(SPE 156394) A Numerical Model for Predicting the Rate of Sand Production in Injector Wells

Abstract In this paper, a numerical model for volumetric prediction of sand production in injector wells is presented. Sanding in injector wells is mainly associated with the backflow and crossflow generated during shutin in addition to the waterhammer pressure pulsing in the wellbore due to fast flow rate changes. Emphasis is given to the geomechanical aspects of sanding such as rock fatigue due to cyclic pressure changes and the concomitant degradation of bonding between the sand grains. This model is robust in capturing the key parameters in the sandstone behavior such as stressdependent elasticity, hardening, softening and dilatancy. Rock degradation is considered to be the necessary condition for sand production which is assumed to obey the erosion mechanics. The model is calibrated and validated using physical model tests carried out under various stresses and fluid flow conditions. The numerical model has been utilized to analyze sanding potential in a cased and perforated injector which will be presented to demonstrate the field application of the proposed concepts.

SPE 156394

A Numerical Model for Predicting the Rate of Sand Production in Injector Wells

Azadbakht, S., Jafarpour, M., Rahmati, H., Nouri, A.; University of Alberta; Vaziri, H., BP America Inc.; Chan D., University of Alberta

Copyright 2012, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Deepwater Drilling and Completions Conference held in Galveston, Texas, USA, 20–21 June 2012

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s) Contents

of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s) The material does not necessar ily reflect any position of the Society of Petroleum Engineers, its officers, or members Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied The abstract must contain conspicuous acknowledgment of SPE copyright

Abstract

In this paper, a numerical model for volumetric prediction

of sand production in injector wells is presented Sanding

in injector wells is mainly associated with the back-flow

and cross-flow generated during shut-in in addition to the

waterhammer pressure pulsing in the wellbore due to fast

flow rate changes Emphasis is given to the

geomechanical aspects of sanding such as rock fatigue

due to cyclic pressure changes and the concomitant

degradation of bonding between the sand grains This

model is robust in capturing the key parameters in the

sandstone behavior such as stress-dependent elasticity,

hardening, softening and dilatancy Rock degradation is

considered to be the necessary condition for sand

production which is assumed to obey the erosion

mechanics The model is calibrated and validated using

physical model tests carried out under various stresses and

fluid flow conditions The numerical model has been

utilized to analyze sanding potential in a cased and

perforated injector which will be presented to demonstrate

the field application of the proposed concepts

Introduction

Sand production is a common problem in production and

injection wells Extensive research has been carried out in

the past couple of decades to identify the key parameters

affecting initiation and severity of this phenomenon

Detection and management of sand production is more

obscure when it comes to injection wells as there is no

fluid production and hence no indication of sand initiation

and severity

Practical problems associated with sand production

include erosion of pipelines and surface facilities,

reduction in productivity, intervention costs and

complexities and other environmental effects These

problems cost the oil industry billions of dollars annually (Nouri, 2004) On the other hand, a controllable amount

of sand production may omit the need for installing more complex active sand controls involving use of gravel packs which have been used extensively to reduce and avoid sand production from unconsolidated formations (Saucier, 1974) Therefore, understanding the sand production mechanisms and the ability to predict and manage the rate of sand production are beneficial

A linked finite difference-finite element (FE-FD) code is used for the sanding assessment of an injection well This model can simulate the impact of injection pressure and shut-in cycles, including the effects of inflow differential pressure (DP) (due to cross-flow or back-flow) and waterhammer (WH) pulses on sanding It also accounts for the in situ strength and its gradual degradation due to stress and pressure changes By simulating the shut-in cycles, the main effects of well operation over the wellbore life can be accounted for The model incorporates the essential physics in water injection operations and accounts for the critical factors with respect to the rock behavior and sanding mechanisms, including the influence of flow rate on sand production

Brief Physics of Sand production

When a fluid is injected into a reservoir, the following occur depending on the formation consistency:

 Increase in pressure leads to reduction in effective stress and hence reduction in particle-to-particle frictional resistance which particularly impacts the sanding response in unconsolidated

or disaggregated materials Under high injection pressure and/or waterhammer pressure pulsing, particularly if exceeding the overburden stress,

sand may reach a fluidized state This condition

may exacerbate sanding right after shut-in

depending on the injection magnitude and

shut-in rate

 In weakly consolidated formations, injection and

shut-in cycles particularly if combined with

waterhammer pulses may result in the

destruction of cementation and turn the material

into an unconsolidated sand mass with

consequences as discussed above

 In competent and cemented formations, high

injection may result in development of fractures

but no major sanding is expected as a result of

this phenomenon

Every cycle of shut-in and start-up will promote sand face

fatigue (gradual weakening of cementation due to strain

cycles) and this may eventually breakdown all

cementation The extent and acceleration of degradation

depend on the magnitude of injection, shut-in rate

(impacting magnitude of cross-and back-flow and

waterhammer intensity) and strength properties of the

formation

In water injectors, the injection pressure and water are

likely to destroy the multi-grain structure and hence create

a high potential and volume of sanding

Numerical Model Description

A numerical tool that links a finite difference (FD) code

with a finite element (FE) code is used for this study The

distinguishing features of this model that are of critical

importance to the proposed study include:

 The drilling phase is simulated in a fully coupled

manner to capture the critical processes that

happen during drilling more accurately

 The injection/shut-in cycles are performed using

sequential coupling In that, the FD code

performs the mechanical calculations and the FE

code does the fluid flow calculations This is

done to take advantage of fast fluid flow

calculation of the FE code

 The model allows for rock strength degradation

with changes in stress and strain associated with

the injection cycles and the waterhammer

pressure pulses

 The model is capable of computing sand

production

 The model accounts for the very rapid inflows

following shut-ins and hence can capture the

effects of the shut-in rate

 It can track changes in hydro-mechanical

properties (e.g., stiffness, permeability) as a

result of sanding and changes in effective stress

with injection

Two main components of this numerical model are the

constitutive model and sanding criterion which are briefly

described below

Constitutive model The importance of precise and

descriptive modeling of constitutive behavior of rocks can not be overemphasized This part plays a vital role in any sand production simulation

As shown schematically in Fig 1, results of laboratory experiments indicate that granular materials usually demonstrate strain softening at Low Effective Confining Stress (LECS) and strain hardening at the state of High Effective Confining Stress (HECS) (Vermeer & de Borst, 1984; Sulem et al., 1999) These facts are taken into account in formulation of the yield surface which expands (strain hardening) or contracts (strain softening) as a function of the hardening parameter which will be introduced later

Fig 1: Different stress-strain regimes at various confining stresses (Vaziri et al., 2007)

Elasto-palstic constitutive models have shown to model sandstone behavior with adequate accuracy In this paper the same approach as Sulem et al (1999) and Nouri et al (2009) is undertaken in which a bilinear Mohr-Coulomb (MC) model is calibrated using laboratory tests

This model involves the calibration of elastic properties, initial and peak yield surfaces, friction hardening, cohesion softening and mobilized dilation angle The last three parameters are expressed as a function of the hardening parameter that is itself a function of principal plastic strains For the sake of brevity, the details of the constitutive modeling are not presented here but the interested reader can refer to the above references

Sanding criterion The sanding criterion used in this

study is based on erosion mechanics (Detournay, 2006)

In this logic, it is assumed that sanding will start when both of the following conditions are met:

a) All cohesion (which represents cementation) is lost; that is, real cohesion degrades to zero, and b) The totally disaggregated or cohesionless sand particles are broken away from sand mass and carried into the perforation/wellbore by the action of hydrodynamic forces (erosion process) This process causes the porosity of the elements

to increase (as a result of sand removal) until it reaches the critical porosity, i.e., the porosity at

which the rock matrix collapses (Rahmati et al,

2011)

There are almost no field cases involving injectors where

the sanding events have been recorded as they occurred

This makes validation or even calibration of any modeling

effort difficult

Having said that, the following measures are undertaken

into considerations to deal with the general uncertainties:

• Maximize value from laboratory tests performed

on formation rock samples The rigor involved

in back-analyzing properties that capture the

rock behavior is shown later along with

validation

• Use rigorous numerical analyses In this case,

we have used a complex numerical model along

with a fine mesh and complex procedures to

capture very short duration events, such as

pressure waves, relatively short events such as

shut-in and injection build up and longer term

operations during steady injection

Numerical Model Calibration

Calibration of the numerical model involves calibration of

both constitutive model and the sanding criterion

Constitutive Model Calibration

The bilinear MC with combined hardening/softening

model was calibrated using a series of uniaxial and

triaxial tests Fig 2 shows this model schematically

Fig 2: a) Hardening and b) softening of the bilinear

Mohr-Coulomb model (Sulem et al., 1999)

Fig 2-a shows the hardening behavior wherein line (0) stands for the initial yield surface Once a stress state reaches line (0), plastic deformation begins Further loading increases the friction coefficient or the slope of the line up to the peak yield surface (line 1) This is shown by the upward arrows from line (0) to line (1) Up

to this point, the tension cut-off is approximately constant both for the low and high effective confining stresses ( and ) Additional deformation after the peak results in the softening of the material and shrinkage

of the yield surface This is demonstrated in Fig 2-b by the downward arrows from line (1) to line (2) During softening, tension cut-off shrinks to the residual value ( ), and it is equal to zero for fully degraded sandstone,

as depicted in Fig 2-b However, the friction coefficient remains constant That is, the line is lowered to the residual state with the same slope as that of the peak Line (2) is the new yield surface during softening when the residual tension cut-off gradually decreases to zero leading to the development of shear bands (Jafarpour et al., 2012)

Tension cut-off can be related to the mobilized cohesion,

C, by the following relationship:

(1)

where is the friction angle of the rock

In this work, the hardening parameter is Equivalent Plastic Strain (EPS), which is defined by the following (Vermeer and de Borst, 1984):

(2)

where

(3)

are the principal plastic shear strain increments

The MC envelope as shown in Fig 2 varies as a function

of (EPS) and hence can simulate the strength degradation

Fig 3 shows the elastic properties of a pay sandstone layer with UCS of 1,250 psi as a function of the confining stresses Shear and bulk moduli increase with increase in confining stress which is due to the closing of pre-existing micro-cracks As plastic deformations start, friction and dilation are mobilized as a function of EPS until they reach the peak values after which they remain constant

Fig 4 shows the mobilized friction and dilation angles for the same sandstone



tan /

C



EPS = 1

2 ( D e1ps- D em ps)2

+ 1

2 ( ) D em ps 2

+ 1

2 ( D e3ps- D em ps)2

æ è

1 2

ps

3





3 , 1 , 

e j ps j

Fig 3: Bulk and shear moduli as a function of confining

stresses

Fig 4: Mobilized friction and dilation angles as a function of

EPS

After reaching the peak stress state, the rock enters the

softening stage, which is demonstrated through cohesion

degradation as shown in Fig 5

Fig 5: Mobilized cohesion during hardening and softening

Fig 6 shows the comparison between the triaxial test

measurements and the numerical results for the sandstone

at a certain confining pressure As seen, the constitutive

model predicts the rock behavior with reasonable

accuracy

Fig 6: Comparison of numerical and experimental stress-strain response for a triaxial test

Sanding Model Calibration

The main parameters in the sanding model are critical porosity, critical flow rate and erosion rate coefficient

These parameters are calibrated using laboratory data obtained by testing perforated rock samples

The rate of the produced sand mass is proportional to the specific flow rate (Detournay et al., 2006):

( )( ) (4) where is the specific mass flux, is the specific discharge normal to the boundary, is the critical value

of specific discharge, λ is the erosion rate coefficient, is the rock porosity and is the grain density

Papamichos et al (2001) showed that using a constant erosion coefficient may result in physically unrealistic behavior To improve prediction of sanding rate, they suggested an erosion coefficient λ as a function of EPS as follows:

{ (

) (5)

where EPSres stands for residual equivalent plastic strain and is calculated at residual strength state

Using the Law of Conservation of Mass, the rate of the change of porosity is related to the rate of generated sand

mass by the equation (Detournay et al., 2006):

(6) where is the boundary surface area of the element and

is the volume of the element

Sand is produced at a rate given by Eq 4 until the porosity of the element reaches a critical value

As the porosity increases the material becomes less competent This phenomenon is represented in the model

by degrading the bulk and shear modulus with the

increasing porosity as follows:

(7)

(8)

After the element reaches the critical porosity, it is kept in

the mesh at residual stiffness properties to represent infill

materials

Further details about sanding model calibration can be

found in Rahmati et al (2011)

Finite difference mesh and boundary conditions

Fig 7 shows a close-up of the FE mesh near the wellbore

Two types of injection shut-downs are expected: planned

down (PSD) and unplanned (or emergency)

shut-down (UPSD) Fig 8 shows the schematics of an

injection cycle with planned shut down

Fig 7: Close up of the FD mesh

Fig 8: Schematic of PSD with cross-flow

Fig 9 shows a schematic of a cycle with unplanned shut

down The time intervals for various sections of the

injection/shut down cycles are selected in a way to assure

establishment of steady state flow conditions The

pressure and time axes in Fig 8 and Fig 9 are not to scale

In case of a PSD, the shut-in rate is in such a way that minimizes the resulting waterhammer pressure pulses whereas during a UPSD the shut down periods are rapid enough to create considerable WH pulses Some field-scale transient analysis data were used to estimate the WH pressure pulses for the well under study

Fig 9: Schematic of UPSD with cross-flow

Depending on reservoir heterogeneity, fluid injection can create different pressure gradients in different layers due

to differences in permeability, porosity, compressibility, etc Upon injection shut down, fluid may flow from high pressure layers with low permeability to layers having a lower pressure and usually higher permeability; a process which is known as interlayer cross-flow Another type of cross-flow is the flow of fluid from the high-pressure layer to the low-pressure layer through the wellbore This

is called intra-well cross-flow

Fig 10 shows schematically a possible scenario for intra-well cross-flow Upper layer has lower permeability and will retain the injection pressure, which upon shutdown will become the driving force to squeeze the injection fluid into the lower layer with higher permeability

Fig 10: Schematic of intra-well cross-flow

Depending on differential pressures and magnitude of fluid flow, cross-flow can have a serious impact on sanding behavior of a well Cross-flow is incorporated in the injection/shut down cycles by applying a drawdown (DD) after the shut-down period Also, cases are examined without the cross-flow effect, i.e., zero DD after the shut down

Basic Simulation Steps

First, in-situ stresses are initiated and the model is solved

for equilibrium Then, reservoir pressure is reduced to

simulate the production induced depletion in the reservoir

After depletion, multiple injection and shut-in cycles are

applied to the perforation cavity

The model computes changes in stress, strain and any

associated degradation in strength, which along with

seepage forces may result in sand failure and production

Water weakening effect

Experimental observations indicate that water contact can

have a high impact on rock strength and hence on sanding

potential of a well (Santarelli et al., 2000; Han and

Dusseault, 2002) Data from literature are used to

correlate the UCS of samples with non-native water

saturation to the UCS of dry samples This correlation is

then used to reduce rock strength parameters accordingly

to account for water weakening effect in the numerical

simulations

Results

The numerical model has been used to perform some

sensitivity analysis to assess the effect of various

parameters on sanding Figures 11-13 show the results of

sensitivity analysis for one of the rocks in this study In all

the cases, only one parameter has been changed at a time

and everything else is kept the same In all these plots, the

horizontal axis shows the time after the start of injection

operations

Fig 11 shows the effect of perforation size on sanding

response In case of a larger perforation size, sanding

starts earlier and has a considerably higher magnitude

This points out the importance of proper perforation size

selection Generally, the smaller the perforation size, the

less the risk and severity of sanding but care should be

taken not to compromise the well productivity in this

process A more in-depth description of the effect of this

factor on sanding is yet to be investigated

Fig 11: Effect of perforation size on sanding

Fig 12 shows the effect of water induced weakening on

sanding This parameter appears to have the highest

impact on sanding response of the rock Sanding starts

very early and picks up quickly in this case Although water weakening effect might seem inevitable in injector wells; proper measures can be taken to ensure the compatibility of the injected water and the formation rock

to minimize the impact of this factor

Fig 12: Effect of water induced weakening on sanding

Fig 13 demonstrates the effect of cross-flow on sanding Cross-flow appears to have the same effect as perforation size in terms of pattern and magnitude This exemplifies the impact cross-flow can have on sanding response of wellbores

Fig 13: Effect of cross-flow on sanding

The effect of rock strength on sanding behavior is shown

in Fig 14 Three different rock types were used in this

part having different UCS values and the result are plotted for the first, second and third year after the start of injection operations Such a parametric study can help the production engineers in selecting a rock strength cut-off below which the formation shouldn’t be perforated

Fig 14: Sanding response as a function of rock UCS

Discussion and Conclusions

To provide an insight into sanding behavior of injector

wells and the effect of different parameters that contribute

to sanding, a comprehensive numerical model is

proposed Using geomechanics principles and basic

physics of sand production, a set of criteria were

incorporated into this model for describing conditions

required for sanding initiation and propagation

This numerical model can take into account the effects of

the following parameters on sanding behavior of

injectors:

Rock strength Different rock types were used in this

work having different UCS values The outcome can help

the production engineers in selecting a rock strength

cut-off below which the formation shouldn’t be perforated

Water-weakening effect In water injectors, formation

rock may lose some of its strength due to chemical and/or

mechanical effects of water contact This study shows that

this factor had a significant impact on sanding severity

Cross-flow Effect In heterogeneous reservoirs,

cross-flow is a very likely phenomenon after the injection

shut-down Results of this work show that this factor can have

a considerable impact on sanding behavior of wellbores

One should note that in this work, only one parameter is

changed at a time to study the corresponding effect on

sanding The fact is that in a real case scenario, all this

factors are present and the overall sanding response of the

formation is determined by taking into account the effect

of all the individual parameters

Results of this study shows that considering the following

factors can be helpful in reducing the risk and severity of

sanding in injector wells:

 Reducing the number of unplanned shut-ins in

order to reduce the waterhammer events

 In case of inevitable unplanned shut-ins, it is

ideal to make the shut-in time (time required to

shut down the pumps and/or close the

wellhead) as long as possible to reduce the

severity of waterhammer pulses

 Assuring the compatibility of the injected water with formation rock to reduce the water weakening effect

 Optimizing the perforation size as far as it doesn’t compromise well productivity

 Avoiding perforation of weak layers or using sand control devices in such layers

Acknowledgment

We thank BP for the permission to publish this paper The financial support provided by NSERC is also acknowledged

Nomenclature

C&P = Cased and Perforated DD= Drawdown

DP= Differential pressure EPS = Equivalent Plastic Strain EVS = Effective Vertical Stress HECS = High Effective Confining Stress LECS = Low Effective Confining Stress

MC = Mohr-Coulomb PSD = Planned Shut-down

P = mean stress

q = tension cut-off

= initial yield tension cut-off

= peak tension cut-off

= residual tension cut-off

= tension cut-off at HECS

= tension cut-off at LECS

T= square root of the second invariant of the

deviatoric stress

UCS = Unconfined Compressive Strength USD = Unplanned Shut-down

WH = Waterhammer

= Principal plastic shear strain increments = friction coefficient

= friction coefficient at HECS

= friction coefficient at LECS = Friction angle

= grain density

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

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



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