Research on the Damage of Porosity and Permeability due to Perforation on Sandstone in the Compaction Zone

Shifeng Xue,Xiuxing Zhu,Lin Zhang,Shenghu Zhu,Guigen Ye,Xuejun Fan

Research on the Damage of Porosity and Permeability due to Perforation on Sandstone in the Compaction Zone

Shifeng Xue1,2,Xiuxing Zhu1,2,Lin Zhang3,Shenghu Zhu4,Guigen Ye1,Xuejun Fan5

A perforating hole is a channel through which the oil and gas in a reservoir pass into the production well bore.During the process of perforating due to explosion,the surrounding sandstone will be damaged to a certain extent,which will increase the well bore skin and lead to the decrease of production consequently.In this work a mechanical model of perforating damage is developed to describe the influences of perforating due to explosion on the porosity and permeability of the surrounding sandstone near the compaction zone.Based on this developed model,the important data related to the damage of sandstone,such as matrix effective stress,plastic deformation,volumetric strain,and so on,can be numerically simulated.Especially the behaviors of plasticity kinematic hardening at high strain rate due to impact loads,which are the important characteristics in the sandstone,is taken into account in this developed model.Both numerical and testing results show that the damage due to perforation in the sandstone can be accurately predicted by the developed model together with the porosity and permeability evolving model of perforation in a compaction zone.As a practical application,a methodology for the analysis of damage of porosity and permeability around a perforation tunnel is supposed based on the developed model and the core flow efficiency test of inter-particle pore spaced sandstone target in China Shengli Oilfield and the computed tomography test.

1 Introduction

During the perforation process,the metal jet with high energy formed by the explosion of perforating charges penetrates the casing and sandstone formations,andforms a channel of perforating hole that connects reservoirs to a well bore.The sandstone matrix and particles around a perforation channel are extruded by pressure loads due to the explosion,and constitute the complex crushed and compacted zone around the perforation channel,which damages the porosity and permeability of sandstone and leads to the sharp declining of the productivity of oil and gas wells[Swift(1998);Baumann and Brinsden(2014)].It is important to explain the mechanism of porosity and permeability damaging due to perforation in a compaction zone and develop a model to describe the porosity and permeability damaging for improving the production of oil-gas well[Nabipour,Sarmadivaleh,and Asadi(2010);Glenn,Serra,and Rodgers(2014);Heiland,Grove,and Harvey(2009)].

Untiltoday some work about the mechanism of sandstone damage under the load of low strain rate have been published and available[Bostrom,Chertov,and Pagels(2014);Veronika et al.,(2010);Thurman et al.,(1991);Sheldon(2006);Wong,David,and Zhu(1997)].According to above work,the porosity reduction is caused by the plastic volume deformation of sandstone matrix.Based on this mechanism,Morris,Lomov,and Glenn(2001,2003)established the porosity and permeabilityevolving model of sandstone damaging zone.

The sandstone damage under the impact load of the perforation pressure at a high strain rate is mainly accorded to the standard of API RP 19B to broadly and qualitatively evaluate the extent of porosity and permeability damage[API RP 19B-2006(2006);Hiltl,Hagelberg,and Swift(1999);Karacan and Halleck(2003);Halleck,karacan,and Hardesty(2004);Karacan,Grader,and Halleck(2001);Swift and Hagelberg(2000);Karacan and Halleck(2002)].

The structure and physical characteristics of sandstone particles in the compaction zone after perforation are evaluated based on the data of core flow ef fi ciency(CFE)obtained by using the means of NMR,CT MPR and electron microscope,etc..However,research is relatively scarce on the mechanics and on the quantitative model of macroscopic mechanical parameters that are related to the sandstone damage[James,Brooks,and Solutions(2011)].

In this paper,the preliminary mechanical model to evaluate the compaction damage of perforation in a sandstone reservoir is established based on the information available in literature,and presents a method of quantitative evaluation that combines the efficiency test of perforation with a numerical simulation[Karacan and Halleck(2003);Zhu,Zhang,and Xue(2014)]in order to further solve the problems of cost and time-effectiveness during the damage evaluation experiment of perforation.The method offers an effective way for the porosity and permeability damage evaluation of sandstone perforation.

The dynamic response of sandstone in the perforation compacting zone is simulated by nonlinear finite element method.The key data of matrix pressure,plastic deformation and volumetric strain,which are related to the sandstone damage of perforation compacting zone,can be obtained to quantitatively evaluate the extent of the porosity and permeability damage of a perforation compacting zone.

The applicability of the method proposed in this paper is proven by comparing the numerical simulation results of sandstone damage in a perforation compacting zone with the test data of CFE,CT MPR,and electron microscope analysis.

2 The model of porosity and permeability damage of the sandstone target

According to the dynamic response of sandstone under the impact load of perforation,the porosity and permeability-evolving model of sandstone under low strain rate is modified,and the preliminary mechanical model for evaluating the compaction damage of perforation in a sandstone reservoir is established.

Morris gives out the porosity and permeability-evolving model of sandstone under low strain rate,as seen in Formula(1)–(6)[Morris,Lomov,and Glenn(2003)].The effective stress,plastic strain and damage variable of a sandstone unit are the parameters for the model calculation,which can be obtained by means of numerical analysis and will facilitate the analysis of porosity and permeability evolution.

In Morris'model,the parameter Φ?is defined as the fluid porosity intercept,which refers to the porosity of sandstone when the effective stress reaches the critical hydrostatic pressure under a static loading.During the loading process of the impact load of perforation,the effective stress of sandstone will rapidly exceed the critical hydrostatic stress,thus the fluid porosity intercept Φ?will change little comparing with the reference fluid porosity ΦI,then it can be considered that Peff≥ P? and Φ?=ΦI,so Formula(3)can be modified as:

Based on the above analysis,the mechanical model to evaluate the damage of porosity and permeability in the compaction zone of sandstone perforation can be established as equation(8)–(10).

Where the damage variable A can be derived according to Formula(5)as:

Where εpis the plastic strain of sandstone unit andεvis the volumetric strain of the sandstone unit.c1is the loss factor of plastic strain with unit of GPa?1and c2is the damage factor of hydrostatic pressure.Furthermore,is the maximum damaging stress of plastic strain with unit of GPa,p?is the critical pressure of hydrostatic pressing compaction with unit of GPa,and Ph2is the maximum damaging stress of hydrostatic pressing compaction with unit of GPa.H()is a step function.The values of the parameters are shown in Table 1.

This study defined the evaluation indexes φcdand kcdof porosityamp;permeability damage extent as:

Where φcdis the damaged extent of porosity.kcdis the damage extent of permeability.ˉφ is the average porosity of compaction zone.ˉk is the average permeability of compaction zone,with the unit of m D.

3 Quantitative analysis of the macro-mechanical parameter of sandstone in compaction zone

During the perforation process,sandstone is subjected to the impact load that generates plastic shear,squeezing and complex deformations associated with inter-particle dislocation.This paper applied the deformation characteristic of sandstone from the characterized high strain rate of plasticity in a kinematic hardening material model under impact loads[Yang(2012)].The numerical simulation of the dynamic response of sandstone in the compaction zone of perforation was performed,which provided the key macroscopic mechanical parameters of sandstone matrix pressure,plastic deformation and volumetric strain.It also provided the essential data for the quantitative evaluation of the porosity and permeability damage in a compaction zone.

3.1 Essential data

Based on the performance test of sandstone target core,properties needed for perforating damage in the compaction zone were obtained as shown in Table 1.

3.2 Numerical model

Considering that the structure of a sandstone target is axially symmetric,the numerical model of sandstone under the impact load of perforation was established as shown in Figure 1.

Figure 1:The numerical model of sandstone under the impact load of perforation.

In order to simulate the actions of shear,friction and squeezing on the sandstone of the inner wall in the perforating hole by the impact load during the perforating process of a perforating metal jet,this study assumed a rigid unit block in the model,which experienced an axial motion along the perforating hole.The pressure pulse load was forced on the Boundary 1.Boundary 2 came in contact with the boundaryCD of the inner wall in the perforating hole.The velocity v0of the rigid unit block was set same as the forming speed of the perforating hole.

Table 1:Essential Properties of Sandstone.

3.3 Dynamic response analysis of sandstone close to a perforating hole

3.3.1 Pressure impact load

After the charge perforation is detonated,the resulting particle flow of metal with high speed will penetrate the casing,impact sandstone and form a huge impact load.Then,the load will induce a huge lateral pressure in the sandstone,squeeze the particles of sandstone and form a perforating hole.

So far,the lateral pressure pulsing load in the inner wall and the forming speed of a perforating hole cannot be obtained through experiments.They can only be estimated by the method of numerical simulation.Literature suggests that for the conventional perforating charge,the peak value of pressure pulse load in the inlet end of a perforating hole is 10 GPa[Baumann and Brinsden(2004)].The peak value at the bottom is declined to 1/10 of that in the inlet end of the perforating hole.The forming speed of the perforating hole is about 2000 m/s.

Assuming that the pressure impact load is linearly decreased,the peak load along the axis of perforating hole can be obtained as:

Where:Pfis the peak value,with unit of GPa.Ldis the length from the inlet end of the perforating hole,with unit of m.D is the depth of the perforating hole,which was selected as 0.56 m in this study.

3.3.2 Algorithm of contact-impact interface

Contact-impact interface involves complex nonlinear deformation.To solve such a complex issue,the selection of interfacial algorithm is very important[Rubin,Vorobiev,and Glenn(2000);Rbewi and Tiab(2012)].

In this paper,the penalty function method was adopted to process the contactimpact interface.The ultimate principle can be described as follows.

In every time step,the first step is to check whether the principal plane is penetrated from each slave node.No treatment is needed if there is no penetration.If there is a penetration,a contact force will exist at the interface between the slave node and the penetrated principal plane.The force is proportional to the depth of penetration and the rigidity of the principal plane,and is represented by a function named as the penalty function.That is:

Where fsis the contact force vector at the contact node and normal to the contact plane,lcis the depth of penetration,with unit of mm.niis the outer normal unit vector of the master slice Siat the point of contact.kiis the stiffness factor of the master slice Si.

3.3.3 Material model

Under the impact load of perforation,the strain rate of sandstone increases.Meanwhile,the relationship of stress-strain tends to be complex.Some essential parameters,such as yield limit and instant aneousstress,change in varying degrees.All of them influence on the dynamics response of sandstone.In order to represent the mechanical property of sandstone under the condition of high strain rate,the plastic-kinematics hardening nonlinear material model,which contains the strain rate effect,was adopted.The following expression represents the relationship between the dynamic-limit yield stress and the strain rate of material in the model:

Where σdis the dynamic-limit yield stress,with unit of MPa,˙ε is the strain rate,with unit of s?1,and Cqand P1are the constants which are related to the material property.σyis the static yield stress with unit of MPa.β is the hardening parameter,which is zero for the plastic-kinematics hardening model.Epis the plastic-kinematics hardening modulus,with unit of GPa.is the equivalent plastic strain.For the inter-particle pore of sandstone,the values of relevant parameters can be seen in Table 1.

3.3.4 Mesh generation

Two-dimensional physical unit with four nodes of PLANE 162 was used to map the mesh of the model.The mesh density was determined according to the accuracy requirement.

A trial mesh generation suggested that the length AB in the model(Figure 1)be divided into 900 segments and the width AC be divided into 90 segments.Then,each unit could be expressed by serial number rij(i=1,2,...m;j=1,2,...,n)where m=900 and n=90.The resulting data for a unit can be expressed by an array R,as seen in Formula(16).

3.3.5 The macroscopic mechanical parameter analysis of sandstone in the compaction zone

Dynamic response of sandstone in the compacting zone during the process of perforation was simulated numerically.The radial displacement of sandstone in the process of perforating and compacting was obtained as illustrated in Figure 2.

Mechanical parameters,such as displacement,stress and plastic strain,were extracted from each node,and were assigned to the corresponding arrays of Rux,Ruy,and Rεpy.According to formulas(17)and(18),the effective stress and volumetric strain of a sandstone unit can be computed as:

Figure 2:The radial deformation of sandstone in the process of perforating and compacting(mm).

where σxis the x directional stress with a unit of GPa,σyis the y directional stress with a unit of GPa,P0is the pore pressure with unit of GPa,and εvis the volumetric strain of the sandstone unit.εxand εyare,respectively,the x and y directional strains of the unit.

The damage variable A of the sandstone unit can be solved according to Formula(10),then,according to Formula(8)and(9),the distribution of porosity and permeability of sandstone can be obtained.

4 Test on sandstone target in a perforation compaction zone

By using a CFE detection device with ultra-temperature and high pressure from the logging company of Shengli Oil Field as a platform,and following the standard procedure of API RP 19B,this study simulated the perforating process of sandstone target between inter-particle pores under the environments of reservoir and wellbore(bottom hole temperature,pressure of well-bore,pore pressure,and pressure of overlying strata[API RP 19B-2006(2006)].

Meanwhile,the sandstone target and the perforating gun and charges were designed according to the experimental flow recommended in the API RP 19B Standard as[API RP 19B-2006(2006)].

The outer diameter of sandstone target was 160 mm with the length of 800 mm;the porosity was 10.8%and the permeability was 3.4 mD.

The charge type was DP44RDX32-5,with the explosive of RDX and the charge amount of 32 g.

During the whole perforating process,the parameters were kept as below:the temperature of sandstone target was 80°C.The pressure of well bore was 10 MPa.The con fining pressure was 35 MPa,and the pore pressure was 20 MPa.

Figure 3:Core holder and simulated well bore assembly.

Table 2:CFE Test Data of Sandstone Target.

4.1 CFE test of sandstone target

During the experiment,by maintaining the con fining pressure and pore pressure constant after perforation,the fluid flow(kerosene)in the sandstone target perforated channel under the constant pressure difference was obtained,which can be seen in Table 2.

The CFE of perforation can be calculated from Formula(19)as API RP 19B-2006(2006)

Where r is the perforating radius with unit of m,R is the core radius with unit of m,Qcis the theoretical calculation flow with unit of m3/s,and Qmis the measured flow with unit of m3/s.

Figure 4:The shape of a sandstone target perforating hole.

Qccan be calculated as

Where?P is the pressure difference between confining pressure and effective stress of sandstone target,with unit of Pa,μis the viscosity of kerosene with unit of Pa.s,D is the perforating depth with unit of m.While K1is the horizontal(axial direction of perforating hole)permeability,with unit of m2,K2is the vertical permeability,with unit of m2.K1and K2can be obtained by testing the core sample of sandstone target with a Hassler Permeameter.

4.2 CT scan of perforating sandstone target

Figure 4 shows the shape of a sandstone target perforating hole after perforation.By analysis,the perforating depth D was 560 mm,with the average perforating radius r of 12.2 mm.

This study selected the cross sections of six sandstone targets,which are located with different lengths(Ld)from the entrance of the perforating hole,as shown as Figure 5.CT scan was performed for the sandstone target after drying and for the sandstone target saturated with water to obtain the CT distributions in the different cross sections.The porosity distributions of sandstone targets in different cross sections are obtained according to Formula(21)as Karacan and Halleck(2003)

Figure 5:The radial CT scanning imageries of sandstone targets in different locations.

Where λrwis the CT number of sandstone targets with saturated water in the cross sections,λrais the CT number of sandstone targets after drying in the cross sections,λwis the CT number of water,and λais the CT number of air.The CT number of saturated water that was selected in the experiment was λw=52 H.While,the CT number of air was λa= ?1000 H.

CT scanning data of the sandstone target cross-sections were statistically analyzed to obtain the distribution of porosity on the cross sections as shown in Figure 6[Karacan and Halleck(2003);Karacan,Grader,and Halleck(2001)].Based on this porosity distribution,the permeability distribution on the cross-sections was determined according to the calculation method as proposed by Karacan,Grader,and Halleck(2001),the permeability distribution is shown in Figure 7[Karacan,Grader,and Halleck(2001)].

5 Evaluation of the porosity and permeability damage of the sandstone target

5.1 Distribution of porosity

Figure 6:The radial porosity distribution of sandstone target.

Figure 7:The radial permeability distribution of sandstone target.

Figure 8(a)and Figure 8(b)present the experimental data and numerical analysis results of the radial porosity distribution for the sandstone target cross sections located at distances of Ld=20 mm and Ld=100 mm to the inlet end of the perforating hole.The results of numerical calculation well agree with the results of experiments.

Due to the strong shear action on the inner wall of perforation hole by the impact load,the shear swelling for sandstone was formed on the surface layer of the inner wall in the perforation hole,thus,increasing the porosity of sandstone(corresponding to the experimental observation that sandstone in the internal surface of a perforation hole was fractured forming fine map cracks as shown in Figure 9.In about 3 mm from the inner wall of perforation hole,the porosity of sandstone was greater than the initial porosity.

Figure 8:The porosity distribution of sandstone perforation in the compaction zone.

Figure 9:Scanning electron microscope images.(a)Uncompacted status;(b)Micro-fracture of the inner wall in the perforation hole.

5.2 Distribution of permeability

Figure 10 shows the radial permeability distribution of sandstone corresponding to Ld=20 mm and Ld=100 mm.It is known from the experimental data and the results from the numerical analysis that the severest damage of permeability occurred on the surface of the inner wall in the perforation hole.The extent of damage declined with an increase in radial distance.

Since the map cracks were not only narrow,but also offered a greater resistance and a poorer flow ability than that by the initial pore structure,the superficial sandstone permeability of the inner wall in the perforation hole didnot follow the change law same as that by porosity.

During the experiment,the negative pressure of well bore after the completion of perforation lowered down the damage extent and improved the permeability and flow capacity for cleaning the chippings of perforation hole.This point was not considered in the numerical model.Therefore,the sandstone permeability near the perforation hole was lower in the results of numerical analysis.

5.3 Evaluation of the Extent of Damage

Based on the CFE tested data of sandstone target perforation(see Table 2),CFE=60.1%can be obtained from Formula(19)and Formula(20).Where r is 6.1 mm,R is 80 mm,?P is 20 MPa,μ is 1.7 MPa.s,and D is 560 mm.K1is 2.6 mD.K2is 3.4mD.Qcis 0.00747 ml/s and Qmis 0.00283 ml/s.

Figure 10:The radial permeability distribution of sandstone.

The extent of damage can be evaluated by combining the results of the numerical analysis and by means of quantitatively describing the porosity distribution of sandstone.

Twenty measuring points were selected along perforaction hole distance.Radial porosity and permeability data of sandstone were extracted from the corresponding positions,and solved for their average value.According to Formulas(11)and(12),the extent of damage due to perforation in a compaction zone was quantitatively evaluated.The numerical results are shown in Table 3.The curve of thickness of compaction versus perforaction hole distance,the curves of average porosity versus perforaction hole distance,and the curves of average permeability versus perforaction hole distance are ploted in Figures 11,12,and 13,respectively.

Table 3:The damage extent evaluation of perforating porosity and permeability in the compaction zone.

Table 3 shows that the average compacted thickness of perforation hole is 6.5 mm.The average porosity in the compacted zone is 0.089,and the damage extent is 16.7%.While,the average permeability in the compacted zone is 1.53 mD,the damage extent is 54.85%.According to figures 11,12 and 13,the sandstone matrix was subjected to the squeezing action of the impact load in a 400 mm range along the axis of the sandstone-perforating hole,which generated the compacted zone with a thickness of 7 mm.

Figure 11:Curve of thickness of compaction versus perforaction hole distance.

Figure 12:Curves of average porosity versus perforaction hole distance.

The deceased percentages of porosity and permeability are calculated based on the results of numerical caltionlation in Table 3.The deceased values of porosity and permeability are 16.7%and 54.85%,respectively.

Comparison with the test results of CFE provided that the error of numerical analysis for permeability damage extent was 8.7%,which further verified the reliability of the quantitative method.

According to this method,the porosity and permeability damage extents of perforation in the compacted zone under different operating conditions can be evaluated.Meanwhile,the corresponding map can be drawn.All of these can serve as a ref-erence to match the perforating charge with the stratum and to optimize the design parameters of perforation.

Figure 13:Curves of average permeability versus perforaction hole distance.

6 Conclusion

(1)A plastic-kinematics hardening nonlinear model which contained the strain rate effect was proposed to describe the mechanical behaviors of sandstone upon impact loads with high strain rate of perforation.The dynamic response of the sandstone in the compacted zone of perforation was quantitatively evaluated by numerical simulation.This work provided an important numerical simulation method for the quantitative evaluation of damage in the compacted zone.The numerical results well agreed with the experimental results,which verified the effectiveness of the method.

(2)The supposed numerical simulation model can be used to efficiently evaluate the porosity and permeability damage extents of perforation in the compacted zone under different operating conditions together with the experimental data.Therefore it can serve as a reference to match the perforating charge with the stratum and to optimize the design parameters of perforation.

(3)The results of numerical analysis show that two important mechanisms related to the damage of sandstone in the compacted zone caused by the highpressure metal jet are plastic squeezing and shear swelling.The damage extents of sandstone thickness in the compacted zone,porosity and permeability declined with the decline of pressure and load along the axial direction of the perforation hole.

(4)During the process of charge perforation,the actions of shear,friction and squeezing by the impact load caused the slippage/breaking of sandstone particles and the compaction/fragmentation of matrix[Rubin,Vorobiev,and Glenn(2000);Rbewi and Tiab(2012)].The damage mechanism of sandstone reservoir by perforation is very complex,which needs to be further researched in future.

Acknowledgement:This work was supported by the National Key Technologies Major Program of China(2011ZX05006-002).

Thanks to engineers Yungang Liu and Maiquan Zhang from the Super-high Temperatureamp;High Pressure CFE Lab of SLOF Logging Company for the help in the aspects of core essential data and CFE test of perforating target.

Also,thanks to Dr.Han Zhidong of Liver more Software Technology Corp,USA,and Dr.Jie Wu of San Ying Precision Engineering Research Center for providing the technical supports in the aspects of Dyna numerical simulation and CT analysis of sandstone.

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1Department of Engineering Mechanics,China University of Petroleum,Qingdao 266580,China

2Corresponding author:Prof.ShifengXue,sfeng@upc.edu.cn Doc.Xiuxing Zhu,zhuxx99@upc.edu.cn

3Well Logging Company,ShengliOil Engineering Co.,Ltd.,Dongying 257000,China

4Schlumberger Drilling Dynamic Solution,Beijing 100022,China

5Department of Mechanical Engineering,Lamar University,Beaumont,TX 77710,USA