Prediction of Sand Production on Pressure versus Flow Rate (IPR graphs)

1. Applying Geo-mechanics to Well Modelling and Inflow Performance Relationships
Daniel Chia, SPE, Chia Up Engineering.
Abstract:
By the application of a mathematical substitution of the sanding lines (safe or critical
drawdown relationship - Pwf vs Pres) into an inflow performance relationship results in an
equation which projects the sanding line on the graph of Pwf (bottom-hole flowing
pressure) vs Q (flow rate) called the “Chia Line.” The “Chia Line” can help determine the
operational conditions of a well bore or visually understand when the well bore will
produce sand relative to reservoir performance, tubing design or surface wellhead
pressure conditions and impacts due to sand screens.
Introduction:
Geo-mechanics and Petroleum
Engineering are currently considered
two distinct specialties. Simple
mathematics and models are used to try
to demonstrate the usefulness
application of predictive models to help
bridge the gap between Geo-mechanics
and Petroleum Engineering to
understand sand production and when
sand control is required.
Gas Deliverability
The inflow performance can be
estimated for a gas reservoir using the
C & n empirical back pressure equation
by Rawlins and Schellhardt (ref. 6):
Q = C(Pres2-Pwf2)n eqn. 1
Q is the flow rate (MMscf/day), C is the
flow coefficient and n is the deliverability
exponent. Pwf is the bottom-hole flowing
pressure at the current conditions (psia)
and Pres is the reservoir pressure at the
current conditions (psia). Other inflow
models can be used to substitute the
sanding lines. But in this paper Rawlins
and Schellhardt was used due to the
simplicity.
Sanding Lines - Safe or Critical
Drawdown Relationships
Safe or critical drawdown relationships
ideally require rock properties from core,
pore pressure, stress magnitudes,
stress direction, well bore azimuth, and
well bore deviation to help predict when
a well will produce sand.
Substituting the following sanding line
(Pwf vs Pres) into the C & n empirical
back pressure equation (ref. 1):
Pwf = mPres + k eqn. 2
Where m is the gradient of the sanding
line and k is the intercept of the sanding
line.
The result is the following (eqn. 1 + eqn.
2):
QC = C[Pres2-(mPres + k)2]n
or in Pwf form

2. QC = C
eqn. 3
The resultant equation projects the
sanding line on to the Q vs P graph
resulting in the “Chia Line” (QC).
Fault Regimes
In a three dimensional space (x, y, z)
where there are three perpendicular
forces/stresses acting on a body (Shmin,
SHmax, Sv), given Sv is the vertical stress
(z direction). Shmin is the minimum
horizontal stress (x direction). SHmax is
the maximum horizontal stress (y
direction). In this space there are
fundamentally 3 modes of failure or
faulting in a three dimensional space.
Normal, Strike-slip or Reverse faulting
(ref. 5).
Normal Faulting is when Sv is greater
than SHmax and SHmax is greater than
Shmin.
Strike-slip Faulting is when SHmax is
greater than Sv and Sv is greater than
Shmin.
Reverse Faulting is when SHmax is
greater than Shmin and Shmin is greater
than Sv.
Assuming the rock body is the Earth and
considering a vertical well, Sv is
obtained from the integral of the density
log over the given depth. Shmin is from a
leak of test (LOT) (x direction). SHmax is
estimated from the “hoop stress”
equation (SHmax ≥ 3Shmin - Pp - Pmud);
where the well bore is at a static
pressure condition and the pore
pressure equals the drilling mud weight
(Pp = Pmud), this estimated maximum
horizontal stress is the minimum stress
required to cause drilling induced tensile
fractures, so in essence the real
maximum horizontal stress could be
g r e a t e r t h a n t h i s e s t i m a t e d
mathematical derived value.
Failure around a well bore is known as
borehole breakouts and tensile fractures
which are orthogonal to each other.
Tensile fractures occur in the direction of
SHmax while borehole breakouts occur at
Shmin. For deviated wells the cross
product in three dimensions is used to
determine the new SHmax and new Shmin
relative to the deviated well bore and
then solve for stress around the well
bore (e.g. every 5 degrees) to find the
new SHmax and new Shmin.
Mohr-Coloumb Failure Theory
Mohr-Coloumb Failure is the theory
used to understand simple rock failure
which leads to sanding:
CFF = τ – µσ eqn. 4
CFF is the Coloumb Failure Function, µ
= tan(φ) coefficient of friction, σn is the
normal force to the plane and τ is the
shear stress.
To understand this theory consider an
angled plane with an angle φ and a
block of wood with a normal force (it’s
own weight) exerted on the plane σn and
(
Pwf − k
m )
2
− Pwf
2
n

3. the friction acting between the wood and
surface of the plane τ. When the angle
is at a high enough angle to overcome
the friction, the wooden block will slide
down the plane. This is fundamentally
how we describe rock failure which
produces sand. So forces act on a body
until failure then a new equilibrium is
formed after the stress has been
released e.g. the block of wood is lying
flat at rest some distance from the
origin. Sliding occurs when µ = τ/σn.
Fig. 1 – Wood block on an angled plane
The three dimensional stresses can be
drawn on a Mohr circle (Sv, SHmax, Shmin),
figure 2. When any of these three
stresses increase to where the Mohr
circle circumference exceeds the rock
strength (e.g. line with slope µ) then the
rock will break, e.g., the red dashed
semicircle. The least and greatest stress
magnitudes are used for failure criteria
described by the following equation:
% = %
S1 and S2 are the extreme stress fields
acting on a rock or can be drawn on the
Mohr circle.
Fig. 2 – Mohr Circle Diagram Rock
Strength (y axis) vs Stress (x axis)
Typically rocks have a µ = tan(φ)
coefficient of friction between 0.6 and 1
(ref. 5). Given the tectonic environment
estimates of stress magnitudes can be
used (ref. 5). For extensional Shmin =
0.6 Sv, for reverse faulting SHmax = 2.3
Sv, for strike slip environments Sv =
½(SHmax + Shmin), SHmax = 2.2
Shmin.
For a practical case which consists of
consolidated rock or core, an extra
element of cohesion is added.
CFF = τ – µσ + CO eqn. 5
CO is the cohesion with in the rock.
Note some rocks do not conform to pure
linear failure as described by Mohr-
Coloumb Failure. Other models can be
used from Material Failure Theory to
model deviations.
Effective Stress
The effective stress (σ’) acting on a
porous medium is the difference of the
total stress (σT) and the pore pressure
(Pp) of the fluid in the medium. Given by:
σ’ = σT - δijαPp
S1 − Pp
S2 − Pp
[(μ^2 + 1)
1
2 +μ ]^2
σn
µ = slope
Co = Cohesion
= y-intercept
S1 S3 S2
φ
τ

4. Where α is Biot’s constant α = 1 – %
and δij is the Kronecker delta. When
Biot’s constant is equal 1 then maximum
pore pressure influence occurs on the
porous solid, while for Biot’s constant
equal to 0 then there is no pore
pressure influence on the solid e.g. no
pore space in the rock. Usually the Biot
constant is assumed to be 1. And the
Kronecker delta is the tensor or vectors
in space. This effective stress can be
thought of as the force that keeps the
particles rigid. When injecting fluid into a
well the effective stress decreases but
when fluid is withdrawn the effective
stress increases. This relationship is
used to derive the stresses in the well
bore and for changes due to pore
pressure.
Tangential Stresses at the Well Bore
For a vertical well in Earth with
perforation tunnels the sanding line
(critical drawdown pressure relationship
- CDR) can be described by (ref. 1)
CDR = x [2Pres – (3SHmax – Shmin
–U)] eqn. 6
Note that equation 6 is different to fault
regimes. Where A is the Poroelastic
constant A = % , ν is Poisson’s
constant and α is Biot’s constant. Typical
Poisson’s constant are as follows cork is
0, rubber is 0.5, concrete is 0.2, steel is
0.3 and sands are typically assumed to
be 0.25 or calculated from seismic
velocities (A% ), where
Vp and Vs are p-wave and s-wave
velocities respectively (ref. 5). U is
effective formation strength, a loading
factor is used which increases the rock
strength due to the configuration of the
perforation tunnels relative to the well. U
increases with increasing stress given
by the figure 2 with a slope of µ.
Unconfined strength tests (UCS) and
thick walled cylinder tests (TWC) are
used to determine the effective
formation strength. A UCS test is used
to determine the rock strength with one
dimension of stress applied to the rock
core. With side wall confining pressure a
tri-axial shear test can help determine
the Mohr Circle of the rock core, with
increasing side wall applied pressure
the strength of the rock increases. As a
rule of thumb TWC which best
represents a perforation tunnel in a well
with steel casing can be 3 to 7 times the
UCS (ref. 1).
Since CDR = Pres – Pwf eqn. 7
Substituting equation 6 into equation 7
the result is
P w f = P r e s +
% eqn. 8
Equation 8 can be written in the form of
equation 2 and be substituted into
equation 3.
Cr
Cb
1
2 − A
α(1 − 2v)
1−v
=
Vp
2
− 2Vs
2
2(Vp
2
−Vs
2
)
−A
(2 − A)
3 SHmax – Shmin – U
2 – A

5. QC = C[Pres2-(mPres + k)2]n
or in terms of Pwf
QC = C
m = % eqn. 9
k = % eqn. 10
C is the flow coefficient and n is the
deliverability exponent.
Application of Geo-mechanics to
Inflow Relationships
Two cases are shown below to illustrate
the applications of Geo-mechanics to
Petroleum Engineering. Assumptions
are used in the following cases where
there is no experimental data however it
is always better to take measurements
to model the formation as accurately as
possible or carry ranges of probable
numbers to help understand what could
happen down hole and tune/calibrate to
real time data. In the following
examples, assumptions have been
used, and as such the temperature
changes are assumed negligible and do
not affect the stresses.
Given a well bore with a UCS = 10000
psia at Pres, depth = 10000 ft and Pres =
5000 psia with C = 0.0001 MMscf/day/
psi, n = 0.9, α = 1 and ν = 0.25. Assume
that no changes in stresses due to pore
pressure, a rock density of 2.8 gm/cc
(1.21342 psi/ft) and that the well is in an
extensional system.
Sv = 10000 ft x 1.21342 psi/ft = 12134
psig = 12149 psia
Shmin = S2 = 0.6 x Sv = 0.6 x 12149 psia
= 7289 psia
SHmax = S1 = 3 x Shmin – 2 x Pres = 3 x
7289 psia – 2 x 5000 psia = 11868 psia
Consider Case 1 open-hole well and
assume the UCS = U, then
QC = C[Pres2-(mPres + k)2]n
QC = C
A = % = 1(1 – 2 x 0.25)/(1 –
0.25)
A = 0.5/0.75 = 2/3 = 0.66
m = % = -0.66/(1.33) = -0.5
k = %
k = (3 x 11868 – 7289 – 10000)/(1.33)
k = 13736
QC = 0.0001[Pres2-(-1/2Pres + 13736)2]0.9
And the sanding line is given by:
Pwf = mPres + k = -0.5Pres + 13736
(
Pwf − k
m )
2
− Pwf
2
n
−A
(2 − A)
3 SHmax – Shmin – U
2 – A
(
Pwf − k
m )
2
− Pwf
2
n
α(1 − 2)
1−
−A
(2 − A)
3 SHmax – Shmin – U
2 – A

6. %
Fig. 3 – Case 1 Sanding Line for an
Open-hole Well Pwf = -0.5Pres + 13736
In figure 3, the red area (sand
production area) is where the open-hole
well will fail and produce sand while the
green area (safe drawdown area) is
where the well will not produce sand.
We can see that the operating range is
in the red area and will sand on first
production i.e. reservoir pressure is
5000 psi.
Consider Case 2 cased-hole well with
perforations and the TWC = 3 x UCS =
U
k = %
k = (3 x 11868 – 7289 – 3 x 10000)/
(1.33)
k = -1264
QC = 0.0001[Pres2-(-1/2Pres - 1264)2]0.9
Pwf = mPres + k = -0.5Pres -1264
In figure 4, the red area is nowhere to
be seen and the cased-hole well will not
produce sand in regards to sensible
operating combinations.
%
Fig. 4 – Case 2 Sanding Line for Cased-
hole Well with Perforations Pwf = -0.5Pres
-1264
!
Fig. 5 – Q vs BHP relationships Case 1
vs Case 2
In figure 5, the “Chia Line’s” are green
for U = 30000 psia and red for U =
10000 psia on the bottom hole flowing
pressure (BHP) vs gas flow rate (Q). We
3 SHmax – Shmin – U
2 – A

7. can see where sanding will occur
relative to the blue inflow performance
relationship (IPR). Because it is below
the red line (U = 10000 psia)
representing the sanding area and will
produce sand for any given operating
range. But for the green line due to the
quadratic nature of the C & n equation
there is an artefact, as on figure 4 no
sanding occurs.
!
Fig. 6 – Q vs BHP relationships
Extended Case 2
Figure 6 shows both solution curves
extended down into negative numbers
meaning additional pressure/force is
required to fail the rock as seen from
Figure 4. With a line of symmetry on the
bottom-hole pressure (y-axis) at about
-2000 psia the top part of the graph is
removed shown in figure 7.
Figure 7 shows what the two completion
styles compare relative to the IPR,
where the true cased-hole well with
perforations (green line) are out of the
scale of the graph below the sensible
operating ranges. Hence for this case
and assumptions used it is better to
case and perforate the well and it is not
required to install sand screens.
!
Fig. 7 – Q vs BHP relationships
Corrected
Failure to understand the failure mode
and the correct application of the
equations of the well will cause
unwarranted capital and have an impact
on skin caused by incorrectly installing
sand screens. I.e. Case 1 represents a
bare foot completion while Case 2
represents a well cased and perforated.
Case 2 does not require sand screens.
Other Considerations and Scenarios
The example shown has a clear
definition between open-hole vs cased-
hole with perforations. Logically one
would always case and perforate the
well. But there may be instances where
cased-hole with perforations and
operating at lower surface pressures vs
sand screens with skin may offer an
economic trade-off or even producibility
of the reservoir.

8. If no UCS testing was performed on
core then the correlation can be used as
an estimate (ref S. Bruce):
%
; w h e r e % ;
% ; and %
Where Kb is the bulk modulus, E is the
Young’s Modulus, v is Poisson’s ratio,
dtsm is the shear sonic travel time, % is
the bulk density and % is the angle of
which the rock shear occurs, similar to φ
in the Mohr Coulomb Criteria. Note that
Young’s Modulus and other variables
presented in this paper maybe derived
in other ways given different data sets.
Calibration of the data to these curves
and the assumed rock properties,
stresses, load factor, etc. will ensure the
accuracy of the sanding curves. If
deviations are seen in actual measured
vs modelled of the occurrence of sand,
then a linear regression or some other
higher order regression or a mixed
variable regression maybe required.
Liquids surging are not considered in
the equations described in previous
sections and assume a single phase
fluid. Theoretically determination of the
change in flow rate or velocity due to
fluid change (e.g. surge) using D’Arcys
equation or similar flow relationship then
the additional pressure surge could be
added in to the sanding line as a
potential “buffer” or “engineering fudge
factor”.
Water/fluid hammering is also not
considered but theoretically may also be
modelled using Joukowsky equations
and can be compared with fluid
production rates measured at the
separator to determine if fluid
hammering is occurring. And then
determining proper management of
sand production rates vs erosional limits
and operating rates for a given fluid
combinations, i.e. 100 bbls/day of water
with 1 MMscf/day of gas is all the
formation can handle at a given
reservoir pressure and drawdown at a
given acceptable sanding rate.
As previously discussed applications to
completion design with cost benefits can
be applied using and understanding the
“Chia Line.” For example, comparing the
completion skin of sand screens vs
operating within the safe drawdown area
with no sand screens. Here something
like the Houpeurt’s equation can be
used as a skin comparison (ref. 7).
Another application is to see the impacts
of a strike slip vs normal vs reverse
stress and the well bore positioning and
the impact on sanding. This could
change the angle of attack to the
reservoir or the location of a platform or
drill pad.
Overlaying tubing performance
relationships (TPR) e.g. surface
pressures, tubing, etc., and the IPR and
“Chia line” will show holistically the
operating area and when or if sanding
UCS =
0.026aE
Cb106 [0.008Vsh + 0.0045(1 − Vsh)]
a =
2cosθ
1 − sinθ
E =
2ρb(1 + v)
dtsm
Cb =
1
Kb
ρb
θ

9. will occur. Or only consider points where
positive pressures are calculated for Pres
and Pwf and when Pres is less than Pwf.
Projection of the Q vs BHP lines (e.g.
IPR and TPR) can be imposed onto the
Pwf vs Pres sanding lines plot. To
demonstrate the value to Geo-
mechanics subject matter experts
(SME) and can help remove the
confusion over the second “Chia Line”
artefact.
Having simplified the variables that go
into the “Chia Line,” may suffice
however the U actually varies with
stress and is in a linear form assuming
Mohr-Coloumb Failure Theory. Also
changing the Pore Pressure also
changes the effective stress on the
stresses.
Poro-elastic Effect of a Change of
Pore Pressure
Consider formations where changes in
reservoir pressure change the stress
fields (ref. 2). Given L is the length of the
reservoir and h is the height of the
reservoir and assume that L is much
greater than h. Assume that the vertical
displacement is prevented by the
overlying and underlying formations
then the apparent tensile strength
around the well bore can be described
by
σθθ = 2 G (e – eij)
σθθ = -2 G cm [p(r) –
Where G is the bulk shear or rigidity
modulus, eij is the strain component; e is
the relative volume change of the bulk
material.
At r = a
σθθ = Δp = AΔp
where A is the Poro-elastic constant
A = %
This stress change in pressure is added
to the static stress to predict the effects
of stress change due to a reduction in
pressure.
SHmax’ = SHmax - AΔp eqn. 11
Shmin’ = Shmin - AΔp eqn. 12
U’ = z Pwf + y eqn. 13
Where Δp is the change in bottom-hole
pressure from current reservoir
pressure. And z is the gradient of the U’
relationship and y is the y-intercept of
the U’ relationship. SHmax’ and Shmin’
are the changed stresses due to a
change in bottom-hole pressure.
Bottom-hole pressure is used to be
conservative and assumes the near well
bore failure of the formation.
Using equations 3, 9, 10, 11, 12 and 13.
And reapplying to Case 2 assuming U’ =
3(7000 + 0.6 Pwf) and stress changes
due to a change in pressure, then Case
2.1 is as follows:
1
r2 ∫
r
a
p(p)p dp
α(1 − 2v)
1− v
α(1 − 2v)
1− v

10. QC = C[Pres2-(mPres + k)2]n or in terms of
Pwf
QC = C
k = %
Taking into account the variable
pressure effects on rock strength and
stresses the result causes Case 2 to be
modelled as a weaker formation
compared than previously first modelled
see Case 2.1 on Figure 8.
%
Fig. 8 – Sanding Line Case 2 vs Case
2.1
It is important to note that these
equations are models and have inherent
limitations and assumptions and may or
may not represent the reservoir.
!
Fig. 9 – Q vs BHP relationships Case 2
vs Case 2.1
Conclusions:
1. The application of the “Chia Line” to
project the sanding line on the typical
flow rate (Q) vs bottom-hole
pressure (BHP) plot can assist the
P e t r o l e u m E n g i n e e r i n
understanding the sanding regions
and the changes relative to the IPR
and TPR.
2. Impacts on the IPR and TPR in
regards to completion design on
skin, surface operations, etc. can
studied with sanding lines on one
plot.
3. In some cases there maybe two
“Chia Line’s” it is important to
recognise this from the sanding lines
on the Pwf vs Pres (e.g., Figures 1
&2).
4. The completion style changes the
equations and the “Chia Line’s.”
(
Pwf − k
m )
2
− Pwf
2
n
3 SHmax′ – Shmin′ – U′
2 – A

11. Acknowledgements:
I would like to thank Sithu Moe Myint,
Daniel Leon and Henrik Lundin for peer
reviewing this paper. I also would like to
thank colleagues Rob Creswell and Joe
Tasch for their time to review this work.
Nomenclature:
Q is the flow rate (MMscf/day)
C is the flow coefficient
n is the deliverability exponent
Pwf is the bottom-hole flowing pressure
at the current conditions (psia)
Pres is the reservoir pressure at the
current conditions (psia)
m is the gradient of the sanding line
k is the intercept of the sanding line
Sv is the vertical stress (z direction)
Shmin is the minimum horizontal stress
SHmax is the maximum horizontal stress
Pp is the pore pressure
Pmud is the drilling mud weight
µ = tan(φ) is the coefficient of friction
φ is the angle of the plane at which
failure will occur
σ is the normal force to the plane
τ is the shear stress
CO is the cohesion in the rock
σ’ is the effective stress acting on a
porous medium
σT is the total stress in the medium.
δij is the Kronecker delta
CDR is the Critical Drawdown Pressure
Relationship
A is the Poro-elastic constant
ν is Poisson’s constant
α is Biot’s constant
Vp is the p-wave velocity
Vs is the s-wave velocity
S1 is the maximum stress on a Mohr
Circle
S2 is the minimum stress on a Mohr
Circle
S3 is the intermediate stress on a Mohr
Circle
U is the effective formation strength
UCS is the unconfined rock strength
tests
TWC is the Thick walled cylinder tests
L is the length of the reservoir
h is the height of the reservoir
G is the bulk shear or rigidity modulus
eij is the strain component
e is the relative volume change of the
bulk material
Δp is the change in reservoir pressure

12. z is the gradient of the U relationship
y is the y-intercept of the U relationship
SHmax’ and Shmin’ are the changed
stresses due to a change in pore
pressure
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