Research ArticleEstimation of Sand Production Rate Using Geomechanical andHydromechanical Models

Son Tung Pham

Faculty of Geology amp Petroleum Engineering Department of Drilling amp Production EngineeringHo Chi Minh City University of Technology-Vietnam National University Ho Chi Minh City Vietnam

Correspondence should be addressed to Son Tung Pham phamsontunghcmuteduvn

Received 16 May 2017 Revised 8 August 2017 Accepted 24 August 2017 Published 17 October 2017

Academic Editor Antonio Riveiro

Copyright copy 2017 Son Tung Pham This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper aims to develop a numerical model that can be used in sand control during production phase of an oil and gaswell The model is able to predict not only the onset of sand production using critical bottom hole pressure inferred fromgeomechanical modelling but also the mass of sand produced versus time as well as the change of porosity versus space andtime using hydromechanical modelling A detailed workflow of the modelling was presented with each step of calculations Theempirical parameters were calibrated using laboratory data Then the modelling was applied in a case study of an oilfield in CuuLong basin In addition a sensitivity study of the effect of drawdown pressure was presented in this paper Moreover a comparisonbetween results of different hydromechanical models was also addressed The outcome of this paper demonstrated the possibilityof modelling the sand production mass in real cases opening a new approach in sand control in petroleum industry

1 Introduction

Sand production occurs in many oil fields across the worldand it is especially common in the porous sediments Sandproduction observed on the surface occurs as a series of threeevents that happen at downhole area (1) formation failure (2)sand erosion due to flow and (3) sand transport

(1) Formation Failure In situ stresses and pore pressure act onformation sands and under certain conditions the criteria forfailure are metThe presence of the wellbore and perforationscauses a concentration of stresses near these cavities anddeformation and failure can occur under certain well-knownconditions This criterion is bottomhole pressure that makesthe maximum effective tangential compressive stress equalor higher than the rock strength (failure criteria) significantsanding begins at some point (the onset)

(2) Sand Erosion due to Flow Damaged regions that havefailed (meeting the failure criteria) face additional stressescaused by pore pressure gradients The process of sanderosion is essential for the sand to be removed from the failedregion and then to be entrained with the fluid

(3) Sand Transport Sand erosion detaches sand grains into aperforating cavity or wellbore Some of the grains are trans-ported to the surface while others settle into the perforationtunnel or into the well hole

During and after sand production wells can sand-up andthat has different effects on the productivity At first theproductivity seems to increase due to the increase of perme-abilityHowever after awhile the sandproduced can obstructthe entrance of hydrocarbon into the wellbore especially forcased perforated wells using sand screens or gravel packDisposal of produced sand is also a significant cost associatedwith sand production Finally sand can be transported to thesurface which causes erosion of pipe lines joints chokes andvalves So if the prediction of sand production is identifiedit will help operators to manage the situation properlyand prepare suitable treatment methods for the well

This study focuses on the first and second steps ofsand production because they have the most impact andthey are not well understood The main reasons why inpetroleum industry nowadays we still do not predict themass of sand production are because of the complexity ofnumerical models (hence the lack of professional software in

HindawiAdvances in Materials Science and EngineeringVolume 2017 Article ID 2195404 10 pageshttpsdoiorg10115520172195404

2 Advances in Materials Science and Engineering

this domain) and the unavailable real data of sand productiondue to the difficulty in collecting this kind of data in the oilfield Therefore most of the studies predicting the mass ofproduced sand still stay at the laboratory step In real life inpetroleum companiesrsquo reports only Geomechanical modelsare being used to predict the onset of sand production Thispapers aims to bring the application of the Hydromechanicalmodels into a real case in petroleum industry and to combinethe use of Geomechanical model which predicts the onsetof sand production and the Hydromechanical model whichpredicts the mass of produced sand

2 Literature Review

Several studies released models predicting the onset of sandproduction and the amount of sand produced Parametersaffecting sand production have been discussed for decadesHowever there is no clear consensus In this section a briefreview ofmajor conclusions of these past studies is presented

Willson et al [1] developed a model to predict the sandproduction rate from the onset of sanding model The onsetof sanding is predicted using a stressndashbased model of shearfailure around a perforation or an open hole Sand productionis assumed to occur once the maximum value of the effectivetangential stress around the perforation exceeds the apparentrock strength (rock strength is 119887119891 sdot 155 sdot TWC strength with119887119891 is boost factor for cased perforated wells 119887119891 = 2 TWC istheThickWall Cylinder which is a measure of rockrsquos strengthand is used in sand production study instead of UnconfinedCompressive Strength because TWC reflects more closelythe reality of in situ stress sustained by the borehole or theperforation channels than the UCS) No consideration wasgiven to sand transport by drag forces The model of Willsonet al [1] requires TWC test data for hole size the same asthe perforation size If the perforation is of different size theapplication of the model must be careful Hence the 119887119891 boostfactor exists to compensate for the difference between the realdata and the test data

The criterion for sanding is

CBHP ge 31205901 minus 1205902 minus UCS2 minus 119860 minus 119901119900 119860

2 minus 119860 (1)

where CBHP is critical bottomhole pressure 1205901 is max-imum principal stress 1205902 is minimum principal stress 119901119900 ispore pressure 119860 is poroelastic constant given by 119860 = (1 minus2V)120572(1 minus V) with ] Poisson ratio and 120572 Biotrsquos constant

The model predicts the rate of sand production byutilizing the nondimensionalized concepts of Loading FactorLF (near-wellbore formation stress normalized by strength)Reynolds number (Re) and water production factor Anempirical relationship between Loading Factor Reynoldsnumber and the rate of sand production incorporating theeffect of water production was proposed as follows

SPR = 119891 (LFReWater-cut) (2)

(see [1])In this formula given by Willson et al [1] SPR is the

sand production rate water cut is the ratio of water producedcompared to the volume of total liquids produced

Although the Willson et al model [1] takes into accountthe different phases of the fluid via water cut the model didnot give clear expression of the Sand Production Rate SPR infunction of the Loading Factor LF the Reynolds number andthe water cut

In 1996 Vardoulakis et al [2] proposed the followingsand production model usingmixture theory assuming thatthe sand in place is fully degraded from the beginning andthe production is due only to the hydrodynamic forcesEquilibrium equation for solid phase is often ignored Theprocess is initiatedwith a very small solid concentration givenas a boundary condition The results are insensitive to thisvalue as long as it is small

119889120601119889119905 =

120588119904 = 120582 (1 minus 120601) 119888 10038171003817100381710038171003817119902119894fl10038171003817100381710038171003817 (3)

where 120601 is the porosity is the rate of eroded solid mass perunit volume 120588119904 is the solid density 120582 is the experimentallyevaluated sand production coefficient 119888 is the transportconcentration 119902119894fl is the specific discharge in the 119894th directionand is the notation representing the norm of a vector

The Vardoulakis model is difficult to solve because of thecomplexity of the equations Moreover the model does nottake into account the different phases of the fluid so the fluidis considered as single phase which does not reflect the realityof petroleum fluid Furthermore the coefficients were notcalibrated due to lack of experimental data

Papamichos et al [3] developed the following modelbased on the assumption that failure is due to erosion andporosity increases until it reaches unity Below is the relationbetween sand mass and the porosity

120588119904 =

120597120601120597119905 (4)

The dimension of the above equation is that is the rateof eroded solid mass per unit volume (gsdotsminus1sdotmminus3) 120588119904 is thesolid density (gsdotmminus3) 120601 is the porosity and 119905 is time in second(s)

Variation of sand mass due to erosion is given as120588119904 = 120582 (1 minus 120601) 10038171003817100381710038171003817119902119894fl10038171003817100381710038171003817 (5)

where 120582 is sand production coefficient 120582 = 120582(120576119901) and 120576119901 isplastic shear strain

Themain advantage of Papamichos et al model [3] is thatthe authors provided experimental data in order to calibratethe experimental parameters In this model not all the plasticdeformation areas will produce sand The sand is producedonly when the plastic deformation reaches a limit value(120576119901 gt 120576peak) However it is practically difficult to determinethis limit value in reality Besides the system of differentialequations of this model is extremely complex with plenty ofempirical parameters

Chin and Ramos [4] developed a sand production modelconsidering that erosionoccurs during sand production asfollows where V119904 is the solid velocity

119889120601119889119905 = (1 minus 120601) nabla sdot V119904 (6)

Advances in Materials Science and Engineering 3

The main inconvenient of this model [4] is that it wasdeveloped only for weak formation Besides the modelretains only primary physics of rock failure and coupled rockdeformation and fluid flow The model does not include theeffects of well configuration and completion wellbore stor-age erosion interaction of disaggregated solid and flowingfluid and solid transport through the porous medium

The analytical model developed by Fjaeligr et al [5] com-bined a theoretical model with an empirical relationship asshown below

120597119898sand120597119905 = 120582sand1205831 minus 1206011206013 (119902fl minus 119902crfl ) (7)

where the porosity increases with time

120601 = 1206010 [1 + 4120582sand12058312058811990412060140 (119902fl minus 119902crfl ) (119905 minus 1199050)]14

(8)

where 119902fl is the flow rate 119902crfl is the critical flow rate 1206010is the initial porosity 1199050 is the initial time and 120582sand isproportionality constant 120582sand has the dimension of sm3

Gravanis et al [6] developed a coupled stress-fluid flowerosion model

120588119904 = 120582120582119901 (119903 119905) (1 minus 120601)120573 119902fl (9)

where 120582 with dimension of inverse length represents thestrength of the erosive processes that lead to sand production120573 is an exponent parameter of themodel120582119901(119903 119905) is an erosionfunction defined as follows

120582119901 (119903 119905) = exp [minusΛ (119905)minus119886 [Γ (1 + 1119886)]119886 times (119903 minus 119903in)119886] (10)

Γ(119909) is the usual Euler Gamma function and Λ is thedepth of the plastic region The profile function 120582119901(119903 119905)approaches a step function as 119886 increases The exponent 119886 isfixed at the beginning of the analysis and can be tuned furtheras part of the calibration procedure In this work we will fix119886 = 2 which is the same value used in the work of Fjaeligr et al[5]

In 2010 Isehunwa and Olanrewaju [7] proposed a newmodel for sand production considering the effects of flowrate fluid viscosity and density grain size and cavity heightSand is produced by drag and buoyancy forces which pre-dominantly act on the sand particles The radius of sandproduction cavity is

119877119886 = 119902fl120583(49) 1198772119904120587119867120588119891119892 (11)

The volume of sand produced can be expressed as

119881sp = 1205871198772119886119867 (12)

where 119902fl is the fluid flow rate 119877119904 is the grain radius119867 is thecavity height and 119892 is the gravitational acceleration

Among these models the ones of Fjaeligr et al [5] andGravanis et al [6] were chosen for this study because of their

Pout

out

rout

Pin

rin

Λ

Figure 1 Schematic of the hollow cylinder [6]

clarity in the explanation and equations which are necessaryfor us to be able to numerically solve the problems In addi-tion these models have some advantages in comparison withothers They proposed the basic theory for hydrodynamicerosion of sandstone which is based on filtration theoryThey adopted full strength hardeningsoftening behavior ofreservoir stone They also do take into account the grain sizethe gradient elastoplasticity for thick wall cylinders and thecavity failure around boreholes

The models were solved using the workflow developedin Section 3 of this paper then the coding was made usingMATLAB software

3 Workflow for Calculation and Calibration

31 Hydromechanical Erosion Model of Gravanis et al [6]According to Gravanis et al [6] the basic assumptions areas follows

(1) Fluid flow can be described by Darcyrsquos law(2) We define the mathematical time 119879 by 119889119879 =(120582Δ119903)119860(119905)119889119905 for simple calculations 119905 is the real time

while 119879 is the mathematical time which is introducedto facilitate the resolution of the problem Δ119903 = 119903out minus119903in (Figure 1)

The function 119860(119905) is related to pressure dropΔ119901 = 119901out minus 119901in as follows119860 (119905) = Δ119901

120583int119903out119903in

(119889119903119896 (120601) 119903) (13)

119896 is the permeability

(3) The whole region is divided into a plastic region andan elastic region In elastic region we apply Hookersquoslaw and in plastic region we consider the Mohr-Coulomb failure criterion

4 Advances in Materials Science and Engineering

(4) Under the condition that plasticity of the material isdamaged and subject to decohesion it can be erodedunder weak hydrodynamic forces It happens whendrawdown pressure (DP) exceeds a critical drawdownpressure (CDP)

311 Calculation Workflow

Step 1 (at 119879 = 0)(i) Calculate function 119860(119879)

119860 (119879) = Δ119901120583int119903out119903in

(119889119903119896 (120601) 119903) (14)

where 119860(119879) is related to the flow rate 119902fl(119879) by therelation 119902fl(119879) = 2120587119867119860(119879) Δ119901 = 119901out minus 119901in

(ii) Calculate pressure 119901(119903 119879)119901 (119903 119879) = 119901in + 120583

1198960119860 (119879)int119903119903in

(1 minus 120601)21206013

119889119903119903 (15)

(iii) The depth of plastic region Λ the elastic and plasticregions are presented in Figure 1 according to Gra-vanis et al [6] In elastic region combining with theboundary condition 120590119903 elastic(119903out) = 120590out we have

120590119903 = 120590out + 11986222119866 [ 11199032out minus

11199032 ]

+ 1205722119866120582Lame + 2119866 [ 1

1199032 int119903

119903out

1199031015840119901 (1199031015840) 1198891199031015840]

120590120579 = 120590out + 11986222119866 [ 11199032out +

11199032 ]

+ 1205722119866120582Lame + 2119866 [119901 (119903) minus 1

1199032 int119903

119903out

1199031015840119901 (1199031015840) 1198891199031015840]

(16)

119903 varies from 119903in to 119903outIn plastic region we consider the Mohr-Coulombfailure criterion and combine with the boundarycondition 120590(119903in) = 120590in

120590119903 = 120590in 119903119870minus1

119903119870minus1inminus 1198780119870 minus 1 [1 minus

119903119870minus1119903119870minus1in

] minus 120572 (119870 minus 1)

sdot [119903119870minus1 int119903119903in

(1199031015840)minus119870 119901 (119903)1015840 1198891199031015840]

120590120579 = 120572119901 (1 minus 119870) + UCS + 119870[120590in 119903119870minus1

119903119870minus1in

minus 1198780119870 minus 1 (1 minus119903119870minus1119903119870minus1in

)

minus 120572 (119870 minus 1) 119903119870minus1 int119903119903in

(1199031015840)minus119870 119901 (119903)1015840 1198891199031015840]

(17)

where 1198780 is material cohesion [MPa] 120572 is Biotrsquosconstant variant from0 to 1 herewe shall set this valueto 1 meaning that we will neglect any compressibilityeffects119870 is the principal stress ratio which is equal to120590ℎ120590V with 120590ℎ being the horizontal stress and 120590V thevertical stressAt this stage there are two continuity conditions andtwo unknowns the integration constant 1198622 and thelocation of the plastic zone boundary location 119877 Theconditions are shown below

120590119903 elastic (119877) = 120590119903 plastic (119877)120590120579 elastic (119877) = 120590120579 plastic (119877) (18)

1198622 is given explicitly in terms of 119877 which is solvednumerically for 119877 [6] Note that (18) are purelyalgebraicThus the plastic region depth is determinedas Λ = 119877 minus 119903in

Step 2 (at time 119879 = 120575119879)(i) The porosity field is calculated from erosion model

equation (9)

120601 (119903 119879) = 1minus 1 minus 1206010(1 + (120573 minus 1) (1 minus 1206010)120573minus1 (Δ119903119903) int1198790 120582119901 (119903 119879) 119889119879)

1(120573minus1)

if 120573 = 1120601 (119903 119879) = 1 minus (1 minus 1206010) exp [minusΔ119903119903 int119879

0120582119901 (119903 119879) 119889119879]

if 120573 = 1

(19)

The formula involves the integral int1198790120582119901(119903 119879)119889119879

which is approximated by 120582119901(119903 0)120575119879 and 120582119901(119903 0)Λ 0 = Λ (119879 = 0) is known from the previous time stepwhere 120573 is an exponent parameter of the model andmust be tuned by calibration

(ii) Calculation of porosity and radius of sand productioncavity

120601 (119879) = Λ (119879)minus1 int119903out119903in

120582119901 (119903 119879) 120601 (119903 119879) 119889119903 (20)

Radius of sand production cavity 119903119886 is calculated fromwellbore to the distance where porosity of formationis constant and equal to initial porosity 1206010

(iii) Calculate function 119860(119879) pressure 119901(119903 119879) and theplastic region depth Λ 1 from (14) to (17)

(iv) The erosion strength is 120582

120582 (Λ) = 1205820 + (1205822 minus 1205820) Λ minus Λ0Λ 119901 minus Λ 0 Λ le Λ 1199011205822 Λ ge Λ119901

(21)

Advances in Materials Science and Engineering 5

where 1205820 1205822 andΛ 119901 aremodel parameters that mustbe tuned by calibration 1205820 is initial erosion strength[mminus1] 1205822 is maximum erosion strength [mminus1] andΛ 119901 is threshold depth of the 120582 plateau

(v) Calculate sand production mass rate

= int 119889119881 = 120588119904 sdot 120597120601120597119905 sdot 120587 sdot (1199032119886 minus 1199032in) sdot 119871 (22)

Note that this equation must be used in the next timestep along with Λ 119901

Step 3 (at the general time 119879 = 119894120575119879)(i) The porosity field is calculated from (20) and (21)

where the integral int1198790120582119901(119903 119879)119889119879 is approximated by120582119901(119903 0) + sdot sdot sdot + 120582119901(119903 (119894 minus 1)120575119879)120575119879

This involves the values of the depth Λ at all theprevious time steps which are known As explainedabove the function119860 the pressure and the new valueof the plastic region depthΛ 119894 = Λ (119879 = 119894120575Τ) can thenbe calculated

32 Analytical Model of Fjaeligr et al [5] The analytical modelis based on these assumptions

(1) The driving mechanism for continuous sand produc-tion is erosion from plastified material in the vicinityof the production cavity

(2) The sand production rate depends on (1) how muchthe well pressure is reduced below the critical sandproduction pressure (2) the fluid flow rate and thefluid viscosity and (3) the cementation of the rock

(3) The sand in place is fully degraded from the beginningand the production is due only to the hydrodynamicforces These forces are proportional to the fluidpressure drop over the volume element and thepressure drop is proportional to the fluid flow rate asspecified by Darcyrsquos law moreover the permeability isgiven by the Kozeny-Carman equation we have

120597119898sand120597119905 = 120582sand1205831 minus 1206011206013 (119902fl minus 119902crfl ) (23)

where 120583 is the fluid viscosity The proportionalityconstant 120582sand has the dimension of sm3

(4) Firstly it is required that DP gt CDP which expressesthe fact that stress induced damage of the rock is anecessary condition for sand production Secondlyit is required that 119902fl gt 119902crfl (critical fluid flow rate)which expresses that the hydrodynamic forces mustbe strong enough to uphold the erosion process

321 Calculation Workflow Following Fjaeligr et al [5] wesimply assume that the stiffness of the rock remains constantuntil the porosity has reached a critical value 120601cr At thatpoint the entire part of the rock that has been producing

Rc

Rcr

Rp

Figure 2 Illustration of the plastified zone and the sand producingzone around a cavity [8]

sand collapses and the remaining solid material in that partis produced in one burst The stresses around the cavity arethen redistributed and the situation is similar to the initialstate at 119905 = 119905119900 except the cavity has become a little larger

Step 1 (continuous sand production lt 120601cr)(i) Calculate porosity from (8)

120601 = 1206010 [1 + 4120582sand12058312058811990412060140 (119902fl minus 119902crfl ) (119905 minus 1199050)]14

(24)

(ii) Calculate volumetric sand production119881sp we assumethat sand production is related to plastification of therock so that only parts of the rock that have sufferedsome plastic deformations are in a state where sandcan be produced

119877cr = 119877119888119890(DPminusCDP)UCS asymp 119877119888 (1 + DP minus CDPUCS

)119881sp = 120587 (1198772cr minus 1198772119888) 119871 asymp 21205871198711198772119888 DP minus CDP

UCS

(25)

where 119877119888 is radius of the cavity and 119877cr is sandproduction zone (Figure 2)

(iii) Sand production mass

119872 = 120588119904 (120601 minus 1206010) 119881sp (26)

Step 2 (instantaneous sand production 120601 = 120601cr) Once theentire part of the rock that has been producing sand collapsesthe remaining solid material in that part is produced in oneburst

6 Advances in Materials Science and Engineering

Table 1 Experimental data of Papamichos et al [3]

Variable ValueCylinder internal radius 119903in [m] 001Cylinder external radius 119903out [m] 01Cylinder height119867 [m] 02Internal pore pressure 119875in [Mpa] 0External pore pressure 119875out [Mpa] 015Inner radial stress 119878in [Mpa] 0Outer radial stress 119878out [Mpa] 11Flow rate 119902fl [lmin] 05Ratio 1198861 003Youngrsquos modulus 119864 [MPa] 6750Poisson ratio ] [mdash] 019Biotrsquos ratio 120572 [mdash] 1Cohesion 119878119900 [MPa] 37Friction angle Φ [∘] 374Initial porosity 1206010 [mdash] 03Permeability 119896 [mD] 500Kozeny-Carman parameter 119896119900 [m2] 896E minus 12Solids density 120588solid [kgm3] 2640Dynamic viscosity 120583 [MPasdots] 5E09

The collapse of the sand producing zone implies that theradius of the cavity increases from 119877119888 to (1 + 1198861) sdot 119877119888 where1198861 can be found from (27)

1198861 = 119877cr119877119888 minus 1 asymp DP minus CDPUCS (27)

119877119888 997888rarr (1 + 1198861) 119877119888119877cr 997888rarr (1 + 1198861) 119877cr (28)

(i) Cumulative sand production sand mass is givenas the initial amount of solid material in the sandproducing zone that is

119872 = 119881sp120588119904 (1 minus 1206010) (29)

Now the stresses around the cavity are redistributedand the situation is the same as it was at 119905 = 119905119900 Thusthe erosion process starts a new cycle

33 Calibration of Empirical Parameters of the Models ofFjaeligr et al [5] and Gravanis et al [6] We use experimentaldata profile of Papamichos et al [3] to calibrate empiricalparameters (Table 1)

331 Calibration of Empirical Parameters of the Gravanis etal Model We have empirical parameters 120573 1205820 1205822 and Λ 119901and we slightly varied these parameters around the valuesobtained by trial and error from calibrating with test data ofPapamichos et al 2001 [3] The calibration results are shownfrom Figures 3 to 6 For each calibration the model curve iscompared with the experimental curve

007800820086

009094Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 8000 100000Time (s)

Figure 3 Sand mass produced over time while maximum erosionstrength 1205822 varies from 0078 to 094mminus1 120573 = 1 1205820 = 0421205822 andΛ 119901 = 11Λ 0

09111

1213Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 80000Time (s)

Figure 4 Sand mass produced over time while changing exponentcoefficient 120573 from 09 to 13 1205822 = 0088 1205820 = 0421205822 and Λ 119901 =11Λ 0

From the results presented from Figures 3ndash6 we choosesuitable values of empirical parameters which are the expo-nent coefficient (120573 = 1) maximum erosion strength (1205822 =0088mminus1) initial erosion strength (1205820 = 0421205822mminus1) andthreshold depth of the 120582 plateau (Λ 119901 = 11Λ 119900m) The smalldiscrepancy with the experimental data can be neglected

332 Calibration of Empirical Parameters of Fjaeligr et al ModelThe model of Fjaeligr et al has three empirical parameters 119902crfl 120601cr and 120582sand Similar to the model of Gravanis et al [6]we use experimental data of Papamichos et al 2001 [3] andthen run the model and compare with the correspondingexperimental curve to calibrate empirical parameters Resultsare presented in Figures 7 8 and 9

Figures 7 8 and 9 indicate that the erosional processgradually increases until critical porosity is reached Each stepis followed by a long period of continuous production at a lowrateWhen the porosity has reached a critical value120601cr at thatpoint the collapsed zone of the remained solids is produced

Advances in Materials Science and Engineering 7

00

03034038

042046Measured

2

4

6

8

Sand

pro

duct

ion

(g)

4000 6000 8000 100002000Time (s)

Figure 5 Sand mass produced over time while changing ratio ofinitial to maximum erosion strength 12058201205822 from 03 to 046 1205822 =0088 120573 = 1 and Λ 119901 = 11Λ 0

108111114

11712Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 80000Time (s)

Figure 6 Sand mass produced over time while changing ratio ofthreshold depth and initial plastic region Λ 119901Λ 0 from 108 to 12120573 = 1 1205822 = 0088 and 1205820 = 0421205822

and process of sand production is immediate and rapidThenthe erosion process starts on a new cycle

In fact the result of the model seems very different thanthe experimental result because Fjaerrsquos model considers astep of ldquocollapserdquo (Step 2 instantaneous sand production)but overall the sand production mass over time is matchedbetween themodel and the experimental data and that is whywe could calibrate the parameters

These results also allow us to choose suitable values ofempirical parameters which are critical porosity (120601cr = 04)critical fluid flux (119902crfl = 0001ms) and sand productioncoefficient (120582sand = 600m3s)

4 Case Study

An application for the oilfield X in Cuu Long basin usingGeomechanicalmodel ofWillson et al [1] to predict the onsetof sand production condition is now presented In additionit was combined with the Hydromechanical model of Fjaeligr etal [5] and Gravanis et al [6] to predict sand production ratethen the results of the models will be compared

Measured036038

04042

0

5

10

15

20

Sand

pro

duct

ion

(g)

10000 20000 300000Time (s)

Figure 7 Sand mass produced over time while changing criticalporosity 120601cr from 036 to 042

Measured00000800001

000012000014

0

5

10

15

20

Sand

pro

duct

ion

(g)

5000 10000 15000 20000 250000Time (s)

Figure 8 Sand mass produced over time while changing criticalfluid flux 119902crfl from 000008 to 000014ms

Measured500550

600650

0

5

10

15

20

Sand

pro

duct

ion

(g)

5000 10000 15000 20000 250000Time (s)

Figure 9 Sand mass produced over time coefficient sand produc-tion 120582sand from 500 to 650m3s

After collecting and processing data by analyzing logand other parameters input data are shown in Table 2 Thepetroleum industry for historical reasons has been landedwith a mixture of US British and SI units which is oftenreferred to by ldquooilfield unitsrdquo In Vietnam they follow this

8 Advances in Materials Science and Engineering

Table 2 Data of Well X1

Input data ValueTVD (ft) 94795119868 (degree) 52119882az (degree) 244120590] (psia) 9300120590ℎ (psia) 8670120590119867 (psia) 9448] 03UCS (psia) 2450120572 1120590hz (degree) 193120601perf (degree) 90TWC (psia) 4936119896 (mD) 60120601119900 03120583 (cP) 3119903perf = 119903in (ft) 00328119897perf (ft) 16405119864 (psia) 979004

tradition of using oilfield units which is a requirement by thegovernment hence in Table 2 we do not use SI units

41 Calculate Critical Bottomhole Pressure Using Geomechan-ical Model of Willson et al [1] The critical bottomholepressure CBHP is important information during productionphase because it indicates the lowest bottomhole pressurefor sand production to not occur For a specific well inoil field the only production data that we control is thebottomhole pressure which is adjusted using surface chokesThe pressure is controlled hence the flowrate is controlledFor this reason before predicting the sand production usingHydromechanical models of Gravanis et al and Fjaeligr et alwe firstly calculate the CBHP using Geomechanical model ofWillson et al

The testing showed that a relationship between the effec-tive in situ strength of the formation U and the TWC (Thickwall cylinder) strength relative to a specimen with an ODIDratio (OuterInner diameters) would be equivalent to

UCS = 119887119891 lowast 155 lowast TWC (30)

where 119887119891 the boost factor is often taken as 119887119891 = 2 forcasedndashhole perforated wells Equation (1) considers effect ofreservoir pressure decline [9] which can be accounted for byupdating in situ stresses while the vertical stresses are usuallykept constant

120590ℎ = 120590ℎ + 119860 sdot Δ119875120590119867 = 120590119867 + 119860 sdot Δ119875 (31)

where 119860 = 120572((1 minus 2])(1 minus ])) with 120572 is the Biotrsquos constantand ] is coefficient of Poisson Δ119875 = 119875119888 minus119875119894 with 119875119888 is currentreservoir pressure and 119875119894 is initial reservoir pressure

Sand free

Sand production

2000 4000 6000 800000

1000

2000

3000

4000

5000

6000

7000

8000

CBHP_UCS = 3160 psiaCBHP_UCS = 2450 psiaCBHP_UCS = 1950 psia

Pr (psia)

Pw = Pr

Pw

(psia

)

Figure 10 Prediction of sand production with variation of UCS(Unconfined Compressive Strength)

Figure 10 presents the bottomhole pressure 119875119908 versusreservoir pressure 119875119903 In Figure 10 we also present the CBHPwhen the UCS changes For example for UCS = 1950 psiaif reservoir pressure 119875119903 is 6000 psia the CBHP is about 1500psia if the bottomhole pressure 119875119908 is lower than this value1500 psia sand production will occur

The results showed that if UCS gets smaller the sandproduction window is greater This can be explained by thereason that the formation rocks around the wellbore areweakenedwhenUCSdecreasesThus risk of sandproductionincreases

We consider a case study with UCS = 2450 psi 119875119903 =4000 psi From Figure 10 we see that if the current reservoirpressure is 4000 psi then the critical bottomhole pressure is2020 psia For sand production to not occur the bottomholepressuremust be greater than 2020 psi Otherwise to increasethe flow rate we must increase DP and if it is greater thanCDP it means that bottomhole pressure must be lower thancritical bottomhole pressure (1000 psi) then the sand willbe produced So there is a compromise between increasingflowrate (production requirement) and sand production

42 Calculate Sand Production Mass Using the Models ofGravanis et al and Fjaeligr et al We use HydromechanicalErosion models of Fjaeligr et al [5] and Gravanis et al [6]to calculate sand production mass versus time Howeverwe do not have experimental data of Well X1 to calibrateempirical parameters so we must use empirical parametersalready calibrated in Section 3 In Figure 11 the results ofsand mass produced over time calculated by the two modelsare presented We also did a sensitivity study of the effectof drawdown pressure Drawdown pressure which is thedifference between the reservoir pressure and the bottomholepressure is the most important data that we control duringproduction phase We control the drawdown consequentlywe control the production rate The drawdown is typicallycontrolled by surface chokesThemore the choke is open the

Advances in Materials Science and Engineering 9

0

2000

4000

6000

8000

Sand

pro

duct

ion

(g)

5000 10000 15000 200000Time (s)

Gravanis_Pw = 1800 psiaGravanis_Pw = 1800 psia

= 1800 psiaFjaer_Pw

= 1500 psiaFjaer_Pw

Figure 11 Results of two models in the case study of Well X1

bigger the bottomhole pressure is and hence the smaller thedrawdown is and vice versa

Figure 11 shows that sand production mass increaseswhen drawdown pressure increases However results givenby the model of Gravanis et al are smaller than results givenby Fjaeligr et al At each bottomhole pressure sand mass givenby the model of Gravanis et al increases until it reachesconstant value On the other hand the one of Fjaeligr et alincreases continuously because the process of erosion repeatswhich causes the radius of cavity and production zone toincrease so sand production mass increases continuouslyThese twomodels showdifferent results although they use thesame physical mechanisms the fluid flow can be describedby Darcyrsquos law and the driving mechanism for continuoussand production is erosion from plastified material in thevicinity of the production cavity under hydrodynamic forcesHowever the model of Fjaeligr et al uses a step of instantaneoussand production which is not considered in the model ofGravanis et al Moreover while Gravanis et al divide thewhole region into a plastic region where Mohr-Coulombfailure criterion is considered and an elastic region whereHookersquos law is applied Fjaeligr et al consider only the plasticregion Fjaeligr et al simply assume that the stiffness of therock remains constant until the porosity has reached a criticalvalue 120601cr At that point the entire part of the rock thathas been producing sand collapses and the remaining solidmaterial in that part is produced in one burst As a result theequations used in these two models as well as the associatedexperimental parameters are quite different and eventuallyled to quite different results

Finally it is important to recall that due to the lack ofvalidation data in reality it is impossible to choose whichmodel to use in practical use At this state these results canonly be used for illustration in study and to demonstrate thepossibility of using Hydromechanical models in real casesto predict eroded sand mass over time In the future if realdata is available thesemodels can be revised and recalibratedUnfortunately until now measuring sand mass rate is tech-nically impossible in the oilfields An idea was proposed tosolve this problem the determination of real sand rate maybe based on the erosion rate of the choke However this study

has never been realized andmay constitute a new objective inour future study

5 Conclusion

This study combines Geomechanical model and the twoHydromechanical Erosion models of Fjaeligr et al [5] andGravanis et al [6] aiming to predict sand production Theproposed model can not only predict the critical pressure foronset of sand production but also estimate sand productionmass and the variation of porosity in time and space Inaddition the study also gives detailed calculation steps forthe two Hydromechanical models Although each modelhas assumptions and empirical parameters we can adjustthem if we have experimental data The results of this studywill help to make an effective production planning that canavoid sand production when reservoir pressure declinesHydromechanical model is still very new and not yet widelyused due to several reasons such as complicated calculationmethods and parameters that need to be experimentallyadjusted in practice or in the laboratoryTherefore this initialcomputational workflow using MATLAB will significantlycontribute to the development of future research in thisdomain in Petroleum Engineering

The experimental data (sand productionrsquos mass in func-tion of time) is rarely done not only in Vietnam but also inthe world (only some laboratories and authors mentionedin the paper have done this kind of experience because oftheir research in this field) Hence it is currently impossibleto obtain empirical data for Cuu Long basin (sand massin function of time) not only data in laboratory but alsoproduction data due to difficulties in collecting and mea-suring sand production rate Therefore application of thecalibrated model for Cuu Long basin in this study is destinedto demonstrate the possibility of using Hydromechanicalmodels in real cases and the result can be used for illustrationin research In the next studies when we have available datafrom Cuu Long basin we can do a revisionrecalibration ofthese models We already thought about the determinationof real sand rate data based on the erosion rate of the chokeThis idea will be the objective of our future study

Notations

CBHP Critical bottomhole pressure (psia)DP Drawdown pressure (psia)LF Loading Factor (near-wellbore

formation stress normalized bystrength)

SPR Sand Production Rate (gs)TVD True vertical depth (ft)TWC Thick wall cylinder strength (psia)UCS Unconfined compressive strength

(psia) Notation representing the norm ofa vector120572 Biotrsquos constant120573 Exponent parameter120576119901 Plastic shear strain

10 Advances in Materials Science and Engineering

Γ(119909) Usual Euler Gamma function120582 Experimentally evaluated sandproduction coefficient (1m)120582sand Proportionality constant (sm3)1205820 Initial erosion strength (mminus1)1205822 Maximum erosion strength (mminus1)Λ Depth of the plastic region (m)Λ 0 Λ at 119905 = 0 (m)Λ 119901 Threshold depth of the 120582 plateau120583 Viscosity (MPasdots)120601 Porosity120601119900 Initial porosity120601cr Critical porosity120601perf Perforation orientation from thetop of the wellbore in deviatedwells (degree)120588119904 Solid density (gm3)1205901 Maximum principal stress (psia)1205902 Minimum principal stress (psia)120590] Vertical stress (psia)120590ℎ Minimum principal horizontalstress (psia)120590119867 Maximum principal horizontalstress (psia)120590hz Minimum horizontal stressdirection (degree)

] Poisson ratio1198622 Integration constant119864 Youngrsquos modulus119868 Well inclination (degree)119896 Permeability (mD)119870 Principal stress ratio119897perf Perforation length (ft)119898sand Mass of produced sand Rate of eroded solid mass per unitvolume (gssdotm3) Rate of eroded solid mass (gs)119872 Initial amount of solid material inthe sand producing zone (g)119901119900 Pore pressure (psia)119875in Internal pore pressure (Mpa)119875out External pore pressure (Mpa)119875119908 Bottomhole pressure119875119903 Reservoir Pressure119902fl Flow rate (m3s)119902119894fl Specific flow rate in the 119894thdirection (gs)119902crfl Critical flow rate (m3s)119903perf Perforation radius (ft)119903in Cylinder internal radius (m)119903out Cylinder external radius (m)119877 Location of the plastic zoneboundary (m)119877119888 Radius of the cavity (m)119877cr Radius of the sand productionzone (m)119877119901 Radius of the plastic zone (m)

Re Reynolds number

119878119900 Material cohesion (MPa)119905 Real time119879 Mathematical time119905119900 Initial timeV119904 Solid velocity (ms)119881sp Volumetric sand production (m3)119882az Well azimuth (degree)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] S Willson Z A Moschovidis J Cameron and I Palmer ldquoNewmodel for predicting the rate of sand productionrdquo inProceedingsof the SPEISRM Rock Mechanics Conference Irving Tex USAOctober 2002

[2] I Vardoulakis M Stavropoulou and P Papanastasiou ldquoHydro-mechanical aspects of the sand production problemrdquo Transportin Porous Media vol 22 no 2 pp 225ndash244 1996

[3] E Papamichos I Vardoulakis J Tronvoll and A SkjrsteinldquoVolumetric sand production model and experimentrdquo Inter-national Journal for Numerical and Analytical Methods inGeomechanics vol 25 no 8 pp 789ndash808 2001

[4] L Y Chin and G G Ramos ldquoPredicting volumetric sandproduction in weak reservoirrdquo in Proceedings of the SPEISRMRock Mechanics Conference Irving Tex USA October 2002

[5] E Fjaeligr P Cerasi L Li andE Papamichos ldquoModeling the rate ofsand productionrdquo in Proceedings of the 6th North America RockMechanics Symposium (NARMS rsquo04) Houston Tex USA June2004 httpswwwonepetroorgconference-paperARMA-04-588

[6] E Gravanis E Sarris and P Papanastasiou ldquoHydro-mechanicalerosion models for sand productionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 39 no18 pp 2017ndash2036 2015

[7] S O Isehunwa and O M Olanrewaju ldquoA simple analyticalmodel for predicting sand production in a niger delta oil fieldrdquoInternational Journal of Engineering Science and Technology vol2 no 9 pp 4379ndash4387 2010

[8] E Fjaeligr R M Holt P Horsrud A M Raaen and R RisnesPetroleum Related Rock Mechanics Elsevier 2008

[9] K Rahman A Khaksar and T J Kayes ldquoAn integrated geome-chanical and passive sand-control approach to minimizingsanding risk from openhole and cased-and-perforated wellsrdquoSociety of Petroleum Engineers vol 25 no 2 pp 155ndash167 2010

Advances in Materials Science and Engineering 3

The main inconvenient of this model [4] is that it wasdeveloped only for weak formation Besides the modelretains only primary physics of rock failure and coupled rockdeformation and fluid flow The model does not include theeffects of well configuration and completion wellbore stor-age erosion interaction of disaggregated solid and flowingfluid and solid transport through the porous medium

The analytical model developed by Fjaeligr et al [5] com-bined a theoretical model with an empirical relationship asshown below

120597119898sand120597119905 = 120582sand1205831 minus 1206011206013 (119902fl minus 119902crfl ) (7)

where the porosity increases with time

120601 = 1206010 [1 + 4120582sand12058312058811990412060140 (119902fl minus 119902crfl ) (119905 minus 1199050)]14

(8)

where 119902fl is the flow rate 119902crfl is the critical flow rate 1206010is the initial porosity 1199050 is the initial time and 120582sand isproportionality constant 120582sand has the dimension of sm3

Gravanis et al [6] developed a coupled stress-fluid flowerosion model

120588119904 = 120582120582119901 (119903 119905) (1 minus 120601)120573 119902fl (9)

where 120582 with dimension of inverse length represents thestrength of the erosive processes that lead to sand production120573 is an exponent parameter of themodel120582119901(119903 119905) is an erosionfunction defined as follows

120582119901 (119903 119905) = exp [minusΛ (119905)minus119886 [Γ (1 + 1119886)]119886 times (119903 minus 119903in)119886] (10)

Γ(119909) is the usual Euler Gamma function and Λ is thedepth of the plastic region The profile function 120582119901(119903 119905)approaches a step function as 119886 increases The exponent 119886 isfixed at the beginning of the analysis and can be tuned furtheras part of the calibration procedure In this work we will fix119886 = 2 which is the same value used in the work of Fjaeligr et al[5]

In 2010 Isehunwa and Olanrewaju [7] proposed a newmodel for sand production considering the effects of flowrate fluid viscosity and density grain size and cavity heightSand is produced by drag and buoyancy forces which pre-dominantly act on the sand particles The radius of sandproduction cavity is

119877119886 = 119902fl120583(49) 1198772119904120587119867120588119891119892 (11)

The volume of sand produced can be expressed as

119881sp = 1205871198772119886119867 (12)

where 119902fl is the fluid flow rate 119877119904 is the grain radius119867 is thecavity height and 119892 is the gravitational acceleration

Among these models the ones of Fjaeligr et al [5] andGravanis et al [6] were chosen for this study because of their

Pout

out

rout

Pin

rin

Λ

Figure 1 Schematic of the hollow cylinder [6]

clarity in the explanation and equations which are necessaryfor us to be able to numerically solve the problems In addi-tion these models have some advantages in comparison withothers They proposed the basic theory for hydrodynamicerosion of sandstone which is based on filtration theoryThey adopted full strength hardeningsoftening behavior ofreservoir stone They also do take into account the grain sizethe gradient elastoplasticity for thick wall cylinders and thecavity failure around boreholes

The models were solved using the workflow developedin Section 3 of this paper then the coding was made usingMATLAB software

3 Workflow for Calculation and Calibration

31 Hydromechanical Erosion Model of Gravanis et al [6]According to Gravanis et al [6] the basic assumptions areas follows

(1) Fluid flow can be described by Darcyrsquos law(2) We define the mathematical time 119879 by 119889119879 =(120582Δ119903)119860(119905)119889119905 for simple calculations 119905 is the real time

while 119879 is the mathematical time which is introducedto facilitate the resolution of the problem Δ119903 = 119903out minus119903in (Figure 1)

The function 119860(119905) is related to pressure dropΔ119901 = 119901out minus 119901in as follows119860 (119905) = Δ119901

120583int119903out119903in

(119889119903119896 (120601) 119903) (13)

119896 is the permeability

(3) The whole region is divided into a plastic region andan elastic region In elastic region we apply Hookersquoslaw and in plastic region we consider the Mohr-Coulomb failure criterion

4 Advances in Materials Science and Engineering

(4) Under the condition that plasticity of the material isdamaged and subject to decohesion it can be erodedunder weak hydrodynamic forces It happens whendrawdown pressure (DP) exceeds a critical drawdownpressure (CDP)

311 Calculation Workflow

Step 1 (at 119879 = 0)(i) Calculate function 119860(119879)

119860 (119879) = Δ119901120583int119903out119903in

(119889119903119896 (120601) 119903) (14)

where 119860(119879) is related to the flow rate 119902fl(119879) by therelation 119902fl(119879) = 2120587119867119860(119879) Δ119901 = 119901out minus 119901in

(ii) Calculate pressure 119901(119903 119879)119901 (119903 119879) = 119901in + 120583

1198960119860 (119879)int119903119903in

(1 minus 120601)21206013

119889119903119903 (15)

(iii) The depth of plastic region Λ the elastic and plasticregions are presented in Figure 1 according to Gra-vanis et al [6] In elastic region combining with theboundary condition 120590119903 elastic(119903out) = 120590out we have

120590119903 = 120590out + 11986222119866 [ 11199032out minus

11199032 ]

+ 1205722119866120582Lame + 2119866 [ 1

1199032 int119903

119903out

1199031015840119901 (1199031015840) 1198891199031015840]

120590120579 = 120590out + 11986222119866 [ 11199032out +

11199032 ]

+ 1205722119866120582Lame + 2119866 [119901 (119903) minus 1

1199032 int119903

119903out

1199031015840119901 (1199031015840) 1198891199031015840]

(16)

119903 varies from 119903in to 119903outIn plastic region we consider the Mohr-Coulombfailure criterion and combine with the boundarycondition 120590(119903in) = 120590in

120590119903 = 120590in 119903119870minus1

119903119870minus1inminus 1198780119870 minus 1 [1 minus

119903119870minus1119903119870minus1in

] minus 120572 (119870 minus 1)

sdot [119903119870minus1 int119903119903in

(1199031015840)minus119870 119901 (119903)1015840 1198891199031015840]

120590120579 = 120572119901 (1 minus 119870) + UCS + 119870[120590in 119903119870minus1

119903119870minus1in

minus 1198780119870 minus 1 (1 minus119903119870minus1119903119870minus1in

)

minus 120572 (119870 minus 1) 119903119870minus1 int119903119903in

(1199031015840)minus119870 119901 (119903)1015840 1198891199031015840]

(17)

where 1198780 is material cohesion [MPa] 120572 is Biotrsquosconstant variant from0 to 1 herewe shall set this valueto 1 meaning that we will neglect any compressibilityeffects119870 is the principal stress ratio which is equal to120590ℎ120590V with 120590ℎ being the horizontal stress and 120590V thevertical stressAt this stage there are two continuity conditions andtwo unknowns the integration constant 1198622 and thelocation of the plastic zone boundary location 119877 Theconditions are shown below

120590119903 elastic (119877) = 120590119903 plastic (119877)120590120579 elastic (119877) = 120590120579 plastic (119877) (18)

1198622 is given explicitly in terms of 119877 which is solvednumerically for 119877 [6] Note that (18) are purelyalgebraicThus the plastic region depth is determinedas Λ = 119877 minus 119903in

Step 2 (at time 119879 = 120575119879)(i) The porosity field is calculated from erosion model

equation (9)

120601 (119903 119879) = 1minus 1 minus 1206010(1 + (120573 minus 1) (1 minus 1206010)120573minus1 (Δ119903119903) int1198790 120582119901 (119903 119879) 119889119879)

1(120573minus1)

if 120573 = 1120601 (119903 119879) = 1 minus (1 minus 1206010) exp [minusΔ119903119903 int119879

0120582119901 (119903 119879) 119889119879]

if 120573 = 1

(19)

The formula involves the integral int1198790120582119901(119903 119879)119889119879

which is approximated by 120582119901(119903 0)120575119879 and 120582119901(119903 0)Λ 0 = Λ (119879 = 0) is known from the previous time stepwhere 120573 is an exponent parameter of the model andmust be tuned by calibration

(ii) Calculation of porosity and radius of sand productioncavity

120601 (119879) = Λ (119879)minus1 int119903out119903in

120582119901 (119903 119879) 120601 (119903 119879) 119889119903 (20)

Radius of sand production cavity 119903119886 is calculated fromwellbore to the distance where porosity of formationis constant and equal to initial porosity 1206010

(iii) Calculate function 119860(119879) pressure 119901(119903 119879) and theplastic region depth Λ 1 from (14) to (17)

(iv) The erosion strength is 120582

120582 (Λ) = 1205820 + (1205822 minus 1205820) Λ minus Λ0Λ 119901 minus Λ 0 Λ le Λ 1199011205822 Λ ge Λ119901

(21)

Advances in Materials Science and Engineering 5

where 1205820 1205822 andΛ 119901 aremodel parameters that mustbe tuned by calibration 1205820 is initial erosion strength[mminus1] 1205822 is maximum erosion strength [mminus1] andΛ 119901 is threshold depth of the 120582 plateau

(v) Calculate sand production mass rate

= int 119889119881 = 120588119904 sdot 120597120601120597119905 sdot 120587 sdot (1199032119886 minus 1199032in) sdot 119871 (22)

Note that this equation must be used in the next timestep along with Λ 119901

Step 3 (at the general time 119879 = 119894120575119879)(i) The porosity field is calculated from (20) and (21)

where the integral int1198790120582119901(119903 119879)119889119879 is approximated by120582119901(119903 0) + sdot sdot sdot + 120582119901(119903 (119894 minus 1)120575119879)120575119879

This involves the values of the depth Λ at all theprevious time steps which are known As explainedabove the function119860 the pressure and the new valueof the plastic region depthΛ 119894 = Λ (119879 = 119894120575Τ) can thenbe calculated

32 Analytical Model of Fjaeligr et al [5] The analytical modelis based on these assumptions

(1) The driving mechanism for continuous sand produc-tion is erosion from plastified material in the vicinityof the production cavity

(2) The sand production rate depends on (1) how muchthe well pressure is reduced below the critical sandproduction pressure (2) the fluid flow rate and thefluid viscosity and (3) the cementation of the rock

(3) The sand in place is fully degraded from the beginningand the production is due only to the hydrodynamicforces These forces are proportional to the fluidpressure drop over the volume element and thepressure drop is proportional to the fluid flow rate asspecified by Darcyrsquos law moreover the permeability isgiven by the Kozeny-Carman equation we have

120597119898sand120597119905 = 120582sand1205831 minus 1206011206013 (119902fl minus 119902crfl ) (23)

where 120583 is the fluid viscosity The proportionalityconstant 120582sand has the dimension of sm3

(4) Firstly it is required that DP gt CDP which expressesthe fact that stress induced damage of the rock is anecessary condition for sand production Secondlyit is required that 119902fl gt 119902crfl (critical fluid flow rate)which expresses that the hydrodynamic forces mustbe strong enough to uphold the erosion process

321 Calculation Workflow Following Fjaeligr et al [5] wesimply assume that the stiffness of the rock remains constantuntil the porosity has reached a critical value 120601cr At thatpoint the entire part of the rock that has been producing

Rc

Rcr

Rp

Figure 2 Illustration of the plastified zone and the sand producingzone around a cavity [8]

sand collapses and the remaining solid material in that partis produced in one burst The stresses around the cavity arethen redistributed and the situation is similar to the initialstate at 119905 = 119905119900 except the cavity has become a little larger

Step 1 (continuous sand production lt 120601cr)(i) Calculate porosity from (8)

120601 = 1206010 [1 + 4120582sand12058312058811990412060140 (119902fl minus 119902crfl ) (119905 minus 1199050)]14

(24)

(ii) Calculate volumetric sand production119881sp we assumethat sand production is related to plastification of therock so that only parts of the rock that have sufferedsome plastic deformations are in a state where sandcan be produced

119877cr = 119877119888119890(DPminusCDP)UCS asymp 119877119888 (1 + DP minus CDPUCS

)119881sp = 120587 (1198772cr minus 1198772119888) 119871 asymp 21205871198711198772119888 DP minus CDP

UCS

(25)

where 119877119888 is radius of the cavity and 119877cr is sandproduction zone (Figure 2)

(iii) Sand production mass

119872 = 120588119904 (120601 minus 1206010) 119881sp (26)

Step 2 (instantaneous sand production 120601 = 120601cr) Once theentire part of the rock that has been producing sand collapsesthe remaining solid material in that part is produced in oneburst

6 Advances in Materials Science and Engineering

Table 1 Experimental data of Papamichos et al [3]

Variable ValueCylinder internal radius 119903in [m] 001Cylinder external radius 119903out [m] 01Cylinder height119867 [m] 02Internal pore pressure 119875in [Mpa] 0External pore pressure 119875out [Mpa] 015Inner radial stress 119878in [Mpa] 0Outer radial stress 119878out [Mpa] 11Flow rate 119902fl [lmin] 05Ratio 1198861 003Youngrsquos modulus 119864 [MPa] 6750Poisson ratio ] [mdash] 019Biotrsquos ratio 120572 [mdash] 1Cohesion 119878119900 [MPa] 37Friction angle Φ [∘] 374Initial porosity 1206010 [mdash] 03Permeability 119896 [mD] 500Kozeny-Carman parameter 119896119900 [m2] 896E minus 12Solids density 120588solid [kgm3] 2640Dynamic viscosity 120583 [MPasdots] 5E09

The collapse of the sand producing zone implies that theradius of the cavity increases from 119877119888 to (1 + 1198861) sdot 119877119888 where1198861 can be found from (27)

1198861 = 119877cr119877119888 minus 1 asymp DP minus CDPUCS (27)

119877119888 997888rarr (1 + 1198861) 119877119888119877cr 997888rarr (1 + 1198861) 119877cr (28)

(i) Cumulative sand production sand mass is givenas the initial amount of solid material in the sandproducing zone that is

119872 = 119881sp120588119904 (1 minus 1206010) (29)

Now the stresses around the cavity are redistributedand the situation is the same as it was at 119905 = 119905119900 Thusthe erosion process starts a new cycle

33 Calibration of Empirical Parameters of the Models ofFjaeligr et al [5] and Gravanis et al [6] We use experimentaldata profile of Papamichos et al [3] to calibrate empiricalparameters (Table 1)

331 Calibration of Empirical Parameters of the Gravanis etal Model We have empirical parameters 120573 1205820 1205822 and Λ 119901and we slightly varied these parameters around the valuesobtained by trial and error from calibrating with test data ofPapamichos et al 2001 [3] The calibration results are shownfrom Figures 3 to 6 For each calibration the model curve iscompared with the experimental curve

007800820086

009094Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 8000 100000Time (s)

Figure 3 Sand mass produced over time while maximum erosionstrength 1205822 varies from 0078 to 094mminus1 120573 = 1 1205820 = 0421205822 andΛ 119901 = 11Λ 0

09111

1213Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 80000Time (s)

Figure 4 Sand mass produced over time while changing exponentcoefficient 120573 from 09 to 13 1205822 = 0088 1205820 = 0421205822 and Λ 119901 =11Λ 0

From the results presented from Figures 3ndash6 we choosesuitable values of empirical parameters which are the expo-nent coefficient (120573 = 1) maximum erosion strength (1205822 =0088mminus1) initial erosion strength (1205820 = 0421205822mminus1) andthreshold depth of the 120582 plateau (Λ 119901 = 11Λ 119900m) The smalldiscrepancy with the experimental data can be neglected

332 Calibration of Empirical Parameters of Fjaeligr et al ModelThe model of Fjaeligr et al has three empirical parameters 119902crfl 120601cr and 120582sand Similar to the model of Gravanis et al [6]we use experimental data of Papamichos et al 2001 [3] andthen run the model and compare with the correspondingexperimental curve to calibrate empirical parameters Resultsare presented in Figures 7 8 and 9

Figures 7 8 and 9 indicate that the erosional processgradually increases until critical porosity is reached Each stepis followed by a long period of continuous production at a lowrateWhen the porosity has reached a critical value120601cr at thatpoint the collapsed zone of the remained solids is produced

Advances in Materials Science and Engineering 7

00

03034038

042046Measured

2

4

6

8

Sand

pro

duct

ion

(g)

4000 6000 8000 100002000Time (s)

Figure 5 Sand mass produced over time while changing ratio ofinitial to maximum erosion strength 12058201205822 from 03 to 046 1205822 =0088 120573 = 1 and Λ 119901 = 11Λ 0

108111114

11712Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 80000Time (s)

Figure 6 Sand mass produced over time while changing ratio ofthreshold depth and initial plastic region Λ 119901Λ 0 from 108 to 12120573 = 1 1205822 = 0088 and 1205820 = 0421205822

and process of sand production is immediate and rapidThenthe erosion process starts on a new cycle

In fact the result of the model seems very different thanthe experimental result because Fjaerrsquos model considers astep of ldquocollapserdquo (Step 2 instantaneous sand production)but overall the sand production mass over time is matchedbetween themodel and the experimental data and that is whywe could calibrate the parameters

These results also allow us to choose suitable values ofempirical parameters which are critical porosity (120601cr = 04)critical fluid flux (119902crfl = 0001ms) and sand productioncoefficient (120582sand = 600m3s)

4 Case Study

An application for the oilfield X in Cuu Long basin usingGeomechanicalmodel ofWillson et al [1] to predict the onsetof sand production condition is now presented In additionit was combined with the Hydromechanical model of Fjaeligr etal [5] and Gravanis et al [6] to predict sand production ratethen the results of the models will be compared

Measured036038

04042

0

5

10

15

20

Sand

pro

duct

ion

(g)

10000 20000 300000Time (s)

Figure 7 Sand mass produced over time while changing criticalporosity 120601cr from 036 to 042

Measured00000800001

000012000014

0

5

10

15

20

Sand

pro

duct

ion

(g)

5000 10000 15000 20000 250000Time (s)

Figure 8 Sand mass produced over time while changing criticalfluid flux 119902crfl from 000008 to 000014ms

Measured500550

600650

0

5

10

15

20

Sand

pro

duct

ion

(g)

5000 10000 15000 20000 250000Time (s)

Figure 9 Sand mass produced over time coefficient sand produc-tion 120582sand from 500 to 650m3s

After collecting and processing data by analyzing logand other parameters input data are shown in Table 2 Thepetroleum industry for historical reasons has been landedwith a mixture of US British and SI units which is oftenreferred to by ldquooilfield unitsrdquo In Vietnam they follow this

8 Advances in Materials Science and Engineering

Table 2 Data of Well X1

Input data ValueTVD (ft) 94795119868 (degree) 52119882az (degree) 244120590] (psia) 9300120590ℎ (psia) 8670120590119867 (psia) 9448] 03UCS (psia) 2450120572 1120590hz (degree) 193120601perf (degree) 90TWC (psia) 4936119896 (mD) 60120601119900 03120583 (cP) 3119903perf = 119903in (ft) 00328119897perf (ft) 16405119864 (psia) 979004

tradition of using oilfield units which is a requirement by thegovernment hence in Table 2 we do not use SI units

41 Calculate Critical Bottomhole Pressure Using Geomechan-ical Model of Willson et al [1] The critical bottomholepressure CBHP is important information during productionphase because it indicates the lowest bottomhole pressurefor sand production to not occur For a specific well inoil field the only production data that we control is thebottomhole pressure which is adjusted using surface chokesThe pressure is controlled hence the flowrate is controlledFor this reason before predicting the sand production usingHydromechanical models of Gravanis et al and Fjaeligr et alwe firstly calculate the CBHP using Geomechanical model ofWillson et al

The testing showed that a relationship between the effec-tive in situ strength of the formation U and the TWC (Thickwall cylinder) strength relative to a specimen with an ODIDratio (OuterInner diameters) would be equivalent to

UCS = 119887119891 lowast 155 lowast TWC (30)

where 119887119891 the boost factor is often taken as 119887119891 = 2 forcasedndashhole perforated wells Equation (1) considers effect ofreservoir pressure decline [9] which can be accounted for byupdating in situ stresses while the vertical stresses are usuallykept constant

120590ℎ = 120590ℎ + 119860 sdot Δ119875120590119867 = 120590119867 + 119860 sdot Δ119875 (31)

where 119860 = 120572((1 minus 2])(1 minus ])) with 120572 is the Biotrsquos constantand ] is coefficient of Poisson Δ119875 = 119875119888 minus119875119894 with 119875119888 is currentreservoir pressure and 119875119894 is initial reservoir pressure

Sand free

Sand production

2000 4000 6000 800000

1000

2000

3000

4000

5000

6000

7000

8000

CBHP_UCS = 3160 psiaCBHP_UCS = 2450 psiaCBHP_UCS = 1950 psia

Pr (psia)

Pw = Pr

Pw

(psia

)

Figure 10 Prediction of sand production with variation of UCS(Unconfined Compressive Strength)

Figure 10 presents the bottomhole pressure 119875119908 versusreservoir pressure 119875119903 In Figure 10 we also present the CBHPwhen the UCS changes For example for UCS = 1950 psiaif reservoir pressure 119875119903 is 6000 psia the CBHP is about 1500psia if the bottomhole pressure 119875119908 is lower than this value1500 psia sand production will occur

The results showed that if UCS gets smaller the sandproduction window is greater This can be explained by thereason that the formation rocks around the wellbore areweakenedwhenUCSdecreasesThus risk of sandproductionincreases

We consider a case study with UCS = 2450 psi 119875119903 =4000 psi From Figure 10 we see that if the current reservoirpressure is 4000 psi then the critical bottomhole pressure is2020 psia For sand production to not occur the bottomholepressuremust be greater than 2020 psi Otherwise to increasethe flow rate we must increase DP and if it is greater thanCDP it means that bottomhole pressure must be lower thancritical bottomhole pressure (1000 psi) then the sand willbe produced So there is a compromise between increasingflowrate (production requirement) and sand production

42 Calculate Sand Production Mass Using the Models ofGravanis et al and Fjaeligr et al We use HydromechanicalErosion models of Fjaeligr et al [5] and Gravanis et al [6]to calculate sand production mass versus time Howeverwe do not have experimental data of Well X1 to calibrateempirical parameters so we must use empirical parametersalready calibrated in Section 3 In Figure 11 the results ofsand mass produced over time calculated by the two modelsare presented We also did a sensitivity study of the effectof drawdown pressure Drawdown pressure which is thedifference between the reservoir pressure and the bottomholepressure is the most important data that we control duringproduction phase We control the drawdown consequentlywe control the production rate The drawdown is typicallycontrolled by surface chokesThemore the choke is open the

Advances in Materials Science and Engineering 9

0

2000

4000

6000

8000

Sand

pro

duct

ion

(g)

5000 10000 15000 200000Time (s)

Gravanis_Pw = 1800 psiaGravanis_Pw = 1800 psia

= 1800 psiaFjaer_Pw

= 1500 psiaFjaer_Pw

Figure 11 Results of two models in the case study of Well X1

bigger the bottomhole pressure is and hence the smaller thedrawdown is and vice versa

Figure 11 shows that sand production mass increaseswhen drawdown pressure increases However results givenby the model of Gravanis et al are smaller than results givenby Fjaeligr et al At each bottomhole pressure sand mass givenby the model of Gravanis et al increases until it reachesconstant value On the other hand the one of Fjaeligr et alincreases continuously because the process of erosion repeatswhich causes the radius of cavity and production zone toincrease so sand production mass increases continuouslyThese twomodels showdifferent results although they use thesame physical mechanisms the fluid flow can be describedby Darcyrsquos law and the driving mechanism for continuoussand production is erosion from plastified material in thevicinity of the production cavity under hydrodynamic forcesHowever the model of Fjaeligr et al uses a step of instantaneoussand production which is not considered in the model ofGravanis et al Moreover while Gravanis et al divide thewhole region into a plastic region where Mohr-Coulombfailure criterion is considered and an elastic region whereHookersquos law is applied Fjaeligr et al consider only the plasticregion Fjaeligr et al simply assume that the stiffness of therock remains constant until the porosity has reached a criticalvalue 120601cr At that point the entire part of the rock thathas been producing sand collapses and the remaining solidmaterial in that part is produced in one burst As a result theequations used in these two models as well as the associatedexperimental parameters are quite different and eventuallyled to quite different results

Finally it is important to recall that due to the lack ofvalidation data in reality it is impossible to choose whichmodel to use in practical use At this state these results canonly be used for illustration in study and to demonstrate thepossibility of using Hydromechanical models in real casesto predict eroded sand mass over time In the future if realdata is available thesemodels can be revised and recalibratedUnfortunately until now measuring sand mass rate is tech-nically impossible in the oilfields An idea was proposed tosolve this problem the determination of real sand rate maybe based on the erosion rate of the choke However this study

has never been realized andmay constitute a new objective inour future study

5 Conclusion

This study combines Geomechanical model and the twoHydromechanical Erosion models of Fjaeligr et al [5] andGravanis et al [6] aiming to predict sand production Theproposed model can not only predict the critical pressure foronset of sand production but also estimate sand productionmass and the variation of porosity in time and space Inaddition the study also gives detailed calculation steps forthe two Hydromechanical models Although each modelhas assumptions and empirical parameters we can adjustthem if we have experimental data The results of this studywill help to make an effective production planning that canavoid sand production when reservoir pressure declinesHydromechanical model is still very new and not yet widelyused due to several reasons such as complicated calculationmethods and parameters that need to be experimentallyadjusted in practice or in the laboratoryTherefore this initialcomputational workflow using MATLAB will significantlycontribute to the development of future research in thisdomain in Petroleum Engineering

The experimental data (sand productionrsquos mass in func-tion of time) is rarely done not only in Vietnam but also inthe world (only some laboratories and authors mentionedin the paper have done this kind of experience because oftheir research in this field) Hence it is currently impossibleto obtain empirical data for Cuu Long basin (sand massin function of time) not only data in laboratory but alsoproduction data due to difficulties in collecting and mea-suring sand production rate Therefore application of thecalibrated model for Cuu Long basin in this study is destinedto demonstrate the possibility of using Hydromechanicalmodels in real cases and the result can be used for illustrationin research In the next studies when we have available datafrom Cuu Long basin we can do a revisionrecalibration ofthese models We already thought about the determinationof real sand rate data based on the erosion rate of the chokeThis idea will be the objective of our future study

Notations

CBHP Critical bottomhole pressure (psia)DP Drawdown pressure (psia)LF Loading Factor (near-wellbore

formation stress normalized bystrength)

SPR Sand Production Rate (gs)TVD True vertical depth (ft)TWC Thick wall cylinder strength (psia)UCS Unconfined compressive strength

(psia) Notation representing the norm ofa vector120572 Biotrsquos constant120573 Exponent parameter120576119901 Plastic shear strain

10 Advances in Materials Science and Engineering

Γ(119909) Usual Euler Gamma function120582 Experimentally evaluated sandproduction coefficient (1m)120582sand Proportionality constant (sm3)1205820 Initial erosion strength (mminus1)1205822 Maximum erosion strength (mminus1)Λ Depth of the plastic region (m)Λ 0 Λ at 119905 = 0 (m)Λ 119901 Threshold depth of the 120582 plateau120583 Viscosity (MPasdots)120601 Porosity120601119900 Initial porosity120601cr Critical porosity120601perf Perforation orientation from thetop of the wellbore in deviatedwells (degree)120588119904 Solid density (gm3)1205901 Maximum principal stress (psia)1205902 Minimum principal stress (psia)120590] Vertical stress (psia)120590ℎ Minimum principal horizontalstress (psia)120590119867 Maximum principal horizontalstress (psia)120590hz Minimum horizontal stressdirection (degree)

] Poisson ratio1198622 Integration constant119864 Youngrsquos modulus119868 Well inclination (degree)119896 Permeability (mD)119870 Principal stress ratio119897perf Perforation length (ft)119898sand Mass of produced sand Rate of eroded solid mass per unitvolume (gssdotm3) Rate of eroded solid mass (gs)119872 Initial amount of solid material inthe sand producing zone (g)119901119900 Pore pressure (psia)119875in Internal pore pressure (Mpa)119875out External pore pressure (Mpa)119875119908 Bottomhole pressure119875119903 Reservoir Pressure119902fl Flow rate (m3s)119902119894fl Specific flow rate in the 119894thdirection (gs)119902crfl Critical flow rate (m3s)119903perf Perforation radius (ft)119903in Cylinder internal radius (m)119903out Cylinder external radius (m)119877 Location of the plastic zoneboundary (m)119877119888 Radius of the cavity (m)119877cr Radius of the sand productionzone (m)119877119901 Radius of the plastic zone (m)

Re Reynolds number

119878119900 Material cohesion (MPa)119905 Real time119879 Mathematical time119905119900 Initial timeV119904 Solid velocity (ms)119881sp Volumetric sand production (m3)119882az Well azimuth (degree)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] S Willson Z A Moschovidis J Cameron and I Palmer ldquoNewmodel for predicting the rate of sand productionrdquo inProceedingsof the SPEISRM Rock Mechanics Conference Irving Tex USAOctober 2002

[2] I Vardoulakis M Stavropoulou and P Papanastasiou ldquoHydro-mechanical aspects of the sand production problemrdquo Transportin Porous Media vol 22 no 2 pp 225ndash244 1996

[3] E Papamichos I Vardoulakis J Tronvoll and A SkjrsteinldquoVolumetric sand production model and experimentrdquo Inter-national Journal for Numerical and Analytical Methods inGeomechanics vol 25 no 8 pp 789ndash808 2001

[4] L Y Chin and G G Ramos ldquoPredicting volumetric sandproduction in weak reservoirrdquo in Proceedings of the SPEISRMRock Mechanics Conference Irving Tex USA October 2002

[5] E Fjaeligr P Cerasi L Li andE Papamichos ldquoModeling the rate ofsand productionrdquo in Proceedings of the 6th North America RockMechanics Symposium (NARMS rsquo04) Houston Tex USA June2004 httpswwwonepetroorgconference-paperARMA-04-588

[6] E Gravanis E Sarris and P Papanastasiou ldquoHydro-mechanicalerosion models for sand productionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 39 no18 pp 2017ndash2036 2015

[7] S O Isehunwa and O M Olanrewaju ldquoA simple analyticalmodel for predicting sand production in a niger delta oil fieldrdquoInternational Journal of Engineering Science and Technology vol2 no 9 pp 4379ndash4387 2010

[8] E Fjaeligr R M Holt P Horsrud A M Raaen and R RisnesPetroleum Related Rock Mechanics Elsevier 2008

[9] K Rahman A Khaksar and T J Kayes ldquoAn integrated geome-chanical and passive sand-control approach to minimizingsanding risk from openhole and cased-and-perforated wellsrdquoSociety of Petroleum Engineers vol 25 no 2 pp 155ndash167 2010

4 Advances in Materials Science and Engineering

(4) Under the condition that plasticity of the material isdamaged and subject to decohesion it can be erodedunder weak hydrodynamic forces It happens whendrawdown pressure (DP) exceeds a critical drawdownpressure (CDP)

311 Calculation Workflow

Step 1 (at 119879 = 0)(i) Calculate function 119860(119879)

119860 (119879) = Δ119901120583int119903out119903in

(119889119903119896 (120601) 119903) (14)

where 119860(119879) is related to the flow rate 119902fl(119879) by therelation 119902fl(119879) = 2120587119867119860(119879) Δ119901 = 119901out minus 119901in

(ii) Calculate pressure 119901(119903 119879)119901 (119903 119879) = 119901in + 120583

1198960119860 (119879)int119903119903in

(1 minus 120601)21206013

119889119903119903 (15)

(iii) The depth of plastic region Λ the elastic and plasticregions are presented in Figure 1 according to Gra-vanis et al [6] In elastic region combining with theboundary condition 120590119903 elastic(119903out) = 120590out we have

120590119903 = 120590out + 11986222119866 [ 11199032out minus

11199032 ]

+ 1205722119866120582Lame + 2119866 [ 1

1199032 int119903

119903out

1199031015840119901 (1199031015840) 1198891199031015840]

120590120579 = 120590out + 11986222119866 [ 11199032out +

11199032 ]

+ 1205722119866120582Lame + 2119866 [119901 (119903) minus 1

1199032 int119903

119903out

1199031015840119901 (1199031015840) 1198891199031015840]

(16)

119903 varies from 119903in to 119903outIn plastic region we consider the Mohr-Coulombfailure criterion and combine with the boundarycondition 120590(119903in) = 120590in

120590119903 = 120590in 119903119870minus1

119903119870minus1inminus 1198780119870 minus 1 [1 minus

119903119870minus1119903119870minus1in

] minus 120572 (119870 minus 1)

sdot [119903119870minus1 int119903119903in

(1199031015840)minus119870 119901 (119903)1015840 1198891199031015840]

120590120579 = 120572119901 (1 minus 119870) + UCS + 119870[120590in 119903119870minus1

119903119870minus1in

minus 1198780119870 minus 1 (1 minus119903119870minus1119903119870minus1in

)

minus 120572 (119870 minus 1) 119903119870minus1 int119903119903in

(1199031015840)minus119870 119901 (119903)1015840 1198891199031015840]

(17)

where 1198780 is material cohesion [MPa] 120572 is Biotrsquosconstant variant from0 to 1 herewe shall set this valueto 1 meaning that we will neglect any compressibilityeffects119870 is the principal stress ratio which is equal to120590ℎ120590V with 120590ℎ being the horizontal stress and 120590V thevertical stressAt this stage there are two continuity conditions andtwo unknowns the integration constant 1198622 and thelocation of the plastic zone boundary location 119877 Theconditions are shown below

120590119903 elastic (119877) = 120590119903 plastic (119877)120590120579 elastic (119877) = 120590120579 plastic (119877) (18)

1198622 is given explicitly in terms of 119877 which is solvednumerically for 119877 [6] Note that (18) are purelyalgebraicThus the plastic region depth is determinedas Λ = 119877 minus 119903in

Step 2 (at time 119879 = 120575119879)(i) The porosity field is calculated from erosion model

equation (9)

120601 (119903 119879) = 1minus 1 minus 1206010(1 + (120573 minus 1) (1 minus 1206010)120573minus1 (Δ119903119903) int1198790 120582119901 (119903 119879) 119889119879)

1(120573minus1)

if 120573 = 1120601 (119903 119879) = 1 minus (1 minus 1206010) exp [minusΔ119903119903 int119879

0120582119901 (119903 119879) 119889119879]

if 120573 = 1

(19)

The formula involves the integral int1198790120582119901(119903 119879)119889119879

which is approximated by 120582119901(119903 0)120575119879 and 120582119901(119903 0)Λ 0 = Λ (119879 = 0) is known from the previous time stepwhere 120573 is an exponent parameter of the model andmust be tuned by calibration

(ii) Calculation of porosity and radius of sand productioncavity

120601 (119879) = Λ (119879)minus1 int119903out119903in

120582119901 (119903 119879) 120601 (119903 119879) 119889119903 (20)

Radius of sand production cavity 119903119886 is calculated fromwellbore to the distance where porosity of formationis constant and equal to initial porosity 1206010

(iii) Calculate function 119860(119879) pressure 119901(119903 119879) and theplastic region depth Λ 1 from (14) to (17)

(iv) The erosion strength is 120582

120582 (Λ) = 1205820 + (1205822 minus 1205820) Λ minus Λ0Λ 119901 minus Λ 0 Λ le Λ 1199011205822 Λ ge Λ119901

(21)

Advances in Materials Science and Engineering 5

where 1205820 1205822 andΛ 119901 aremodel parameters that mustbe tuned by calibration 1205820 is initial erosion strength[mminus1] 1205822 is maximum erosion strength [mminus1] andΛ 119901 is threshold depth of the 120582 plateau

(v) Calculate sand production mass rate

= int 119889119881 = 120588119904 sdot 120597120601120597119905 sdot 120587 sdot (1199032119886 minus 1199032in) sdot 119871 (22)

Note that this equation must be used in the next timestep along with Λ 119901

Step 3 (at the general time 119879 = 119894120575119879)(i) The porosity field is calculated from (20) and (21)

where the integral int1198790120582119901(119903 119879)119889119879 is approximated by120582119901(119903 0) + sdot sdot sdot + 120582119901(119903 (119894 minus 1)120575119879)120575119879

This involves the values of the depth Λ at all theprevious time steps which are known As explainedabove the function119860 the pressure and the new valueof the plastic region depthΛ 119894 = Λ (119879 = 119894120575Τ) can thenbe calculated

32 Analytical Model of Fjaeligr et al [5] The analytical modelis based on these assumptions

(1) The driving mechanism for continuous sand produc-tion is erosion from plastified material in the vicinityof the production cavity

(2) The sand production rate depends on (1) how muchthe well pressure is reduced below the critical sandproduction pressure (2) the fluid flow rate and thefluid viscosity and (3) the cementation of the rock

(3) The sand in place is fully degraded from the beginningand the production is due only to the hydrodynamicforces These forces are proportional to the fluidpressure drop over the volume element and thepressure drop is proportional to the fluid flow rate asspecified by Darcyrsquos law moreover the permeability isgiven by the Kozeny-Carman equation we have

120597119898sand120597119905 = 120582sand1205831 minus 1206011206013 (119902fl minus 119902crfl ) (23)

where 120583 is the fluid viscosity The proportionalityconstant 120582sand has the dimension of sm3

(4) Firstly it is required that DP gt CDP which expressesthe fact that stress induced damage of the rock is anecessary condition for sand production Secondlyit is required that 119902fl gt 119902crfl (critical fluid flow rate)which expresses that the hydrodynamic forces mustbe strong enough to uphold the erosion process

321 Calculation Workflow Following Fjaeligr et al [5] wesimply assume that the stiffness of the rock remains constantuntil the porosity has reached a critical value 120601cr At thatpoint the entire part of the rock that has been producing

Rc

Rcr

Rp

Figure 2 Illustration of the plastified zone and the sand producingzone around a cavity [8]

sand collapses and the remaining solid material in that partis produced in one burst The stresses around the cavity arethen redistributed and the situation is similar to the initialstate at 119905 = 119905119900 except the cavity has become a little larger

Step 1 (continuous sand production lt 120601cr)(i) Calculate porosity from (8)

120601 = 1206010 [1 + 4120582sand12058312058811990412060140 (119902fl minus 119902crfl ) (119905 minus 1199050)]14

(24)

(ii) Calculate volumetric sand production119881sp we assumethat sand production is related to plastification of therock so that only parts of the rock that have sufferedsome plastic deformations are in a state where sandcan be produced

119877cr = 119877119888119890(DPminusCDP)UCS asymp 119877119888 (1 + DP minus CDPUCS

)119881sp = 120587 (1198772cr minus 1198772119888) 119871 asymp 21205871198711198772119888 DP minus CDP

UCS

(25)

where 119877119888 is radius of the cavity and 119877cr is sandproduction zone (Figure 2)

(iii) Sand production mass

119872 = 120588119904 (120601 minus 1206010) 119881sp (26)

Step 2 (instantaneous sand production 120601 = 120601cr) Once theentire part of the rock that has been producing sand collapsesthe remaining solid material in that part is produced in oneburst

6 Advances in Materials Science and Engineering

Table 1 Experimental data of Papamichos et al [3]

Variable ValueCylinder internal radius 119903in [m] 001Cylinder external radius 119903out [m] 01Cylinder height119867 [m] 02Internal pore pressure 119875in [Mpa] 0External pore pressure 119875out [Mpa] 015Inner radial stress 119878in [Mpa] 0Outer radial stress 119878out [Mpa] 11Flow rate 119902fl [lmin] 05Ratio 1198861 003Youngrsquos modulus 119864 [MPa] 6750Poisson ratio ] [mdash] 019Biotrsquos ratio 120572 [mdash] 1Cohesion 119878119900 [MPa] 37Friction angle Φ [∘] 374Initial porosity 1206010 [mdash] 03Permeability 119896 [mD] 500Kozeny-Carman parameter 119896119900 [m2] 896E minus 12Solids density 120588solid [kgm3] 2640Dynamic viscosity 120583 [MPasdots] 5E09

The collapse of the sand producing zone implies that theradius of the cavity increases from 119877119888 to (1 + 1198861) sdot 119877119888 where1198861 can be found from (27)

1198861 = 119877cr119877119888 minus 1 asymp DP minus CDPUCS (27)

119877119888 997888rarr (1 + 1198861) 119877119888119877cr 997888rarr (1 + 1198861) 119877cr (28)

(i) Cumulative sand production sand mass is givenas the initial amount of solid material in the sandproducing zone that is

119872 = 119881sp120588119904 (1 minus 1206010) (29)

Now the stresses around the cavity are redistributedand the situation is the same as it was at 119905 = 119905119900 Thusthe erosion process starts a new cycle

33 Calibration of Empirical Parameters of the Models ofFjaeligr et al [5] and Gravanis et al [6] We use experimentaldata profile of Papamichos et al [3] to calibrate empiricalparameters (Table 1)

331 Calibration of Empirical Parameters of the Gravanis etal Model We have empirical parameters 120573 1205820 1205822 and Λ 119901and we slightly varied these parameters around the valuesobtained by trial and error from calibrating with test data ofPapamichos et al 2001 [3] The calibration results are shownfrom Figures 3 to 6 For each calibration the model curve iscompared with the experimental curve

007800820086

009094Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 8000 100000Time (s)

Figure 3 Sand mass produced over time while maximum erosionstrength 1205822 varies from 0078 to 094mminus1 120573 = 1 1205820 = 0421205822 andΛ 119901 = 11Λ 0

09111

1213Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 80000Time (s)

Figure 4 Sand mass produced over time while changing exponentcoefficient 120573 from 09 to 13 1205822 = 0088 1205820 = 0421205822 and Λ 119901 =11Λ 0

From the results presented from Figures 3ndash6 we choosesuitable values of empirical parameters which are the expo-nent coefficient (120573 = 1) maximum erosion strength (1205822 =0088mminus1) initial erosion strength (1205820 = 0421205822mminus1) andthreshold depth of the 120582 plateau (Λ 119901 = 11Λ 119900m) The smalldiscrepancy with the experimental data can be neglected

332 Calibration of Empirical Parameters of Fjaeligr et al ModelThe model of Fjaeligr et al has three empirical parameters 119902crfl 120601cr and 120582sand Similar to the model of Gravanis et al [6]we use experimental data of Papamichos et al 2001 [3] andthen run the model and compare with the correspondingexperimental curve to calibrate empirical parameters Resultsare presented in Figures 7 8 and 9

Figures 7 8 and 9 indicate that the erosional processgradually increases until critical porosity is reached Each stepis followed by a long period of continuous production at a lowrateWhen the porosity has reached a critical value120601cr at thatpoint the collapsed zone of the remained solids is produced

Advances in Materials Science and Engineering 7

00

03034038

042046Measured

2

4

6

8

Sand

pro

duct

ion

(g)

4000 6000 8000 100002000Time (s)

Figure 5 Sand mass produced over time while changing ratio ofinitial to maximum erosion strength 12058201205822 from 03 to 046 1205822 =0088 120573 = 1 and Λ 119901 = 11Λ 0

108111114

11712Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 80000Time (s)

Figure 6 Sand mass produced over time while changing ratio ofthreshold depth and initial plastic region Λ 119901Λ 0 from 108 to 12120573 = 1 1205822 = 0088 and 1205820 = 0421205822

and process of sand production is immediate and rapidThenthe erosion process starts on a new cycle

In fact the result of the model seems very different thanthe experimental result because Fjaerrsquos model considers astep of ldquocollapserdquo (Step 2 instantaneous sand production)but overall the sand production mass over time is matchedbetween themodel and the experimental data and that is whywe could calibrate the parameters

These results also allow us to choose suitable values ofempirical parameters which are critical porosity (120601cr = 04)critical fluid flux (119902crfl = 0001ms) and sand productioncoefficient (120582sand = 600m3s)

4 Case Study

An application for the oilfield X in Cuu Long basin usingGeomechanicalmodel ofWillson et al [1] to predict the onsetof sand production condition is now presented In additionit was combined with the Hydromechanical model of Fjaeligr etal [5] and Gravanis et al [6] to predict sand production ratethen the results of the models will be compared

Measured036038

04042

0

5

10

15

20

Sand

pro

duct

ion

(g)

10000 20000 300000Time (s)

Figure 7 Sand mass produced over time while changing criticalporosity 120601cr from 036 to 042

Measured00000800001

000012000014

0

5

10

15

20

Sand

pro

duct

ion

(g)

5000 10000 15000 20000 250000Time (s)

Figure 8 Sand mass produced over time while changing criticalfluid flux 119902crfl from 000008 to 000014ms

Measured500550

600650

0

5

10

15

20

Sand

pro

duct

ion

(g)

5000 10000 15000 20000 250000Time (s)

Figure 9 Sand mass produced over time coefficient sand produc-tion 120582sand from 500 to 650m3s

After collecting and processing data by analyzing logand other parameters input data are shown in Table 2 Thepetroleum industry for historical reasons has been landedwith a mixture of US British and SI units which is oftenreferred to by ldquooilfield unitsrdquo In Vietnam they follow this

8 Advances in Materials Science and Engineering

Table 2 Data of Well X1

Input data ValueTVD (ft) 94795119868 (degree) 52119882az (degree) 244120590] (psia) 9300120590ℎ (psia) 8670120590119867 (psia) 9448] 03UCS (psia) 2450120572 1120590hz (degree) 193120601perf (degree) 90TWC (psia) 4936119896 (mD) 60120601119900 03120583 (cP) 3119903perf = 119903in (ft) 00328119897perf (ft) 16405119864 (psia) 979004

tradition of using oilfield units which is a requirement by thegovernment hence in Table 2 we do not use SI units

41 Calculate Critical Bottomhole Pressure Using Geomechan-ical Model of Willson et al [1] The critical bottomholepressure CBHP is important information during productionphase because it indicates the lowest bottomhole pressurefor sand production to not occur For a specific well inoil field the only production data that we control is thebottomhole pressure which is adjusted using surface chokesThe pressure is controlled hence the flowrate is controlledFor this reason before predicting the sand production usingHydromechanical models of Gravanis et al and Fjaeligr et alwe firstly calculate the CBHP using Geomechanical model ofWillson et al

The testing showed that a relationship between the effec-tive in situ strength of the formation U and the TWC (Thickwall cylinder) strength relative to a specimen with an ODIDratio (OuterInner diameters) would be equivalent to

UCS = 119887119891 lowast 155 lowast TWC (30)

where 119887119891 the boost factor is often taken as 119887119891 = 2 forcasedndashhole perforated wells Equation (1) considers effect ofreservoir pressure decline [9] which can be accounted for byupdating in situ stresses while the vertical stresses are usuallykept constant

120590ℎ = 120590ℎ + 119860 sdot Δ119875120590119867 = 120590119867 + 119860 sdot Δ119875 (31)

where 119860 = 120572((1 minus 2])(1 minus ])) with 120572 is the Biotrsquos constantand ] is coefficient of Poisson Δ119875 = 119875119888 minus119875119894 with 119875119888 is currentreservoir pressure and 119875119894 is initial reservoir pressure

Sand free

Sand production

2000 4000 6000 800000

1000

2000

3000

4000

5000

6000

7000

8000

CBHP_UCS = 3160 psiaCBHP_UCS = 2450 psiaCBHP_UCS = 1950 psia

Pr (psia)

Pw = Pr

Pw

(psia

)

Figure 10 Prediction of sand production with variation of UCS(Unconfined Compressive Strength)

Figure 10 presents the bottomhole pressure 119875119908 versusreservoir pressure 119875119903 In Figure 10 we also present the CBHPwhen the UCS changes For example for UCS = 1950 psiaif reservoir pressure 119875119903 is 6000 psia the CBHP is about 1500psia if the bottomhole pressure 119875119908 is lower than this value1500 psia sand production will occur

The results showed that if UCS gets smaller the sandproduction window is greater This can be explained by thereason that the formation rocks around the wellbore areweakenedwhenUCSdecreasesThus risk of sandproductionincreases

We consider a case study with UCS = 2450 psi 119875119903 =4000 psi From Figure 10 we see that if the current reservoirpressure is 4000 psi then the critical bottomhole pressure is2020 psia For sand production to not occur the bottomholepressuremust be greater than 2020 psi Otherwise to increasethe flow rate we must increase DP and if it is greater thanCDP it means that bottomhole pressure must be lower thancritical bottomhole pressure (1000 psi) then the sand willbe produced So there is a compromise between increasingflowrate (production requirement) and sand production

42 Calculate Sand Production Mass Using the Models ofGravanis et al and Fjaeligr et al We use HydromechanicalErosion models of Fjaeligr et al [5] and Gravanis et al [6]to calculate sand production mass versus time Howeverwe do not have experimental data of Well X1 to calibrateempirical parameters so we must use empirical parametersalready calibrated in Section 3 In Figure 11 the results ofsand mass produced over time calculated by the two modelsare presented We also did a sensitivity study of the effectof drawdown pressure Drawdown pressure which is thedifference between the reservoir pressure and the bottomholepressure is the most important data that we control duringproduction phase We control the drawdown consequentlywe control the production rate The drawdown is typicallycontrolled by surface chokesThemore the choke is open the

Advances in Materials Science and Engineering 9

0

2000

4000

6000

8000

Sand

pro

duct

ion

(g)

5000 10000 15000 200000Time (s)

Gravanis_Pw = 1800 psiaGravanis_Pw = 1800 psia

= 1800 psiaFjaer_Pw

= 1500 psiaFjaer_Pw

Figure 11 Results of two models in the case study of Well X1

bigger the bottomhole pressure is and hence the smaller thedrawdown is and vice versa

Figure 11 shows that sand production mass increaseswhen drawdown pressure increases However results givenby the model of Gravanis et al are smaller than results givenby Fjaeligr et al At each bottomhole pressure sand mass givenby the model of Gravanis et al increases until it reachesconstant value On the other hand the one of Fjaeligr et alincreases continuously because the process of erosion repeatswhich causes the radius of cavity and production zone toincrease so sand production mass increases continuouslyThese twomodels showdifferent results although they use thesame physical mechanisms the fluid flow can be describedby Darcyrsquos law and the driving mechanism for continuoussand production is erosion from plastified material in thevicinity of the production cavity under hydrodynamic forcesHowever the model of Fjaeligr et al uses a step of instantaneoussand production which is not considered in the model ofGravanis et al Moreover while Gravanis et al divide thewhole region into a plastic region where Mohr-Coulombfailure criterion is considered and an elastic region whereHookersquos law is applied Fjaeligr et al consider only the plasticregion Fjaeligr et al simply assume that the stiffness of therock remains constant until the porosity has reached a criticalvalue 120601cr At that point the entire part of the rock thathas been producing sand collapses and the remaining solidmaterial in that part is produced in one burst As a result theequations used in these two models as well as the associatedexperimental parameters are quite different and eventuallyled to quite different results

Finally it is important to recall that due to the lack ofvalidation data in reality it is impossible to choose whichmodel to use in practical use At this state these results canonly be used for illustration in study and to demonstrate thepossibility of using Hydromechanical models in real casesto predict eroded sand mass over time In the future if realdata is available thesemodels can be revised and recalibratedUnfortunately until now measuring sand mass rate is tech-nically impossible in the oilfields An idea was proposed tosolve this problem the determination of real sand rate maybe based on the erosion rate of the choke However this study

has never been realized andmay constitute a new objective inour future study

5 Conclusion

This study combines Geomechanical model and the twoHydromechanical Erosion models of Fjaeligr et al [5] andGravanis et al [6] aiming to predict sand production Theproposed model can not only predict the critical pressure foronset of sand production but also estimate sand productionmass and the variation of porosity in time and space Inaddition the study also gives detailed calculation steps forthe two Hydromechanical models Although each modelhas assumptions and empirical parameters we can adjustthem if we have experimental data The results of this studywill help to make an effective production planning that canavoid sand production when reservoir pressure declinesHydromechanical model is still very new and not yet widelyused due to several reasons such as complicated calculationmethods and parameters that need to be experimentallyadjusted in practice or in the laboratoryTherefore this initialcomputational workflow using MATLAB will significantlycontribute to the development of future research in thisdomain in Petroleum Engineering

The experimental data (sand productionrsquos mass in func-tion of time) is rarely done not only in Vietnam but also inthe world (only some laboratories and authors mentionedin the paper have done this kind of experience because oftheir research in this field) Hence it is currently impossibleto obtain empirical data for Cuu Long basin (sand massin function of time) not only data in laboratory but alsoproduction data due to difficulties in collecting and mea-suring sand production rate Therefore application of thecalibrated model for Cuu Long basin in this study is destinedto demonstrate the possibility of using Hydromechanicalmodels in real cases and the result can be used for illustrationin research In the next studies when we have available datafrom Cuu Long basin we can do a revisionrecalibration ofthese models We already thought about the determinationof real sand rate data based on the erosion rate of the chokeThis idea will be the objective of our future study

Notations

CBHP Critical bottomhole pressure (psia)DP Drawdown pressure (psia)LF Loading Factor (near-wellbore

formation stress normalized bystrength)

SPR Sand Production Rate (gs)TVD True vertical depth (ft)TWC Thick wall cylinder strength (psia)UCS Unconfined compressive strength

(psia) Notation representing the norm ofa vector120572 Biotrsquos constant120573 Exponent parameter120576119901 Plastic shear strain

10 Advances in Materials Science and Engineering

Γ(119909) Usual Euler Gamma function120582 Experimentally evaluated sandproduction coefficient (1m)120582sand Proportionality constant (sm3)1205820 Initial erosion strength (mminus1)1205822 Maximum erosion strength (mminus1)Λ Depth of the plastic region (m)Λ 0 Λ at 119905 = 0 (m)Λ 119901 Threshold depth of the 120582 plateau120583 Viscosity (MPasdots)120601 Porosity120601119900 Initial porosity120601cr Critical porosity120601perf Perforation orientation from thetop of the wellbore in deviatedwells (degree)120588119904 Solid density (gm3)1205901 Maximum principal stress (psia)1205902 Minimum principal stress (psia)120590] Vertical stress (psia)120590ℎ Minimum principal horizontalstress (psia)120590119867 Maximum principal horizontalstress (psia)120590hz Minimum horizontal stressdirection (degree)

] Poisson ratio1198622 Integration constant119864 Youngrsquos modulus119868 Well inclination (degree)119896 Permeability (mD)119870 Principal stress ratio119897perf Perforation length (ft)119898sand Mass of produced sand Rate of eroded solid mass per unitvolume (gssdotm3) Rate of eroded solid mass (gs)119872 Initial amount of solid material inthe sand producing zone (g)119901119900 Pore pressure (psia)119875in Internal pore pressure (Mpa)119875out External pore pressure (Mpa)119875119908 Bottomhole pressure119875119903 Reservoir Pressure119902fl Flow rate (m3s)119902119894fl Specific flow rate in the 119894thdirection (gs)119902crfl Critical flow rate (m3s)119903perf Perforation radius (ft)119903in Cylinder internal radius (m)119903out Cylinder external radius (m)119877 Location of the plastic zoneboundary (m)119877119888 Radius of the cavity (m)119877cr Radius of the sand productionzone (m)119877119901 Radius of the plastic zone (m)

Re Reynolds number

119878119900 Material cohesion (MPa)119905 Real time119879 Mathematical time119905119900 Initial timeV119904 Solid velocity (ms)119881sp Volumetric sand production (m3)119882az Well azimuth (degree)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] S Willson Z A Moschovidis J Cameron and I Palmer ldquoNewmodel for predicting the rate of sand productionrdquo inProceedingsof the SPEISRM Rock Mechanics Conference Irving Tex USAOctober 2002

[2] I Vardoulakis M Stavropoulou and P Papanastasiou ldquoHydro-mechanical aspects of the sand production problemrdquo Transportin Porous Media vol 22 no 2 pp 225ndash244 1996

[3] E Papamichos I Vardoulakis J Tronvoll and A SkjrsteinldquoVolumetric sand production model and experimentrdquo Inter-national Journal for Numerical and Analytical Methods inGeomechanics vol 25 no 8 pp 789ndash808 2001

[4] L Y Chin and G G Ramos ldquoPredicting volumetric sandproduction in weak reservoirrdquo in Proceedings of the SPEISRMRock Mechanics Conference Irving Tex USA October 2002

[5] E Fjaeligr P Cerasi L Li andE Papamichos ldquoModeling the rate ofsand productionrdquo in Proceedings of the 6th North America RockMechanics Symposium (NARMS rsquo04) Houston Tex USA June2004 httpswwwonepetroorgconference-paperARMA-04-588

[6] E Gravanis E Sarris and P Papanastasiou ldquoHydro-mechanicalerosion models for sand productionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 39 no18 pp 2017ndash2036 2015

[7] S O Isehunwa and O M Olanrewaju ldquoA simple analyticalmodel for predicting sand production in a niger delta oil fieldrdquoInternational Journal of Engineering Science and Technology vol2 no 9 pp 4379ndash4387 2010

[8] E Fjaeligr R M Holt P Horsrud A M Raaen and R RisnesPetroleum Related Rock Mechanics Elsevier 2008

[9] K Rahman A Khaksar and T J Kayes ldquoAn integrated geome-chanical and passive sand-control approach to minimizingsanding risk from openhole and cased-and-perforated wellsrdquoSociety of Petroleum Engineers vol 25 no 2 pp 155ndash167 2010

Advances in Materials Science and Engineering 5

where 1205820 1205822 andΛ 119901 aremodel parameters that mustbe tuned by calibration 1205820 is initial erosion strength[mminus1] 1205822 is maximum erosion strength [mminus1] andΛ 119901 is threshold depth of the 120582 plateau

(v) Calculate sand production mass rate

= int 119889119881 = 120588119904 sdot 120597120601120597119905 sdot 120587 sdot (1199032119886 minus 1199032in) sdot 119871 (22)

Note that this equation must be used in the next timestep along with Λ 119901

Step 3 (at the general time 119879 = 119894120575119879)(i) The porosity field is calculated from (20) and (21)

where the integral int1198790120582119901(119903 119879)119889119879 is approximated by120582119901(119903 0) + sdot sdot sdot + 120582119901(119903 (119894 minus 1)120575119879)120575119879

This involves the values of the depth Λ at all theprevious time steps which are known As explainedabove the function119860 the pressure and the new valueof the plastic region depthΛ 119894 = Λ (119879 = 119894120575Τ) can thenbe calculated

32 Analytical Model of Fjaeligr et al [5] The analytical modelis based on these assumptions

(1) The driving mechanism for continuous sand produc-tion is erosion from plastified material in the vicinityof the production cavity

(2) The sand production rate depends on (1) how muchthe well pressure is reduced below the critical sandproduction pressure (2) the fluid flow rate and thefluid viscosity and (3) the cementation of the rock

(3) The sand in place is fully degraded from the beginningand the production is due only to the hydrodynamicforces These forces are proportional to the fluidpressure drop over the volume element and thepressure drop is proportional to the fluid flow rate asspecified by Darcyrsquos law moreover the permeability isgiven by the Kozeny-Carman equation we have

120597119898sand120597119905 = 120582sand1205831 minus 1206011206013 (119902fl minus 119902crfl ) (23)

where 120583 is the fluid viscosity The proportionalityconstant 120582sand has the dimension of sm3

(4) Firstly it is required that DP gt CDP which expressesthe fact that stress induced damage of the rock is anecessary condition for sand production Secondlyit is required that 119902fl gt 119902crfl (critical fluid flow rate)which expresses that the hydrodynamic forces mustbe strong enough to uphold the erosion process

321 Calculation Workflow Following Fjaeligr et al [5] wesimply assume that the stiffness of the rock remains constantuntil the porosity has reached a critical value 120601cr At thatpoint the entire part of the rock that has been producing

Rc

Rcr

Rp

Figure 2 Illustration of the plastified zone and the sand producingzone around a cavity [8]

sand collapses and the remaining solid material in that partis produced in one burst The stresses around the cavity arethen redistributed and the situation is similar to the initialstate at 119905 = 119905119900 except the cavity has become a little larger

Step 1 (continuous sand production lt 120601cr)(i) Calculate porosity from (8)

120601 = 1206010 [1 + 4120582sand12058312058811990412060140 (119902fl minus 119902crfl ) (119905 minus 1199050)]14

(24)

(ii) Calculate volumetric sand production119881sp we assumethat sand production is related to plastification of therock so that only parts of the rock that have sufferedsome plastic deformations are in a state where sandcan be produced

119877cr = 119877119888119890(DPminusCDP)UCS asymp 119877119888 (1 + DP minus CDPUCS

)119881sp = 120587 (1198772cr minus 1198772119888) 119871 asymp 21205871198711198772119888 DP minus CDP

UCS

(25)

where 119877119888 is radius of the cavity and 119877cr is sandproduction zone (Figure 2)

(iii) Sand production mass

119872 = 120588119904 (120601 minus 1206010) 119881sp (26)

Step 2 (instantaneous sand production 120601 = 120601cr) Once theentire part of the rock that has been producing sand collapsesthe remaining solid material in that part is produced in oneburst

6 Advances in Materials Science and Engineering

Table 1 Experimental data of Papamichos et al [3]

Variable ValueCylinder internal radius 119903in [m] 001Cylinder external radius 119903out [m] 01Cylinder height119867 [m] 02Internal pore pressure 119875in [Mpa] 0External pore pressure 119875out [Mpa] 015Inner radial stress 119878in [Mpa] 0Outer radial stress 119878out [Mpa] 11Flow rate 119902fl [lmin] 05Ratio 1198861 003Youngrsquos modulus 119864 [MPa] 6750Poisson ratio ] [mdash] 019Biotrsquos ratio 120572 [mdash] 1Cohesion 119878119900 [MPa] 37Friction angle Φ [∘] 374Initial porosity 1206010 [mdash] 03Permeability 119896 [mD] 500Kozeny-Carman parameter 119896119900 [m2] 896E minus 12Solids density 120588solid [kgm3] 2640Dynamic viscosity 120583 [MPasdots] 5E09

The collapse of the sand producing zone implies that theradius of the cavity increases from 119877119888 to (1 + 1198861) sdot 119877119888 where1198861 can be found from (27)

1198861 = 119877cr119877119888 minus 1 asymp DP minus CDPUCS (27)

119877119888 997888rarr (1 + 1198861) 119877119888119877cr 997888rarr (1 + 1198861) 119877cr (28)

(i) Cumulative sand production sand mass is givenas the initial amount of solid material in the sandproducing zone that is

119872 = 119881sp120588119904 (1 minus 1206010) (29)

Now the stresses around the cavity are redistributedand the situation is the same as it was at 119905 = 119905119900 Thusthe erosion process starts a new cycle

33 Calibration of Empirical Parameters of the Models ofFjaeligr et al [5] and Gravanis et al [6] We use experimentaldata profile of Papamichos et al [3] to calibrate empiricalparameters (Table 1)

331 Calibration of Empirical Parameters of the Gravanis etal Model We have empirical parameters 120573 1205820 1205822 and Λ 119901and we slightly varied these parameters around the valuesobtained by trial and error from calibrating with test data ofPapamichos et al 2001 [3] The calibration results are shownfrom Figures 3 to 6 For each calibration the model curve iscompared with the experimental curve

007800820086

009094Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 8000 100000Time (s)

Figure 3 Sand mass produced over time while maximum erosionstrength 1205822 varies from 0078 to 094mminus1 120573 = 1 1205820 = 0421205822 andΛ 119901 = 11Λ 0

09111

1213Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 80000Time (s)

Figure 4 Sand mass produced over time while changing exponentcoefficient 120573 from 09 to 13 1205822 = 0088 1205820 = 0421205822 and Λ 119901 =11Λ 0

From the results presented from Figures 3ndash6 we choosesuitable values of empirical parameters which are the expo-nent coefficient (120573 = 1) maximum erosion strength (1205822 =0088mminus1) initial erosion strength (1205820 = 0421205822mminus1) andthreshold depth of the 120582 plateau (Λ 119901 = 11Λ 119900m) The smalldiscrepancy with the experimental data can be neglected

332 Calibration of Empirical Parameters of Fjaeligr et al ModelThe model of Fjaeligr et al has three empirical parameters 119902crfl 120601cr and 120582sand Similar to the model of Gravanis et al [6]we use experimental data of Papamichos et al 2001 [3] andthen run the model and compare with the correspondingexperimental curve to calibrate empirical parameters Resultsare presented in Figures 7 8 and 9

Figures 7 8 and 9 indicate that the erosional processgradually increases until critical porosity is reached Each stepis followed by a long period of continuous production at a lowrateWhen the porosity has reached a critical value120601cr at thatpoint the collapsed zone of the remained solids is produced

Advances in Materials Science and Engineering 7

00

03034038

042046Measured

2

4

6

8

Sand

pro

duct

ion

(g)

4000 6000 8000 100002000Time (s)

Figure 5 Sand mass produced over time while changing ratio ofinitial to maximum erosion strength 12058201205822 from 03 to 046 1205822 =0088 120573 = 1 and Λ 119901 = 11Λ 0

108111114

11712Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 80000Time (s)

Figure 6 Sand mass produced over time while changing ratio ofthreshold depth and initial plastic region Λ 119901Λ 0 from 108 to 12120573 = 1 1205822 = 0088 and 1205820 = 0421205822

and process of sand production is immediate and rapidThenthe erosion process starts on a new cycle

In fact the result of the model seems very different thanthe experimental result because Fjaerrsquos model considers astep of ldquocollapserdquo (Step 2 instantaneous sand production)but overall the sand production mass over time is matchedbetween themodel and the experimental data and that is whywe could calibrate the parameters

These results also allow us to choose suitable values ofempirical parameters which are critical porosity (120601cr = 04)critical fluid flux (119902crfl = 0001ms) and sand productioncoefficient (120582sand = 600m3s)

4 Case Study

An application for the oilfield X in Cuu Long basin usingGeomechanicalmodel ofWillson et al [1] to predict the onsetof sand production condition is now presented In additionit was combined with the Hydromechanical model of Fjaeligr etal [5] and Gravanis et al [6] to predict sand production ratethen the results of the models will be compared

Measured036038

04042

0

5

10

15

20

Sand

pro

duct

ion

(g)

10000 20000 300000Time (s)

Figure 7 Sand mass produced over time while changing criticalporosity 120601cr from 036 to 042

Measured00000800001

000012000014

0

5

10

15

20

Sand

pro

duct

ion

(g)

5000 10000 15000 20000 250000Time (s)

Figure 8 Sand mass produced over time while changing criticalfluid flux 119902crfl from 000008 to 000014ms

Measured500550

600650

0

5

10

15

20

Sand

pro

duct

ion

(g)

5000 10000 15000 20000 250000Time (s)

Figure 9 Sand mass produced over time coefficient sand produc-tion 120582sand from 500 to 650m3s

After collecting and processing data by analyzing logand other parameters input data are shown in Table 2 Thepetroleum industry for historical reasons has been landedwith a mixture of US British and SI units which is oftenreferred to by ldquooilfield unitsrdquo In Vietnam they follow this

8 Advances in Materials Science and Engineering

Table 2 Data of Well X1

Input data ValueTVD (ft) 94795119868 (degree) 52119882az (degree) 244120590] (psia) 9300120590ℎ (psia) 8670120590119867 (psia) 9448] 03UCS (psia) 2450120572 1120590hz (degree) 193120601perf (degree) 90TWC (psia) 4936119896 (mD) 60120601119900 03120583 (cP) 3119903perf = 119903in (ft) 00328119897perf (ft) 16405119864 (psia) 979004

tradition of using oilfield units which is a requirement by thegovernment hence in Table 2 we do not use SI units

41 Calculate Critical Bottomhole Pressure Using Geomechan-ical Model of Willson et al [1] The critical bottomholepressure CBHP is important information during productionphase because it indicates the lowest bottomhole pressurefor sand production to not occur For a specific well inoil field the only production data that we control is thebottomhole pressure which is adjusted using surface chokesThe pressure is controlled hence the flowrate is controlledFor this reason before predicting the sand production usingHydromechanical models of Gravanis et al and Fjaeligr et alwe firstly calculate the CBHP using Geomechanical model ofWillson et al

The testing showed that a relationship between the effec-tive in situ strength of the formation U and the TWC (Thickwall cylinder) strength relative to a specimen with an ODIDratio (OuterInner diameters) would be equivalent to

UCS = 119887119891 lowast 155 lowast TWC (30)

where 119887119891 the boost factor is often taken as 119887119891 = 2 forcasedndashhole perforated wells Equation (1) considers effect ofreservoir pressure decline [9] which can be accounted for byupdating in situ stresses while the vertical stresses are usuallykept constant

120590ℎ = 120590ℎ + 119860 sdot Δ119875120590119867 = 120590119867 + 119860 sdot Δ119875 (31)

where 119860 = 120572((1 minus 2])(1 minus ])) with 120572 is the Biotrsquos constantand ] is coefficient of Poisson Δ119875 = 119875119888 minus119875119894 with 119875119888 is currentreservoir pressure and 119875119894 is initial reservoir pressure

Sand free

Sand production

2000 4000 6000 800000

1000

2000

3000

4000

5000

6000

7000

8000

CBHP_UCS = 3160 psiaCBHP_UCS = 2450 psiaCBHP_UCS = 1950 psia

Pr (psia)

Pw = Pr

Pw

(psia

)

Figure 10 Prediction of sand production with variation of UCS(Unconfined Compressive Strength)

Figure 10 presents the bottomhole pressure 119875119908 versusreservoir pressure 119875119903 In Figure 10 we also present the CBHPwhen the UCS changes For example for UCS = 1950 psiaif reservoir pressure 119875119903 is 6000 psia the CBHP is about 1500psia if the bottomhole pressure 119875119908 is lower than this value1500 psia sand production will occur

The results showed that if UCS gets smaller the sandproduction window is greater This can be explained by thereason that the formation rocks around the wellbore areweakenedwhenUCSdecreasesThus risk of sandproductionincreases

We consider a case study with UCS = 2450 psi 119875119903 =4000 psi From Figure 10 we see that if the current reservoirpressure is 4000 psi then the critical bottomhole pressure is2020 psia For sand production to not occur the bottomholepressuremust be greater than 2020 psi Otherwise to increasethe flow rate we must increase DP and if it is greater thanCDP it means that bottomhole pressure must be lower thancritical bottomhole pressure (1000 psi) then the sand willbe produced So there is a compromise between increasingflowrate (production requirement) and sand production

42 Calculate Sand Production Mass Using the Models ofGravanis et al and Fjaeligr et al We use HydromechanicalErosion models of Fjaeligr et al [5] and Gravanis et al [6]to calculate sand production mass versus time Howeverwe do not have experimental data of Well X1 to calibrateempirical parameters so we must use empirical parametersalready calibrated in Section 3 In Figure 11 the results ofsand mass produced over time calculated by the two modelsare presented We also did a sensitivity study of the effectof drawdown pressure Drawdown pressure which is thedifference between the reservoir pressure and the bottomholepressure is the most important data that we control duringproduction phase We control the drawdown consequentlywe control the production rate The drawdown is typicallycontrolled by surface chokesThemore the choke is open the

Advances in Materials Science and Engineering 9

0

2000

4000

6000

8000

Sand

pro

duct

ion

(g)

5000 10000 15000 200000Time (s)

Gravanis_Pw = 1800 psiaGravanis_Pw = 1800 psia

= 1800 psiaFjaer_Pw

= 1500 psiaFjaer_Pw

Figure 11 Results of two models in the case study of Well X1

bigger the bottomhole pressure is and hence the smaller thedrawdown is and vice versa

Figure 11 shows that sand production mass increaseswhen drawdown pressure increases However results givenby the model of Gravanis et al are smaller than results givenby Fjaeligr et al At each bottomhole pressure sand mass givenby the model of Gravanis et al increases until it reachesconstant value On the other hand the one of Fjaeligr et alincreases continuously because the process of erosion repeatswhich causes the radius of cavity and production zone toincrease so sand production mass increases continuouslyThese twomodels showdifferent results although they use thesame physical mechanisms the fluid flow can be describedby Darcyrsquos law and the driving mechanism for continuoussand production is erosion from plastified material in thevicinity of the production cavity under hydrodynamic forcesHowever the model of Fjaeligr et al uses a step of instantaneoussand production which is not considered in the model ofGravanis et al Moreover while Gravanis et al divide thewhole region into a plastic region where Mohr-Coulombfailure criterion is considered and an elastic region whereHookersquos law is applied Fjaeligr et al consider only the plasticregion Fjaeligr et al simply assume that the stiffness of therock remains constant until the porosity has reached a criticalvalue 120601cr At that point the entire part of the rock thathas been producing sand collapses and the remaining solidmaterial in that part is produced in one burst As a result theequations used in these two models as well as the associatedexperimental parameters are quite different and eventuallyled to quite different results

Finally it is important to recall that due to the lack ofvalidation data in reality it is impossible to choose whichmodel to use in practical use At this state these results canonly be used for illustration in study and to demonstrate thepossibility of using Hydromechanical models in real casesto predict eroded sand mass over time In the future if realdata is available thesemodels can be revised and recalibratedUnfortunately until now measuring sand mass rate is tech-nically impossible in the oilfields An idea was proposed tosolve this problem the determination of real sand rate maybe based on the erosion rate of the choke However this study

has never been realized andmay constitute a new objective inour future study

5 Conclusion

This study combines Geomechanical model and the twoHydromechanical Erosion models of Fjaeligr et al [5] andGravanis et al [6] aiming to predict sand production Theproposed model can not only predict the critical pressure foronset of sand production but also estimate sand productionmass and the variation of porosity in time and space Inaddition the study also gives detailed calculation steps forthe two Hydromechanical models Although each modelhas assumptions and empirical parameters we can adjustthem if we have experimental data The results of this studywill help to make an effective production planning that canavoid sand production when reservoir pressure declinesHydromechanical model is still very new and not yet widelyused due to several reasons such as complicated calculationmethods and parameters that need to be experimentallyadjusted in practice or in the laboratoryTherefore this initialcomputational workflow using MATLAB will significantlycontribute to the development of future research in thisdomain in Petroleum Engineering

The experimental data (sand productionrsquos mass in func-tion of time) is rarely done not only in Vietnam but also inthe world (only some laboratories and authors mentionedin the paper have done this kind of experience because oftheir research in this field) Hence it is currently impossibleto obtain empirical data for Cuu Long basin (sand massin function of time) not only data in laboratory but alsoproduction data due to difficulties in collecting and mea-suring sand production rate Therefore application of thecalibrated model for Cuu Long basin in this study is destinedto demonstrate the possibility of using Hydromechanicalmodels in real cases and the result can be used for illustrationin research In the next studies when we have available datafrom Cuu Long basin we can do a revisionrecalibration ofthese models We already thought about the determinationof real sand rate data based on the erosion rate of the chokeThis idea will be the objective of our future study

Notations

CBHP Critical bottomhole pressure (psia)DP Drawdown pressure (psia)LF Loading Factor (near-wellbore

formation stress normalized bystrength)

SPR Sand Production Rate (gs)TVD True vertical depth (ft)TWC Thick wall cylinder strength (psia)UCS Unconfined compressive strength

(psia) Notation representing the norm ofa vector120572 Biotrsquos constant120573 Exponent parameter120576119901 Plastic shear strain

10 Advances in Materials Science and Engineering

Γ(119909) Usual Euler Gamma function120582 Experimentally evaluated sandproduction coefficient (1m)120582sand Proportionality constant (sm3)1205820 Initial erosion strength (mminus1)1205822 Maximum erosion strength (mminus1)Λ Depth of the plastic region (m)Λ 0 Λ at 119905 = 0 (m)Λ 119901 Threshold depth of the 120582 plateau120583 Viscosity (MPasdots)120601 Porosity120601119900 Initial porosity120601cr Critical porosity120601perf Perforation orientation from thetop of the wellbore in deviatedwells (degree)120588119904 Solid density (gm3)1205901 Maximum principal stress (psia)1205902 Minimum principal stress (psia)120590] Vertical stress (psia)120590ℎ Minimum principal horizontalstress (psia)120590119867 Maximum principal horizontalstress (psia)120590hz Minimum horizontal stressdirection (degree)

] Poisson ratio1198622 Integration constant119864 Youngrsquos modulus119868 Well inclination (degree)119896 Permeability (mD)119870 Principal stress ratio119897perf Perforation length (ft)119898sand Mass of produced sand Rate of eroded solid mass per unitvolume (gssdotm3) Rate of eroded solid mass (gs)119872 Initial amount of solid material inthe sand producing zone (g)119901119900 Pore pressure (psia)119875in Internal pore pressure (Mpa)119875out External pore pressure (Mpa)119875119908 Bottomhole pressure119875119903 Reservoir Pressure119902fl Flow rate (m3s)119902119894fl Specific flow rate in the 119894thdirection (gs)119902crfl Critical flow rate (m3s)119903perf Perforation radius (ft)119903in Cylinder internal radius (m)119903out Cylinder external radius (m)119877 Location of the plastic zoneboundary (m)119877119888 Radius of the cavity (m)119877cr Radius of the sand productionzone (m)119877119901 Radius of the plastic zone (m)

Re Reynolds number

119878119900 Material cohesion (MPa)119905 Real time119879 Mathematical time119905119900 Initial timeV119904 Solid velocity (ms)119881sp Volumetric sand production (m3)119882az Well azimuth (degree)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] S Willson Z A Moschovidis J Cameron and I Palmer ldquoNewmodel for predicting the rate of sand productionrdquo inProceedingsof the SPEISRM Rock Mechanics Conference Irving Tex USAOctober 2002

[2] I Vardoulakis M Stavropoulou and P Papanastasiou ldquoHydro-mechanical aspects of the sand production problemrdquo Transportin Porous Media vol 22 no 2 pp 225ndash244 1996

[3] E Papamichos I Vardoulakis J Tronvoll and A SkjrsteinldquoVolumetric sand production model and experimentrdquo Inter-national Journal for Numerical and Analytical Methods inGeomechanics vol 25 no 8 pp 789ndash808 2001

[4] L Y Chin and G G Ramos ldquoPredicting volumetric sandproduction in weak reservoirrdquo in Proceedings of the SPEISRMRock Mechanics Conference Irving Tex USA October 2002

[5] E Fjaeligr P Cerasi L Li andE Papamichos ldquoModeling the rate ofsand productionrdquo in Proceedings of the 6th North America RockMechanics Symposium (NARMS rsquo04) Houston Tex USA June2004 httpswwwonepetroorgconference-paperARMA-04-588

[6] E Gravanis E Sarris and P Papanastasiou ldquoHydro-mechanicalerosion models for sand productionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 39 no18 pp 2017ndash2036 2015

[7] S O Isehunwa and O M Olanrewaju ldquoA simple analyticalmodel for predicting sand production in a niger delta oil fieldrdquoInternational Journal of Engineering Science and Technology vol2 no 9 pp 4379ndash4387 2010

[8] E Fjaeligr R M Holt P Horsrud A M Raaen and R RisnesPetroleum Related Rock Mechanics Elsevier 2008

[9] K Rahman A Khaksar and T J Kayes ldquoAn integrated geome-chanical and passive sand-control approach to minimizingsanding risk from openhole and cased-and-perforated wellsrdquoSociety of Petroleum Engineers vol 25 no 2 pp 155ndash167 2010

6 Advances in Materials Science and Engineering

Table 1 Experimental data of Papamichos et al [3]

Variable ValueCylinder internal radius 119903in [m] 001Cylinder external radius 119903out [m] 01Cylinder height119867 [m] 02Internal pore pressure 119875in [Mpa] 0External pore pressure 119875out [Mpa] 015Inner radial stress 119878in [Mpa] 0Outer radial stress 119878out [Mpa] 11Flow rate 119902fl [lmin] 05Ratio 1198861 003Youngrsquos modulus 119864 [MPa] 6750Poisson ratio ] [mdash] 019Biotrsquos ratio 120572 [mdash] 1Cohesion 119878119900 [MPa] 37Friction angle Φ [∘] 374Initial porosity 1206010 [mdash] 03Permeability 119896 [mD] 500Kozeny-Carman parameter 119896119900 [m2] 896E minus 12Solids density 120588solid [kgm3] 2640Dynamic viscosity 120583 [MPasdots] 5E09

The collapse of the sand producing zone implies that theradius of the cavity increases from 119877119888 to (1 + 1198861) sdot 119877119888 where1198861 can be found from (27)

1198861 = 119877cr119877119888 minus 1 asymp DP minus CDPUCS (27)

119877119888 997888rarr (1 + 1198861) 119877119888119877cr 997888rarr (1 + 1198861) 119877cr (28)

(i) Cumulative sand production sand mass is givenas the initial amount of solid material in the sandproducing zone that is

119872 = 119881sp120588119904 (1 minus 1206010) (29)

Now the stresses around the cavity are redistributedand the situation is the same as it was at 119905 = 119905119900 Thusthe erosion process starts a new cycle

33 Calibration of Empirical Parameters of the Models ofFjaeligr et al [5] and Gravanis et al [6] We use experimentaldata profile of Papamichos et al [3] to calibrate empiricalparameters (Table 1)

331 Calibration of Empirical Parameters of the Gravanis etal Model We have empirical parameters 120573 1205820 1205822 and Λ 119901and we slightly varied these parameters around the valuesobtained by trial and error from calibrating with test data ofPapamichos et al 2001 [3] The calibration results are shownfrom Figures 3 to 6 For each calibration the model curve iscompared with the experimental curve

007800820086

009094Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 8000 100000Time (s)

Figure 3 Sand mass produced over time while maximum erosionstrength 1205822 varies from 0078 to 094mminus1 120573 = 1 1205820 = 0421205822 andΛ 119901 = 11Λ 0

09111

1213Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 80000Time (s)

Figure 4 Sand mass produced over time while changing exponentcoefficient 120573 from 09 to 13 1205822 = 0088 1205820 = 0421205822 and Λ 119901 =11Λ 0

From the results presented from Figures 3ndash6 we choosesuitable values of empirical parameters which are the expo-nent coefficient (120573 = 1) maximum erosion strength (1205822 =0088mminus1) initial erosion strength (1205820 = 0421205822mminus1) andthreshold depth of the 120582 plateau (Λ 119901 = 11Λ 119900m) The smalldiscrepancy with the experimental data can be neglected

332 Calibration of Empirical Parameters of Fjaeligr et al ModelThe model of Fjaeligr et al has three empirical parameters 119902crfl 120601cr and 120582sand Similar to the model of Gravanis et al [6]we use experimental data of Papamichos et al 2001 [3] andthen run the model and compare with the correspondingexperimental curve to calibrate empirical parameters Resultsare presented in Figures 7 8 and 9

Figures 7 8 and 9 indicate that the erosional processgradually increases until critical porosity is reached Each stepis followed by a long period of continuous production at a lowrateWhen the porosity has reached a critical value120601cr at thatpoint the collapsed zone of the remained solids is produced

Advances in Materials Science and Engineering 7

00

03034038

042046Measured

2

4

6

8

Sand

pro

duct

ion

(g)

4000 6000 8000 100002000Time (s)

Figure 5 Sand mass produced over time while changing ratio ofinitial to maximum erosion strength 12058201205822 from 03 to 046 1205822 =0088 120573 = 1 and Λ 119901 = 11Λ 0

108111114

11712Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 80000Time (s)

Figure 6 Sand mass produced over time while changing ratio ofthreshold depth and initial plastic region Λ 119901Λ 0 from 108 to 12120573 = 1 1205822 = 0088 and 1205820 = 0421205822

and process of sand production is immediate and rapidThenthe erosion process starts on a new cycle

In fact the result of the model seems very different thanthe experimental result because Fjaerrsquos model considers astep of ldquocollapserdquo (Step 2 instantaneous sand production)but overall the sand production mass over time is matchedbetween themodel and the experimental data and that is whywe could calibrate the parameters

These results also allow us to choose suitable values ofempirical parameters which are critical porosity (120601cr = 04)critical fluid flux (119902crfl = 0001ms) and sand productioncoefficient (120582sand = 600m3s)

4 Case Study

An application for the oilfield X in Cuu Long basin usingGeomechanicalmodel ofWillson et al [1] to predict the onsetof sand production condition is now presented In additionit was combined with the Hydromechanical model of Fjaeligr etal [5] and Gravanis et al [6] to predict sand production ratethen the results of the models will be compared

Measured036038

04042

0

5

10

15

20

Sand

pro

duct

ion

(g)

10000 20000 300000Time (s)

Figure 7 Sand mass produced over time while changing criticalporosity 120601cr from 036 to 042

Measured00000800001

000012000014

0

5

10

15

20

Sand

pro

duct

ion

(g)

5000 10000 15000 20000 250000Time (s)

Figure 8 Sand mass produced over time while changing criticalfluid flux 119902crfl from 000008 to 000014ms

Measured500550

600650

0

5

10

15

20

Sand

pro

duct

ion

(g)

5000 10000 15000 20000 250000Time (s)

Figure 9 Sand mass produced over time coefficient sand produc-tion 120582sand from 500 to 650m3s

After collecting and processing data by analyzing logand other parameters input data are shown in Table 2 Thepetroleum industry for historical reasons has been landedwith a mixture of US British and SI units which is oftenreferred to by ldquooilfield unitsrdquo In Vietnam they follow this

8 Advances in Materials Science and Engineering

Table 2 Data of Well X1

Input data ValueTVD (ft) 94795119868 (degree) 52119882az (degree) 244120590] (psia) 9300120590ℎ (psia) 8670120590119867 (psia) 9448] 03UCS (psia) 2450120572 1120590hz (degree) 193120601perf (degree) 90TWC (psia) 4936119896 (mD) 60120601119900 03120583 (cP) 3119903perf = 119903in (ft) 00328119897perf (ft) 16405119864 (psia) 979004

tradition of using oilfield units which is a requirement by thegovernment hence in Table 2 we do not use SI units

41 Calculate Critical Bottomhole Pressure Using Geomechan-ical Model of Willson et al [1] The critical bottomholepressure CBHP is important information during productionphase because it indicates the lowest bottomhole pressurefor sand production to not occur For a specific well inoil field the only production data that we control is thebottomhole pressure which is adjusted using surface chokesThe pressure is controlled hence the flowrate is controlledFor this reason before predicting the sand production usingHydromechanical models of Gravanis et al and Fjaeligr et alwe firstly calculate the CBHP using Geomechanical model ofWillson et al

The testing showed that a relationship between the effec-tive in situ strength of the formation U and the TWC (Thickwall cylinder) strength relative to a specimen with an ODIDratio (OuterInner diameters) would be equivalent to

UCS = 119887119891 lowast 155 lowast TWC (30)

where 119887119891 the boost factor is often taken as 119887119891 = 2 forcasedndashhole perforated wells Equation (1) considers effect ofreservoir pressure decline [9] which can be accounted for byupdating in situ stresses while the vertical stresses are usuallykept constant

120590ℎ = 120590ℎ + 119860 sdot Δ119875120590119867 = 120590119867 + 119860 sdot Δ119875 (31)

where 119860 = 120572((1 minus 2])(1 minus ])) with 120572 is the Biotrsquos constantand ] is coefficient of Poisson Δ119875 = 119875119888 minus119875119894 with 119875119888 is currentreservoir pressure and 119875119894 is initial reservoir pressure

Sand free

Sand production

2000 4000 6000 800000

1000

2000

3000

4000

5000

6000

7000

8000

CBHP_UCS = 3160 psiaCBHP_UCS = 2450 psiaCBHP_UCS = 1950 psia

Pr (psia)

Pw = Pr

Pw

(psia

)

Figure 10 Prediction of sand production with variation of UCS(Unconfined Compressive Strength)

Figure 10 presents the bottomhole pressure 119875119908 versusreservoir pressure 119875119903 In Figure 10 we also present the CBHPwhen the UCS changes For example for UCS = 1950 psiaif reservoir pressure 119875119903 is 6000 psia the CBHP is about 1500psia if the bottomhole pressure 119875119908 is lower than this value1500 psia sand production will occur

The results showed that if UCS gets smaller the sandproduction window is greater This can be explained by thereason that the formation rocks around the wellbore areweakenedwhenUCSdecreasesThus risk of sandproductionincreases

We consider a case study with UCS = 2450 psi 119875119903 =4000 psi From Figure 10 we see that if the current reservoirpressure is 4000 psi then the critical bottomhole pressure is2020 psia For sand production to not occur the bottomholepressuremust be greater than 2020 psi Otherwise to increasethe flow rate we must increase DP and if it is greater thanCDP it means that bottomhole pressure must be lower thancritical bottomhole pressure (1000 psi) then the sand willbe produced So there is a compromise between increasingflowrate (production requirement) and sand production

42 Calculate Sand Production Mass Using the Models ofGravanis et al and Fjaeligr et al We use HydromechanicalErosion models of Fjaeligr et al [5] and Gravanis et al [6]to calculate sand production mass versus time Howeverwe do not have experimental data of Well X1 to calibrateempirical parameters so we must use empirical parametersalready calibrated in Section 3 In Figure 11 the results ofsand mass produced over time calculated by the two modelsare presented We also did a sensitivity study of the effectof drawdown pressure Drawdown pressure which is thedifference between the reservoir pressure and the bottomholepressure is the most important data that we control duringproduction phase We control the drawdown consequentlywe control the production rate The drawdown is typicallycontrolled by surface chokesThemore the choke is open the

Advances in Materials Science and Engineering 9

0

2000

4000

6000

8000

Sand

pro

duct

ion

(g)

5000 10000 15000 200000Time (s)

Gravanis_Pw = 1800 psiaGravanis_Pw = 1800 psia

= 1800 psiaFjaer_Pw

= 1500 psiaFjaer_Pw

Figure 11 Results of two models in the case study of Well X1

bigger the bottomhole pressure is and hence the smaller thedrawdown is and vice versa

Figure 11 shows that sand production mass increaseswhen drawdown pressure increases However results givenby the model of Gravanis et al are smaller than results givenby Fjaeligr et al At each bottomhole pressure sand mass givenby the model of Gravanis et al increases until it reachesconstant value On the other hand the one of Fjaeligr et alincreases continuously because the process of erosion repeatswhich causes the radius of cavity and production zone toincrease so sand production mass increases continuouslyThese twomodels showdifferent results although they use thesame physical mechanisms the fluid flow can be describedby Darcyrsquos law and the driving mechanism for continuoussand production is erosion from plastified material in thevicinity of the production cavity under hydrodynamic forcesHowever the model of Fjaeligr et al uses a step of instantaneoussand production which is not considered in the model ofGravanis et al Moreover while Gravanis et al divide thewhole region into a plastic region where Mohr-Coulombfailure criterion is considered and an elastic region whereHookersquos law is applied Fjaeligr et al consider only the plasticregion Fjaeligr et al simply assume that the stiffness of therock remains constant until the porosity has reached a criticalvalue 120601cr At that point the entire part of the rock thathas been producing sand collapses and the remaining solidmaterial in that part is produced in one burst As a result theequations used in these two models as well as the associatedexperimental parameters are quite different and eventuallyled to quite different results

Finally it is important to recall that due to the lack ofvalidation data in reality it is impossible to choose whichmodel to use in practical use At this state these results canonly be used for illustration in study and to demonstrate thepossibility of using Hydromechanical models in real casesto predict eroded sand mass over time In the future if realdata is available thesemodels can be revised and recalibratedUnfortunately until now measuring sand mass rate is tech-nically impossible in the oilfields An idea was proposed tosolve this problem the determination of real sand rate maybe based on the erosion rate of the choke However this study

has never been realized andmay constitute a new objective inour future study

5 Conclusion

This study combines Geomechanical model and the twoHydromechanical Erosion models of Fjaeligr et al [5] andGravanis et al [6] aiming to predict sand production Theproposed model can not only predict the critical pressure foronset of sand production but also estimate sand productionmass and the variation of porosity in time and space Inaddition the study also gives detailed calculation steps forthe two Hydromechanical models Although each modelhas assumptions and empirical parameters we can adjustthem if we have experimental data The results of this studywill help to make an effective production planning that canavoid sand production when reservoir pressure declinesHydromechanical model is still very new and not yet widelyused due to several reasons such as complicated calculationmethods and parameters that need to be experimentallyadjusted in practice or in the laboratoryTherefore this initialcomputational workflow using MATLAB will significantlycontribute to the development of future research in thisdomain in Petroleum Engineering

The experimental data (sand productionrsquos mass in func-tion of time) is rarely done not only in Vietnam but also inthe world (only some laboratories and authors mentionedin the paper have done this kind of experience because oftheir research in this field) Hence it is currently impossibleto obtain empirical data for Cuu Long basin (sand massin function of time) not only data in laboratory but alsoproduction data due to difficulties in collecting and mea-suring sand production rate Therefore application of thecalibrated model for Cuu Long basin in this study is destinedto demonstrate the possibility of using Hydromechanicalmodels in real cases and the result can be used for illustrationin research In the next studies when we have available datafrom Cuu Long basin we can do a revisionrecalibration ofthese models We already thought about the determinationof real sand rate data based on the erosion rate of the chokeThis idea will be the objective of our future study

Notations

CBHP Critical bottomhole pressure (psia)DP Drawdown pressure (psia)LF Loading Factor (near-wellbore

formation stress normalized bystrength)

SPR Sand Production Rate (gs)TVD True vertical depth (ft)TWC Thick wall cylinder strength (psia)UCS Unconfined compressive strength

(psia) Notation representing the norm ofa vector120572 Biotrsquos constant120573 Exponent parameter120576119901 Plastic shear strain

10 Advances in Materials Science and Engineering

Γ(119909) Usual Euler Gamma function120582 Experimentally evaluated sandproduction coefficient (1m)120582sand Proportionality constant (sm3)1205820 Initial erosion strength (mminus1)1205822 Maximum erosion strength (mminus1)Λ Depth of the plastic region (m)Λ 0 Λ at 119905 = 0 (m)Λ 119901 Threshold depth of the 120582 plateau120583 Viscosity (MPasdots)120601 Porosity120601119900 Initial porosity120601cr Critical porosity120601perf Perforation orientation from thetop of the wellbore in deviatedwells (degree)120588119904 Solid density (gm3)1205901 Maximum principal stress (psia)1205902 Minimum principal stress (psia)120590] Vertical stress (psia)120590ℎ Minimum principal horizontalstress (psia)120590119867 Maximum principal horizontalstress (psia)120590hz Minimum horizontal stressdirection (degree)

] Poisson ratio1198622 Integration constant119864 Youngrsquos modulus119868 Well inclination (degree)119896 Permeability (mD)119870 Principal stress ratio119897perf Perforation length (ft)119898sand Mass of produced sand Rate of eroded solid mass per unitvolume (gssdotm3) Rate of eroded solid mass (gs)119872 Initial amount of solid material inthe sand producing zone (g)119901119900 Pore pressure (psia)119875in Internal pore pressure (Mpa)119875out External pore pressure (Mpa)119875119908 Bottomhole pressure119875119903 Reservoir Pressure119902fl Flow rate (m3s)119902119894fl Specific flow rate in the 119894thdirection (gs)119902crfl Critical flow rate (m3s)119903perf Perforation radius (ft)119903in Cylinder internal radius (m)119903out Cylinder external radius (m)119877 Location of the plastic zoneboundary (m)119877119888 Radius of the cavity (m)119877cr Radius of the sand productionzone (m)119877119901 Radius of the plastic zone (m)

Re Reynolds number

119878119900 Material cohesion (MPa)119905 Real time119879 Mathematical time119905119900 Initial timeV119904 Solid velocity (ms)119881sp Volumetric sand production (m3)119882az Well azimuth (degree)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] S Willson Z A Moschovidis J Cameron and I Palmer ldquoNewmodel for predicting the rate of sand productionrdquo inProceedingsof the SPEISRM Rock Mechanics Conference Irving Tex USAOctober 2002

[2] I Vardoulakis M Stavropoulou and P Papanastasiou ldquoHydro-mechanical aspects of the sand production problemrdquo Transportin Porous Media vol 22 no 2 pp 225ndash244 1996

[3] E Papamichos I Vardoulakis J Tronvoll and A SkjrsteinldquoVolumetric sand production model and experimentrdquo Inter-national Journal for Numerical and Analytical Methods inGeomechanics vol 25 no 8 pp 789ndash808 2001

[4] L Y Chin and G G Ramos ldquoPredicting volumetric sandproduction in weak reservoirrdquo in Proceedings of the SPEISRMRock Mechanics Conference Irving Tex USA October 2002

[5] E Fjaeligr P Cerasi L Li andE Papamichos ldquoModeling the rate ofsand productionrdquo in Proceedings of the 6th North America RockMechanics Symposium (NARMS rsquo04) Houston Tex USA June2004 httpswwwonepetroorgconference-paperARMA-04-588

[6] E Gravanis E Sarris and P Papanastasiou ldquoHydro-mechanicalerosion models for sand productionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 39 no18 pp 2017ndash2036 2015

[7] S O Isehunwa and O M Olanrewaju ldquoA simple analyticalmodel for predicting sand production in a niger delta oil fieldrdquoInternational Journal of Engineering Science and Technology vol2 no 9 pp 4379ndash4387 2010

[8] E Fjaeligr R M Holt P Horsrud A M Raaen and R RisnesPetroleum Related Rock Mechanics Elsevier 2008

[9] K Rahman A Khaksar and T J Kayes ldquoAn integrated geome-chanical and passive sand-control approach to minimizingsanding risk from openhole and cased-and-perforated wellsrdquoSociety of Petroleum Engineers vol 25 no 2 pp 155ndash167 2010

Advances in Materials Science and Engineering 7

00

03034038

042046Measured

2

4

6

8

Sand

pro

duct

ion

(g)

4000 6000 8000 100002000Time (s)

Figure 5 Sand mass produced over time while changing ratio ofinitial to maximum erosion strength 12058201205822 from 03 to 046 1205822 =0088 120573 = 1 and Λ 119901 = 11Λ 0

108111114

11712Measured

0

2

4

6

8

Sand

pro

duct

ion

(g)

2000 4000 6000 80000Time (s)

Figure 6 Sand mass produced over time while changing ratio ofthreshold depth and initial plastic region Λ 119901Λ 0 from 108 to 12120573 = 1 1205822 = 0088 and 1205820 = 0421205822

and process of sand production is immediate and rapidThenthe erosion process starts on a new cycle

In fact the result of the model seems very different thanthe experimental result because Fjaerrsquos model considers astep of ldquocollapserdquo (Step 2 instantaneous sand production)but overall the sand production mass over time is matchedbetween themodel and the experimental data and that is whywe could calibrate the parameters

These results also allow us to choose suitable values ofempirical parameters which are critical porosity (120601cr = 04)critical fluid flux (119902crfl = 0001ms) and sand productioncoefficient (120582sand = 600m3s)

4 Case Study

An application for the oilfield X in Cuu Long basin usingGeomechanicalmodel ofWillson et al [1] to predict the onsetof sand production condition is now presented In additionit was combined with the Hydromechanical model of Fjaeligr etal [5] and Gravanis et al [6] to predict sand production ratethen the results of the models will be compared

Measured036038

04042

0

5

10

15

20

Sand

pro

duct

ion

(g)

10000 20000 300000Time (s)

Figure 7 Sand mass produced over time while changing criticalporosity 120601cr from 036 to 042

Measured00000800001

000012000014

0

5

10

15

20

Sand

pro

duct

ion

(g)

5000 10000 15000 20000 250000Time (s)

Figure 8 Sand mass produced over time while changing criticalfluid flux 119902crfl from 000008 to 000014ms

Measured500550

600650

0

5

10

15

20

Sand

pro

duct

ion

(g)

5000 10000 15000 20000 250000Time (s)

Figure 9 Sand mass produced over time coefficient sand produc-tion 120582sand from 500 to 650m3s

After collecting and processing data by analyzing logand other parameters input data are shown in Table 2 Thepetroleum industry for historical reasons has been landedwith a mixture of US British and SI units which is oftenreferred to by ldquooilfield unitsrdquo In Vietnam they follow this

8 Advances in Materials Science and Engineering

Table 2 Data of Well X1

Input data ValueTVD (ft) 94795119868 (degree) 52119882az (degree) 244120590] (psia) 9300120590ℎ (psia) 8670120590119867 (psia) 9448] 03UCS (psia) 2450120572 1120590hz (degree) 193120601perf (degree) 90TWC (psia) 4936119896 (mD) 60120601119900 03120583 (cP) 3119903perf = 119903in (ft) 00328119897perf (ft) 16405119864 (psia) 979004

tradition of using oilfield units which is a requirement by thegovernment hence in Table 2 we do not use SI units

41 Calculate Critical Bottomhole Pressure Using Geomechan-ical Model of Willson et al [1] The critical bottomholepressure CBHP is important information during productionphase because it indicates the lowest bottomhole pressurefor sand production to not occur For a specific well inoil field the only production data that we control is thebottomhole pressure which is adjusted using surface chokesThe pressure is controlled hence the flowrate is controlledFor this reason before predicting the sand production usingHydromechanical models of Gravanis et al and Fjaeligr et alwe firstly calculate the CBHP using Geomechanical model ofWillson et al

The testing showed that a relationship between the effec-tive in situ strength of the formation U and the TWC (Thickwall cylinder) strength relative to a specimen with an ODIDratio (OuterInner diameters) would be equivalent to

UCS = 119887119891 lowast 155 lowast TWC (30)

where 119887119891 the boost factor is often taken as 119887119891 = 2 forcasedndashhole perforated wells Equation (1) considers effect ofreservoir pressure decline [9] which can be accounted for byupdating in situ stresses while the vertical stresses are usuallykept constant

120590ℎ = 120590ℎ + 119860 sdot Δ119875120590119867 = 120590119867 + 119860 sdot Δ119875 (31)

where 119860 = 120572((1 minus 2])(1 minus ])) with 120572 is the Biotrsquos constantand ] is coefficient of Poisson Δ119875 = 119875119888 minus119875119894 with 119875119888 is currentreservoir pressure and 119875119894 is initial reservoir pressure

Sand free

Sand production

2000 4000 6000 800000

1000

2000

3000

4000

5000

6000

7000

8000

CBHP_UCS = 3160 psiaCBHP_UCS = 2450 psiaCBHP_UCS = 1950 psia

Pr (psia)

Pw = Pr

Pw

(psia

)

Figure 10 Prediction of sand production with variation of UCS(Unconfined Compressive Strength)

Figure 10 presents the bottomhole pressure 119875119908 versusreservoir pressure 119875119903 In Figure 10 we also present the CBHPwhen the UCS changes For example for UCS = 1950 psiaif reservoir pressure 119875119903 is 6000 psia the CBHP is about 1500psia if the bottomhole pressure 119875119908 is lower than this value1500 psia sand production will occur

The results showed that if UCS gets smaller the sandproduction window is greater This can be explained by thereason that the formation rocks around the wellbore areweakenedwhenUCSdecreasesThus risk of sandproductionincreases

We consider a case study with UCS = 2450 psi 119875119903 =4000 psi From Figure 10 we see that if the current reservoirpressure is 4000 psi then the critical bottomhole pressure is2020 psia For sand production to not occur the bottomholepressuremust be greater than 2020 psi Otherwise to increasethe flow rate we must increase DP and if it is greater thanCDP it means that bottomhole pressure must be lower thancritical bottomhole pressure (1000 psi) then the sand willbe produced So there is a compromise between increasingflowrate (production requirement) and sand production

42 Calculate Sand Production Mass Using the Models ofGravanis et al and Fjaeligr et al We use HydromechanicalErosion models of Fjaeligr et al [5] and Gravanis et al [6]to calculate sand production mass versus time Howeverwe do not have experimental data of Well X1 to calibrateempirical parameters so we must use empirical parametersalready calibrated in Section 3 In Figure 11 the results ofsand mass produced over time calculated by the two modelsare presented We also did a sensitivity study of the effectof drawdown pressure Drawdown pressure which is thedifference between the reservoir pressure and the bottomholepressure is the most important data that we control duringproduction phase We control the drawdown consequentlywe control the production rate The drawdown is typicallycontrolled by surface chokesThemore the choke is open the

Advances in Materials Science and Engineering 9

0

2000

4000

6000

8000

Sand

pro

duct

ion

(g)

5000 10000 15000 200000Time (s)

Gravanis_Pw = 1800 psiaGravanis_Pw = 1800 psia

= 1800 psiaFjaer_Pw

= 1500 psiaFjaer_Pw

Figure 11 Results of two models in the case study of Well X1

bigger the bottomhole pressure is and hence the smaller thedrawdown is and vice versa

Figure 11 shows that sand production mass increaseswhen drawdown pressure increases However results givenby the model of Gravanis et al are smaller than results givenby Fjaeligr et al At each bottomhole pressure sand mass givenby the model of Gravanis et al increases until it reachesconstant value On the other hand the one of Fjaeligr et alincreases continuously because the process of erosion repeatswhich causes the radius of cavity and production zone toincrease so sand production mass increases continuouslyThese twomodels showdifferent results although they use thesame physical mechanisms the fluid flow can be describedby Darcyrsquos law and the driving mechanism for continuoussand production is erosion from plastified material in thevicinity of the production cavity under hydrodynamic forcesHowever the model of Fjaeligr et al uses a step of instantaneoussand production which is not considered in the model ofGravanis et al Moreover while Gravanis et al divide thewhole region into a plastic region where Mohr-Coulombfailure criterion is considered and an elastic region whereHookersquos law is applied Fjaeligr et al consider only the plasticregion Fjaeligr et al simply assume that the stiffness of therock remains constant until the porosity has reached a criticalvalue 120601cr At that point the entire part of the rock thathas been producing sand collapses and the remaining solidmaterial in that part is produced in one burst As a result theequations used in these two models as well as the associatedexperimental parameters are quite different and eventuallyled to quite different results

Finally it is important to recall that due to the lack ofvalidation data in reality it is impossible to choose whichmodel to use in practical use At this state these results canonly be used for illustration in study and to demonstrate thepossibility of using Hydromechanical models in real casesto predict eroded sand mass over time In the future if realdata is available thesemodels can be revised and recalibratedUnfortunately until now measuring sand mass rate is tech-nically impossible in the oilfields An idea was proposed tosolve this problem the determination of real sand rate maybe based on the erosion rate of the choke However this study

has never been realized andmay constitute a new objective inour future study

5 Conclusion

This study combines Geomechanical model and the twoHydromechanical Erosion models of Fjaeligr et al [5] andGravanis et al [6] aiming to predict sand production Theproposed model can not only predict the critical pressure foronset of sand production but also estimate sand productionmass and the variation of porosity in time and space Inaddition the study also gives detailed calculation steps forthe two Hydromechanical models Although each modelhas assumptions and empirical parameters we can adjustthem if we have experimental data The results of this studywill help to make an effective production planning that canavoid sand production when reservoir pressure declinesHydromechanical model is still very new and not yet widelyused due to several reasons such as complicated calculationmethods and parameters that need to be experimentallyadjusted in practice or in the laboratoryTherefore this initialcomputational workflow using MATLAB will significantlycontribute to the development of future research in thisdomain in Petroleum Engineering

The experimental data (sand productionrsquos mass in func-tion of time) is rarely done not only in Vietnam but also inthe world (only some laboratories and authors mentionedin the paper have done this kind of experience because oftheir research in this field) Hence it is currently impossibleto obtain empirical data for Cuu Long basin (sand massin function of time) not only data in laboratory but alsoproduction data due to difficulties in collecting and mea-suring sand production rate Therefore application of thecalibrated model for Cuu Long basin in this study is destinedto demonstrate the possibility of using Hydromechanicalmodels in real cases and the result can be used for illustrationin research In the next studies when we have available datafrom Cuu Long basin we can do a revisionrecalibration ofthese models We already thought about the determinationof real sand rate data based on the erosion rate of the chokeThis idea will be the objective of our future study

Notations

CBHP Critical bottomhole pressure (psia)DP Drawdown pressure (psia)LF Loading Factor (near-wellbore

formation stress normalized bystrength)

SPR Sand Production Rate (gs)TVD True vertical depth (ft)TWC Thick wall cylinder strength (psia)UCS Unconfined compressive strength

(psia) Notation representing the norm ofa vector120572 Biotrsquos constant120573 Exponent parameter120576119901 Plastic shear strain

10 Advances in Materials Science and Engineering

Γ(119909) Usual Euler Gamma function120582 Experimentally evaluated sandproduction coefficient (1m)120582sand Proportionality constant (sm3)1205820 Initial erosion strength (mminus1)1205822 Maximum erosion strength (mminus1)Λ Depth of the plastic region (m)Λ 0 Λ at 119905 = 0 (m)Λ 119901 Threshold depth of the 120582 plateau120583 Viscosity (MPasdots)120601 Porosity120601119900 Initial porosity120601cr Critical porosity120601perf Perforation orientation from thetop of the wellbore in deviatedwells (degree)120588119904 Solid density (gm3)1205901 Maximum principal stress (psia)1205902 Minimum principal stress (psia)120590] Vertical stress (psia)120590ℎ Minimum principal horizontalstress (psia)120590119867 Maximum principal horizontalstress (psia)120590hz Minimum horizontal stressdirection (degree)

] Poisson ratio1198622 Integration constant119864 Youngrsquos modulus119868 Well inclination (degree)119896 Permeability (mD)119870 Principal stress ratio119897perf Perforation length (ft)119898sand Mass of produced sand Rate of eroded solid mass per unitvolume (gssdotm3) Rate of eroded solid mass (gs)119872 Initial amount of solid material inthe sand producing zone (g)119901119900 Pore pressure (psia)119875in Internal pore pressure (Mpa)119875out External pore pressure (Mpa)119875119908 Bottomhole pressure119875119903 Reservoir Pressure119902fl Flow rate (m3s)119902119894fl Specific flow rate in the 119894thdirection (gs)119902crfl Critical flow rate (m3s)119903perf Perforation radius (ft)119903in Cylinder internal radius (m)119903out Cylinder external radius (m)119877 Location of the plastic zoneboundary (m)119877119888 Radius of the cavity (m)119877cr Radius of the sand productionzone (m)119877119901 Radius of the plastic zone (m)

Re Reynolds number

119878119900 Material cohesion (MPa)119905 Real time119879 Mathematical time119905119900 Initial timeV119904 Solid velocity (ms)119881sp Volumetric sand production (m3)119882az Well azimuth (degree)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] S Willson Z A Moschovidis J Cameron and I Palmer ldquoNewmodel for predicting the rate of sand productionrdquo inProceedingsof the SPEISRM Rock Mechanics Conference Irving Tex USAOctober 2002

[2] I Vardoulakis M Stavropoulou and P Papanastasiou ldquoHydro-mechanical aspects of the sand production problemrdquo Transportin Porous Media vol 22 no 2 pp 225ndash244 1996

[3] E Papamichos I Vardoulakis J Tronvoll and A SkjrsteinldquoVolumetric sand production model and experimentrdquo Inter-national Journal for Numerical and Analytical Methods inGeomechanics vol 25 no 8 pp 789ndash808 2001

[4] L Y Chin and G G Ramos ldquoPredicting volumetric sandproduction in weak reservoirrdquo in Proceedings of the SPEISRMRock Mechanics Conference Irving Tex USA October 2002

[5] E Fjaeligr P Cerasi L Li andE Papamichos ldquoModeling the rate ofsand productionrdquo in Proceedings of the 6th North America RockMechanics Symposium (NARMS rsquo04) Houston Tex USA June2004 httpswwwonepetroorgconference-paperARMA-04-588

[6] E Gravanis E Sarris and P Papanastasiou ldquoHydro-mechanicalerosion models for sand productionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 39 no18 pp 2017ndash2036 2015

[7] S O Isehunwa and O M Olanrewaju ldquoA simple analyticalmodel for predicting sand production in a niger delta oil fieldrdquoInternational Journal of Engineering Science and Technology vol2 no 9 pp 4379ndash4387 2010

[8] E Fjaeligr R M Holt P Horsrud A M Raaen and R RisnesPetroleum Related Rock Mechanics Elsevier 2008

[9] K Rahman A Khaksar and T J Kayes ldquoAn integrated geome-chanical and passive sand-control approach to minimizingsanding risk from openhole and cased-and-perforated wellsrdquoSociety of Petroleum Engineers vol 25 no 2 pp 155ndash167 2010

8 Advances in Materials Science and Engineering

Table 2 Data of Well X1

Input data ValueTVD (ft) 94795119868 (degree) 52119882az (degree) 244120590] (psia) 9300120590ℎ (psia) 8670120590119867 (psia) 9448] 03UCS (psia) 2450120572 1120590hz (degree) 193120601perf (degree) 90TWC (psia) 4936119896 (mD) 60120601119900 03120583 (cP) 3119903perf = 119903in (ft) 00328119897perf (ft) 16405119864 (psia) 979004

tradition of using oilfield units which is a requirement by thegovernment hence in Table 2 we do not use SI units

41 Calculate Critical Bottomhole Pressure Using Geomechan-ical Model of Willson et al [1] The critical bottomholepressure CBHP is important information during productionphase because it indicates the lowest bottomhole pressurefor sand production to not occur For a specific well inoil field the only production data that we control is thebottomhole pressure which is adjusted using surface chokesThe pressure is controlled hence the flowrate is controlledFor this reason before predicting the sand production usingHydromechanical models of Gravanis et al and Fjaeligr et alwe firstly calculate the CBHP using Geomechanical model ofWillson et al

The testing showed that a relationship between the effec-tive in situ strength of the formation U and the TWC (Thickwall cylinder) strength relative to a specimen with an ODIDratio (OuterInner diameters) would be equivalent to

UCS = 119887119891 lowast 155 lowast TWC (30)

where 119887119891 the boost factor is often taken as 119887119891 = 2 forcasedndashhole perforated wells Equation (1) considers effect ofreservoir pressure decline [9] which can be accounted for byupdating in situ stresses while the vertical stresses are usuallykept constant

120590ℎ = 120590ℎ + 119860 sdot Δ119875120590119867 = 120590119867 + 119860 sdot Δ119875 (31)

where 119860 = 120572((1 minus 2])(1 minus ])) with 120572 is the Biotrsquos constantand ] is coefficient of Poisson Δ119875 = 119875119888 minus119875119894 with 119875119888 is currentreservoir pressure and 119875119894 is initial reservoir pressure

Sand free

Sand production

2000 4000 6000 800000

1000

2000

3000

4000

5000

6000

7000

8000

CBHP_UCS = 3160 psiaCBHP_UCS = 2450 psiaCBHP_UCS = 1950 psia

Pr (psia)

Pw = Pr

Pw

(psia

)

Figure 10 Prediction of sand production with variation of UCS(Unconfined Compressive Strength)

Figure 10 presents the bottomhole pressure 119875119908 versusreservoir pressure 119875119903 In Figure 10 we also present the CBHPwhen the UCS changes For example for UCS = 1950 psiaif reservoir pressure 119875119903 is 6000 psia the CBHP is about 1500psia if the bottomhole pressure 119875119908 is lower than this value1500 psia sand production will occur

The results showed that if UCS gets smaller the sandproduction window is greater This can be explained by thereason that the formation rocks around the wellbore areweakenedwhenUCSdecreasesThus risk of sandproductionincreases

We consider a case study with UCS = 2450 psi 119875119903 =4000 psi From Figure 10 we see that if the current reservoirpressure is 4000 psi then the critical bottomhole pressure is2020 psia For sand production to not occur the bottomholepressuremust be greater than 2020 psi Otherwise to increasethe flow rate we must increase DP and if it is greater thanCDP it means that bottomhole pressure must be lower thancritical bottomhole pressure (1000 psi) then the sand willbe produced So there is a compromise between increasingflowrate (production requirement) and sand production

42 Calculate Sand Production Mass Using the Models ofGravanis et al and Fjaeligr et al We use HydromechanicalErosion models of Fjaeligr et al [5] and Gravanis et al [6]to calculate sand production mass versus time Howeverwe do not have experimental data of Well X1 to calibrateempirical parameters so we must use empirical parametersalready calibrated in Section 3 In Figure 11 the results ofsand mass produced over time calculated by the two modelsare presented We also did a sensitivity study of the effectof drawdown pressure Drawdown pressure which is thedifference between the reservoir pressure and the bottomholepressure is the most important data that we control duringproduction phase We control the drawdown consequentlywe control the production rate The drawdown is typicallycontrolled by surface chokesThemore the choke is open the

Advances in Materials Science and Engineering 9

0

2000

4000

6000

8000

Sand

pro

duct

ion

(g)

5000 10000 15000 200000Time (s)

Gravanis_Pw = 1800 psiaGravanis_Pw = 1800 psia

= 1800 psiaFjaer_Pw

= 1500 psiaFjaer_Pw

Figure 11 Results of two models in the case study of Well X1

bigger the bottomhole pressure is and hence the smaller thedrawdown is and vice versa

Figure 11 shows that sand production mass increaseswhen drawdown pressure increases However results givenby the model of Gravanis et al are smaller than results givenby Fjaeligr et al At each bottomhole pressure sand mass givenby the model of Gravanis et al increases until it reachesconstant value On the other hand the one of Fjaeligr et alincreases continuously because the process of erosion repeatswhich causes the radius of cavity and production zone toincrease so sand production mass increases continuouslyThese twomodels showdifferent results although they use thesame physical mechanisms the fluid flow can be describedby Darcyrsquos law and the driving mechanism for continuoussand production is erosion from plastified material in thevicinity of the production cavity under hydrodynamic forcesHowever the model of Fjaeligr et al uses a step of instantaneoussand production which is not considered in the model ofGravanis et al Moreover while Gravanis et al divide thewhole region into a plastic region where Mohr-Coulombfailure criterion is considered and an elastic region whereHookersquos law is applied Fjaeligr et al consider only the plasticregion Fjaeligr et al simply assume that the stiffness of therock remains constant until the porosity has reached a criticalvalue 120601cr At that point the entire part of the rock thathas been producing sand collapses and the remaining solidmaterial in that part is produced in one burst As a result theequations used in these two models as well as the associatedexperimental parameters are quite different and eventuallyled to quite different results

Finally it is important to recall that due to the lack ofvalidation data in reality it is impossible to choose whichmodel to use in practical use At this state these results canonly be used for illustration in study and to demonstrate thepossibility of using Hydromechanical models in real casesto predict eroded sand mass over time In the future if realdata is available thesemodels can be revised and recalibratedUnfortunately until now measuring sand mass rate is tech-nically impossible in the oilfields An idea was proposed tosolve this problem the determination of real sand rate maybe based on the erosion rate of the choke However this study

has never been realized andmay constitute a new objective inour future study

5 Conclusion

This study combines Geomechanical model and the twoHydromechanical Erosion models of Fjaeligr et al [5] andGravanis et al [6] aiming to predict sand production Theproposed model can not only predict the critical pressure foronset of sand production but also estimate sand productionmass and the variation of porosity in time and space Inaddition the study also gives detailed calculation steps forthe two Hydromechanical models Although each modelhas assumptions and empirical parameters we can adjustthem if we have experimental data The results of this studywill help to make an effective production planning that canavoid sand production when reservoir pressure declinesHydromechanical model is still very new and not yet widelyused due to several reasons such as complicated calculationmethods and parameters that need to be experimentallyadjusted in practice or in the laboratoryTherefore this initialcomputational workflow using MATLAB will significantlycontribute to the development of future research in thisdomain in Petroleum Engineering

The experimental data (sand productionrsquos mass in func-tion of time) is rarely done not only in Vietnam but also inthe world (only some laboratories and authors mentionedin the paper have done this kind of experience because oftheir research in this field) Hence it is currently impossibleto obtain empirical data for Cuu Long basin (sand massin function of time) not only data in laboratory but alsoproduction data due to difficulties in collecting and mea-suring sand production rate Therefore application of thecalibrated model for Cuu Long basin in this study is destinedto demonstrate the possibility of using Hydromechanicalmodels in real cases and the result can be used for illustrationin research In the next studies when we have available datafrom Cuu Long basin we can do a revisionrecalibration ofthese models We already thought about the determinationof real sand rate data based on the erosion rate of the chokeThis idea will be the objective of our future study

Notations

CBHP Critical bottomhole pressure (psia)DP Drawdown pressure (psia)LF Loading Factor (near-wellbore

formation stress normalized bystrength)

SPR Sand Production Rate (gs)TVD True vertical depth (ft)TWC Thick wall cylinder strength (psia)UCS Unconfined compressive strength

(psia) Notation representing the norm ofa vector120572 Biotrsquos constant120573 Exponent parameter120576119901 Plastic shear strain

10 Advances in Materials Science and Engineering

Γ(119909) Usual Euler Gamma function120582 Experimentally evaluated sandproduction coefficient (1m)120582sand Proportionality constant (sm3)1205820 Initial erosion strength (mminus1)1205822 Maximum erosion strength (mminus1)Λ Depth of the plastic region (m)Λ 0 Λ at 119905 = 0 (m)Λ 119901 Threshold depth of the 120582 plateau120583 Viscosity (MPasdots)120601 Porosity120601119900 Initial porosity120601cr Critical porosity120601perf Perforation orientation from thetop of the wellbore in deviatedwells (degree)120588119904 Solid density (gm3)1205901 Maximum principal stress (psia)1205902 Minimum principal stress (psia)120590] Vertical stress (psia)120590ℎ Minimum principal horizontalstress (psia)120590119867 Maximum principal horizontalstress (psia)120590hz Minimum horizontal stressdirection (degree)

] Poisson ratio1198622 Integration constant119864 Youngrsquos modulus119868 Well inclination (degree)119896 Permeability (mD)119870 Principal stress ratio119897perf Perforation length (ft)119898sand Mass of produced sand Rate of eroded solid mass per unitvolume (gssdotm3) Rate of eroded solid mass (gs)119872 Initial amount of solid material inthe sand producing zone (g)119901119900 Pore pressure (psia)119875in Internal pore pressure (Mpa)119875out External pore pressure (Mpa)119875119908 Bottomhole pressure119875119903 Reservoir Pressure119902fl Flow rate (m3s)119902119894fl Specific flow rate in the 119894thdirection (gs)119902crfl Critical flow rate (m3s)119903perf Perforation radius (ft)119903in Cylinder internal radius (m)119903out Cylinder external radius (m)119877 Location of the plastic zoneboundary (m)119877119888 Radius of the cavity (m)119877cr Radius of the sand productionzone (m)119877119901 Radius of the plastic zone (m)

Re Reynolds number

119878119900 Material cohesion (MPa)119905 Real time119879 Mathematical time119905119900 Initial timeV119904 Solid velocity (ms)119881sp Volumetric sand production (m3)119882az Well azimuth (degree)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] S Willson Z A Moschovidis J Cameron and I Palmer ldquoNewmodel for predicting the rate of sand productionrdquo inProceedingsof the SPEISRM Rock Mechanics Conference Irving Tex USAOctober 2002

[2] I Vardoulakis M Stavropoulou and P Papanastasiou ldquoHydro-mechanical aspects of the sand production problemrdquo Transportin Porous Media vol 22 no 2 pp 225ndash244 1996

[3] E Papamichos I Vardoulakis J Tronvoll and A SkjrsteinldquoVolumetric sand production model and experimentrdquo Inter-national Journal for Numerical and Analytical Methods inGeomechanics vol 25 no 8 pp 789ndash808 2001

[4] L Y Chin and G G Ramos ldquoPredicting volumetric sandproduction in weak reservoirrdquo in Proceedings of the SPEISRMRock Mechanics Conference Irving Tex USA October 2002

[5] E Fjaeligr P Cerasi L Li andE Papamichos ldquoModeling the rate ofsand productionrdquo in Proceedings of the 6th North America RockMechanics Symposium (NARMS rsquo04) Houston Tex USA June2004 httpswwwonepetroorgconference-paperARMA-04-588

[6] E Gravanis E Sarris and P Papanastasiou ldquoHydro-mechanicalerosion models for sand productionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 39 no18 pp 2017ndash2036 2015

[7] S O Isehunwa and O M Olanrewaju ldquoA simple analyticalmodel for predicting sand production in a niger delta oil fieldrdquoInternational Journal of Engineering Science and Technology vol2 no 9 pp 4379ndash4387 2010

[8] E Fjaeligr R M Holt P Horsrud A M Raaen and R RisnesPetroleum Related Rock Mechanics Elsevier 2008

[9] K Rahman A Khaksar and T J Kayes ldquoAn integrated geome-chanical and passive sand-control approach to minimizingsanding risk from openhole and cased-and-perforated wellsrdquoSociety of Petroleum Engineers vol 25 no 2 pp 155ndash167 2010

Advances in Materials Science and Engineering 9

0

2000

4000

6000

8000

Sand

pro

duct

ion

(g)

5000 10000 15000 200000Time (s)

Gravanis_Pw = 1800 psiaGravanis_Pw = 1800 psia

= 1800 psiaFjaer_Pw

= 1500 psiaFjaer_Pw

Figure 11 Results of two models in the case study of Well X1

bigger the bottomhole pressure is and hence the smaller thedrawdown is and vice versa

Figure 11 shows that sand production mass increaseswhen drawdown pressure increases However results givenby the model of Gravanis et al are smaller than results givenby Fjaeligr et al At each bottomhole pressure sand mass givenby the model of Gravanis et al increases until it reachesconstant value On the other hand the one of Fjaeligr et alincreases continuously because the process of erosion repeatswhich causes the radius of cavity and production zone toincrease so sand production mass increases continuouslyThese twomodels showdifferent results although they use thesame physical mechanisms the fluid flow can be describedby Darcyrsquos law and the driving mechanism for continuoussand production is erosion from plastified material in thevicinity of the production cavity under hydrodynamic forcesHowever the model of Fjaeligr et al uses a step of instantaneoussand production which is not considered in the model ofGravanis et al Moreover while Gravanis et al divide thewhole region into a plastic region where Mohr-Coulombfailure criterion is considered and an elastic region whereHookersquos law is applied Fjaeligr et al consider only the plasticregion Fjaeligr et al simply assume that the stiffness of therock remains constant until the porosity has reached a criticalvalue 120601cr At that point the entire part of the rock thathas been producing sand collapses and the remaining solidmaterial in that part is produced in one burst As a result theequations used in these two models as well as the associatedexperimental parameters are quite different and eventuallyled to quite different results

Finally it is important to recall that due to the lack ofvalidation data in reality it is impossible to choose whichmodel to use in practical use At this state these results canonly be used for illustration in study and to demonstrate thepossibility of using Hydromechanical models in real casesto predict eroded sand mass over time In the future if realdata is available thesemodels can be revised and recalibratedUnfortunately until now measuring sand mass rate is tech-nically impossible in the oilfields An idea was proposed tosolve this problem the determination of real sand rate maybe based on the erosion rate of the choke However this study

has never been realized andmay constitute a new objective inour future study

5 Conclusion

This study combines Geomechanical model and the twoHydromechanical Erosion models of Fjaeligr et al [5] andGravanis et al [6] aiming to predict sand production Theproposed model can not only predict the critical pressure foronset of sand production but also estimate sand productionmass and the variation of porosity in time and space Inaddition the study also gives detailed calculation steps forthe two Hydromechanical models Although each modelhas assumptions and empirical parameters we can adjustthem if we have experimental data The results of this studywill help to make an effective production planning that canavoid sand production when reservoir pressure declinesHydromechanical model is still very new and not yet widelyused due to several reasons such as complicated calculationmethods and parameters that need to be experimentallyadjusted in practice or in the laboratoryTherefore this initialcomputational workflow using MATLAB will significantlycontribute to the development of future research in thisdomain in Petroleum Engineering

The experimental data (sand productionrsquos mass in func-tion of time) is rarely done not only in Vietnam but also inthe world (only some laboratories and authors mentionedin the paper have done this kind of experience because oftheir research in this field) Hence it is currently impossibleto obtain empirical data for Cuu Long basin (sand massin function of time) not only data in laboratory but alsoproduction data due to difficulties in collecting and mea-suring sand production rate Therefore application of thecalibrated model for Cuu Long basin in this study is destinedto demonstrate the possibility of using Hydromechanicalmodels in real cases and the result can be used for illustrationin research In the next studies when we have available datafrom Cuu Long basin we can do a revisionrecalibration ofthese models We already thought about the determinationof real sand rate data based on the erosion rate of the chokeThis idea will be the objective of our future study

Notations

CBHP Critical bottomhole pressure (psia)DP Drawdown pressure (psia)LF Loading Factor (near-wellbore

formation stress normalized bystrength)

SPR Sand Production Rate (gs)TVD True vertical depth (ft)TWC Thick wall cylinder strength (psia)UCS Unconfined compressive strength

(psia) Notation representing the norm ofa vector120572 Biotrsquos constant120573 Exponent parameter120576119901 Plastic shear strain

10 Advances in Materials Science and Engineering

Γ(119909) Usual Euler Gamma function120582 Experimentally evaluated sandproduction coefficient (1m)120582sand Proportionality constant (sm3)1205820 Initial erosion strength (mminus1)1205822 Maximum erosion strength (mminus1)Λ Depth of the plastic region (m)Λ 0 Λ at 119905 = 0 (m)Λ 119901 Threshold depth of the 120582 plateau120583 Viscosity (MPasdots)120601 Porosity120601119900 Initial porosity120601cr Critical porosity120601perf Perforation orientation from thetop of the wellbore in deviatedwells (degree)120588119904 Solid density (gm3)1205901 Maximum principal stress (psia)1205902 Minimum principal stress (psia)120590] Vertical stress (psia)120590ℎ Minimum principal horizontalstress (psia)120590119867 Maximum principal horizontalstress (psia)120590hz Minimum horizontal stressdirection (degree)

] Poisson ratio1198622 Integration constant119864 Youngrsquos modulus119868 Well inclination (degree)119896 Permeability (mD)119870 Principal stress ratio119897perf Perforation length (ft)119898sand Mass of produced sand Rate of eroded solid mass per unitvolume (gssdotm3) Rate of eroded solid mass (gs)119872 Initial amount of solid material inthe sand producing zone (g)119901119900 Pore pressure (psia)119875in Internal pore pressure (Mpa)119875out External pore pressure (Mpa)119875119908 Bottomhole pressure119875119903 Reservoir Pressure119902fl Flow rate (m3s)119902119894fl Specific flow rate in the 119894thdirection (gs)119902crfl Critical flow rate (m3s)119903perf Perforation radius (ft)119903in Cylinder internal radius (m)119903out Cylinder external radius (m)119877 Location of the plastic zoneboundary (m)119877119888 Radius of the cavity (m)119877cr Radius of the sand productionzone (m)119877119901 Radius of the plastic zone (m)

Re Reynolds number

119878119900 Material cohesion (MPa)119905 Real time119879 Mathematical time119905119900 Initial timeV119904 Solid velocity (ms)119881sp Volumetric sand production (m3)119882az Well azimuth (degree)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] S Willson Z A Moschovidis J Cameron and I Palmer ldquoNewmodel for predicting the rate of sand productionrdquo inProceedingsof the SPEISRM Rock Mechanics Conference Irving Tex USAOctober 2002

[2] I Vardoulakis M Stavropoulou and P Papanastasiou ldquoHydro-mechanical aspects of the sand production problemrdquo Transportin Porous Media vol 22 no 2 pp 225ndash244 1996

[3] E Papamichos I Vardoulakis J Tronvoll and A SkjrsteinldquoVolumetric sand production model and experimentrdquo Inter-national Journal for Numerical and Analytical Methods inGeomechanics vol 25 no 8 pp 789ndash808 2001

[4] L Y Chin and G G Ramos ldquoPredicting volumetric sandproduction in weak reservoirrdquo in Proceedings of the SPEISRMRock Mechanics Conference Irving Tex USA October 2002

[5] E Fjaeligr P Cerasi L Li andE Papamichos ldquoModeling the rate ofsand productionrdquo in Proceedings of the 6th North America RockMechanics Symposium (NARMS rsquo04) Houston Tex USA June2004 httpswwwonepetroorgconference-paperARMA-04-588

[6] E Gravanis E Sarris and P Papanastasiou ldquoHydro-mechanicalerosion models for sand productionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 39 no18 pp 2017ndash2036 2015

[7] S O Isehunwa and O M Olanrewaju ldquoA simple analyticalmodel for predicting sand production in a niger delta oil fieldrdquoInternational Journal of Engineering Science and Technology vol2 no 9 pp 4379ndash4387 2010

[8] E Fjaeligr R M Holt P Horsrud A M Raaen and R RisnesPetroleum Related Rock Mechanics Elsevier 2008

[9] K Rahman A Khaksar and T J Kayes ldquoAn integrated geome-chanical and passive sand-control approach to minimizingsanding risk from openhole and cased-and-perforated wellsrdquoSociety of Petroleum Engineers vol 25 no 2 pp 155ndash167 2010

10 Advances in Materials Science and Engineering

Γ(119909) Usual Euler Gamma function120582 Experimentally evaluated sandproduction coefficient (1m)120582sand Proportionality constant (sm3)1205820 Initial erosion strength (mminus1)1205822 Maximum erosion strength (mminus1)Λ Depth of the plastic region (m)Λ 0 Λ at 119905 = 0 (m)Λ 119901 Threshold depth of the 120582 plateau120583 Viscosity (MPasdots)120601 Porosity120601119900 Initial porosity120601cr Critical porosity120601perf Perforation orientation from thetop of the wellbore in deviatedwells (degree)120588119904 Solid density (gm3)1205901 Maximum principal stress (psia)1205902 Minimum principal stress (psia)120590] Vertical stress (psia)120590ℎ Minimum principal horizontalstress (psia)120590119867 Maximum principal horizontalstress (psia)120590hz Minimum horizontal stressdirection (degree)

] Poisson ratio1198622 Integration constant119864 Youngrsquos modulus119868 Well inclination (degree)119896 Permeability (mD)119870 Principal stress ratio119897perf Perforation length (ft)119898sand Mass of produced sand Rate of eroded solid mass per unitvolume (gssdotm3) Rate of eroded solid mass (gs)119872 Initial amount of solid material inthe sand producing zone (g)119901119900 Pore pressure (psia)119875in Internal pore pressure (Mpa)119875out External pore pressure (Mpa)119875119908 Bottomhole pressure119875119903 Reservoir Pressure119902fl Flow rate (m3s)119902119894fl Specific flow rate in the 119894thdirection (gs)119902crfl Critical flow rate (m3s)119903perf Perforation radius (ft)119903in Cylinder internal radius (m)119903out Cylinder external radius (m)119877 Location of the plastic zoneboundary (m)119877119888 Radius of the cavity (m)119877cr Radius of the sand productionzone (m)119877119901 Radius of the plastic zone (m)

Re Reynolds number

119878119900 Material cohesion (MPa)119905 Real time119879 Mathematical time119905119900 Initial timeV119904 Solid velocity (ms)119881sp Volumetric sand production (m3)119882az Well azimuth (degree)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] S Willson Z A Moschovidis J Cameron and I Palmer ldquoNewmodel for predicting the rate of sand productionrdquo inProceedingsof the SPEISRM Rock Mechanics Conference Irving Tex USAOctober 2002

[2] I Vardoulakis M Stavropoulou and P Papanastasiou ldquoHydro-mechanical aspects of the sand production problemrdquo Transportin Porous Media vol 22 no 2 pp 225ndash244 1996

[3] E Papamichos I Vardoulakis J Tronvoll and A SkjrsteinldquoVolumetric sand production model and experimentrdquo Inter-national Journal for Numerical and Analytical Methods inGeomechanics vol 25 no 8 pp 789ndash808 2001

[4] L Y Chin and G G Ramos ldquoPredicting volumetric sandproduction in weak reservoirrdquo in Proceedings of the SPEISRMRock Mechanics Conference Irving Tex USA October 2002

[5] E Fjaeligr P Cerasi L Li andE Papamichos ldquoModeling the rate ofsand productionrdquo in Proceedings of the 6th North America RockMechanics Symposium (NARMS rsquo04) Houston Tex USA June2004 httpswwwonepetroorgconference-paperARMA-04-588

[6] E Gravanis E Sarris and P Papanastasiou ldquoHydro-mechanicalerosion models for sand productionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 39 no18 pp 2017ndash2036 2015

[7] S O Isehunwa and O M Olanrewaju ldquoA simple analyticalmodel for predicting sand production in a niger delta oil fieldrdquoInternational Journal of Engineering Science and Technology vol2 no 9 pp 4379ndash4387 2010

[8] E Fjaeligr R M Holt P Horsrud A M Raaen and R RisnesPetroleum Related Rock Mechanics Elsevier 2008

[9] K Rahman A Khaksar and T J Kayes ldquoAn integrated geome-chanical and passive sand-control approach to minimizingsanding risk from openhole and cased-and-perforated wellsrdquoSociety of Petroleum Engineers vol 25 no 2 pp 155ndash167 2010

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