IMPERIAL COLLEGE LONDON Department of Earth Science and

IMPERIAL COLLEGE LONDON

Department of Earth Science and Engineering

Centre for Petroleum Studies

Analysis of Pressure Fall-Off Tests for Polymer Injection

By

Mark Thomas

A report submitted in partial fulfillment of the requirements for

the MSc and/or the DIC.

September 2016

ii

DECLARATION OF OWN WORK

I declare that this thesis, “Analysis of Pressure Fall-Off Tests for Polymer Injection”, is entirely my own work and

that where any material could be constructed as the work of others, it is fully cited and referenced, and/or with

appropriate acknowledgement given.

Signature: ……………

Name of student: Mark Thomas

Name of supervisor: Alain Gringarten (Imperial College), Loriane Augustin & Vincent Jaffrezic (TOTAL)

Analysis of Pressure Fall-Off Tests for Polymer Injection

Mark Thomas

Alain Gringarten, Imperial College

Loriane Augustin & Vincent Jaffrezic, TOTAL

Abstract Injection of polymer solutions can increase recovery as a secondary and tertiary production technique. Predicting the in-situ

rheology of the polymer is key to ensuring successful design of polymer floods. Polymers commonly injected are non-

Newtonian fluids, exhibiting shear thinning and shear thickening behaviour in the reservoir. Therefore, a different

interpretation methodology needs to be applied than conventional well test analysis (WTA) techniques.

For a Pressure Fall-Off (PFO) test, the pressure signature is distorted as the polymer approaches the Newtonian limit,

characterised by a “bump” in the derivative. This distortion could lead to inaccurate estimations of the polymer rheology and

position of the polymer front. Results indicate the “bump” begins to appear when the shear rate drops below twice the value of

the Newtonian limit. By understanding the input parameters, it is possible to design an injection test to ensure that one is far

from the Newtonian limit.

The application of WTA to polymer floods has been extended to secondary recovery. Simulation results indicate that non-

Newtonian interpretation can be applied to both water-wet and mixed-wet reservoirs independently of the mobility ratio.

However, the pressure signature in an oil-wet reservoir is largely affected by the oil saturation. An alternative Newtonian

interpretation method using a sequence of fall-offs is proposed.

Introduction Polymer injection is an enhanced oil recovery (EOR) method commonly used in oil fields containing oil with high viscosities.

Poor mobility ratios can lead to low sweep efficiencies in the reservoir, subsequently leading to low final recovery factors.

Mixing a polymer with water can lead to better mobility control and hence higher recoveries. There are many field examples

where the injection of a polymer has been implemented for tertiary recovery techniques (Han et al. 2006, Dong et al. 2008,

Delamaide et al. 2013). More recently, secondary polymer flooding has been considered (Fabbri et al. 2013, Delamaide 2016).

One drawback of polymer injection is the significant increase of cost compared to conventional waterflooding. Therefore,

it is important to be able to predict the in-situ rheology and position of the polymer bank to predict the additional production

achieved by injection. One method of water injection surveillance is Pressure Fall-Off (PFO) testing (Abbaszadeh 1989,

Bratvold 1990). Similar to a pressure build-up test, the well is shut in after a period of injection and the subsequent pressure

response is interpreted. However, there are complications when applied to a polymer flood. Firstly, the polymers commonly

injected are non-Newtonian fluids. Therefore, they may exhibit shear-thinning or shear-thickening behaviour on the viscosity

in the reservoir, depending on the shear rate. Secondly, polymers injectivity is typically lower than water and can lead to the

formation of induced fractures in the reservoir (Seright et al. 2009). Polymers may also start to degrade under high shear rates.

There have been several conceptual studies into using well test analysis (WTA) for non-Newtonian fluids. Papers by Ikoku

et al. (1978) and Odeh et al. (1979) derive the partial differential diffusivity equation for power law fluids. Using a pseudo-

steady state assumption, this equation is linearised to provide a solution to analyse pressure data. Vongvuthipornchai et al.

(1987) investigate the validity of the superposition principle. They find that a correction factor must be applied to the fall-off

derivative time to correctly represent the injection period. Mahani et al. (2011) take forward their methodology, proposing

type curves for radial flow for power law fluids. A methodology is proposed to be applied to field examples including induced

fractures. More recently, van den Hoek et al. (2012) proposed a non-Newtonian methodology to estimate the “transition time”

between the polymer bank and preceding water bank. This methodology is then applied to a field case to estimate the in-situ

polymer rheology.

However, there are still many important aspects of applying WTA to non-Newtonian fluids that have not been investigated.

One of these is the effect the Newtonian limit, or viscosity cut-off value, has on the pressure derivative. As previously

documented by Mahani et al. (2011) and van den Hoek et al. (2012), as the polymer solution tends towards the maximum

viscosity, a distortion or bump can be observed in the derivative. However, the full effect it has on the non-Newtonian

Imperial College

London

2

interpretation is yet to be investigated. Secondly, the use of WTA on polymer flooding has only ever been considered for

tertiary recovery following a waterflood. This is despite the increase of interest for use as a secondary recovery technique. The

effect of oil saturation and relative permeabilities on non-Newtonian interpretation is yet unknown.

This paper looks to i) investigate the effect of the Newtonian Limit on fall-off tests and propose an injection design ii)

extend the application to secondary recovery techniques.

Methodology Polymer Rheology in Porous Media

Most polymers used in polymer floods behave as non-Newtonian fluids, meaning that the viscosity of the polymer is a

function of the shear rate. At low shear rates (typically below 1s-1

), the polymer viscosity may be considered as constant and is

independent of the shear rate (Delshad et al. 2008). This is the Newtonian region which occurs when the shear rate drops

below the Newtonian limit, also referred to as the viscosity cut-off value. At intermediate shear rates (between 1 to 100 s-1

),

polymers exhibit pseudo-plastic or shear thinning behavior, showing a decrease in polymer viscosity with an increase of shear

rate. At higher shear rates (between 100 to 1000 s-1

), polymers can exhibit visco-elastic effects or shear thickening. At shear

rates higher than 1000 s-1

, the polymer will start to degrade. These high shear rates are only common near the wellbore region.

This work is focused on the shear thinning behaviour of the polymer in the reservoir and can be described with a power

law function:

� � �� (1)

where � is the viscosity, � is the consistency index, �� is the shear rate and the power law index.

Typical viscosity versus shear rate profiles for Newtonian fluids, power law fluids and polymers with shear thinning, shear

thickening and degradation are shown in Figure 1.

During the injection period, the shear rate (due to the flow rate) decreases with

distance from the well, meaning that the viscosity of the polymer is a function of

distance. The viscosity at the well reaches its lowest value due to the shear thinning

effect, gradually increasing away from the well until reaching the Newtonian limit

and the maximum viscosity value.

During the shut in period, the shear rate quickly drops to zero around the well

meaning that the viscosity distribution changes rapidly, before becoming flat at a

constant value throughout the reservoir (Figure 2). Due non-linear effects, classical

pressure transient analysis (PTA) methodology cannot be applied. As shown by

Vongvuthipornchai et al. (1987), differences in the pressure transient behavior

between the injection period and fall-off periods means that the superposition

principle is not applicable in the same way as for Newtonian fluids. A correction

factor must be applied to the shut in derivative time. Moreover, the radial flow

period is a straight line slope dependent on the power law index, contrary to the flat

stabilisation seen for Newtonian fluids.

0

10

20

30

40

50

60

70

80

0.1 1 10 100 1000 10000

Po

lym

er

vis

cois

ty (

cP)

Shear rate (s-1)

Newtonian Polymer Power Law

Newtonian Shear-thinning Shear-thickening Degradation

Figure 1: Typical viscosity profiles for Newtonian, polymers (with shear thickening and degradation) and power law fluids.

Figure 2: Typical viscosity profile after

shut in.

3

Shear Rate Calculation

Shear rate in porous media is estimated based on formation properties and fluid actual velocity. Several formulas exist in the

literature (see Sun et al. (2012) for an exhaustive review). However, in this paper, a relationship given by Canella et al. (1988)

is used:

γ� � √2 �� /(��) ��∅ (2)

where �� is the water permeability in Darcy, �� is the water saturation, is the power law index, ∅ is the porosity and is the

Darcy velocity in ft/d. is the shear rate coefficient, which is a function of the network parameters (permeability and

porosity). However, a value of 6.0 was reported to fit well a wide variety of coreflood data (Delshad et al. 2008).

Diffusivity Equation for non-Newtonian Fluid

This section describes the method used to derive the dimensionless diffusivity equation for a non-Newtonian fluid. The

method is largely similar to those found in literature (Odeh et al. 1979, Vongvuthipornchai et al. 1987, Mahani et al. 2011).

The single phase radial flow equation for a slightly compressible fluid can be expressed as:

�!""! �# �

$"%"!� � ∅&' "%"' (3)

where r is the radius, p is the pressure and ct is the total compressibility.

Darcy’s law in radial coordinates is as follows:

� �$"%"! (4)

Equation 5 is a viscosity formula which relates viscosity and fluid velocity :

� � ( �� (5)

where ( is a function of formation properties. It can be derived using equation 1 and the shear rate formulae shown by

equation 2. ( is sometimes referred to as the effective viscosity µeff (Ikoku et al. 1978) or the bed factor (Vongvuthipornchai et

al. 1987).

Darcy law combined to the viscosity equation 5 leads the following relationship between fluid viscosity and pressure drop:

� � (�/� �� "%"!�

��/� (6)

This expression can then be introduced into the radial flow equation 3 to provide the diffusivity equation for power-law fluid

fluids:

�!""! # ��) "%

"!�*+ � ∅&' "%"' (7)

Using the following dimensionless variables:

,- � ./�012$∗ 4,5 − ,�7(8)9 (8) 8- � �'

∅:;$∗!�< (9) #- � !!� (10)

with �∗ defined as the “characteristic viscosity” that corresponds to the estimated viscosity at the sand face:

�∗ � ( = 12./0!�=

�� (11)

Equation 7 can be rewritten as the dimensionless diffusivity equation for power law fluids:

�!>

""!> #- ="%>"!>=

*+ � "%>

?"'>@ (12)

4

Pseudo-Steady State Assumption

In order to linearise equation 12, Odeh et al. (1979) introduce a pseudo-steady state assumption to estimate the fluid viscosity

in the reservoir:

� � ( �� ≈ ( � 12./0!�

�� (13)

Non-Newtonian Interpretation Workflow

Equation 12 can be solved using equation 13 in the Laplace form to give (full solution shown in Appendix A-3):

�) � B C(�)

D'EFE;G;H*IJKL+< D 21./0G

*M+< D �

:;∅G*L+<

with N() � (��)L*M+KL+

O� <KL+�

!+L*KL+ (14)

where ! is the correction factor applied to the derivative time as proposed by Vongvuthipornchai et al. (1987).

Using this solution, the non-Newtonian interpretation workflow is as follows:

1. Plot the logarithmic derivative versus injection or fall-off time on a log-log plot

2. Identify the non Newtonian radial flow regime and find the best straight line fit. The slope will provide an estimate of

s = (1-n)/(3-n).

3. Extrapolate the straight line to t=1s and pick the value of the pressure derivative at this point: D8 "%"'G'P�Q

4. Estimate the ratio �) using the equation.

5. If the permeability is known, the effective viscosity or bed factor ( can then be computed.

Radius of Investigation

van den Hoek et al. (2012) use numerical models to generate type-curves of the “transition time” between the polymer bank

and the preceding water bank. In their calculations, the transition time was defined as the moment that the slope of the

logarithmic derivative started to drop below zero.

An estimate of the investigation radius in the polymer zone can be derived from these dimensionless curves:

#5�R � 2.701� 12./0�

*L+KL+ D �'

∅):;G*

KL+ (15)

Contrary to Newtonian fluid, the radius of investigation of non-Newtonian fluids is dependant of the injection rate; the higher

the flow rate, the deeper the investigation.

Results This section is split into three parts. The first investigates the effect the Newtonian limit has on the pressure derivative. The

second part looks at what implications this may have when applying non-Newtonian interpretation to field cases. The third

investigates the application of non-Newtonian PTA to secondary recovery.

All numerical simulations are carried out using an in-house simulation software. A 1D radial grid of 1500m was used with

a single injection well at the centre. The cell sizes increase logarithmically from 0.001m near the well bore to 25m at the outer

boundary which was modeled at a constant pressure. This fine gridding around the wellbore means that the pressure transient

during the fall-off is properly captured. Polymer solutions were injected for a long enough period to ensure pseudo-steady

state. No skin or wellbore storage was considered. Initially no relative permeabilities were added in the model.

Part 1: Effect of the Newtonian Limit This section investigates the effect the Newtonian limit has on the pressure derivative.

Analytical Solution

The steady state laplace solution of equation 7 was solved and the solution is plotted, shown in Figure 3 (full solution shown in

Appendix A-3). When radial flow is reached, a slope of s = (1-n)/(3-n) can be seen. This slope allows the power law index n to

be estimated. The smaller the power law index, the larger the slope. However, the analytical solution only describes the

behavior of the fluid during shear thinning and does not take the Newtonian limit into account.

5

0.01

0.1

1

10

100

0.01 0.1 1 10 100 1000 10000 100000

Dim

en

sio

nle

ss P

ress

ure

Dimensionless Time

n=1.0

n=0.8

n=0.6

n=0.4

n=0.2

Effect of the Newtonian Limit on Fall-off Signature The objective of Figure 4 is to show the nature of a fall-off derivative for a polymer flood. The green line shows the case of

injecting polymer into polymer. The blue line shows the case of injecting polymer into water, with a characteristic mobility

change in the derivative. The characteristic non-Newtonian slope for radial flow is shown by the red dotted line. However, as

can be seen in the case of injecting polymer into polymer, a “bump” can be seen in the derivative at late times when

approaching the Newtonian limit, before reaching a constant viscosity. Using Equation 15, the distance into the reservoir

where the “bump” begins to appear was calculated. The corresponding point on the viscosity profile, shown on Figure 5,

confirms that the “bump” in the derivative appears before the Newtonian limit is reached. Table 1 shows the input parameters

for the simulations shown in Figure 4 and Figure 5.

0.0001

0.001

0.01

0.1

0.01 0.1 1 10 100

Pre

ssu

re d

eri

va

tiv

e (

ba

r/m

3/d

)

Time (hours)

Figure 3: Plot of the analytical solution for power law fluids.

Figure 4: Fall-off pressure derivative of polymer injection showing

bump in the derivative. Figure 5: Viscosity profile in reservoir at shut-in.

Bump begins to

appear in derivative

Newtonian limit is

reached

Table 1: Reservoir and fluid properties for reservoir

simulation.

Parameter Value

Initial Water Saturation, Swi 1

Initial Pressure, pi (bar) 42

Permeability, k (mD) 500

Porosity, Φ 0.25

Thickness, h (m) 10

Well Radius, rw (m) 0.0768

System Compressibility, ct (bar-1) 1.24×10-4

Water Viscosity, uw (cP) 0.53

Polymer Concentration (kg/m3) 1.551

n 0.62

Newtonian Limit (s-1) 1

Viscosity @ 1 s-1 (cP) 13

Radial Flow

s = (1-n)/(3-n)

6

By varying the Newtonian limit, the effect it has on the pressure derivative can be seen. Figure 6 shows the results when

the Newtonian limit is varied from 1 s-1

to 0.1 s-1

. The concentration was increased from 1.5 to 2.5 g/l for the case with a lower

Newtonian limit to ensure that the polymers have the same viscosity. As shown in Figure 7 and Figure 8, both the viscosity

and shear rate profiles are identical until the Newtonian limit is reached and the power law is obeyed. Despite this, the pressure

signatures do not match, and different non-Newtonian slopes can be interpreted from the derivatives.

Figure 9 shows the pressure derivative for both the injection and fall-off periods for polymer injected into polymer. Both

curves show an initial non-Newtonian slope. For the injection period, as the Newtonian limit is approached, the pressure

derivative leaves the non-Newtonian slope before stabilising at a constant viscosity value, exhibiting a classical transition

effect. However the fall-off derivative is distorted before stabilising at the same final value. The reason for this distortion is

still unclear. This may be due to the significant differences between the injection and fall off viscosity profiles or combined

effect of rate history and non-linearity.

Further analysis shows that this distortion occurs even when no fluid in the reservoir has yet reached the Newtonian limit.

Injection of polymer into water for 1000 days was simulated. The derivative curves match up until the mobility change and the

“bump” is visible. The viscosity profile (Figure 11) shows that the polymer at the polymer bank is far from the Newtonian

limit. Therefore, for the case of injecting polymer into water, no polymer exists in the reservoir that has reached the limit. This

means that the distortion in the derivative cannot be due to polymer further into the reservoir that has already reached the

Newtonian regime.

0.001

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Pre

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ba

r/m

3/d

)

Time (hours)

NL=1 NL=0.1

Figure 6: Pressure derivative for Newtonian limit 1 s-1 & 0.1 s-1.

Figure 7: Shear rate profile for Newtonian limit 1 s-1 & 0.1 s-1. Figure 8: Viscosity profile for Newtonian limit 1 s-1 & 0.1 s-1.

7

0.002

0.02

0.01 0.1 1 10 100

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ba

r/m

3/d

)

Time (hours)

1000 Days

Polymer-Polymer

Power (Non-

Newtonian Slope)

Shear Rate Effect

The relationship between the shear rate in the reservoir and the distortion in the derivative was investigated. Using equation

15, the corresponding distance in the reservoir at which the “bump” begins to appear on the derivative was estimated for

different cases, with varying input parameters. From the shear rate profiles, the corresponding value of the shear rate was

estimated. A ratio of shear rate to the Newtonian limit was calculated shown in Table 2 (refer to Table A-4 shown in Appendix

A-4 for more detail on simulation cases). For the few simulations carried out, the distortion in the derivative begins to occur

when the value of the shear rate drops below twice the Newtonian limit. Despite being a value derived from observation, it

indicates that the “bump” is a function of the shear rate, and gives a rough guide on when to expect the distortion.

Variable Rate (sm3/d) Viscosity @ 1 s-1 (cP) Newtonian limit (s-1) n Permeability (mD)

Case 75 100 500 13 113 436 1 0.8 0.6 0.4 0.2 0.1 0.2 0.4 0.6 0.8 1000 500 100

SR (s-1) 2.1 1.9 2.0 2.1 2.0 2.0 2.1 1.8 1.3 0.9 0.4 0.2 1.5 2.0 2.1 2.2 2.1 2.1 2.1

SR/NL 2.1 1.9 2.0 2.1 2.0 2.0 2.1 2.2 2.1 2.1 2.1 2.0 1.2 1.5 2.1 2.1 2.1 2.1 2.1

0.006

0.06

0.01 0.1 1 10 100 1000

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ba

r/m

3/d

)

Time (hours)

Injection period

Fall off

Newtonian Slope

Power (Non-Newtonian Slope)

Figure 9: Pressure derivative for both injection and fall-off period (polymer into polymer).

Figure 10: Fall-off with polymer injected into water for 1000 days

Newtonian

limit has not

yet been

reached

Figure 11: Viscosity profile for 1000 days injection of polymer

into water and polymer into polymer.

Table 2: Value of shear rate/Newtonian limit where bump begins to appear in derivative.

8

Part 2: Sensitivities It has been shown that as a polymer solution approaches the Newtonian limit, there is a “bump” in the pressure signature.

There is now a real risk of taking the “bump” into account when applying non-Newtonian interpretation techniques to field

cases. This could mean incorrectly estimating the in-situ polymer rheology i.e. i) the power law index n, ii) the effective

viscosity H, and iii) the location of the polymer bank. With certain sensitivities, it is possible to better understand which

parameters affect the PTA, and to propose a design of injection tests to avoid incorrect characterisation of the polymer

rheology.

For visual clarity, the case of injecting polymer into polymer is predominantly shown. It is important to reiterate that the

pressure responses of injecting polymer into polymer or water are in fact identical up until the mobility change. Therefore, the

risk of the “bump” on the non-Newtonian interpretation results are the same. Table 3 is a summary of the input parameters

used for the base case in the following simulations. Only the considered parameter is changed.

Table 3: Input parameters for base case.

Parameter Value

Initial Water Saturation, Swi 1



Porosity, Φ 0.25

Thickness, h (m) 10





n 0.616



Injection Rate (sm3 /day) 100

Injection Length (days) 100

Reservoir Characteristics

i) Permeability

Simulations were carried out with permeabilities of 50, 100, 500 and 1000 mD. The results are displayed on Figure 12 and in

Table 4. Reducing the permeability leads to higher accuracy in the polymer rheology estimation. This can be explained by the

increase of shear rate in the reservoir. Therefore, PTA may lead to more accurate results in lower permeability reservoirs.

Figure 12: Pressure derivative for permeabilies 50 mD, 100 mD, 500 mD & 1000 mD.

Table 4: Results for polymer rheology estimation for various permeabilities.

Permeability (mD) 50 100 500 1000

Power law index 0.62 0.62 0.62 0.62

Estimated power law index 0.62 0.61 0.56 0.51

Difference (%) 0 1 9 17

Effective viscosity H (cPo*(m/j)^(1-n)) 1.77 1.95 2.66 3.04

Estimated H (cPo*(m/j)^(1-n)) 1.77 1.95 2.53 2.68


y = 0.0654x0.1671

y = 0.0145x0.1807

y = 0.0412x0.1689

y = 0.0097x0.2009

0.001

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DP

& d

eri

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ba

r/m

3/d

)

Time (hours)

k=50mD

k=100mD

k=500mD

k=1000mD

9

ii) Porosity

Figure 14 shows the fall-off derivative with reservoir

porosities of 10%, 20% and 30%. The corresponding

bottom hole pressure profiles can be seen on Figure 13.

Despite changing the bottom hole pressure, varying the

porosity does not affect the pressure derivative or the

accuracy of the polymer rheology estimation (Table 5).

The porosity does influence the shear rate, as shown in

equation 2. However, unlike permeability, the range of

porosity variation is less than one order of magnitude and

any influence is likely to be insignificant.

Rate

The injection rate of the polymer solution was varied between 50 and 1000 sm3/d. Table 6 displays the results of the non-

Newtonian interpretation. A higher injection rate leads to more accurate estimation of the polymer rheology. The higher the

rate, the higher the shear rate in the reservoir and the further away the polymer is from the Newtonian limit.

Table 6: Results for polymer rheology estimation for various rates.

Polymer Characteristics

i) Polymer Concentration

Polymer solution injection was simulated with concentrations of 0.5 kg/m3, 1.5 kg/m

3, 5kg/m

3 and 10 kg/m

3. As it can be

observed from Figure 15, increasing the polymer concentration results in the increase of the viscosity at zero shear rate. Value

as high as 10 kg/m3 was considered as it generates viscosity behaviour similar to solutions used on some polymer injection

tests.

Non-Newtonian interpretation was carried out and the results are shown in Table 7. Higher concentrations of polymer

solutions lead to more accurate estimations of the polymer rheology. When injected at the same rate, the higher concentrations

experience a higher shear rate due to the increase of viscosity. This in turn means that the solution takes longer to reach the

Newtonian limit.

Porosity 10% 20% 30%

Power law index 0.62 0.62 0.62

Estimated power law index 0.61 0.56 0.51

Difference (%) 1 9 17

Effective viscosity H (cPo*(m/j)^(1-n)) 1.64 1.87 2.02

Estimated H (cPo*(m/j)^(1-n)) 1.62 1.87 2.03


Injection rate (sm3/d) 50 100 200 500 1000

Power law index 0.62 0.62 0.62 0.62 0.62

Estimated power law index 0.45 0.51 0.54 0.58 0.60

Difference (%) 27 17 12 5 3

Effective viscosity H (cPo*(m/j)^(1-n)) 3.04 3.04 3.04 3.04 3.04

Estimated H (cPo*(m/j)^(1-n)) 2.33 2.68 2.85 3.06 3.05

Difference (%) 23 12 6 1 0

40

50

60

70

0 50 100

Bo

ttm

Ho

le P

ress

ure

(b

ar)

Time (days)

ɸ=10%

ɸ=20%

y = 0.0412x0.169

y = 0.0412x0.1686

y = 0.0411x0.168

0.01

0.1

0.01 0.1 1 10 100 1000

Pre

ssu

re D

eri

va

tiv

e (

ba

r/m

3/d

)

Time (hours)

ɸ=10%

ɸ=20%

ɸ=30%

Figure 14: Pressure derivative for various porosities. Figure 13: Bottom hole pressure profiles for various porosities.

Table 5: Interpretation results for porosity simulations.

10

Table 7: Results for polymer rheology estimation for various polymer concentrations.

Polymer Concentration (kg/m3) 0.5 1.5 5 10

Power law index 0.62 0.62 0.62 0.62

Estimated power law index 0.46 0.52 0.58 0.60


Effective viscosity H (cPo*(m/j)^(1-n)) 0.54 3.04 26 102

Estimated H (cPo*(m/j)^(1-n)) 0.44 2.68 26.3 103.03


ii) Newtonian Limit

Figure 16 shows the results when the Newtonian limit is varied from 1 s-1

to 0.1 s-1

. In order to keep the viscosity profiles the

same, the concentration was increased from 1.5 to 2.5 g/l for the case with a lower Newtonian limit. The case of the smaller

Newtonian limit gives an accurate estimation of the polymer viscosity in the reservoir. When the Newtonian limit is at 1s-1

,

both the estimation of the power law and effective viscosity differs significantly from the input value.

Polymer Bank Extension

Estimating the position of the polymer bank in the reservoir is a key element when using WTA for polymer flooding.

Therefore it is important to understand how the Newtonian limit affects the analysis. By injecting polymer into water, it is

possible to see the position of the polymer bank on the pressure derivative. The distance to front can then be estimated using

equation 15 and compared to the calculated volumetric radius. Polymer was injected into water for increasing length of time

(100-1000 days). Figure 17 displays the pressure derivative curves of the various injection periods, the polymer bank

propagating further into the reservoir. Table 9 shows the results of the non-Newtonian method to estimate the position of the

polymer bank. As this analysis is to see how the “bump” affects just the value of radius calculated, the correct input values of n

and H were used in equation 15. As the bank approaches the Newtonian limit, there is an increase in error when estimating the

distance to the polymer front, up to a 20% difference for 1000 days of injection. Therefore, the distortion in the derivative not

only affects estimation of the polymer rheology, but also the position of the polymer in the reservoir.

Newtonian Limit (s-1) 0.1 1

Power law index 0.62 0.62

Estimated power law index 0.6 0.53

Difference (%) 3 14

Effective viscosity H (cPo*(m/j)^(1-n)) 3.04 3.04

Estimated H (cPo*(m/j)^(1-n)) 3.02 2.75

Difference (%) 1 10

Injection Period (days) 50 100 250 500 1000

Injection rate (Sm3/d) 100 100 100 100 100

Cumulative injection (Sm3) 4902 9804 24510 49020 98039

Polymer bank – Volumetric Radius (m) 25 36 56 80 113

Estimated polymer bank (m) 25.2 35.2 59.5 92.4 135.7

Difference (%) 0 1 5 16 20

1

10

100

1000

0.1 1 10 100

Vis

cosi

ty (

cP)

Shear rate (s-1)

C = 0.5 kg/m3 C = 1.5 kg/m3 C = 5 kg/m3 C = 10 kg/m3

Figure 15: Viscosity profiles with shear rate for different concentrations.

y = 0.0095x0.19

y = 0.0086x0.1665

0.001

0.01

0.1

0.01 0.1 1 10 100 1000

Pre

ssu

re d

eri

va

tiv

e

(ba

r/m

3/d

)

Time (hours)

NL=1

NL=0.1

Figure 16: Pressure derivative for Newtonian limit 1 s-1 & 0.1 s-1.

Table 8: non-Newtonian interpretation results for varying

Newtonian Limit.

Table 9: Results for polymer bank position estimation for various injection times.

11

Recommended Injection Design

There is a risk of taking the “bump” into account when applying non-Newtonian interpretation techniques to field cases.

However, it is possible to design injection fall-off tests to ensure that one is far from the “bump”. The results of the

sensitivities carried out are summarised in Figure 18. The graph shows whether increasing the concerned parameter has a

positive or negative effect on the interpretation.

Additionally, early analysis (results shown in Appendix A-4) identified that the bump begins to appear in the derivative

when the shear rate decreases roughly below twice the Newtonian limit. One can use formulas proposed in literature, such as

Canella et al. (1988) shown by equation 16, to estimate whether the polymer will reach the Newtonian limit in the reservoir.

This in turn allows one to design a test to ensure that one is far from the Newtonian limit, or predict whether the analysis may

be affected by it.

γ� � √2 �� /(��) ��∅ (16)

Part 3: Application to Secondary Recovery The use of well test analysis on polymer floods has so far been restricted to tertiary recovery (injection of polymer after a

water flood). This part investigates the use the well test analysis for polymer floods designed for secondary recovery. The

inclusion of oil leads to a multiphase system where relative permeabilities are taken into account. The objective is to identify

whether non-Newtonian interpretation techniques can be applied to aid the design of secondary polymer floods.

0.0001

0.001

0.01

0.1

0.01 0.1 1 10 100

Pre

ssu

re d

eri

va

tiv

e (

ba

r/m

3/d

)

Time (hours)

100 Days

50 Days

250 Days

500 Days

1000 Days

Figure 17: Pressure derivatives for various polymer bank extensions.

Negative

Positive

Permeability Porosity Rate Polymer

Concentration

Newtonian Limit

Figure 18: Summary of the results from sensitivity simulations (effect of increase of parameter).

12

Effect of Residual Resistance Factor

The reduction on the water relative permeability due to the polymer can be captured with the Residual Resistance Factor

(RRF). The impact of this was studied by injecting a polymer solution into water for 100 days with RRF of 1, 1.5 & 2. This

means that the water relative permeability is reduced by a factor of 1, 1.5 & 2, respectively. Non-Newtonian interpretation was

carried out on the pressure derivatives. Figure 19 shows the pressure derivative for each case.

Due to the reduction in water permeability, the higher RRF values leads to high injection pressures. Table 10 shows the results

of the interpretation. The RRF does not affect the value of n estimated. However, due to the change in mobility, the value of

the estimated effective viscosity is affected. Correct values can be calculated if the RRF is determined in the laboratory.

Table 10: Results of non-Newtonian interpretation for various values of RRF.

RRF 1 1.5 2

Power law index 0.62 0.62 0.62

Estimated power law index 0.60 0.60 0.60


Effective viscosity H (cPo*(m/j)^(1-n)) 1.95 1.95 1.95

Estimated H (cPo*(m/j)^(1-n)) 1.95 2.93 3.91

H/RRF 1.95 1.95 1.96


Polymer Injection for Secondary Recovery

A polymer solution was injected for 100 days into a reservoir initialised with an oil saturation Soi of 0.9. An irreducible water

saturation, Sirr of 0.1 ensured that there was no movement of water ahead of the polymer bank. The viscosity of the oil was

varied to give mobility ratios of M = 4, M = 1 and M = 0.25 (Figure 20). For each mobility ratio, three different relative

permeability curves were used; water-wet, mixed wet, and oil-wet (relative permeability curves are shown in Appendix A-5).

After 100 days of injection, a fall-off test was carried out. The pressure derivatives for each mobility ratio are shown in Figure

21, Figure 22 and Figure 23. Table 11 displays the input parameters used for simulation. The RRF was not taken into account.

Table 11: Input parameters for simulation (secondary recovery).

Parameter Value

Initial Water Saturation, Swi 0.1

Initial Water Saturation, Soi 0.9

Irreducible Water Saturation, Swirr 0.1

Residual Oil Saturation, Sor 0.1



Porosity, Φ 0.25

Thickness, h (m) 10





n 0.616



0.001

0.01

0.1

0.01 0.1 1 10 100

pre

ssu

re d

eri

va

tiv

e (

ba

r/m

3/d

)

Time (hours)

RRF=1 RRF=1.5 RRF=2

Figure 19: Pressure derivative for various values of RRF.

13

y = 0.0877x0.1684

y = 0.0498x0.1826

y = 0.1168x0.1601

0.01

0.1

1

0.01 0.1 1 10 100

Pre

ssu

re D

eri

va

tiv

e (

ba

r/m

3/d

)

Time (hours)

Water-Wet

Mixed-Wet

Oil-Wet

Figure 20 shows the water saturation profiles for each relative permeability when M=4. For the case of the mixed-wet

reservoir, a piston like displacement of the front can be seen. This means that the movement behind the front is mostly due to

the polymer solution. The water-wet case displacement profile behind the front has a larger gradient, and a steeper slope for

the oil-wet case. This behaviour can be seen on the corresponding pressure derivative (Figure 21). For the mixed-wet case, the

slope on the derivative directly corresponds to the correct power law index. For the water-wet case, despite a slightly larger

saturation gradient behind the front, the very low water permeability means that the polymer solution also dominates the

mobility. However, for the oil-wet case, a combination of the steep saturation gradient behind the front and the higher water

permeability means that the straight line slope is less obvious and is affected by the increase of oil saturation behind the front.

As the mobility ratio is reduced, this behaviour becomes more apparent. Figure 22 and Figure3 show the derivatives for M

= 1 and M = 0.25. Despite the large reduction in the mobility ratio, for both the mixed-wet and water-wet case the straight line

slope representative of the power law index can be easily identified. However, the oil-wet slope is dominated by the effect the

oil saturation has on the mobility.

Whether non-Newtonian interpretation can be applied to an oil saturated reservoir is dependent on the saturation profile

behind the front and the relative permeability curves. These results indicate that the non-Newtonian interpretation can be

applied to both mixed-wet and water-wet reservoirs. However, the straight line slope is hard to identify for oil-wet cases, even

Figure 20: Saturation profiles for rel perms; M = 4. Figure 21: Pressure derivatives for rel perms, M = 4.

y = 0.0867x0.1557

y = 0.0541x0.1886

y = 0.1205x0.1696

0.01

0.1

1

0.01 0.1 1 10 100

Pre

ssu

re D

eri

va

tiv

e (

ba

r/m

3/d

)

Time (hours)

Water-Wet

Mixed-Wet

Oil-Wet

y = 0.0881x0.1634

y = 0.0914x0.2915

y = 0.1195x0.1694

0.01

0.1

1

0.01 0.1 1 10 100

Pre

ssu

re D

eri

va

tiv

e (

ba

r/m

3/d

)

Time (hours)

Water-Wet

Mixed-Wet

Oil-Wet

Figure 22: Pressure derivatives for M = 1. Figure 23: Pressure derivatives for M = 0.25.

14

0.001

0.01

0.1

0.01 0.1 1 10 100

Pre

ssu

re d

eri

va

tiv

e (

ba

r/m

3/d

)

Time (hours)

Q=25

Q=50

Q=100

when the mobility ratio is largely in favour of the polymer. To estimate the value of H, the relative permeabilites must be

obtained from laboratory experiments.

Newtonian Interpretation of n

In the cases where the non-Newtonian slope in the pressure derivative is distorted, such as the oil-wet reservoir, it is desirable

to be able to determine the polymer rheology from other methods. One method that was explored was to use classical

Newtonian interpretation on a series of injection fall-offs, with the viscosity and the rate of subsequent fall-offs having the

relationship: � ∝ X�� (16)

A long injection period and short fall-off was followed by two subsequent short injection periods and short fall-offs, shown

in Table 12 and Figure 22. This long injection period means that the effect on the polymer bank extension by the subsequent

injection periods is negligible (ensuring points correspond to each other). A power law of 0.6 was initially used.

Table 12: Injection sequence flow period information.

Flow Period Injection Time (days) Rate (sm3/d) Cumulative Injection (sm3) Polymer Bank Extension (m)

1 100 100 10000 36.04

2 1 25 25 36.08

3 1 50 50 36.17

Using Newtonian interpretation, the mobility of the fluid at the top of the polymer bank was estimated. From the input

permeability of 100 mD (or estimated from previous well tests), the viscosity at the polymer bank was calculated (results in

table).

Simulation Input Result

Flow Period Injection Time (days) k (mD) µ (cP) k (mD) µ (cP)

1 100 81.79 0.53 100 0.61

2 1 47.40 0.53 100 1.05

3 1 61.96 0.53 100 0.81

30

40

50

60

70

80

0 20 40 60 80 100Bo

ttm

Ho

le P

ress

ure

(ba

r)

Time (days)

Figure 22: Injection sequence for Newtonian interpretation of n.

Figure 23: Pressure derivative for sequence of fall-offs.

Table 13: Results of Newtonian interpretation to series of fall-offs.

15

y = 8.0248x-0.204

y = 5.3723x-0.405

y = 3.3339x-0.596

0.1

1

10

10 100

Vis

cosi

ty (

cP)

Rate (sm3/d)

0.8 0.6 0.4

By plotting the estimated viscosity against rate, the gradient gives − 1. This method was carried out for n = 0.4, 0.6 and

0.8. The results are shown on Figure 24 and Table 14. is estimated to very high accuracy (1% error). It should be noted that

for this method to be used, the position of the polymer bank must be obvious on the derivative.

Summary and Conclusions Injecting polymers into reservoirs is an enhanced oil recovery technique that can lead to better mobility control and higher

recoveries. Estimating the in-situ polymer viscosity and position of the polymer bank is key to ensuring successful design of

polymer floods. Polymers commonly injected behave as non-Newtonian fluids, which means a different interpretation method

needs to be applied than traditional well test analysis techniques. Most notably, the radial flow stabilisation is characterised by

a straight line slope dependant on the non-Newtonian nature.

For a Pressure Fall-Off (PFO) test, the pressure signature is distorted as the polymer approaches the Newtonian limit,

characterised by a “bump” in the derivative. This differs from the derivative of the injection period, which exhibits a classical

transition between the non-Newtonian and Newtonian regimes. By calculating the position of the polymer bank, it can be seen

that the distortion begins to appear when the polymer is far from the Newtonian limit. From the few simulation cases run, it

was observed that this “bump” begins to appear roughly when the shear rate drops below twice the Newtonian limit.

There is a risk of taking the distortion in the derivative into account when interpreting results. Various sensitivities were

carried out in order to better understand which parameters may affect the estimation of the polymer rheology. Using these

results, it is possible to design ones injection test to ensure that one is far from the Newtonian limit, or predict whether the

analysis may be affected by it.

The use of WTA analysis on polymer floods has now been extended to secondary recovery. Whilst the Residual Resistance

Factor (RRF) does not affect the estimation of the power law index, it must be determined from laboratory experiments in

order to estimate the fluid viscosity. For secondary recovery, the applicability of non-Newtonian interpretation is dependent on

both the saturation profile behind the front and the relative permeability curves. The results indicate that the non-Newtonian

interpretation can be applied to both mixed-wet and water-wet reservoirs, although not for oil-wet reservoirs, especially at low

mobility ratios. An alternative Newtonian method has been proposed.

Further work should include i) adding near wellbore effects, such as wellbore storage and skin and temperature effects, ii)

addition of shear thickening and polymer degradation, and ii) application to real field data.

n Estimated n Error

0.8 0.796 1%

0.6 0.595 1%

0.4 0.404 1%

Table 14: Results of Newtonian interpretation.

Figure 24: Results of Newtonian interpretation of n.

16

References Abbaszadeh, M. and Kamal, M., 1989: “Pressure-Transient Testing of Water-Injection Wells”, SPE 16744.

Bratvold, R. and Horne, R., 1990: “Analysis of Pressure-Falloff Tests Following Cold-Water Injection”, SPE 18111, SPE

Annual Technical Conference and Exhibition, Houston, TX.

Cannella, W., Huh, C., and Seright, R., 1988: “Prediction of Xanthan Rheology in Porous Media”, paper SPE 18089,

presented at the SPE Annual Technical Conference and Exhibition, Houston, 2–5 October.

Delamaide, E., Zaitoun, A. et al., 2013: “Pelican lake field: first successful application of polymer flooding in a heavy oil

reservoir”, SPE 165234, SPE enhanced oil recovery conference, 2–4 Jul 2013, Kuala Lumpur, Malaysia.

Delamaide, E., 2016: “Comparison of Primary, Secondary and Tertiary Polymer Flood in Heavy Oil - Field Results”, SPE

180852, SPE Trinidad and Tobago Section Energy Resources Conference, 13-15 June, Port of Spain, Trinidad and Tobago.

Delshad, M. et al, 2008: “Mechanistic Interpretation and Utilization of Viscoelastic Behavior of Polymer Solutions for

Improved Polymer-Flood Efficiency”, SPE 113620, presented at the SPE/DOE Symposium on Improved Oil Recovery, Tulsa,

19–23 April.

Dong, H. Z., Fang, S. F., 2008: “Review of practical experience and management by polymer flooding at Daqing”, SPE

114342, presented at the SPE/DOE Improved Oil Recovery Symposium, Tulsa, 19-23 April 2008.

Fabbri, C. et al, 2014: “Secondary and Tertiary Polymer Flooding in Extra-Heavy Oil: Reservoir Conditions Measurements -

Performance Comparison”, IPTC 17703, International Petroleum Technology Conference, 19-22 January, Doha, Qatar.

Han, M., W. Xiang, J. et al., 2006: "Application of EOR Technology by Means of Polymer Flooding in Bohai Oilfields”, SPE-

104432.

Ikoku, C. and Ramey, H., 1979: “Transient flow of non-Newtonian power-law fluids in porous medium,” SPE 7139.

Mahani, H. et al., 2011: “Injection fall-off analysis of polymer flooding EOR”, SPE 145125, presented in Proceedings of the

SPE Reservoir Characterisation and Simulation Conference and Exhibition, Abu Dhabi, UAE, October 2011.

Odeh, A. and Yang, H., 1979: “Flow of non-Newtonian power-law fluids through in porous medium”, SPE 7150.

Sun, Y., Saleh, L., 2012: “Measurement and Impact Factors of Polymer Rheology in Porous Media”, Rheology, Dr. Juan De

Vicente (Ed.), InTech, DOI: 10.5772/35715. Available from: http://www.intechopen.com/books/rheology/polymer-rheology-

in-porous-media

Seright, R. et al., 2009: “Injectivity Characteristics of EOR Polymers”, SPE Res Eval & Eng, SPE-115142.

van den Hoek, P. et al, 2012: “Application of Injection Fall-Off Analysis in Polymer flooding”, SPE 154376, 74th EAGE

Conference & Exhibition incorporating SPE EUROPEC 4 – 7 June 2012 in Copenhagen, Denmark.

Vongvuthipornchai, S. and Raghavan, R., 1987: “Well test analysis of data dominated by storage and skin: non-Newtonian

power-law fluids,” SPE Formation Evaluation, 618–628, SPE 14454.

Acknowledgements Thank you to TOTAL for providing funding for this project, and Loriane Augustin and Vincent Jaffrezic for their support and

the well test analysis discussions.

Nomenclature

B = formation volume factor r = radius, m

C = shear rate coefficient rD = dimensionless radius

Cr = correction factor rinv = radius of investigation, m

ct = total compressibility, bar-1

rw = well radius, m

H = effective viscosity, cPo*(m/j)^(1-n) S = saturation

K = consistency index, s-1

T = time, s

k = permeability tD = dimensionless time

n = power law index U = darcy velocity, ms

p = pressure, bar µ = viscosity, cP

pi = initial pressure, bar µ* = characteristic viscosity, cP

pwf = flowing bottom hole pressure, bar γ = shear rate, s-1

q = rate, sm3/d φ = porosity

17

Appendix A-1: Critical Milestones in Non-Newtonian PTA Interpretation

SPE

Paper Year Title Authors Contribution

1567 1969 Steady-State and Unsteady-

State Flow of Non-Newtonian

Fluids Through Porous Media

H. K. Van Poollen,

J. R. Jargon

Presents equations for steady-state linear and

radial flow for power law fluids, and the transient

behavior results from a finite difference model of

a radial system and a field test. Showed that

power-law fluid do not exhibit the straight-line

relationship that they exhibit for Newtonian

fluids.

7139 1979 Transient Flow of Non-

Newtonian Power-Law Fluids

in Porous Media

C. U. Ikoku, H. J.

Ramey Jr.

Derived the nonlinear partial differential equation

for power-law fluids through homogenous

reservoir and proposed an analytical solution for

the case of constant rate injection.

7150 1979 Flow of Non-Newtonian Power-

Law Fluids Through Porous

Media

A. S. Odeh, H. T.

Yang

Also derived the partial differential diffusivity

equation for power law fluids and used pseudo-

steady state assumption to linearise to provide an

analytical solution. This solution has been more

accepted than one proposed by Ikoku et al..

13058 1987 Pressure Falloff Behavior in

Vertically Fractured Wells:

Non-Newtonian Power-Law

Fluids

S.

Vongvuthipornchai,

R. Raghavan

Identified that superposition principle is not

applicable in the same way to non-Newtonian

fluids due to non-linear behaviour; proposed

correction factor to be applied to dimensionless

time.

18089 1988 Prediction of Xanthan Rheology

in Porous Media

W. Cannella, C.

Huh, R. Seright

Proposed equation used to estimate shear rate in

porous media based on formation properties and

fluid velocity.

145125 2011 Injection Fall-Off Analysis of

Polymer Flooding EOR

H. Mahani, T. G.

Sorop, P. J. van den

Hoek, A. D. Brooks,

M. Zwaan

1. Proposed interpretation method of polymer

PFO tests with induced fractures 2. First

identification of "bump" in derivative due to

viscosity cut-off value

154376 2012 Application of Injection Fall-

Off Analysis in Polymer

Flooding

P. J. van den Hoek,

H. Mahani, T. G.

Sorop, A. D. Brooks,

M. Zwaan, S. Sen

1. Extension of methods to determine transition

times 2. Application of methods to field examples

18

Appendix A-2: Critical Literature Review

SPE No. 1567

Title: Steady-State and Unsteady-State Flow of Non-Newtonian Fluids Through Porous Media

Authors: H. K. Van Poollen, J. R. Jargon

Year: 1969

Contribution: Presents equations for steady-state linear and radial flow for power law fluids. Showed that power-law fluid do not exhibit the

straight-line relationship that they exhibit for Newtonian fluids.

Objective of the paper:

To obtain relationships and empirical and mathematical descriptions of the flow of non-Newtonian fluids through porous

media.

Methodology used:

Uses a finite difference model to solve the non-linear equations for non-Newtonian fluid.

Conclusions:

1. The drawdown curves for a power-law fluid do not exhibit the straight-line relationship that they exhibit for

Newtonian fluids

2. The slope of pD versus the log of tD could be used to calculate the average viscosity in the non-Newtonian slug

region

3. For a power law fluid, the injectivity index increases with rate

SPE No. 7139

Title: Transient Flow of Non-Newtonian Power-Law Fluids in Porous Media

Authors: C. U. Ikoku, H. J. Ramey Jr.

Year: 1979

Contribution: Derived the nonlinear partial differential equation for power-law fluids through homogenous reservoir and proposed an

analytical solution for the case of constant rate injection


Provide an analytical solution to help interpretation of pressure transient data from field operations involving non-Newtonian

fluids injection

Methodology used:

Mathematically derive and solve diffusivity equation for power law fluids using assumption to linearise equation.

Conclusions:

1. A new technique to analysis non-Newtonian fall-off data is proposed

2. A new expression to the radius of investigation obtained

3. A much higher pressure is observed when injecting pseudo-plastic fluids that Newtonian fluids

SPE No. 7150

Title: Flow of Non-Newtonian Power-Law Fluids Through Porous Media

Authors: A. S. Odeh, H. T. Yang

Year: 1979

Contribution: Also derived the partial differential diffusivity equation for power law fluids and used pseudo-steady state assumption to

linearise to provide an analytical solution. This solution has been more accepted than one proposed by Ikoku et al..


Provide an analytical solution to help interpretation of pressure transient data from field operations involving non-Newtonian

fluids injection

Methodology used:

Solve the partial differential equation for power law fluids and apply method to injection test data. Use theoretical results to

derive equations of flow and drainage radius.

Conclusions:

1. A new technique to analysis non-Newtonian fall-off data is proposed

2. The use of Newtonian methods on non-Newtonian fluids are not satisfactory

3. The shear-rate/viscosity relationship determined by viscometers may not be the governing relationship in the porous

media

19

SPE No. 13058

Title: Pressure Falloff Behavior in Vertically Fractured Wells: Non-Newtonian Power-Law Fluids

Authors: S. Vongvuthipornchai, R. Raghavan

Year: 1987

Contribution: 1. Identified that superposition principle is not applicable in the same way to non-Newtonian fluids due to non-linear

behaviour

2. Proposed correction factor to be applied to dimensionless time


1. Examine the consequences of using the linearised solutions to analyse injection and fall off data

2. Examine the validity of the superposition principle

3. Present solutions to examine fall-off behaviour in vertical fractures and propose methodology to estimate fracture

length

Methodology used:

1. Compare pressure response of the injection and fall off periods

Conclusions:

1. Linearised solution by Odeh and Yang should be used over Ikoku and Ramey

2. Correction factor must be applied to dimensionless time due to limitations of superposition principle for non-

Newtonian fluids

SPE No. 145125

Title: Injection Fall-Off Analysis of Polymer Flooding EOR

Authors: H. Mahani, T. G. Sorop, P. J. van den Hoek, A. D. Brooks, M. Zwaan

Year: 2011

Contribution: 1. Proposed interpretation method of polymer PFO tests with induced fractures

2. First identification of "bump" in derivative due to viscosity cut-off value


1. Deliver a clear workflow on how to apply non-Newtonian interpretation to fall-off tests

2. Extend the application to injection wells with induced fractures

Methodology used:

1. Solve radial diffusivity equation using “semi-analytical” approach

2. Incorporate effect of induced factors into solution

3. Apply methodology to numerical simulations

Conclusions:

1. A “bump” appears in the derivative when the polymer approaches the viscosity cut-off value

2. A new practical methodology of applying non-Newtonian interpretation to reservoirs with induced fracture proposed

3. Generic type curves given

20

Appendix A-3: Constant Rate Solution for Diffusivity Equation for Power Law Fluids

The dimensionless diffusivity equation for power law fluids can be derived as:

�!>

""!> #- ="%>"!>=

*+ � "%>

"'>I (A-1)

In order to linearise equation A-1, Odeh & al. (1979) introduce a pseudo-steady state assumption to estimate the fluid

viscosity:

� � ( �� ≈ ( � 12./0!�

�� (A-2)

Darcy law (equation 4) combined to the viscosity formula (equation 5) leads the following relationship between fluid viscosity

and pressure drop:

� � (�/� �� "%"!�

��/� (A-3)

Combining equations A-2 and A-3 gives the following expression for the pressure gradient:

��) "%"!�

*+�� ./0!12 �� (A-4)

Combining this with the diffusivity equation for power-law fluids (equation 7) gives the following expression:

�!""! Y�./012 �

�� #� �)"%"!Z � ∅&' "%"' (A-5)

which can be developed to give:

"<%"!< +�! . "%"! � ∅:;

� . ( � 12./0!�

�� "%"' (A-6)

which transformed into dimensionless form to give:

"<%>"!>< + �!> "%>

"!> � #-�� "%>"'> (A-7)

Equation A-7 can be solved in a case of constant injection rate and infinite reservoir using a Laplace transform:

,-\\\\(#-, ^) �!>*L+< _*L+KL+`

<KL+√a!>

KL+< baK/<_ <KL+�

<KL+√a�

(A-8)

with Kν(z) are modified Bessel functions of the second kinds of order ν.

At the wellbore, rD = 1, and

,-\\\\(^) �_*L+KL+�

<KL+√a�

aK/<_ <KL+�<

KL+√a� (A-9)

The solution can then be back-transformed in the time domain using the Stefhest algorithm.

Using asymptotic properties of Bessel functions, a long-time approximate inversion can be derived:

21

,�-(8) � (��)<(*L+)KL+

(��)c� <KLd�

8-*L+KL+ − �

�� (A-10)

And its pressure derivative:

?%�>?e�('>) � 8- ?%�>

?'> � (��)L*M+KL+

c� <KLd�

8-*L+KL+ (A-11)

Consequently, once wellbore storage and damage effect have vanished, the pressure derivative will display a straight line with

slope equal to (1-n)/(3-n) on a log-log plot.

Equation A-11 can be combined with dimensionless parameters to give:

�) � B C(�)

D'EFE;G;H*IJKL+< D 21./0G

*M+< D �

:;∅G*L+<

(A-12)

with N() � (��)L*M+KL+

O� <KL+�

!+L*KL+

22

Appendix A-4: Shear Rate Effect Results Table A-4: Table showing simulation cases and results finding the value of shear rate at the beginning of the bump in the derivative

(parameter being changed underlined in bold).

Rate (sm3/d) Viscosity @ 1 s

-1 (cP) k (mD) n NL (s

-1) SR (s

-1) SR/NL

Sensitivity to Rate

75 13 100 0.6 1 2.1 2.1

100 13 100 0.6 1 1.9 1.9

500 13 100 0.6 1 2.0 2.0

Sensitivity to Polymer Concentration

100 13 1000 0.6 1 2.1 2.1

100 113 1000 0.6 1 2.0 2.0

100 436 1000 0.6 1 2.0 2.0

Sensitivity to Permeability

100 13 1000 0.6 1 2.1 2.1

100 13 500 0.6 1 2.1 2.1

100 13 100 0.6 1 2.1 2.1

Sensitivity to n

100 13 100 0.2 1 1.5 1.5

100 13 100 0.4 1 2.0 2.0

100 13 100 0.6 1 2.1 2.1

100 13 100 0.8 1 2.2 2.2

Sensitivity to Newtonian Limit

75 13 100 0.6 1 2.1 2.1

60 12 100 0.6 0.8 1.8 2.2

45 11 100 0.6 0.6 1.3 2.1

30 9 100 0.6 0.4 0.9 2.1

15 7 100 0.6 0.2 0.4 2.1

100 5 100 0.6 0.1 0.2 2.0

23

Appendix A-5: Relative Permeability Curves and Corey Parameters.

Parameter Value

krw0 0.2

nw 2

Swr 0.1

kro0 1

no 2

Sor 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

kr

Sw

Mixed-Wet

oil

water

Graph A-5-1: Mixed-wet relative permeability curves.

Table A-5-1: Mixed-wet relative permeability Corey parameters.

24

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

kr

Sw

Water-Wet

oil

water

Parameter Value

krw0 0.2

nw 3.3

Swr 0.1

kro0 1

no 3.3

Sor 0.1

Graph A-5-2: Water-wet relative permeability curves.

Table A-5-2: Water-wet relative permeability Corey parameters.

25

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

kr

Sw

Oil-Wet

oil

water

Parameter Value

krw0 0.8

nw 3.3

Swr 0.1

kro0 1

no 3.3

Sor 0.1

Graph A-5-3: Oil-wet relative permeability curves.

Table A-5-3: Oil-wet relative permeability Corey parameters.

26

Table of Figures Figure 1: Typical viscosity profiles for Newtonian, polymers (with shear thickening and degradation) and power law fluids. .. 2 Figure 2: Typical viscosity profile after shut in. ........................................................................................................................... 2 Figure 3: Plot of the analytical solution for power law fluids. ...................................................................................................... 5 Figure 4: Pressure derivative of polymer injection showing bump in the derivative .................................................................... 5 Figure 5: Viscosity profile in reservoir at shut-in. ........................................................................................................................ 5 Figure 6: Pressure derivative for Newtonian limit 1 s

-1 & 0.1 s

-1. ................................................................................................. 6

Figure 7: Shear rate profile for Newtonian limit 1 s-1

& 0.1 s-1

. ................................................................................................... 6 Figure 8: Viscosity profile for Newtonian limit 1 s

-1 & 0.1 s

-1. .................................................................................................... 6

Figure 9: Pressure derivative for both injection and fall-off period. ............................................................................................. 7 Figure 10: Fall-off with polymer injected into water for 1000 days ............................................................................................. 7 Figure 11: Viscosity profile for 1000 days injection of polymer into water and polymer into polymer. ...................................... 7 Figure 12: Pressure derivative for permeabilies 50 mD, 100 mD, 500 mD & 1000 mD. ............................................................. 8 Figure 13: Bottom hole pressure profiles for various porosities. .................................................................................................. 9 Figure 14: Pressure derivative for various porosities. ................................................................................................................... 9 Figure 15: Viscosity profiles with shear rate for different concentrations. ..................................................................................10 Figure 16: Pressure derivative for Newtonian limit 1 s

-1 & 0.1 s

-1. ..............................................................................................10

Figure 17: Pressure derivatives for various polymer bank extensions. ........................................................................................11 Figure 18: Summary of the results from sensitivity simulations (effect of increase of parameter). .............................................11 Figure 19: Pressure derivative for various values of RRF. ..........................................................................................................12 Figure 20: Saturation profiles for rel perms; M = 4. ....................................................................................................................13 Figure 21: Pressure derivatives for rel perms, M = 4. ..................................................................................................................13 Figure 22: Pressure derivatives for M = 0.25. ..............................................................................................................................13 Figure 23: Pressure derivatives for M = 1. ...................................................................................................................................13 Figure 24: Injection sequence for Newtonian interpretation of n. ................................................................................................14 Figure 25: Pressure derivative for sequence of fall-offs. .............................................................................................................14 Figure 26: Results of Newtonian interpretation of n. ...................................................................................................................15

27

Table of Contents

Abstract ......................................................................................................................................................................................... 1

Introduction ................................................................................................................................................................................... 1

Methodology ................................................................................................................................................................................. 2

Results ........................................................................................................................................................................................... 4

Part 1: Effect of the Newtonian Limit ........................................................................................................................................... 4

Part 2: Sensitivities........................................................................................................................................................................ 8

Part 3: Application to Secondary Recovery .................................................................................................................................11

Summary and Conclusions ...........................................................................................................................................................15

References ....................................................................................................................................................................................16

Acknowledgements ......................................................................................................................................................................16

Nomenclature ...............................................................................................................................................................................16

Appendix A-1: Critical Milestones in Non-Newtonian PTA Interpretation ................................................................................17

Appendix A-2: Critical Literature Review ...................................................................................................................................18

Appendix A-3: Constant Rate Solution for Diffusivity Equation for Power Law Fluids .............................................................20

Appendix A-4: Shear Rate Effect Results ....................................................................................................................................22

Appendix A-5: Relative Permeability Curves and Corey Parameters. .........................................................................................23

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IMPERIAL COLLEGE LONDON Department of Earth Science and