of geomechanical parameters using equations (4) and (5) in section 2, will be used for training the model.Specifically in ANN, we do the sum of products of inputs (x) and their corresponding weights (w) and apply an activation functionf x( ) to it to get the output of that layer and feed it as an input to the next layer.Hidden layer correlation equation for each neuron from the input layer can be expressed in the following equation [30]:

wheremis number of neurons of input layer,xiis quantity input ofith neure in input layer,yjis potential input of thejth neure in the hidden layer,wijis linkage weight fromith node of input layer to thejth node of hidden layer(s),is threshold of thejth neure.

The output of the neuronkin output layer can be described by the following equation [30]:

wherenis number of neurons in the hidden layer,Dkis output of deformed borehole,f(yj) is output of thejth neure of the hidden layer,wjkis linkage weight fromjth node of hidden layer(s) to thekth node of output layer,is thekth node threshold of output layer.

ANN model performance is evaluated using error of mean square (fe) and coefficient of relationship (R-value) [31].By definition, the error of mean square, which is the average of square differences between ANN prediction and the target value, can be described as following:

whereNis number of specimens,Dgis ANN prediction,Tgis the target value.The coefficient of relationship is can be computed by the following equation:

whereNis the number of specimens,Dgis ANN prediction,Tgis the target value,Daveis the average of ANN prediction,Taveis the average of target value.

Once the relationship between inputs (in-situ stress and geomechanical properties) and outputs (borehole deformation) from training samples has been established, the ANN model forwards the relationships to GA for further estimation of in-situ stress and geomechanical properties based on the desired wellbore displacements.

GA was first introduced by Holland as an abstraction of biological evolution [32].The method has been used for finding optimized parameters from natural selection and natural genetics models.Genetic algorithm starts from a range of values, known as population, to the solution of a problem.There are several potential solutions called chromosomes in the population, which can change to converge on an ideal solution.As chromosomes evolve over time through several successful iterations called generations, stronger chromosomes will be generated, which are evaluated by the objective function (fitness).The chromosomes having stronger fitness are more likely to be chosen in the evolution process [33,34].Once certain chromosomes have been selected, they become parents and are combined with other parents to produce new chromosomes for the next successive generation through the genetic process.The objective function in the case of estimating geomechanics properties from borehole deformation data is defined as the difference between ANN-predicted deformed bore hole size and the caliper measured borehole size as follows:

Fig.6.Workflow of artificial intelligence modeling and probabilistic analysis to estimate geomechanics parameters from borehole deformation data.

whereDiis ANN-predicted deformed bore hole size,Ciis the measured deformed borehole size from four arms caliper logs (C1=C13 ,C2=C24).

4.2.Probabilistic analysis

The results from the above ANN-GA modeling, however, will not give a unique solution due to the large number of total unknown parameters in this case.There will be multiple combinations of in-situ stress, geomechanical parameters and borehole size that all lead to the same borehole deformation.To solve this dilemma, we adopted the Bayes' probabilistic analysis for a number of realizations of the ANN-GA model.

Bayesian probabilistic inversion method enables the overcome of two major difficulties in relating observed data to unknowns, one is observational noise, the other is the underdetermined system due to the number of equations is smaller than the number of unknowns [35].Bayesian and other probabilistic analysis (such as Monte Carlo method) have been applied to solve uncertainty related issues in many geomechanics engineering projects [36,37].In our case, based on the ANN- GA analysis, there are a set of solutions, each of which will give a borehole deformation value that matches the measured deformation.Therefore, there will be a range of Young's modulus values, a range original hole size values, as well as ranges of maximum and minimum horizontal stress values.We need to find from each range the most frequent occurrence of a certain value (or a small range of a certain value), whose combination still leads to the observed borehole deformation.In other words, we need to find the conditional probability value,P(Br|A), from a range of values (B1,B2,…,Bk) of parameter B, whereP(Bi)≠0, fori = 1,2, …,k;andAis the occurrence of other corresponding parameters.According to the Bayes' rule of elimination, the calculation ofP(Br|A) can be described by the following equation [38]:

Table 6 Available known parameters for Well A-006-C/094-O-08.

Table 7 Parameters used for the generation of training and testing data.

For a number of ANN-GA runs, we made statistics analysis and used equation (7) to estimate the combination of geomechanics parameters with highest probability.The overall workflow of combined artificial intelligence modeling and probabilistic analysis is illustrated in Fig.6.

5.Case study

In this section, the proposed GA-ANN modeling and probabilistic analysis method is demonstrated by a field study in Liard basin, western Canada.Borehole deformation data was read from four-arm caliper data of Well A-006-C/094-O-08.Other parameters were taken from the report “In Situ Stress Orientations and Magnitudes in the Liard Basin of Western Canada” by Bell [19].

At depth of 1409 m, top of Fort Simpson of Well A-006-C/094-O-08 in Liard Basin, neither breakout nor drilling induced fractures were reported.The bit size was 8.5 inches while drilling this section.The measured longer borehole diameterC13is 8.7 inches from four-arm caliper log; the measured shorter borehole diameterC24is 8.5 inches.The pore pressure in the area is hydrostatic; drilling is balanced [19]; therefore we adopted 14 MPa as the pore pressure and mud pressure.The vertical in-situ stress and the Poisson's ratio were taken from the analysis report by Bell [19].The parameters are listed in Table 6.

We consider the original borehole size in the range of 8.5-8.8 inches.The input training and testing data for ANN are calculated based on the parameters listed in Table 7.Totally 256 combinations were generated.Thus 256 sets of longer and shorter diameters,C13andC24, were calculated through analytical calculation using equations (4) and (5).We chose 16 sets out of the 256 sets combinations and calculated results as the testing data.

Once the ANN is trained, the inverse analysis modeling is used to characterize the relationship between inputs and outputs.Then fitness (objective function), the borehole longer and shorter diameters in this case, is established and GA is used as an optimization tool to search chromosomes (solutions) from a wide range of inputs that meet the established objective function.The chromosomes that have a stronger fitness are more likely to be selected as the results.Totally 100 realizations ANN-GA modeling for four unknown input parameters (considered original borehole size, Young's modulus, and the two horizontal stresses) are shown in Fig.7.Each realization has been verified by the mean-square error (MSE) to evaluate the convergence of ANN; and has also been verified by the leaner regression between the modeled output and the corresponding target.Fig.8 shows the MSE variations for training, validation, and testing with iterations.The best validation performance is 5.95 × 10-6.Fig.9 shows the linear regression between the modeled and the corresponding target.The regression values are shown on the top of pictures of training, validation, and all samples respectively.All of the regression values are greater than 0.99, which means a very good performance.

Fig.7.Results of hole/bit size ratio, Young's modulus, and horizontal stresses of 100 ANN-GA realizations, top of Fort Simpson in Well A-006-C/094-O-08, Liard Basin, Canada.

Fig.8.Validation of the mean square errors.

Each of the 100 realizations gives a good deformation that matches those measured from caliper logs.However, it can be seen that all the four parameters, Young's modulus, maximum horizontal stress, minimum horizontal stress, and the borehole size to bit size ratio are scattered in ranges without any obvious high frequency values.In order to narrow down the range of these parameters, we applied the Bayes' theorem to find the banded range of the parameters.

Fig.9.Performances of linear regressions.

Histograms for all these four parameters are shown in Fig.10, in which, each parameter was sub-grouped into 5 subsets.The subsets of maximum horizontal stress are: 35.70-38.54 MPa, 38.54-41.38 MPa, 41.38-44.23 MPa, 44.23-47.07 MPa, and 47.07-49.91 MPa; the subsets of minimum horizontal stress are: 20.03-22.81 MPa, 22.81-25.59 MPa, 25.59-28.37 MPa, 28.37-31.15 MPa, and 31.15-33.93 MPa; the subsets of Young's modulus are: 2.01-2.41 GPa, 2.41-2.80 GPa, 2.80-3.20 GPa, 3.20-3.60 GPa, and 3.60-3.99 GPa; the subsets of hole/bit size ratio are:1.021-1.023, 1.023-1.026, 1.026-1.029, 1.029-1.032, and 1.032-1.035.The probability of each subset of a parameter varies.The maximum probability of subsets in maximum horizontal stress (SigH) isP(SigH = 41.38-44.23) = 29%.The maximum probability of subsets in hole over bit size ratio isP(Hole/bit= 1.023-1.026) = 27%.There are two maximum probability of subsets in Young's modulus, which are:P(E= 2.8-3.2) = 25% andP(E= 3.2-3.6) = 25%.If we combine these two subsets together, then the probability of Young's modulus in the range of (2.8-3.6 GPa) is:P(E= 2.8-3.6) = 50%.Minimum horizontal stress doesn't show obvious highest probability subset.Three subsets in the range of 20.03 MPa-28.37 MPa exhibit similar probability.The probability of the 5 subsets of minimum horizontal stress (Sigh) are:P(Sigh = 20.03-22.81) = 27%,P(Sigh = 22.81-25.59) = 25%,P(Sigh = 25.59-28.37) = 26%,P(Sigh = 28.37-31.15) = 19%,P(Sigh = 31.15-33.93) = 3%.For the convenience of probability analysis, we use events A, B, C, and D to represent parameters of maximum horizontal stress, minimum horizontal stress, hole over bit size ratio, and Young's modulus respectively.Each subset has its probability based on the histogram and are listed in Table 8.

Fig.10.Histogram of maximum horizontal stress, minimum horizontal stress, Young's modulus, and hole/bit ratio of 100 ANN-GA realizations.

Therefore, the maximum probability of subset in maximum horizontal stress isP(A3) =P(SigH = 41.38-44.23) = 29%.The maximum probability of subsets in hole over bit size ratio isP(C2) =P(Hole/ bit= 1.023-1.026) = 27%.We combined two equally high probability of subsets in Young's modulus, which are:P(D3) = 25% andP(D4) = 25%, into one subset and calledP(D34) =P(E= 2.8-3.6) = 50%.

We further ran ANN-GA on each subset of minimum horizontal stress (B1 to B5) and achieved probabilities of A3, C2, D34 conditional to B1 to B5 respectively.The highest conditional probability of A3, C2, and D34 are listed in Table 9.By using equation (12), we can find the conditional probability of minimum horizontal stress subsets over A3, C2, and D34 as listed in Table 10.

The highest conditional probability over A3 isP(B2|A3) = 50%; while the highest conditional probability over C2 isP(B1|C2) = 64%, the highest conditional probability over D34 isP(B1|D34) = 32%.

The most possible range for minimum horizontal stress will be 20-26 MPa (subset B1 and B2).Therefore, in the case study of A-006-C/ 094-O-08 in, which was drilled in the Fort Simpson formation with 8.5 inches bit in Liard Basin in Canada, if borehole mud pressure is 14 MPa, vertical stress is 35 MPa, and the Poisson's ratio is 0.2, the borehole size should be estimated as 8.696-8.721 inches (corresponding to the hole over bit size ratio of 1.023-1.026); average borehole size within this range is 8.709 inches.The most possible range of maximum horizontal in-situ stress will be 41-44 MPa, average value is 42.5 MPa; the most possible range for the minimum horizontal in-situ stress will be 20-26 MPa, average value is 23; the most possible range of Yong's modulus will be 2.8-3.6 GPa, average value is 3.2 GPa.The results are listed in Table 11.

6.Discussions

6.1.Comparison with forward deterministic results

For the vertical borehole discussed in this paper, the borehole deformations can be calculated forwardly by deterministic analytical method using equations (4) and (5) if the rock mechanical properties and in-situ stresses values are available.We used the estimated rock mechanical properties and the horizontal in-situ stresses in Table 11 in the previous section to forwardly calculate the borehole deformation.Among the four parameters in Table 11, Young's Modulus ranges and minimum horizontal stress ranges are relatively wider than the other two, because the two parameters cover two subsets.Further finer division of the subsets might be required in future calculations.In the validation calculation of the case study in this paper, the average value of parameter ranges from Table 11 were chosen for the deterministic calculation, and are listed in Table 12 together with other known parameters.

Deterministic analytical borehole deformation calculation resultsare listed in Table 13.Comparison with the measured deformation shows excellent agreement.

Table 8 Probability of subset occurrences in multiple realizations.

Table 9 Conditional probabilities of A3, C2, D34 over subsets of minimum horizontal stress.

Table 10 Conditional probability of minimum horizontal stress subset occurrence.

Table 11 Most possible original borehole size and estimated geomechanical parameters.

Table 12 In-situ stress, rock mechanical properties, and hole size for forward calculation.

Table 13 Deterministic analytical calculation deformation using estimated parameters.

Table 14 Comparison of in-situ horizontal stresses from probabilistic ANN-GA inversion with reported horizontal stresses.

6.2.Comparison with reported field data

In Liard Basin, Canada, in-situ stresses have been analyzed and reported by Bell [19].Table 14 shows the comparison between the results of the probabilistic ANN-GA model inversion analysis and the results reported by Bell.The estimated maximum horizontal stress is in good agreement with the reported maximum horizontal stress.The estimated minimum horizontal stress, however, is a bit away from the reported minimum horizontal stress.The upper boundary of the estimated minimum horizontal stress range is close to the reported stress value.But it should be mentioned that in Bell's report, the minimum horizontal in-situ stress was taken from the leak-off pressure, which was usually regarded as an upper limit for minimum horizontal in-situ stress.The probabilistic ANN-GA method estimated a range of Young's modulus from 2.8 to 3.6 GPa, with an average value of 3.2 GPa.There is no Young's modulus reported in Bell's report, but there are reports of Young's modulus value in similar range for gas shale rock at various depths [20,21,22,23,24].

6.3.Limitations of the work

Although the methodology can be used to estimate rock mechanical properties and the in-situ stresses, it has limitations.Since the requirement of input data is based density logging, four arms caliper data and mud density of drilling data, the availability of all these information is critical.Density logging data and mud information are generally available, the four-arms caliper logging, however, is not always available.Moreover, each parameter needs to be assigned a range in the calculation, which depends on people's experience.If the range is assigned too wide or too narrow, the accuracy of such calculation will be compromised.

7.Conclusions

Borehole deformation can be theoretically calculated from rock mechanical properties and in-situ stresses.Dilemma arises when using bit size as the default original borehole size to calculate borehole deformation.Use of the adjusted original borehole size is essential to obtain the reasonable borehole deformation in the field.

Borehole deformation is more sensitive to some parameters than to others.The influence of Poison's ratio on borehole deformation is the least among in-situ stress and other geomechanical parameters.For this reason, Poisson's ratio is treated as a known parameter and can be reasonably assigned a value by referring to its lithology and/or geological information of the area.As a result, the total number of unknown parameters in the inversion analysis can be reduced.

If neither of rock mechanical properties nor the horizontal in-situ stresses are available, it is possible to estimate both of them from borehole deformation data by probabilistic ANN-GA method.The combined probabilistic analysis with ANN-GA modeling method help narrow down the range of high frequency solutions.

Acknowledgements

The authors are grateful to Dr.J.S.Bell who has provided data of Liard Basin and constructive suggestions for this work.

Nomenclature

μrrDisplacement of the borehole at R distance from the borehole center

μrrADisplacement at the borehole wall along the maximum horizontal stress direction

μrrBDisplacement at the borehole wall along the minimum horizontal stress direction

C13Longer axis of the elliptical borehole

C24Shorter axis of the elliptical borehole

cRatio of shorter axis over longer axis of the elliptical borehole

EYoung's Modulus

Υ Poisson's Ratio

PmBorehole pressure

Vertical stress

Maximum horizontal stress

Minimum horizontal stress

RDistance from borehole center, Coefficient of relationship

rOriginal borehole radius

θAngle from maximum horizontal stress direction

mNumber of neurons of input layer

xiQuantity inputith neure in input layer

yjPotential input of thejth neure in the hidden layer

wijLinkage weight fromith node of input layer to thejth node of hidden layer(s)

Threshold of thejth neure

nNumber of neurons in the hidden layer

DkOutput of deformed borehole

f(yj) Output of thejth neure of the hidden layer

wjkLinkage weight fromjth node of hidden layer(s) to thekth node of output layer

Thekth node threshold of output layer

Number of specimens

DgANN prediction

TgTarget value

feError of mean square

DaveAverage of ANN prediction

TaveAverage of target value

DiANN-predicted deformed bore hole size

CiMeasured deformed borehole size from four arms caliper logs

C1EqualsC13

C2EqualsC24

SigHMaximum horizontal stress

SighMinimum horizontal stress

SigVVertical stress

P(Br|A) Probability ofBrconditional toA

References

[1] B.C.Haimson, G.Clark (Ed.), A Simple Method for Estimating in Situ Stresses as Great Depth.Field Testing and Instrumentation of Rock, STP32150S, ASTM International, West Conshohocken, PA, 1974, pp.156-182.

[2] B.C.Haimson, C.Fairhurst, Initiation and extension of hydraulic fractures in rock, Soc.Petrol.Eng.J.7 (1967) 310-318.

[3] E.M.Anderson, The dynamics of faulting, Trans.Edinb.Geol.Soc.83 (1905) 387-402.

[4] E.G.Kirsch, Die Theorie der Elastizit?t und die Bedürfnisse der Festigkeitslehre, Z.Des.Vereines Dtsch.Ingenieure 42 (1898) 797-807.

[5] P.Peska, M.D.Zoback, Compressive and tensile failure of inclined wellbores and determination of in-situ stress and rock strength, J.Geophys.Res.100 (B7) (1995) 12791-12811.

[6] M.D.Zoback, D.Moos, L.Mastin, R.N.Anderson, Wellbore breakouts and in-situ stress, J.Geophys.Res.90 (1985) 5523-5530.

[7] W.B.Ervine, J.S.Bell, Subsurface in situ stress magnitudes from oil-well drilling records: an example from the venture area, offshore eastern Canada, Can.J.Earth Sci.24 (1987) 1748-1759.

[8] F.H.Cornet, B.Valette, In situ stress determination from hydraulic injection test data, J.Geophys.Res.89 (B13) (1984) pp.11,527-11,537.

[9] C.Chang, M.D.Zoback, A.Khaksar, Empirical relations between rock strength and physical properties in sedimentary rocks, J.Petrol.Sci.Eng.51 (2006) 223-237.

[10] A.Abdulraheem, M.Ahmed, A.Vantala, T.Parvez, Prediction of rock mechanical parameters for hydrocarbon reservoirs using different artificial intelligence techniques, Paper SPE126094 Presented at the 2009 SPE Saudi Arabia Section Technical Symposium and Exhibition.Alkhobar, Saudi Arabia, 09-11, May, 2009.

[11] B.S.Aadnoy, Inversion technique to determine the in-situ stress field from fracturing data, J.Petrol.Sci.Eng.4 (2) (1990) 127-141.

[12] Y.Huang, L.Gao, Z.Yi, K.Tai, P.Kalita, P.Prapainainar, A.Garg, An application of evolutionary system identification algorithm in modelling of energy production system, Measurement 114 (2018) 122-131.

[13] M.Ibrahim, S.Jemei, G.Wimmer, D.Hissel, Nonlinear autoregressive neural network in an energy management strategy for battery/ultra-capacitor hybrid electrical vehicles, Elec.Power Syst.Res.136 (2016) 262-269.

[14] Z.Sabir, M.A.Manzar, M.A.Z.Raja, M.Sheraz, A.M.Wazwaz, Neuro-heuristics for nonlinear singular Thomas-Fermi systems, Appl.Soft Comput.65 (2018) 152-169.April 2018.

[15] H.X.Han, S.Yin, Determination of in-situ stress and geomechanical properties from borehole deformation, Energies 11 (1) (2018) 131.

[16] S.Zhang, S.Yin, Determination of horizontal in-situ stresses and natural fracture properties from wellbore deformation, Int.J.Oil Gas Coal Technol.7 (1) (2014) 1-28.

[17] S.Zhang, S.Yin, Determination of in situ stresses and elastic parameters from hydraulic fracturing tests by geomechanics modeling and soft computing, J.Petrol.Sci.Eng.124 (2014) 484-492.

[18] S.Zhang, S.Yin, Determination of earth stresses using inverse analysis based on coupled numerical modeling and soft computing, Int.J.Comput.Appl.Technol.52 (1) (2015) 18-28.

[19] J.S.Bell, In Situ Stress Orientations and Magnitudes in the Liard Basin of Western Canada, Geological Survey of Canada, 2015, p.410, https://doi.org/10.4095/ 295742 Open File 7049.

[20] D.N.Dewhurst, A.Hennig, Geomechanical properties related to top seal leakage in the Carnarvon Basin, Northwest Shelf, Australia, Petrol.Geosci.9 (2003) 255-263.

[21] M.O.Eshkalak, S.D.Mohaghegh, S.Esmaili, Geomechanical properties of unconventional shale reservoirs, J.Petrol.Eng.2014 (2014) 10.Article ID 961641 https://doi.org/10.1155/2014/961641.

[22] Q.Gao, J.Tao, J.Hu, X.Yu, Laboratory study on the mechanical behaviors of an anisotropic shale rock, J.Rock Mech.Geotech.Eng.7 (2015) 213-219.

[23] Md A.Islam, P.Skalle, An experimental investigation of shale mechanical properties through drained and undrained test mechanisms, Rock Mech.Rock Eng.46 (2013) 1391-1413.

[24] M.Josh, L.Esteban, C.D.Piane, J.Sarout, D.N.Dewhurst, M.B.Clennell, Laboratory characterisation of shale properties, J.Petrol.Sci.Eng.88-89 (2012) 107-124.

[25] A.Garg, B.Panda, K.Shankhwar, Investigation of the joint length of weldment of environmental-friendly magnetic pulse welding process, Int.J.Adv.Manuf.Technol.87 (2016) 2415-2426.

[26] A.Garg, V.Vijayaraghavan, J.Zhang, J.S.L.Lam, Robust model design for evaluation of power characteristics of the cleaner energy system, Renew.Energy 112 (2017) 302-313.

[27] A.Garg, K.Shankhwar, D.Jiang, V.Vijayaraghavan, B.N.Panda, S.S.Panda, An evolutionary framework in modelling of multi-output characteristics of the bone drilling process, Neural Comput.Appl.29 (11) (2018) 1233-1241.

[28] W.S.McCulloch, W.H.Pitts, A logical calculus of the ideas immanent in nervous activity, Bull.Math.Biophys.5 (1943) 115-133.

[29] S.Mohaghegh, Virtual-intelligence applications in petroleum engineering Part 1-artificial neural networks, J.Petrol.Technol.52 (2000) 64-72.

[30] S.Zhang, S.Yin, A hybrid ANN-GA method for analysis of geotechnical parameters, 12th International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery (ICNC-FSKD), IEEE, 2016,https://doi.org/10.1109/FSKD.2016.7603141.

[31] H.Aguir, H.BelHadjSalah, R.Hambli, Parameter identification of an elasto-plastic behavior using artificial neural networks-genetic algorithm method, Mater.Des.32 (2011) 48-53.

[32] J.H.Holland, Adaptation in Natural and Artificial Systems, first ed., University of Michigan Press, 1975.

[33] D.E.Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, Reading, MA, 1989.

[34] J.H.Holland, Adaptation in Natural and Artificial Systems, second ed., MIT Press, Cambridge, MA, 1992.

[35] M.Dashti, A.M.Stuart, The Bayesian Approach to Inverse Problems, Cornell University, 2015 arXiv:1302.6989v4 [math.PR].

[36] R.Arnold, J.Townend, A Bayesian approach to estimation tectonic stress from seismological data, Geophys.J.Int.170 (2007) 1336-1356.

[37] H.EI-Ramly, N.R.Morgenstern, D.M.Cruden, Probabilistic slope stability analysis for practice, Can.Geotech.J.39 (2002) 665-683.

[38] R.E.Walpole, R.H.Mayers, S.L.Mayers, K.Ye, Probability & Statistic for Engineers & Scientists, seventh ed., Prentice-Hall, Inc, New Jersey, USA, 2002.