-
1. INTRODUCTION
The production of formation sand has plagued the oil and gas
industry for decades because of its adverse effects on wellbore
stability and equipment, while it has also been proven to be a most
effective way to increase well productivity. When hydrocarbon
production occurs from shallow and geologically young (or so-called
unconsolidated / weakly consolidated) formations that have little
or no cementation to hold the sand particles together, the
interaction of fluid pressure and stresses within the porous
granular material can lead to the mechanical failure of the
formation and unwanted mobilization of sand. It has been reported
that 10%-40% sand cuts normally stabilize in time to levels less
than 5% in heavy oil reservoirs [1], while an average of 40%
productivity increase was achieved through sand management in light
oil reservoirs [2]. When sand is produced from reservoir
formations, it can cause a number of problems. These include the
instability of wellbores, the erosion of pipes, the plugging of
production liners, the subsidence of
surface ground, and the need for disposal of sand in an
environmentally acceptable manner. Each year, these issues cost the
oil industry hundreds of millions of dollars. Furthermore, sand
production and control becomes extremely crucial in offshore
operations where a very low tolerance to sand production is
allowed. Hence, it is imperative to find an efficient computational
model that has the predictive capability to assist field operators
to understand this unique process. The ultimate goal is to design
an economical well-production strategy in which sand production and
operating costs may be reduced to some extent with maximum
hydrocarbon productivity. It is commonly believed that the
mechanism of sand production can be attributed to geomechanics and
multi-phase or foamy oil effects. However, modelling such a complex
problem is a challenging task since it requires multidisciplinary
physics to capture the whole range of material response from sand
flow initiation to fluidization.
In this paper, sand production is treated as an erosion process
by which a weakly consolidated sand matrix is disaggregated near
perforations of a
ARMA/NARMS 04-494 Sand Production and Instability Analysis in a
Wellbore using a Fully Coupled Reservoir-Geomechanics Model J.
Wang1, R. G. Wan2, A. Settari3, D. Walters4, and Y. N. Liu5 1,4
Taurus Reservoir Solutions Ltd., 2,5 Department of Civil
Engineering, University of Calgary, 3 Department of Chemical and
Petroleum Engineering, University of Calgary
Copyright 2004, ARMA, American Rock Mechanics Association This
paper was prepared for presentation at Gulf Rocks 2004, the 6th
North America Rock Mechanics Symposium (NARMS): Rock Mechanics
Across Borders and Disciplines, held in Houston, Texas, June 5 9,
2004. This paper was selected for presentation by a NARMS Program
Committee following review of information contained in an abstract
submitted earlier by the author(s). Contents of the paper, as
presented, have not been reviewed by ARMA/NARMS and are subject to
correction by the author(s). The material, as presented, does not
necessarily reflect any position of NARMS, ARMA, CARMA, SMMR, their
officers, or members. Electronic reproduction, distribution, or
storage of any part of this paper for commercial purposes without
the written consent of ARMA 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 acknowledgement of where and by whom the paper was
presented.
ABSTRACT: This paper presents a fully coupled
reservoir-geomechanics model with erosion mechanics to address
wellbore instability phenomena associated with sand production
within the framework of mixture theory. A Representative Elementary
Volume (REV) is chosen to comprise of five phases, namely solid
grains (s), fluidized solids (fs), oil fluid (f), water (w) and gas
(g). The particle transport and balance equations are written to
reflect the interactions among phases in terms of mechanical
stresses and hydrodynamics. Constitutive laws (mass generation law,
Darcy's law, and stress-strain relationships) are written to
describe the fundamental behaviour of sand erosion, fluid flow, and
deformation of the solid skeleton respectively. Subsequently, the
resulting governing equations are solved numerically using
Galerkins method with a generic nonlinear Newton-Raphson iteration
scheme. Numerical examples in a typical light oil reservoir are
presented to illustrate the capabilities of the proposed model in
the absence of the gas phase. It is found that there is an intimate
interaction between sand erosion activity and deformation of the
solid matrix. As erosion activity progresses, porosity increases
and in turn degrades the material strength. Strength degradation
leads to an increased propensity for plastic shear failure that
further magnifies the erosion activity. An escalation of plastic
shear deformations will inevitably lead to instability with the
complete erosion of the sand matrix. The self-adjusted mechanism
enables the model to predict both the volumetric sand production
and the propagation of wormholes, and hence instability phenomena
in the wellbore.
-
wellbore due to a combination of stress changes and multiphase
flow. A fully coupled reservoir- geomechanics mathematical model is
presented to account for the effects of multiphase flow and
geomechanics as well as their interaction in a consistent manner.
Numerical solutions, restricted to a typical light oil reservoir
without the influence of the gas phase, are sought to examine the
basic capabilities of this model. As the wellbore pressure is lower
than reservoir pressure, the erosion process begins as a result of
the degradation of the sand matrix strength and the drag force
imposed by fluid pressure gradient. The plastic yielding zones
develop due to the material degradation (erosion) and stress
re-distribution, while the wormholes or cavities form and propagate
in terms of the increasing porosity values. The volumetric oil and
sand productions are also calculated as a function of time,
stresses, and hydrocarbon flow rate.
2. COUPLED MULTIPHASE FLOW AND GEOMECHANICS FORMULATION
2.1. Mass balance equations The single-phase formulation
describing sand production in a deforming sand matrix was derived
in a series of publications [3, 4]. It has been shown to be a
promising method for modeling sand production in terms of matching
numerical calculations with lab test data, both in heavy and light
oil conditions [5, 6, 7]. In this paper, an extension to multiphase
sand production model is presented within the same framework of
mixture theory, i.e., a coupled black-oil/geomechanics sand
production model with erosion mechanics is proposed to further
account for the effects of multiphase flow of three components
(gas, water, oil) and their interaction with geomechanics. The mass
balance equation used in formulating the sand production problem is
typically written as
( ) mt
&& =+ u (1) where state variables , u& are the
density and the absolute velocity respectively, and m& is the
source or sink term to account for the local rate of solid loss or
gain per unit volume due to erosion.
The fluid/gas saturated sand body is idealized as a
Representative Elementary Volume (REV) which comprises of five
phases, namely solid grains (s), fluidized solids (fs), fluid (f),
water (w) and gas (g)
as shown in Figure 1. In reality, the individual distribution
varies discontinuously over space. However, an averaging procedure
in the spirit of mixture theory is used to homogenize each
constituent over the REV volume V such that these individuals are
substituted with continuous ones that fill the whole volume. Each
phase discontinuity in the REV is represented in terms of its own
volume fraction, i.e. saturation and porosity.
fluid
solid
fluidized solid
(f) Mf , f , dVf(fs) Mfs , fs , dVfs(s) Ms , s , dVs
dVvdV
Phase diagram
fluidized solidsfree & disolved gas
fluid
wellbore
REV
sand, oil,
yxwormhole
gas (dg+fg)(g) Mg , g , dVg
solids
gas cavity or
Fig. 1 Phase components of a REV
For solid phase (s), the density of the solid phase averaged out
over a REV of volume dV can be written as the homogenized solid
density (1-)s , where porosity dV
dVV= , and s is the density of the solid phase. The mass
conservation requires that
( )[ ] ( )[ ] mt ss
s && =+ u 11 (2)
where su& is the absolute velocity of the solid phase
boundary, and the negative sign of the right hand side refers to a
solid loss due to erosion since m& is chosen to be the local
rate of solid gain per unit volume as seen from the fluidized solid
phase.
Similarly, for the fluidized solid phase (fs), the mass balance
equation can be written, i.e.
[ ] [ ] mSt
Sfsfsfs
fsfs && =+
u (3)
where the fluidized solid saturation at reservoir condition (RC)
is [ ][ ]RCV RCfsdV
dVfsS = , fsu& is the absolute
velocity of the fluidized solid phase, and fs is the density of
the fluidized solid phase.
The basic assumptions for flow of oil, water and gas phases
follow those used in the classical black-oil model [8]. The oil
phase (o) continuity equation can be derived at stock tank
condition (STC), i.e.
[ ] [ ] 0// o =+
oooooo BS
tBS u& (4)
-
where =o fluid density at stock tank condition, [ ][ ]RCV
RCoVV
oS = = oil saturation in reservoir condition (RC), [ ][ ]STCo
RCdgoV
VVoB
+= = the formation volume factor, and =ou& the absolute
velocity of the oil phase. Furthermore, the averaged density of gas
can be divided into two components: free gas ggg BS / and dissolved
gas og S , where [ ][ ]RCV RCgV
VgS = ,
[ ][ ]STCg
RCg
VV
gB = , gBRg os = , =g the gas density at stock tank condition,
and [ ][ ]STCo STC
dg
VV
sR = . Hence, the mass balance for the gas phase is written,
i.e.
[ ][ ] 0//
//
=+++
oogosgggg
ogosggg
BSRBSt
BSRBS
uu &&
(5)
Since the water is assumed not to partition in either the
hydrocarbon liquid or the gas phase, the mass balance for the water
phase is given as
[ ] [ ] 0// =+
wwwwwww BS
tBS u& (6)
where [ ][ ]RCVRCw
VV
wS = , [ ][ ]STCo RCwVVwB = can be related to a function
involving water phase pressures.
In the above, the velocities ou& , gu& and wu& are
defined somewhat differently from what is customary done in the
multiphase flow literature. They are interstitial velocities, based
on an assumption that the flow area Aj for the any phase j is equal
to the total pore (void) area AV times the phase saturation Sj.
Therefore, the absolute velocity
ju& is related to Darcy velocity jv (see Eq. (9) that follow
in the next section).
2.2. Equilibrium equation for the solid matrix The interaction
between the mechanical behaviour of a deforming solid matrix and
fluid dynamics must be incorporated into the governing equations in
order to describe the coupling effects. The volume-weighted solid
velocity su& provides the linkage between the fluid and
geomechanical aspects of the problem. The latter involves a
deforming sand skeleton under an effective stress field eff and the
volume-averaged pore mixture pressure Pm, which must satisfy
momentum balance, i.e.
( ) 0=+ b1meff P (7) where b are body forces per unit volume,
and is a parameter accounting for the compressibility of the sand
grains. The sign convention adopted is that negative stresses are
compressive and fluid pressures are always positive. The Kronecker
delta tensor is given by 1 such that ijij =1 . The averaged mixture
pressure can be defined as
wwggoom PSPSPSP ++= (8) 2.3. Discharge for each phase In
anticipation for the description of fluid flow through a porous
medium, a volume averaged discharge velocity jv (j= o, w, g) of
each fluid phase relative to the solid matrix (Darcy velocity) is
defined as
)( suuv && = jjj S (9) Both the detachment and
fluidization of solid particles are a dynamic process that is
complex in nature. It is a future research task to define the
interaction between fluidized particle and fluid at a micro/macro
level. However, the discharge of fluidized solid phase can be
related to the average velocity of mixture, i.e.
)( smfsfsfsfs SS uvuv && == (10) where the average
velocity of mixture is
wwggoom SSS vvvv ++= (11) Eqs.(2-6) represent local mass balance
equations for each individual phase. Successively combining these
equations with Eqs.(9-10), the following five governing equations
are obtained for each phase, i.e.
( )[ ] mt s
&& =+ u1 (12)
[ ] ( )[ ] 01)1( =+++ sfsmfsfs SStS uv & (13) 0. =
+
+
o
o
o
so
o
o
BS
tBS
Buv & (14)
0. =
+
+
w
w
w
sw
w BS
tBS
Buvw & (15)
-
0//
=
+
+
+++
g
g
o
os
o
sosoos
g
sggg
BS
BSR
t
BSRBR
BS
B
uvuv && (16)
2.4. Constitutive laws Eqs.(12-16) must be supplemented with
constitutive laws describing sand particle erosion, fluid flow, and
deformation of the sand matrix. It is commonly believed that the
driving force causing the solid detachment from the sand matrix is
due to hydrodynamics and geomechanics. Based on phenomenology, a
possible functional form of mass generation can be obtained from
the inverse of filtration theory as proposed in refs. [9,10],
i.e.
crmm
crmmmfs
s
Sm
vv
vvv
-
potential according to Eq.(18). In return, the erosion process
also weakens the sand matrix through degradation of its strength
properties, see Eq.(22).
In order to complete the derivation of governing equations, we
have to define the capillary pressure Pc relationship. The most
practical method is to use an empirical correlation relating the
capillary pressure and phase saturations [8], i.e.
),(),(
0
0
gowcog
wowcow
SSfPPPSSfPPP
====
(23)
In conclusion, we have eight equations for solving eight field
unknowns, namely,
jfs PS ,, ),,( wgoj = and iu )3,2,1( =i in the three-dimensional
case.
3. STABILIZED FINITE ELEMENT SOLUTIONS
Although the writing of the governing equations is rather
straightforward, both their finite element discretization and
solution are challenging due to the nature of the equations and
field variables. Numerical instability arises in terms of
node-to-node oscillations. Over the past several years, the authors
developed a generic numerical stabilization scheme - an optimized
local mean technique. By enriching main field variables with high
gradient terms, sharp non-local changes can be captured in the
computations to ensure stable solutions. Then, the enriched field
variables enter into the governing equations of physics by way of
averaging of the field values in the neighbourhood of a continuum
point, see details in [11]. Thereafter, the finite element
discretization of the modified governing equations is ready to be
expressed in terms of variables V, i.e. the nodal displacement
)3,2,1( =iiu , phase pressure ),,( wgojj =P , porosity , and
fluidized sand saturation fsS .
)()(),( tNt k VxxV = (24) where V stands for fspS , p , jpp ,
ipu , and pN are respectively fluidized solid saturation, porosity,
fluid pressure, displacement, and interpolation function at node p,
for p=1 to hn , the total number of nodes. It is again recalled
that Einstein index notation is used with repeated indices implying
summation and the index p is dummy. Applying Galerkins method of
weighted residual (with
weighting functions equal to interpolation functions) over the
entire domain to above governing equations in turn together with
discretizing time derivatives by standard finite difference formula
and also linearizing time variables, a system of five non-linear
equations is obtained with its generic form, i.e.
)()( 111 nnnn VHVW +++ = (25) in which W and H are functionals
which originate from Eqs.(12-16) and subscripts n and n+1 refer to
time stations nt and 1+nt respectively. Eq.(25) represents the
standard non-linear matricial equations that can be solved via
iterative schemes such as the Newton-Raphson method. If superscript
k denotes the iteration number during successive attempts to final
solution, then expanding Eq.(25) using the Taylors series leads
to
)()( 111
11knn
kn
k
n
kn
kn VHVV
WVW +++
++ =+ (26)
Hence, the increment of vector V at the end of iteration k
is
[ ] [ ])()( 111111 knknknnknkn +++++ = VWVHJV (27) in which Jn1k
is the Jacobian of the linearized system, i.e.
k
n
kn
11
++
=VWJ (28)
Successive iterations are performed until the convergence
criteria are satisfied, i.e.
-
sub-matrices pertinent to fluid, solid, fluidized solid, and
stress-deformation properties [12]. The procedures of
Newton-Raphson algorithm are listed in Table 1. From a practical
point view, we have to address properly the various coupling
strategies, i.e. decoupling, explicit, and implicit coupling
techniques before proceeding with the fully coupled
reservoir/geomechanics simulation [13]. Table 1. Procedures for
Newton-Raphson scheme
1. Set the initial value k=0 and initial values for each
variable 2. Calculate the Jacobian matrix knJ 1+ according to
Eq.(28) 3. Calculate the right hand side X in Eq.(30) 4. Solve
Eq.(30) 5.Check for convergence IF: Eq.(29) is satisfied THEN Go to
next time step ELSE Go to : 2 with new trial value for each
variable and k=k+1 ENDIF
4. NUMERICAL EXAMPLES
In the following simulation, a numerical example of a light oil
reservoir in North Sea is examined under hydrodynamics and
geomechanics, while examples in heavy oil reservoirs can be found
in a series of publications [5, 6, 7]. In this paper, no gas phase
effect is presented, given the space restriction.
x(m)0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
perforations
extends to 5 m
P1P2
P3
Fig. 3. Mesh layout near wellbore showing perforations.
Figure 3 shows a close-up of the finite element mesh
representing one quarter of a section of a vertical well of inner
radius 1.00 =r m with the outer boundary of the well extending to 5
m. The initial fluidized sand saturation Sfso and porosity 0
are chosen to be 0.001 and 0.25 respectively. The simulation is
conducted as follows. First, the initial state of the reservoir is
computed based on an oil saturation pressure of 27.6 MPa and an
external stress of 42 MPa is imposed on both wellbore and outer
boundaries. Then, the stress around wellbore is changed to a
reservoir pressure of 27.6 MPa to simulate the open-hole
completion. Finally, a 3 MPa drawdown is applied at three
perforations (P1, P2, and P3) as shown in Figure 3. The length of
each perforation is 0.25 m with a 0.012m diameter for P2, and a
0.006m diameter for both P1 and P3. These, in fact, refer to eight
perforations for the full well configuration. The initial porosity
and erosion coefficient in the perforations are set to 0.6 and 3
m-1 respectively to account for the disturbance caused by the
perforation process, while they are set to 0.25 and 2 m-1 in the
remainder part of the reservoir formation. Finally, the entire
finite element grid is comprised of 3840 nodes and 3705 4-nodes
elements and the time step size used in the analysis is 0.005 day
for a total time span of 5 days investigated. Table 2 shows the
material properties (fluid and geomechanics) used in the
simulation.
Table 2. Model parameters 0 = 2 or 3 m-1 s = 2.7 g/cm3 o = 0.8
g/cm3 K0x = 0.5 Darcy K0y = 0.1Darcy = 5 cp C0 = 6 MPa E = 2 GPa =
0.25 0 = 30 ext = 42MPa P0= 27.6 MPa =0.008 =0.1
For the purpose of clarity of illustration, the figures are
plotted in the vicinity of the wellbore, within the first 1 m, 2 m
and 5m as indicated in XY axes respectively.
4.1. Deformations and yielding after open-hole completion and
perforations
In order to examine the wellbore instability and sand
production, it is essential to understand the open-hole completion
and perforation process. The process is simulated by lowering the
initial stresses 42 MPa at inner holes to the initial reservoir
pressure and the outer ones are kept to initial stress conditions
after reservoir initialization.
It is noted that a plastic zone is developed as shown in Figure
4. This is due to the stress re-distribution around wellbore and
the existence of a weakened zone in the perforations (0=0.6) during
the drilling process. It is critical to capture the developed
plastic zones due to drilling and perforation, since the
-
erosion coefficient is linked to plastic shear strain as defined
in Eq.(18) - the larger the plastic shear strains are, the more
intensive the erosion activity is. This enables the simulator to
automatically capture the disturbance caused by open-hole
completion and perforation in terms of the initial values of
erosion coefficient and porosity around wellbore and perforations
at the beginning of the drawdown.
x(m)
y(m
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
P1P2
P3
Plastic yielded zones
Fig. 4 Plastic yielded zones developed after open-hole
completion and perforations (before drawdown).
4.2. Evolution of fluidized sand saturation From this section
on, we look at the field variable profiles due to drawdown. Figures
5-7 illustrate the spatial distribution of the fluidized sand
saturation Sfs at four different times t=0.3 day, 0.6 day, 2 days
and 5 days after drawdown. It is noticed that a sharp rise in
fluidized sand saturation develops in the region near the
perforations P1 and P2 with the remaining part of the well being at
near initial values of Sfso. The amplification factor for fluidized
sand saturation near the perforation, defined as the current
saturation value over the initial one, is about 70 times at
location P1 for time t=0.3 day, 110 times at location P2 for time
t=0.6 day, and 140 times at location P3 for time t=5 days
respectively. These numbers indicate that there is a dramatic
increase in the creation of fluidized sand corresponding to sand
production. In general, an increase in fluidized sand saturation is
governed by the relative rates at which volume of fluidized sand
Vfs and void volume VV are changing, since Sfs = Vfs/VV. This sharp
change is due to the physics of the problem described as follows.
Initially, erosion preferentially occurs in the x-direction
near
x(m)
y(m
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.140.130.120.110.110.100.090.080.070.060.050.040.030.020.01
time=0.3days
P1P2
P3
Fig. 5 Fluidized sand saturation profile at time t=0.3 day.
x(m)
y(m
)
0 0.5 1 1.5 20
0.5
1
1.5
2
0.140.130.120.110.110.100.090.080.070.060.050.040.030.020.01
time=0.6days
Fig. 6 Fluidized sand saturation profile at time t=0.6 day.
x(m)
y(m
)
0 1 2 3 4 50
1
2
3
4
5
0.140.130.120.110.110.100.090.080.070.060.050.040.030.020.01
time=2days
Fig. 7 Fluidized sand saturation profile at time t=2 days.
-
perforation P1 since the horizontal permeability is five times
greater than the vertical one. As most of the sand particles are
mobilized to produce a very loose matrix, further erosion takes
place in regions where more sand particles are available.
Figure 8 shows a decreased fluidized sand saturation profile,
which indicates a decline in erosion activity because there is no
material left for the erosion around wellbore.
x(m)
y(m
)
0 1 2 3 4 50
1
2
3
4
5
0.140.130.120.110.110.100.090.080.070.060.050.040.030.020.01
time=5days
Fig. 8 Fluidized sand saturation profile at time t= 5 days.
4.3. Evolution of erosion coefficient and cavity propagation
As defined in Eq.(17), the erosion coefficient is a function of
plastic shear strain. This indicates that most erosion activity is
confined and intensified in only plastic shearing regions. The
larger the plastic shear is, the more intensive the erosion is. In
other words, the erosion activity aligns itself with the plastic
yielded zones where plastic shearing of the material is most
prevalent. Figures 9-11 show the distribution of erosion
coefficient with time around the wellbore. The erosion activity is
most intense around the wellbore and perforations at the very
beginning, and then propagates further inside the perforations
where the sand matrix has a weak material strength (initial
porosity 0.6), and in the x-direction where the pore pressure
depletion is the fastest due to high permeability in x-direction
initially. This is due to increasing erosion activity taking place
as porosity increases and ultimately degrades the material
strength. These will be discussed in later sections.
Figure 12 shows the initiation of erosion at the perforations at
time t=0.3 day. In fact, at the edges
of wellbore and perforations, very high fluid fluxes prevail,
which in turn give way to high fluidized
x(m)
y(m
)
0 0.5 1 1.5 20
0.5
1
1.5
2
12.0011.2910.57
9.869.148.437.717.006.295.574.864.143.432.712.00
time=0.3days
Fig. 9 Erosion coefficient distribution at time t=0.3 days.
x(m)
y(m
)
0 0.5 1 1.5 20
0.5
1
1.5
2
12.0011.2910.57
9.869.148.437.717.006.295.574.864.143.432.712.00
time=0.6days
Fig. 10 Erosion coefficient distribution at time t=0.6 day.
x(m)
y(m
)
0 0.5 1 1.5 20
0.5
1
1.5
2
12.0011.2910.57
9.869.148.437.717.006.295.574.864.143.432.712.00
time=5days
Fig. 11 Erosion coefficient distribution at time t=5 days.
sand mass fluxes as dictated by the erosion law, see Eq.(16).
However, the maximum erosion activity
-
does not start simultaneously at all perforations as shown in
Figure 12. In fact, the most intensive erosion activity follows
geomechanically yielded zones and a preferential direction of high
flux, i.e. x-direction. Figure 13 shows the coalescence of eroded
zones around perforations P1 and P2 into a ring of loose sand of
about 0.5 m in radius. The porosity values approach 0.77 and
physically correspond to the formation of a cavity and mechanical
failure of the wellbore. Figure 14 shows a snapshot of the fully
developed zone of high porosity that is initiated at the
perforations, and which localizes along the plastic yielded zones
and high flux regions.
x(m)
y(m
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.770.730.700.660.630.590.560.530.490.460.420.390.350.320.28
Porositytime=0.3days
Fig. 12 Porosity profile at time t=0.3 day.
x(m)
y(m
)
0 0.5 1 1.5 20
0.5
1
1.5
2
0.770.730.700.660.630.590.560.530.490.460.420.390.350.320.28
Porositytime=0.6days
Fig. 13 Porosity profile at time t=0.6 day.
4.4. Fluid flux and pressure distribution As the cavity
enlarges, the permeability of the reservoir increases since it is a
function of porosity in Eq.(19). The gradually increased
permeability enhances the well productivity. It is expected
that
x(m)
y(m
)
0 1 2 3 4 50
1
2
3
4
5
0.770.730.700.660.630.590.560.530.490.460.420.390.350.320.28
Porositytime=5days
Fig. 14 Porosity profile at time t=5 days.
x(m)
y(m
)
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
time=0.3days
P2
P1
P3
Fig. 15 Fluid flux profile at time t=0.3 days.
x(m)
y(m
)
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
time=0.6days
P2
P1
P3
Fig. 16 Fluid flux profile at time t=0.6 days.
the high fluid flux dominates in three perforations in Figure 15
at the beginning of drawdown. Then, the
-
direction of large fluid fluxes shows a bias towards high
porosity regions as shown in Figure 16, i.e. mostly x-direction in
anisotropic permeability case. It is also worth to mention that the
erosion process increases the fluid flux by degrading the sand
matrix where more regions progressively yield plastically due to
the high fluid flux and stress redistribution. Figure 17 shows an
increased flux region around the wellbore at time t=5 days.
x(m)
y(m
)
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
time=5days
P2
P1
P3
Fig. 17 Fluid flux profile at time t=5 days.
Due to the initial anisotropic permeability conditions, the
dissipation of fluid pressures around the well also occurs in
regions of high permeabilities, i.e. x-direction. As sand is being
produced, the fluid pressure slowly depletes more from initial
values of 27.6 MPa on the outside boundary to 24.5 MPa than at
perforations P1, P2, and P3 around the wellbore, as shown in Figure
18.
x(m)
y(m
)
0 1 2 3 4 50
1
2
3
4
5
2.74E+072.72E+072.70E+072.69E+072.67E+072.65E+072.63E+072.61E+072.59E+072.57E+072.55E+072.54E+072.52E+072.50E+072.48E+07
(Pa)Time=5days
Fig. 18 Pore pressure distribution at time t=5 days.
4.5. Displacements and stresses In this section, we look at the
plastic shear strain and stresses distribution in the well. The
pressure induced drag forces develop excessive plastic shear
strains around perforations in both x- and y- direction (maximum
value is about 9% after 5 days in Figure 19). It is also noted that
the material strength parameters, i.e. cohesion C and friction
angle follow the same distribution as that of porosity with time
since they are defined as a linear function of porosity in
Eq.(22).
x(m)
y(m
)
0 0.5 1 1.5 20
0.5
1
1.5
2
0.0900.0860.0810.0770.0730.0690.0640.0600.0560.0510.0470.0430.0390.0340.0300.0260.0220.0170.0130.0090.0040.0030.0010.0000.000
time=5days
Fig. 19 Plastic shear strain distribution at time t=5 days.
x(m)
y(m
)
0 1 2 3 4 50
1
2
3
4
5
-7.00E+06-7.53E+06-8.05E+06-8.58E+06-9.11E+06-9.63E+06-1.02E+07-1.07E+07-1.12E+07-1.17E+07-1.23E+07-1.28E+07-1.33E+07-1.38E+07-1.44E+07-1.49E+07-1.54E+07-1.59E+07-1.65E+07-1.70E+07
(Pa)
time= 5 days
Fig. 20 Effective stress xx at time t=5 days.
Considering the wellbore stability, it is very important to look
at the stress distribution after sand production. Figures 20-22
show the distribution of effective stresses xx, yy, xy at 5 days
after drawdown. Due to fluid pressure reduction through three
perforations, drag forces are imposed upon three perforations,
causing a reduced stress xx in P3
-
whereas an increased stress xx around P1 in Figure 20. Also, the
stress yy is reduced in P1 and increased around P3, as shown in
Figure 21. Figure 22 shows the tangential stress profile
distribution. The high stress values indicate a highly sheared
zone. Depending on the re-distribution of pore pressure and stress
during erosion, the high shear stress zone shifts and grows, which
in turn causes the evolution of plastic shear yielded zones.
x(m)
y(m
)
0 1 2 3 4 50
1
2
3
4
5
-7.00E+06-7.53E+06-8.05E+06-8.58E+06-9.11E+06-9.63E+06-1.02E+07-1.07E+07-1.12E+07-1.17E+07-1.23E+07-1.28E+07-1.33E+07-1.38E+07-1.44E+07-1.49E+07-1.54E+07-1.59E+07-1.65E+07-1.70E+07
(Pa)
time=5 days
Fig. 21 Effective stress yy at time t=5 days.
x(m)
y(m
)
0 1 2 3 4 50
1
2
3
4
5
3.00E+062.84E+062.69E+062.53E+062.38E+062.22E+062.07E+061.91E+061.76E+061.60E+061.45E+061.29E+061.14E+069.82E+058.26E+056.71E+055.16E+053.61E+052.05E+055.00E+04
(Pa)
time=5 days
Fig. 22 Effective tangential stress xy at time t=5 days.
4.6. Volumetric sand production and oil rates In the previous
sections, detailed spatial distributions of governing field
variables with time were discussed and the analysis revealed local
phenomena during sand production. From an engineering point of
view, we would be interested in examining the total oil and
volumetric sand production rates as integrated over the total
perforation area S (P1, P2, and P3) of the wellbore. Hence,
dSSqdSqS ffssandS foil == vv ; (31)
Figure 23 gives both the oil and sand rates over the time of
fluid drawdown. We observe that the sand production rate rapidly
increases in an initial phase to reach a peak value in
approximately 0.5 day. During this time period, the oil rate
gradually increases as well. Then, this phase is followed by a
decline in sand production rate corresponding to the decrease in
availability of sand grains. However, the oil rate continues to
increase given the enhancement in permeability of the reservoir
induced by sand production. This trend is also observed in oilwells
under sand production.
0
2000
4000
6000
8000
10000
12000
0 1 2 3 4 5 6
time (days)
oil r
ate
(kg/
day/
m)
0
200
400
600
800
1000
1200
sand
rat
e (k
g/da
y/m
)
oil ratesand rate
Fig. 23 Oil and sand rate history at anisotropic permeability
conditions.
0
5000
10000
15000
20000
25000
0 1 2 3 4 5 6
time (days)
oil r
ate
(kg/
day/
m)
0
500
1000
1500
2000
2500
3000
sand
rat
e (k
g/da
y/m
)
oil ratesand rate
Fig. 24 Oil and sand rate history at isotropic permeability
conditions.
As a comparison, an initial isotropic permeability case is also
computed with kx0=ky0=0.5 Darcies. As expected, more sand and
higher oil rates are obtained as larger initial reservoir
permeability prevails in y-direction, see Figure 24. The same peak
value of fluidized sand saturation is calculated, but a smoother
decline curve of sand rate is obtained in isotropic case, since
there is no erosion lag due to anisotropic permeability
conditions.
-
5. CONCLUSIONS
A fully coupled reservoir/geomechanics numerical model is
presented based on an extension of a theoretical and numerical
model that the authors have developed in the past to address sand
production as an erosion problem coupled with hydro- and
geo-mechanical effects. This is done within the framework of
mixture theory in which mechanics and transport equations are
written for each of the concerned phases, i.e. solid, fluid (oil,
water), gas, and fluidized solid.
Leaving aside gas-related issues, it is found that sand
production is a function of stress, time, and fluid rate. Sand
erosion activity is strongly linked to geomechanics and there is an
intimate interaction between sand erosion activity and deformation
of the solid matrix. As the erosion activity progresses, porosity
increases and in turn degrades the material strength. Strength
degradation leads to an increased propensity for plastic shear
failure that further magnifies the erosion activity. An escalation
of plastic shear deformations will inevitably lead to wellbore
instability with the complete erosion of the sand matrix. The
self-adjusted mechanism enables the model to predict both the
volumetric sand production and the propagation of wormholes.
The multiphase results including gas phase will be presented in
a forthcoming paper. The proposed model can be used for wellbore
stability analysis and design in open-hole completions, perforation
pattern design, as well as volumetric sand prediction at different
pumping strategies in terms of optimization of the hydrocarbon
production.
6. ACKNOWLEDGEMENTS
The authors wish to express their sincere gratitude for funding
provided by Alberta Ingenuity Fund (AIF) and the National Science
and Engineering Research Council of Canada (NSERC).
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