Sensors 16-01077

sensors
Article
Flow Rates Measurement and Uncertainty Analysis in
Multiple-Zone Water-Injection Wells from Fluid
Temperature Profiles
José E. O. Reges 1,*, A. O. Salazar 1,*, Carla W. S. P. Maitelli 2, Lucas G. Carvalho 2
and Ursula J. B. Britto 2
1 Laboratório de Avaliação e Medição em Petróleo, Departamento de Engenharia de Computação e
Automação, Universidade Federal do Rio Grande do Norte, Natal 59072970, Brazil
2 Laboratório de Automação em Petróleo, Departamento de Engenharia de Petróleo,
Universidade Federal do Rio Grande do Norte, Natal 59072970, Brazil;
carlamaitelli@gmail.com (C.W.S.P.M.); lucas.gdcarvalho@bct.ect.ufrn.br (L.G.C.);
ursulabritto@gmail.com (U.J.B.B.)
* Correspondence: jose_edenilson@yahoo.com.br (J.E.O.R.); andres@dca.ufrn.br (A.O.S.);
Tel.: +55-084-3215-3696 (A.O.S.)
Academic Editor: Peter Hauser
Received: 28 March 2016; Accepted: 14 June 2016; Published: 13 July 2016
Abstract: This work is a contribution to the development of flow sensors in the oil and gas industry.
It presents a methodology to measure the flow rates into multiple-zone water-injection wells from
fluid temperature profiles and estimate the measurement uncertainty. First, a method to iteratively
calculate the zonal flow rates using the Ramey (exponential) model was described. Next, this model
was linearized to perform an uncertainty analysis. Then, a computer program to calculate the injected
flow rates from experimental temperature profiles was developed. In the experimental part, a fluid
temperature profile from a dual-zone water-injection well located in the Northeast Brazilian region
was collected. Thus, calculated and measured flow rates were compared. The results proved that
linearization error is negligible for practical purposes and the relative uncertainty increases as the
flow rate decreases. The calculated values from both the Ramey and linear models were very close
to the measured flow rates, presenting a difference of only 4.58 m3/d and 2.38 m3/d, respectively.
Finally, the measurement uncertainties from the Ramey and linear models were equal to 1.22% and
1.40% (for injection zone 1); 10.47% and 9.88% (for injection zone 2). Therefore, the methodology was
successfully validated and all objectives of this work were achieved.
Keywords: flow sensors; oil and gas industry; multiple-zone water-injection wells; wellbore heat
transmission; measurement uncertainty
1. Introduction
The oil reservoirs whose mechanisms are inefficient and retain large quantities of hydrocarbons
after exhaustion of their natural energy are strong candidates for employment of processes aiming to
achieve further recovery. In conventional methods, the oil recovery factor is enhanced by increasing
the reservoir pressure. A fluid is injected inside the reservoir in order to move the oil out of the rock
pores. Among the conventional oil recovery methods is the injection of water [1]. Figure 1 shows
a common scheme for the injection of water in oil fields. The top view of an oil field composed of
water-injection and oil-production wells is presented in Figure 1a. The water flooding process is
illustrated in Figure 1b. The injected water moves the oil up, increasing the oil production.
Sensors 2016, 16, 1077; doi:10.3390/s16071077 www.mdpi.com/journal/sensors

Sensors 2016, 16, 1077 2 of 17
Sensors 2016, 16, 1077 2 of 16
(a) (b)
Figure 1. A common scheme for injection of water in oil fields. (a) Top view of an oil field composed
of water-injection and oil-production wells; (b) The water flooding process. The injected water
moves the oil up, increasing the oil production.
The water for injection may be obtained from: subsurface, through wells drilled for this
purpose; rivers; lakes; the sea; or produced water, a byproduct of oil and gas exploration and
production [1]. The water-injection well may have one or more injection zones. The multiple-zone
injection technique is desirable to reduce investment costs, since it uses a single well for
simultaneous injection in various areas. However, this technique presents operational problems in
relation to the monitoring of individual water-injection flow rates. A typical water-injection system
composed of a multiple-zone water-injection well and a water-production well is shown in Figure 2.
Figure 2. A typical water-injection system composed of a multiple-zone water-injection well and a
water-production well. The total flow of water is measured and controlled on the surface. The
injected flow rates are controlled by mechanical downhole flow regulators and there is no real-time,
in-line flow rates measurement in the injection zones. VSD refers to a variable speed drive. DP refers
to differential pressure.
Figure 1. A common scheme for injection of water in oil fields. (a) Top view of an oil field composed of
water-injection and oil-production wells; (b) The water flooding process. The injected water moves the
oil up, increasing the oil production.
The water for injection may be obtained from: subsurface, through wells drilled for this purpose;
rivers; lakes; the sea; or produced water, a byproduct of oil and gas exploration and production [1].
The water-injection well may have one or more injection zones. The multiple-zone injection technique
is desirable to reduce investment costs, since it uses a single well for simultaneous injection in
various areas. However, this technique presents operational problems in relation to the monitoring of
individual water-injection flow rates. A typical water-injection system composed of a multiple-zone
water-injection well and a water-production well is shown in Figure 2.
Sensors 2016, 16, 1077 2 of 16
(a) (b)
Figure 1. A common scheme for injection of water in oil fields. (a) Top view of an oil field composed
of water-injection and oil-production wells; (b) The water flooding process. The injected water
moves the oil up, increasing the oil production.
The water for injection may be obtained from: subsurface, through wells drilled for this
purpose; rivers; lakes; the sea; or produced water, a byproduct of oil and gas exploration and
production [1]. The water-injection well may have one or more injection zones. The multiple-zone
injection technique is desirable to reduce investment costs, since it uses a single well for
simultaneous injection in various areas. However, this technique presents operational problems in
relation to the monitoring of individual water-injection flow rates. A typical water-injection system
composed of a multiple-zone water-injection well and a water-production well is shown in Figure 2.
water-production well. The total flow of water is measured and controlled on the surface. The
injected flow rates are controlled by mechanical downhole flow regulators and there is no real-time,
in-line flow rates measurement in the injection zones. VSD refers to a variable speed drive. DP refers
to differential pressure.
water-production well. The total flow of water is measured and controlled on the surface. The injected
flow rates are controlled by mechanical downhole flow regulators and there is no real-time, in-line
flow rates measurement in the injection zones. VSD refers to a variable speed drive. DP refers to
differential pressure.

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The total flow of water is easily measured and controlled on the surface. In common approaches,
the measurement of the total flow is performed by electromagnetic or differential pressure (DP) meters.
Moreover, a variable speed drive (VSD) motor may be used to control the flow through a submersible
centrifugal pump installed in the water-production well. On the other hand, it is not simple to control
and measure the injected flow rates at subsurface in the multiple-zone water-injection well. Normally,
the injected flow rates are controlled by mechanical downhole flow regulators into mandrels installed
at each injection zone. However, this approach presents some issues. The mechanical parts have a low
reliability. Therefore, a simple mechanical failure may compromise the injection quotas specified by
the reservoir engineering team.
This problem is compounded by the fact that, in most cases, there is no real-time, in-line flow
rates measurement in the injection zones. Periodically, a flow sensor is inserted within the injection
column to measure the downhole flow rates. Next, the collected data are made available for analysis.
This process is costly and does not offer information in sufficient frequency for a good monitoring of
the water-injector wells. Therefore, faults in the mechanical regulators may stay undetected for a long
time, drastically reducing the efficiency of the recovery method.
Meanwhile, several techniques to measure the temperature profile along the wellbore have
been used. One of these techniques is the optic fiber Distributed Temperature Sensing (DTS). In this
approach, an optic fiber is permanently inserted within the injection column, making possible the
in-line acquisition of the fluid temperature profile along the water-injection well at shorter intervals of
time, or even in real-time. In this context, the following questions arise: is it possible to estimate the
flow rates in each injection zone from the fluid temperature analysis along the water-injection well?
Furthermore, if it is possible, what is the measurement uncertainty?
The main objective of this work is to present a methodology to measure the flow rates into
multiple-zone water-injection wells from fluid temperature profiles and estimate the measurement
uncertainty. In this sense, the specific objectives are: to describe the fluid temperature along wellbore
as a function of the injected flow rates; to obtain a closed-form solution for the injected flow rates as a
function of the fluid temperature profile; to perform an uncertainty analysis to understand how errors
in the measured temperatures influence the calculated flow rates; to develop a computer program
to calculate the injected flow rates; and to validate the methodology from experimental temperature
profiles obtained in the field.
2. Literature Review
The concepts, techniques and applications of downhole temperature measurements have been
discussed in several papers since the 1950s. Qualitative surveys [2] have been applied in the
interpretation of temperature logs through wellbores providing useful information as to the location of
water-intake strata [3] and the direction and amount of fluid that is moving in a bore hole [4].
The first quantitative analysis of wellbore heat transmission was performed by Ramey (1962).
He developed a mathematical model for incompressible liquid and ideal gas, describing the fluid
temperature along a wellbore assuming that heat transfer is steady-state. The solution may be
applied to a large variety of wellbore heat problems involving different types of well completions and
operating methods [5]. Alves et al. (1992) proposed a unified model for predicting flowing temperature
distribution in wellbores and pipelines. This model can be applied to pipelines or production and
injection wells, under single or two-phase flow, over the entire inclination angle range from horizontal
to vertical, with compositional and black-oil fluid models [6]. Maubeuge et al. (1994) described a model
that takes into account the temperature effects due to the decompression of the fluid and the frictional
heating that occurs in the formation. Moreover, the model predicts flow rates and temperatures in the
wellbore and formation in response to the input pressure history [7].
Hasan et al. (2007) presented an analytic model for computing the fluid temperature in complex
wells [8]. Based on this work, Izgec et al. (2010) proposed two methods for flow rate estimation from
wellhead pressure and temperature data. The first method involves the entire wellbore in that both

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pressure-drop and the attendant heat-transfer calculations are done. In the second method, a single
point in the wellbore temperature profile is used to estimate the flow rates [9].
From the last decade, optic fiber Distributed Temperature Sensing (DTS) technology has been
widely used for monitoring injection profile in wellbores [10–12]. Muradov and Davies (2012) proposed
a workflow for zonal flow rate and pressure allocation in wells equipped with DTS. The workflow
is based on the assumption that the depths of the production intervals are known and that the
only measurements taken are the distributed temperatures along the wellbore. The method was
demonstrated on synthetic, production well data [2].
3. Materials and Methods
3.1. Theoretical Development
Initially, the classical model proposed by Ramey (1962) was examined to perform the zonal flow
rates measurement from fluid temperatures. This model describes the fluid temperature along the
wellbore as a transcendental and exponential function of flow. Therefore, the zonal flow rates may be
iteratively calculated from this function. However, it makes an uncertainty analysis difficult.
For this reason, the Ramey equation was expanded in Taylor series and linearized. Then,
a closed-form solution for the zonal flow rates as functions of fluid temperature was obtained
to estimate the measurement uncertainty. In this approach, the systematic error, inherent to the
linearization of the Ramey model, was not taken in account. The random error was estimated from the
propagation of uncertainties for the measured temperatures at each depth along the wellbore.
3.1.1. The Ramey Model for Fluid Temperature along Water-Injection Wells
Consider the flow along a vertical multiple-zone water-injection well whose longitudinal section
is illustrated in Figure 3. The total flow rate is M. The flow rates in the steady-state regions are M1,
M2, . . . , Mn. The injected flow rates are M1
1, M2
1, . . . , Mn
1. The fluid and geothermal temperatures at
surface are T0 and b, respectively. Due to the difference between the fluid and geothermal temperatures
along the wellbore, T(Z) and Tgeo(Z), as water flows down, heat transmission occurs from the formation
to the fluid across the cement, casing, annulus, and tubing. Therefore, the fluid temperature increases
along the wellbore. Into each injection zone the fluid temperature is considered approximately constant
and equal to T1, T2, . . . , Tn. Moreover, as part of the total flow is injected into the upper zones, the flow
rate decreases along the wellbore (M1 > M2 > . . . > Mn) and the fluid temperature gradient increases.
After the last water-injection zone the flow rate is zero and the fluid temperature follows the geothermal
temperature. The Ramey model [5] for fluid temperature as a function of depth (Z) in each steady-state
region is given by:
T pZq “ aZ ` bi ´ aAi ` pT0i ` aAi ´ biq e
´ Z
Ai , i “ 1, 2, . . . , n (1)
where n is the number of injection zones, a is the geothermal gradient, bi is the initial geothermal
temperature, Ai is the relaxation distance, and T0i is the initial fluid temperature in each steady-state
region i. In turn, the relaxation distance Ai [13] is equal to:
Ai “ Mi ˆ
ˆ
cp
2πrtubUc
˙
, i “ 1, 2, . . . , n (2)
where Mi is the fluid injection mass flow rate in each steady-state region i, cp is the specific heat
at constant pressure of fluid, rtub is the inside radius of tubing, and Uc is the over-all heat-transfer
coefficient described in Appendix A.
A qualitative analysis of Equations (1) and (2) indicates that the mass flow rate Mi is proportional
to the relaxation distance Ai. However, the Ramey model for fluid temperature is a transcendental,

Sensors 2016, 16, 1077 5 of 17
exponential function of Ai. Consequently, once the fluid temperature profile T pZq is known, the
calculation of the mass flow rate Mi by Equation (1) has to be iteratively performed.
Moreover, as discussed in Appendix A, the term in parentheses in Equation (2) depends on
thermal properties of fluid and earth; injection time; and constructive characteristics of the well
(dimensions and materials of tubing, casing, and cement). Oftentimes, some of these characteristics are
not easily known. Therefore, to calculate the mass flow rate Mi it is necessary to suppose them.
Sensors 2016, 16, 1077 5 of 16
(a) (b)
Figure 3. Flow and temperature along a vertical multiple-zone water-injection well. (a) Longitudinal
section of the well. At center of wellbore, a fluid (water) flows down along the tubing. Due to the
difference between fluid and geothermal temperatures, heat transmission occurs from the formation
(surround rock) to the fluid across the cement, casing, annulus, and tubing; (b) The plot illustrates
the qualitative behavior of the fluid and geothermal temperatures along the wellbore.
3.1.2. Estimating the Flow Rates from a Fluid Temperature Profile Using the Ramey Model
A method to estimate the flow rates in a multiple-zone water-injection well using the Ramey
model is described below:
1. Identify the injection zones and determine the steady-state regions from the temperature
profile;
2. Determine the total flow at surface, the geothermal temperature at surface, and the time of
injection;
3. Determine density, viscosity, specific heat, and thermal conductivity of fluid;
4. Determine diameters and thermal conductivities of tubing, casing, and cement;
5. Determine thermal diffusivity, outside radius, and thermal conductivity of formation;
6. Calculate, iteratively from (1) and (2), the flow rates M1, M2, …, Mn in each steady-state region;
7. Calculate the flow rates M1’, M2’, …, Mn’ in each injection zone:
= − , = 1, 2, … , (3)
3.1.3. A Linear Model for Fluid Temperature along Water-Injection Wells
As was discussed in Section 3.1.1, because the fluid temperature is a transcendental,
exponential function of the relaxation distance , the mass flow rate calculation has to be
iteratively performed. It makes the flow rates calculations difficult and complicates the analysis of
the measurement uncertainty. Therefore, it would be useful to obtain a closed-form solution for the
injected flow rates as a function of the fluid temperature profile.
Figure 3. Flow and temperature along a vertical multiple-zone water-injection well. (a) Longitudinal
section of the well. At center of wellbore, a fluid (water) flows down along the tubing. Due to the
difference between fluid and geothermal temperatures, heat transmission occurs from the formation
(surround rock) to the fluid across the cement, casing, annulus, and tubing; (b) The plot illustrates the
qualitative behavior of the fluid and geothermal temperatures along the wellbore.
3.1.2. Estimating the Flow Rates from a Fluid Temperature Profile Using the Ramey Model
A method to estimate the flow rates in a multiple-zone water-injection well using the Ramey
model is described below:
1. Identify the injection zones and determine the steady-state regions from the temperature profile;
2. Determine the total flow at surface, the geothermal temperature at surface, and the time
of injection;
3. Determine density, viscosity, specific heat, and thermal conductivity of fluid;
4. Determine diameters and thermal conductivities of tubing, casing, and cement;
5. Determine thermal diffusivity, outside radius, and thermal conductivity of formation;
6. Calculate, iteratively from (1) and (2), the flow rates M1, M2, . . . , Mn in each steady-state region;
7. Calculate the flow rates M1
1, M2
1, . . . , Mn
1 in each injection zone:
Mi
1
“ Mi ´ Mi`1, i “ 1, 2, . . . , n (3)
3.1.3. A Linear Model for Fluid Temperature along Water-Injection Wells
As was discussed in Section 3.1.1, because the fluid temperature is a transcendental, exponential
function of the relaxation distance Ai, the mass flow rate Mi calculation has to be iteratively performed.

Sensors 2016, 16, 1077 6 of 17
It makes the flow rates calculations difficult and complicates the analysis of the measurement
uncertainty. Therefore, it would be useful to obtain a closed-form solution for the injected flow
rates as a function of the fluid temperature profile.
Expanding the Equation (1) in Taylor series (at Z “ Zi) and rearranging the resultant equation to
consider a linear approximation, as described in Appendix B, it follows that fluid temperature as a
depth function is:
T pZq « T pZiq `
„
Tgeo pZiq ´ T pZiq
Ai

pZ ´ Ziq , i “ 1, 2, . . . , n (4)
where T pZiq is the fluid temperature at Zi; Tgeo pZiq is the geothermal temperature at Zi; and the
relaxation distance Ai is given by Equation (2). Solving Equation (4) for Ai:
Ai “
„
Tgeo pZiq ´ T pZiq
T pZq ´ T pZiq

ˆ pZ ´ Ziq , i “ 1, 2, . . . , n (5)
A qualitative analysis of Equation (4) indicates that the relaxation distance Ai is inversely
proportional to the angular coefficient (slope) of the tangent line approaching the fluid temperature
curve at Zi. On the other hand, Equation (5) shows a closed-form solution for Ai.
Consequently, once the fluid temperature profile T pZq is known, it is easily possible to obtain the
tangent line at a region of interest, determine its slope, calculate the relaxation distance Ai and, finally,
estimate the mass flow rate Mi using Equation (2). Therefore, no iterative calculation is necessary.
The only remaining difficulty is in determining the characteristics of the term into parentheses in
Equation (2). A way to eliminate this problem is further discussed in Section 3.1.4. The linear model
simplifies the flow rates calculations and makes available a useful tool for uncertainty estimation as
discussed later in Section 3.1.5.
3.1.4. Estimating the Flow Rates from a Fluid Temperature Profile Using the Linear Model
Analyzing the Equation (2), as discussed in Sections 3.1.1 and 3.1.2, the relaxation distance Ai
is directly proportional to mass flow rate Mi and several properties of the fluid, earth, and wellbore.
However, since the inside radius of tubing remains constant along the wellbore and each region is
subjected to the same mechanisms of heat transfer, the term in parentheses is independent of depth
and can be considered constant for each region. In this case, Equation (2) can be rewritten as:
Ai “ Mi ˆ
ˆ
cp
2πrtubUc
˙
“ Mi ˆ cZ, i “ 1, 2, . . . , n (6)
where cz is a constant along the wellbore.
From Figure 3, the mass flow rate M1 is equal to the total mass flow rate at surface:
M1 “ M (7)
Thus, the mass flowrates Mi and M1 are related to Ai and A1 by Equation (8):
Ai
A1
“
Mi ˆ cZ
M1 ˆ cZ
“
Mi
M1
, i “ 1, 2, . . . , n (8)
Consequently, from Equations (7) and (8), the mass flow rates Mi are:
Mi “
Ai
A1
M1 “
Ai
A1
M, i “ 1, 2, . . . , n (9)
where Ai is given by Equation (5).

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Therefore, once the fluid temperature profile along the wellbore, the geothermal temperature,
and the total mass flow at surface are all known, a second method to estimate the flow rates in a
multiple-zone water-injection well can be used, which is described below:
1. Identify the injection zones and determine the steady-state regions from the temperature profile;
2. Obtain the linear approximation for each steady-state region along the wellbore;
3. Determine the relaxation distances A1, A2, . . . , An from Equation (5);
4. Determine the flow rates M1, M2, . . . , Mn from Equation (9);
5. Calculate the flow rates M1
1, M2
1, . . . , Mn
1:
Mi
1
“ Mi ´ Mi`1, i “ 1, 2, . . . , n (10)
3.1.5. Uncertainty Calculation
Previous research [2,7,9] presented methods for estimating the zonal flow rates in a wellbore from
temperature surveys. A qualitative discussion about the measurement uncertainty [2] was performed
by Muradov and Davies (2012). However, a quantitative analysis of the uncertainties in the flow rates
measurement from the fluid temperature measurement along the well was not found in the literature.
Therefore, a quantitative uncertainty analysis to understand how errors in the measured temperatures
influence the calculated flow rates is presented below.
In Figure 4, a theoretical example [5] of a fluid temperature profile along a single-zone
water-injection well is illustrated. The exponential curve was obtained from Equation (1). The linear
approximation was determined using Equation (4) at Zi = 1700 m.
Sensors 2016, 16, 1077 7 of 16
= − , = 1, 2, … , (10)
3.1.5. Uncertainty Calculation
Previous research [2,7,9] presented methods for estimating the zonal flow rates in a wellbore
from temperature surveys. A qualitative discussion about the measurement uncertainty [2] was
performed by Muradov and Davies (2012). However, a quantitative analysis of the uncertainties in
the flow rates measurement from the fluid temperature measurement along the well was not found
in the literature. Therefore, a quantitative uncertainty analysis to understand how errors in the
measured temperatures influence the calculated flow rates is presented below.
In Figure 4, a theoretical example [5] of a fluid temperature profile along a single-zone
water-injection well is illustrated. The exponential curve was obtained from Equation (1). The linear
approximation was determined using Equation (4) at Zi = 1700 m.
Figure 4. Theoretical Water-Injection Well Fluid Temperature Profile. The blue line represents the
exponential Ramey model. The red dots represent the linear approximation. In this example [5], it is
assumed to be a single-zone water-injection well with the following characteristics; injection rate:
4790 bbl/d (8.80 m³/s); surface water temperature: 58.5 °F (14.7 °C); no tubing; inside diameter of
casing: 6.366 in (0.16 m); geothermal gradient: 0.0083 °F/ft (0.0151 °C/m); geothermal temperature at
surface: 70.0 °F (21.11 °C).
The fluid temperature T(Z) varies exponentially throughout the wellbore. However, as the
depth increases the temperature profile becomes nearly linear. Therefore, it is reasonable to suggest
a linear approximation of the model proposed by Ramey (1962) for the temperature distribution
along the water-injection well [5].
Suppose it is desired to calculate the fluid temperature at Z = 1800 m. By the exponential
Equation (1), T (1800 m) = 65.60 °F (18.67 °C). By the linear Equation (4), T (1800 m) = 65.58 °F (18.66 °C).
Therefore, at this depth, the Linear model is a very close approximation of the Ramey model, and the
systematic error, inherent to the linearization process, is negligible.
From Equation (1), it is difficult to estimate the measurement uncertainty using the Ramey
model because the injected flow rates are transcendental functions of fluid temperature. However,
since the linearization error is negligible, it is easier to estimate the measurement uncertainty from
the Linear model described in Equation (4).
Figure 4. Theoretical Water-Injection Well Fluid Temperature Profile. The blue line represents the
exponential Ramey model. The red dots represent the linear approximation. In this example [5], it is
assumed to be a single-zone water-injection well with the following characteristics; injection rate:
4790 bbl/d (8.80 m3/s); surface water temperature: 58.5 ˝F (14.7 ˝C); no tubing; inside diameter of
casing: 6.366 in (0.16 m); geothermal gradient: 0.0083 ˝F/ft (0.0151 ˝C/m); geothermal temperature at
surface: 70.0 ˝F (21.11 ˝C).
The fluid temperature T(Z) varies exponentially throughout the wellbore. However, as the depth
increases the temperature profile becomes nearly linear. Therefore, it is reasonable to suggest a linear
approximation of the model proposed by Ramey (1962) for the temperature distribution along the
water-injection well [5].
Suppose it is desired to calculate the fluid temperature at Z = 1800 m. By the exponential
Equation (1), T (1800 m) = 65.60 ˝F (18.67 ˝C). By the linear Equation (4), T (1800 m) = 65.58 ˝F

Sensors 2016, 16, 1077 8 of 17
(18.66 ˝C). Therefore, at this depth, the Linear model is a very close approximation of the Ramey model,
and the systematic error, inherent to the linearization process, is negligible.
From Equation (1), it is difficult to estimate the measurement uncertainty using the Ramey model
because the injected flow rates are transcendental functions of fluid temperature. However, since the
linearization error is negligible, it is easier to estimate the measurement uncertainty from the Linear
model described in Equation (4).
Considering the fluid and geothermal temperatures at each depth along the wellbore can be
represented by a measured value and an measurement uncertainty, the temperatures T pZq, T pZiq, and
Tgeo pZiq in Equation (5) can be described by:
T pZq “ T pZq ˘ ∆T pZq (11)
T pZiq “ T pZiq ˘ ∆T pZiq (12)
Tgeo pZiq “ Tgeo pZiq ˘ ∆Tgeo pZiq (13)
where T pZq, T pZiq, and Tgeo pZiq are the measured values and ∆T pZq, ∆T pZiq, and ∆Tgeo pZiq are the
measurement uncertainties.
As described in Appendix B, neglecting the uncertainties in depth measurements and propagating
the temperature uncertainties, the random error for the relaxation distance Ai is:
∆Ai “
„
∆Tgeo pZiq ` ∆T pZiq
Tgeo pZiq ´ T pZiq
`
∆T pZq ` ∆T pZiq
T pZq ´ T pZiq

Ai, i “ 1, 2, . . . , n (14)
Similarly, the random error for the flow rate Mi in each steady-state region is:
∆Mi “
„
∆Ai
Ai
`
∆A1
A1
`
∆M
M

Mi, i “ 1, 2, . . . , n (15)
where ∆M is the measurement uncertainty for the total flow rate at surface.
Finally, the measurement uncertainties for the injected flow rates are:
∆Mi
1
“ ∆Mi ` ∆Mi`1, i “ 1, 2, . . . , n (16)
3.2. Modeling a Flow Rate Meter in Multiple-Zone Water-Injection Wells
After consolidating the theoretical aspects, a computer program that calculates the injected flow
rates in a multiple-zone water-injection well was implemented. To verify that the linearization error is
negligible, both the Ramey and Linear models were used to estimate the water temperature.
First, it runs an iterative process to calculate the geothermal gradient. It starts from the interval
defined between 0.01 and 0.03 ˝C/m. The iterative process is aborted if the error is minor or equal
to 0.001 ˝C/m, or after 10 iterations. Next, it runs another iterative process to calculate the injected
flow rates using the Ramey model. It starts from an interval defined by the program user. The iterative
process is aborted if the error is minor or equal to 0.001 m3/d, or after 10 iterations. Both algorithms
use the Secant Method, a finite difference approximation of the Newton-Rapson Method.
After, it recalculates the injected flow rates using the Linear model and estimates the measurement
uncertainties. Finally, it generates the fluid temperature profile and plots a graph comparing the
theoretical Ramey and Linear models with the experimental profile obtained in the field.
3.3. Data Acquisition
In the experimental part, the fluid temperature profile from a dual-zone water-injection well
located in the Northeast Brazilian region was collected. A temperature sensor was introduced and
moved through the tubing, logging the fluid temperature approximately every 50 m along the wellbore.
In each point, the temperature sensor performed one measurement per second, with a resolution of

Sensors 2016, 16, 1077 9 of 17
0.1 ˝C, for the duration of 5 to 10 min, thus totaling 300 to 600 temperature measurements. Finally,
a flow sensor was also introduced through the well, measuring the flow rates in each injection zone.
Another flow sensor measured the total flow at the surface.
3.4. Data Processing and Analysis
First, the mean value and the standard deviation of the mean from the 300 to 600 measurements
performed at each depth along the wellbore were calculated, thereby building the processed
temperature profile. Next, the processed temperature profile was qualitatively analyzed in order
to identify the injection zones and determine the steady-state regions. After, the flow rates in each
injection zone were calculated by the computer program from both Ramey and Linear models. Finally,
the calculated and measured flow rates were compared.
4. Results and Discussion
The field data collected from a dual-zone water-injection well were processed as described in
Section 3.4. The results are shown in Table 1. The mean fluid temperature and the standard deviation
of the mean were respectively considered the measured value and the measurement uncertainty at
each depth along the wellbore. The experimental fluid temperature profile is illustrated in Figure 5.
From the well schematics, it was known that the water-injection well had two injection zones at 538 m
and 695 m. The changes in the water temperature gradient along the wellbore from 100 m to 500 m
are not clearly observable in Figure 5 because there is an abrupt transition in temperature near the
bottomhole (from 600 m to 700 m). Therefore, to identify the injection zones, determine the steady-state
regions, and perceive the changes in water temperature gradient, it is necessary to stretch out the
graph as viewed in Figure 6.
Sensors 2016, 16, 1077 9 of 16
injection zone were calculated by the computer program from both Ramey and Linear models.
Finally, the calculated and measured flow rates were compared.
4. Results and Discussion
The field data collected from a dual-zone water-injection well were processed as described in
Section 3.4. The results are shown in Table 1. The mean fluid temperature and the standard deviation
of the mean were respectively considered the measured value and the measurement uncertainty at
each depth along the wellbore. The experimental fluid temperature profile is illustrated in Figure 5.
From the well schematics, it was known that the water-injection well had two injection zones at 538
m and 695 m. The changes in the water temperature gradient along the wellbore from 100 m to 500 m
are not clearly observable in Figure 5 because there is an abrupt transition in temperature near the
bottomhole (from 600 m to 700 m). Therefore, to identify the injection zones, determine the
steady-state regions, and perceive the changes in water temperature gradient, it is necessary to
stretch out the graph as viewed in Figure 6.
Table 1. Processed field data for a dual-zone water-injection well.
Depth (m)
Number of
Measurements
Mean Fluid
Temperature (°C)
Standard Deviation of
the Mean (°C)
−4.70 358 36.14 0.05
50.60 593 38.127 0.009
100.61 311 38.400 0.003
150.55 310 38.334 0.003
200.74 308 38.300 0.003
251.13 311 38.301 0.001
300.62 306 38.300 0.003
350.55 320 38.340 0.003
400.56 306 38.400 0.003
450.61 322 38.5006 0.0008
500.55 318 38.612 0.002
600.57 318 39.72 0.02
650.50 311 40.89 0.02
690.55 632 42.19 0.01
Figure 5. Fluid temperature profile built after processing the field data.
Figure 5. Fluid temperature profile built after processing the field data.
Table 1. Processed field data for a dual-zone water-injection well.
Depth (m) Number of Measurements Mean Fluid Temperature (˝C) Standard Deviation of the Mean (˝C)
´4.70 358 36.14 0.05
50.60 593 38.127 0.009
100.61 311 38.400 0.003
150.55 310 38.334 0.003
200.74 308 38.300 0.003
251.13 311 38.301 0.001
300.62 306 38.300 0.003
350.55 320 38.340 0.003
400.56 306 38.400 0.003
450.61 322 38.5006 0.0008
500.55 318 38.612 0.002
600.57 318 39.72 0.02
650.50 311 40.89 0.02
690.55 632 42.19 0.01

Sensors 2016, 16, 1077 10 of 17
Sensors 2016, 16, 1077 10 of 16
(a) (b)
(c) (d)
Figure 6. Fluid temperature profile. (a) The graph was stretched out from 0 m to 250 m; (b) The graph
was stretched out from 300 m to 500 m; (c) The graph was stretched out from 400 m to 500 m; (d) The
graph was stretched out from 600 m to 690 m.
Analyzing Figure 6a, in discarding the temperature measurement taken at surface, there is
practically no heat transmission and the fluid temperature remains approximately constant until 250
m. It occurs because at this depth the annular space between the injection and casing strings is filled
with air, which is a thermal insulator. From 300 m to 500 m (Figure 6b), the temperature starts to
increase in an apparently exponential behavior. From 400 m to 500 m (Figure 6c), immediately before
the first injection zone (538 m), the temperature profile has a clearly linear course. Finally, from 600
m to 690 m (Figure 6d), immediately before the second injection zone (695 m), the temperature
profile also displays linear behavior. To apply the Ramey model and, consequently, the Linear
model, it is necessary to consider a range of depths in steady-state. In this sense, the measurements
taken near the surface were discarded. Moreover, because the temperature measurements at 500 m
and 690 m were taken at depths too close to the injection zones (538 m and 695 m), the steady-state
condition is also not satisfied. Therefore, these two points were also discarded and the ranges from
400 m to 450 m and from 600 m to 650 m were considered as the steady-state regions. The theoretical
(Ramey and Linear) and experimental temperature profiles in the steady-state regions are plotted in
Figure 7.
Figure 6. Fluid temperature profile. (a) The graph was stretched out from 0 m to 250 m; (b) The graph
was stretched out from 300 m to 500 m; (c) The graph was stretched out from 400 m to 500 m;
(d) The graph was stretched out from 600 m to 690 m.
Analyzing Figure 6a, in discarding the temperature measurement taken at surface, there is
practically no heat transmission and the fluid temperature remains approximately constant until 250 m.
It occurs because at this depth the annular space between the injection and casing strings is filled
with air, which is a thermal insulator. From 300 m to 500 m (Figure 6b), the temperature starts to
increase in an apparently exponential behavior. From 400 m to 500 m (Figure 6c), immediately before
the first injection zone (538 m), the temperature profile has a clearly linear course. Finally, from 600 m
to 690 m (Figure 6d), immediately before the second injection zone (695 m), the temperature profile
also displays linear behavior. To apply the Ramey model and, consequently, the Linear model, it is
necessary to consider a range of depths in steady-state. In this sense, the measurements taken near the
surface were discarded. Moreover, because the temperature measurements at 500 m and 690 m were
taken at depths too close to the injection zones (538 m and 695 m), the steady-state condition is also
not satisfied. Therefore, these two points were also discarded and the ranges from 400 m to 450 m and
from 600 m to 650 m were considered as the steady-state regions. The theoretical (Ramey and Linear)
and experimental temperature profiles in the steady-state regions are plotted in Figure 7.

Sensors 2016, 16, 1077 11 of 17
Sensors 2016, 16, 1077 11 of 16
(a)
(b)
Figure 7. Theoretical (Ramey and Linear) and experimental temperature profiles in the steady-state
regions. (a) Region upstream from the first injection zone (538 m). The fluid temperature gradient is
about 0.002 °C/m; (b) Region upstream from the second injection zone (695 m). The fluid temperature
gradient is about 0.02 °C/m.
In Figure 7a the steady-state region upstream from the first injection zone (538 m) is illustrated.
The linear approximation was performed by determining the line passing through the measured
points at 400 m and 450 m. Figure 7b shows the steady-state region upstream from the second injection
zone (695 m). The linear approximation was performed by determining the line passing through the
measured points at 600 m and 650 m. The fluid temperature gradients are about 0.002 °C/m and 0.02
°C/m. In both regions of interest, the exponential and linear curves are overlapping. Therefore, in
these ranges, the linearization error is negligible.
The theoretically calculated and experimentally measured injected flow rates are shown in
Table 2. The uncertainties for the measured flow rates are unknown and were not represented. The
absolute uncertainties for the calculated flow rates from the Ramey and Linear models are about 1.20
m³/d and 1.35 m³/d, respectively. For the first injection zone, this is equivalent to 1.22% and 1.40% of
the calculated values from the Ramey and Linear models, respectively. For the second injection zone,
these absolute uncertainties are equivalent to 10.47% and 9.88% of the calculated values from the
Figure 7. Theoretical (Ramey and Linear) and experimental temperature profiles in the steady-state
regions. (a) Region upstream from the first injection zone (538 m). The fluid temperature gradient is
about 0.002 ˝C/m; (b) Region upstream from the second injection zone (695 m). The fluid temperature
gradient is about 0.02 ˝C/m.
In Figure 7a the steady-state region upstream from the first injection zone (538 m) is illustrated.
The linear approximation was performed by determining the line passing through the measured points
at 400 m and 450 m. Figure 7b shows the steady-state region upstream from the second injection
zone (695 m). The linear approximation was performed by determining the line passing through the
measured points at 600 m and 650 m. The fluid temperature gradients are about 0.002 ˝C/m and
0.02 ˝C/m. In both regions of interest, the exponential and linear curves are overlapping. Therefore,
in these ranges, the linearization error is negligible.
The theoretically calculated and experimentally measured injected flow rates are shown in Table 2.
The uncertainties for the measured flow rates are unknown and were not represented. The absolute
uncertainties for the calculated flow rates from the Ramey and Linear models are about 1.20 m3/d
and 1.35 m3/d, respectively. For the first injection zone, this is equivalent to 1.22% and 1.40% of the
calculated values from the Ramey and Linear models, respectively. For the second injection zone, these
absolute uncertainties are equivalent to 10.47% and 9.88% of the calculated values from the Ramey and
Linear models, respectively. Therefore, the relative uncertainty increases as the flow rate decreases.

Sensors 2016, 16, 1077 12 of 17
Moreover, the calculated flow rates from the Ramey and Linear models presented a difference of
only 4.58 m3/d and 2.38 m3/d, respectively, to the measured flow rates. Therefore, the two methods
presented a satisfactorily accurate measurement. Finally, it is interesting to observe that, in this case,
the calculated flow rates from the Linear model were closer to the measured flow rates than the
calculated flow rates from the Ramey model. A reasonable interpretation of this issue is that, in the
Ramey model, unlike the Linear model, it is necessary to know the thermal properties of fluid and
earth; the injection time; and the constructive characteristics of the well (dimensions and materials
of tubing, casing, and cement). Therefore, to calculate the injected flow rates from Ramey model it is
necessary to suppose these properties, leading to increases in the measurement error.
Table 2. Measured and calculated flow rates in m3/d.
Injection
Zone
Measured Flow Rate
(m3/d)
Calculated Flow Rate from
Ramey Model (m3/d)
Calculated Flow Rate from
Linear Model (m3/d)
1 93.96 98.54 ˘ 1.20 96.34 ˘ 1.35
2 16.04 11.46 ˘ 1.20 13.66 ˘ 1.35
Total 110.00 110.00 110.00
5. Conclusions
This work is a contribution to the development of flow sensors for the oil and gas industry.
It presented two methods to measure the flow rates into multiple-zone water-injection wells from fluid
temperature profiles. First, a method using the Ramey model, which defines the fluid temperature as a
transcendental, exponential function of flow, was described. In this method, the downhole flow rates
have to be iteratively calculated. Moreover, several thermal properties and constructive characteristics
of the well need to be known to perform the flow rate calculations.
Next, a second method using a Linear model that defines a closed-form solution for the injected
flow rates as a function of the fluid temperature was discussed. The obtained results showed that, in
the regions of interest, the exponential and linear curves overlap, as illustrated in Figure 7. Therefore,
under certain conditions, the linearization error is negligible for practical purposes. The possibility
of using a more complex polynomial model to further improve the results is the subject of another
work currently in development. The basic idea is to expand the Ramey equation in Taylor Series and
rearrange the resultant equation in order to consider a quadratic or cubic approximation.
Besides simplifying the downhole flow rates calculations, the second method made available a
useful tool for uncertainty estimation. Thus, an uncertainty analysis was performed. In this approach,
the systematic error, inherent to the linearization of the Ramey model, was not taken into account.
The random error was estimated from the propagation of uncertainties for the measured temperatures
at each depth along the wellbore.
A computer program modeling a flow rate meter in multiple-zone water-injection wells was
implemented. It calculates the injected flow rates using both the Ramey (Exponential) and the Linear
models, estimates the measurement uncertainties, and compares the theoretical Ramey (Exponential)
and Linear models with the experimental profile obtained in the field.
In the experimental part, a fluid temperature profile from a dual-zone water-injection well located
in the Northeast Brazilian region was collected. The total flow rate was 110.00 m3/d. The injected flow
rates were 93.96 m3/d and 16.04 m3/d at zones 1 and 2, respectively. It was observed that the calculated
flow rates from both the Ramey and Linear models presented a satisfactorily accurate measurement.
The calculated values were very close to the measured flow rates, presenting a difference of only
4.58 m3/d and 2.38 m3/d, respectively. It was also observed that the relative uncertainty increases as
the flow rate decreases. For the first injection zone, the estimated uncertainties were equal to 1.22%
and 1.40% of the calculated flow rates from the Ramey and Linear models, respectively. For the second
injection zone, the uncertainties were 10.47% and 9.88% of the calculated values from the Ramey and

Sensors 2016, 16, 1077 13 of 17
Linear models, respectively. Therefore, both methods were successfully validated and all objectives of
this work were achieved.
This approach to measure the downhole flow rates has a limitation due to the temperature
sensor and the field procedure used to collect the fluid temperature profile. In the experimental part,
the temperature sensor had a resolution of 0.1 ˝C and the temperature measurements were taken every
50 m. Each steady-state region also had a length of 50 m. Moreover, the measured fluid temperature
gradients were 0.002 ˝C/m and 0.02 ˝C/m for the steady-state regions 1 and 2, respectively. Thus,
the temperature variations in the steady-state regions 1 and 2 were about 0.1 ˝C and 1 ˝C, respectively.
Consequently, the sensor that was used was able to detect the temperature changes and the obtained
results were satisfactory. However, if the first steady-state region had a length of 40 m, for example,
the temperature change would be only 0.08 ˝C and the sensor would not be able to detect this variation.
This issue is the subject of another paper currently being prepared.
This limitation can be satisfactorily reduced as better temperature sensors are applied.
For example, a typical DTS optic fiber has made downhole temperature measurements with a resolution
of 0.01 ˝C and intervals of 1 m potentially available. Moreover, once the optic fiber is permanently
inserted within the injection column, it makes it possible to have an in-line acquisition of the fluid
temperature profile along the multiple-zone water-injection well at shorter intervals of time, or even in
real-time, thereby offering information in sufficient frequency for the reservoir engineering team.
Acknowledgments: This work was supported by Petróleo Brasileiro S.A (PETROBRAS). The funds for covering
the costs to publish this paper were made available by Universidade Federal do Rio Grande do Norte.
Author Contributions: A. O. Salazar and Carla W. S. P. Maitelli conceived the project. A. O. Salazar led the
research process. Carla W. S. P. Maitelli, Lucas G. Carvalho, and José E. O. Reges performed the literature review.
José E. O. Reges and Lucas G. Carvalho described the method to estimate the flow rates from the Ramey model.
José E. O. Reges described the method to estimate the flow rates from the Linear model and conceived the
uncertainty analysis. José E. O. Reges, Lucas G. Carvalho, and Ursula J. B. Britto implemented the computer
program and processed the field data. All authors discussed the results. José E. O. Reges wrote the paper with
contributions from Carla W. S. P. Maitelli. All authors revised the paper and approved the final manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
As described in Section 3.1.1, the relaxation distance Ai depends on the overall heat-transfer
coefficient Uc. Because of the high thermal conductivity and relatively small radial distance between the
flowing fluids and the borehole wall, heat transfer in this region normally can be considered to be steady
state [13]. Considering the longitudinal section previously illustrated in Figure 3, the steady-state
heat transfer between the flowing fluid and the formation (surrounding rock) is primarily a result
of: convection within the tubing; conduction throughout the tubing wall; convection within the
fluid-filled annulus; conduction throughout the casing and cement walls; and conduction throughout
the formation [13]. Therefore, the overall heat-transfer coefficient Uc [13] is given by:
Uc “
1
rto ˆ
«
1
rtihf
`
ln
´
rto
rti
¯
kt
` 1
rcihan
`
ln
´
rco
rci
¯
kc
`
lnp rw
rco q
kcem
`
fptq
ke
ff (A1)
where rto is the outside radius of the tubing, rti is the inside radius of the tubing, hf is the local film
coefficient of heat transfer for fluid inside tubing, kt is the thermal conductivity of the tubing, rci is the
inside radius of the casing, rco is the outside radius of the casing, han is the local film coefficient of heat
transfer for fluid in annulus, kc is the thermal conductivity of the casing, rw is the outside radius of the
wellbore; kcem is the thermal conductivity of the cement, and f ptq is the transient heat-conduction time
function for earth.

Sensors 2016, 16, 1077 14 of 17
The local film coefficient of heat transfer for fluid inside tubing is equal to:
hf “
Nnuf ˆ kt
2 ˆ rti
(A2)
where Nnuf is the Nusselt number inside tubing.
In turn, the local film coefficient of heat transfer for fluid in annulus is equal to:
han “
Nnuan ˆ kc
2 ˆ rci
(A3)
where Nnuan is the Nusselt number in annulus.
For Reynolds numbers above 104, the Nusselt numbers are related to Reynolds and Prandtl
numbers [13] by:
Nnuf “ 0.023 ˆ Nre0.8
f ˆ Npr0.3
f (A4)
and
Nnuan “ 0.023 ˆ Nre0.8
an ˆ Npr0.3
an (A5)
The Prandtl numbers in tubing and annulus are given by:
Nprf “
cf ˆ µf
kt
(A6)
and
Npran “
can ˆ µan
kc
(A7)
where cf is the specific heat of fluid inside tubing, µf is the viscosity of fluid inside tubing, can is the
specific heat of fluid in annulus and µan is the viscosity of fluid in annulus.
The Reynold numbers in tubing and annulus are given by:
Nref “
2rti Mf
µf πrti
2
(A8)
and
Nrean “
2rci Man
µanπrci
2
(A9)
where Mf is the mass flow rate of fluid inside tubing and Man is the mass flow rate of fluid in annulus.
Finally, tDw is defined as:
tDw “
αt
rw
2
(A10)
where α is the thermal diffusivity of earth and t is the time of injection. The transient heat-conduction
time function for earth f ptq [13] is:
f ptq “ 1.1281
a
tDw
´
1 ´ 0.3
a
tDw
¯
, i f tDw ď 1.5 (A11)
or
f ptq “ r0.4063 ` 0.5ln ptDwqs
ˆ
1 `
0.6
tDw
˙
, i f tDw ą 1.5 (A12)
Therefore, as discussed in Section 3.1.1, the overall heat-transfer coefficient Uc depends on
the thermal properties of fluid and earth; the injection time; and the constructive characteristics
of the well (dimensions and materials of tubing, casing, and cement). Oftentimes, some of these
characteristics are not easily known and it is necessary to suppose them to calculate the mass flow
rates in each steady-state region using the Ramey model. Furthermore, because the Ramey model for
fluid temperature is a transcendental, exponential function of the flow rate, the calculations have to be

Sensors 2016, 16, 1077 15 of 17
iteratively performed, increasing the difficulty to apply the first method to estimate the downhole zonal
flow rates, as described in Section 3.1.2. However, using the Linear model described in Section 3.1.3,
all these properties and constructive characteristics may be ignored and the calculations may be directly
performed as the method discussed in Section 3.1.4.
Appendix B
B.1. Linearization of the Ramey Model
Considering the heat transfer in a single-zone water-injection well is steady state, the Ramey
model [5] for fluid temperature as a function of depth is given by:
T pZq “ aZ ` b ´ aA ` pT0 ` aA ´ bq e´ Z
A (B1)
where a is the geothermal gradient, b is the geothermal temperature at surface, A is the relaxation
distance, and T0 is the fluid temperature at surface.
The exponential term may be expanded in Taylor series at Z = 0 considering a
linear approximation:
e´ Z
A «
”
e´ Z
A
ı
z“0
pZ ´ 0q0
`
„
d
dZ
´
e´ Z
A
¯
Z“0
pZ ´ 0q1
1!
(B2)
where ”
e´ Z
A
ı
Z“0
“ 1 (B3)
and „
d
dZ
´
e´ Z
A
¯
Z“0
“
„
´
1
A
e´ Z
A

Z“0
“ ´
1
A
(B4)
Therefore, the Equation (B1) can be rewritten as:
T pZq « aZ ` b ´ aA ` pT0 ` aA ´ bq
„
1 ´
1
A
Z

« T0 `
ˆ
b ´ T0
A
˙
Z (B5)
Analogously, for a multiple-zone water-injection well, the fluid temperature in each steady-state
region can be approximated at Zi by:
T pZq « T pZiq `
„
Tgeo pZiq ´ T pZiq
Ai

pZ ´ Ziq , i “ 1, 2, . . . , n (B6)
where T pZiq is the fluid temperature at Zi; Tgeo pZiq is the geothermal temperature at Zi; and Ai is the
relaxation distance in each region.
B.2. Uncertainty Calculations from the Linear Model
Solving the Equation (B6) for Ai:
Ai “
„
Tgeo pZiq ´ T pZiq
T pZq ´ T pZiq

ˆ pZ ´ Ziq , i “ 1, 2, . . . , n (B7)
Neglecting the depth uncertainties, from the propagation of errors approach the uncertainty in
the relaxation distance Ai is given by:
∆Ai
Ai
“
∆
“
Tgeo pZiq ´ T pZiq
‰
Tgeo pZiq ´ T pZiq
`
∆ rT pZq ´ T pZiqs
T pZq ´ T pZiq
, i “ 1, 2, . . . , n (B8)

Sensors 2016, 16, 1077 16 of 17
Considering the uncertainties in the measured temperatures:
∆Ai “
„
∆Tgeo pZiq ` ∆T pZiq
Tgeo pZiq ´ T pZiq
`
∆T pZq ` ∆T pZiq
T pZq ´ T pZiq

Ai, i “ 1, 2, . . . , n (B9)
As discussed in Section 3.1.4, the mass flow rates Mi in the steady-state regions are equal to:
Mi “
Ai
A1
M1 “
Ai
A1
M, i “ 1, 2, . . . , n (B10)
Considering the uncertainties in the relaxation distances and in the total flow rate:
∆Mi “
„
∆Ai
Ai
`
∆A1
A1
`
∆M
M

Mi, i “ 1, 2, . . . , n (B11)
Finally, the injected flow rates are given by:
Mi
1
“ Mi ´ Mi`1, i “ 1, 2, . . . , n (B12)
Therefore, the uncertainties in the injected flow rates are equal to:
∆Mi
1
“ ∆Mi ` ∆Mi`1, i “ 1, 2, . . . , n (B13)
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© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access
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