Estimation of gas absorption and diffusion coefficients for dissolved

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Thermodynamics, Transport, and Fluid Mechanics

Estimation of gas absorption and diffusion coefficients for dissolved gases in liquids Petro Babak, and Apostolos Kantzas Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b02343 • Publication Date (Web): 11 Dec 2018 Downloaded from http://pubs.acs.org on December 17, 2018

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Industrial & Engineering Chemistry Research

Estimation of gas absorption and diffusion coefficients for dissolved gases in liquids Petro Babak∗,† and Apostolos Kantzas†,‡ †Department of Chemical and Petroleum Engineering, University of Calgary, 2500 University Dr. NW, Calgary, Alberta, Canada T2N 1N4 ‡PERM Inc, 3956 29 Street NE, Calgary, AB T1Y 6B6 E-mail: [email protected]

Abstract The multifaceted process of gas absorption and diffusion of dissolved gases has several useful applications in engineering, medicine, pharmaceutics and science. A new systematic study for modeling of this process using an inverse problem for concentration dependent diffusion with free boundary at the gas-liquid interface is presented in this work. The study includes the model derivation based on the Fick’s second law of diffusion, the Henry’s absorption law, the Stefan’s principles of free boundary problems, and the development of new techniques for estimation of concentration dependent diffusion coefficient and the gas absorption coefficient governing propagation of gas-liquid interface due to absorption. The developed estimation technique utilized the dissolved gas concentration profiles measured and different time moments and is applied to real x-ray CT measurement data for a binary gas liquid mixture containing dimethyl ether and bitumen.

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1 Introduction Although the problems of absorption of gases into liquids and diffusion of dissolved gases have been attracting the attention of scientists since the early 19th century, they remain even more important in modern life. A number of practical applications such as carbon sequestration, 1,2 enhanced oil recovery, 1–3 food industry, 4 bioengineering, 5 medicine 6 require knowledge about the gas absorption rates and the diffusion coefficient of dissolved gases. The main modeling principles necessary for modeling of gas absorption into liquids and consequent diffusion of dissolved gas were formulated in the 19th century by William Henry (1774–1836), Adolf Eugen Fick (1829– 1901) and Josef Stefan (1835–1893). In 1803 Henry published a manuscript 7 where he examined the quantity of gases absorbed by water at different temperatures and pressures. The distribution of dissolved gases into the liquid was possible to investigate by applying the Fick’s second law 8 according to which the diffusive flux of the dissolved gases was governed by the diffusion coefficient and was proportional to the concentration gradient. The absorption of gases into liquids leads to changes in the liquid volume, and introduces one more unknown into the process called a free boundary defined by the gas-liquid interface position. Josef Stefan 9 systematically studied the free boundary problems obtained from the mass transfer between different phases. To properly identify the unknown boundary Stefan introduced additional boundary conditions often referred to as Stefan conditions. Several works for diffusion estimation of dissolved gases in liquids considered only estimation of the diffusion coefficients under the assumption that the changes of liquid volume are very small and negligible. 10–20 Olander 21 was one of the first who investigated the effect of the volume change due to gas absorption into liquid on the diffusion coefficient. The Olander’s model included a correction for the flux term with respect to the volume change. Renner 22 offered a volumetric correction in the diffusion estimation to account for liquid swelling. Riazi 23 proposed an iterative updating technique for the height of the liquid column assuming that the diffusion is constant at each time step. Upreti and Mehrotra 24 tackled the volume change due to gas absorption using grid adjustments based on the mass conservation principle. To analyze the interface between different 2


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phases governed by constant diffusion coefficients, Grogan et al. 25 developed a boundary condition in a form similar to the Stefan condition for the interface movement using the mass conservation principle. Do and Pinczewski 26,27 applied a moving boundary problem with similar boundary conditions for the estimation of the constant diffusion based on the different types of concentration profiles of the dissolved gases. Jamialahmadi 28 indicated that the mass of gas transferred into √ liquid is proportional to t and proposed a successive substitution method for estimation of the concentration dependent diffusion coefficient from the problem with moving gas-liquid interface. The number of theoretical contributions that study simultaneous identification of unknown diffusion coefficient and free boundary is very limited. 29–32 Majority of works consider either identification of only unknown nonlinear concentration dependent diffusion coefficient, 33–36 or unknown free interface with given nonlinear diffusion coefficients. 37–41 The limited works for identification of both diffusion and free interface at this time are restricted to identification of the time dependent diffusion coefficient and time dependent free boundary. 29–32 In these works the existence and uniqueness conditions of the considered problems with unknown diffusion and free boundary are established. The problems are transformed to integral equations and solved using the Schauder fixed-point theorem. The aim of this paper is to develop a new practical methodology for estimation of gas absorption and concentration dependent diffusion of dissolved gases into liquids. This methodology relies on Fick’s second law for the diffusion of the dissolved gases into the liquids and Henry’s law for characterization of the gas absorption at the gas-liquid interface. To identify the unknown gas-liquid interface propagation, an additional Stefan’s type boundary condition is derived using the mass conservation principle. The derived mathematical model can be considered as an inverse problem for concentration dependent dissolved gas diffusion with free boundary for the gas-liquid interface. The uniqueness of this work is in the possibility of defining both the diffusion coefficient and gas-liquid interface propagation simultaneously using the dissolved gas concentration measurements at different time moments. This work is organized as follows. In Section 2, a model for gas absorption and consequent dif-

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fusion of the dissolved gas into liquid is derived. For relatively short time measurements, the model is approximated over an infinite space interval. In the case of constant diffusion, it is shown that √ √ that the gas-liquid interface, h(t), propagates linearly with respect to t, that is h(t) = h0 + h1 t, where the parameter h1 is the gas absorption coefficient. The same observation is made for any concentration dependent diffusion coefficient by applying the heat integral method. The model is √ reduced to an ordinary differential equation with aid of new rescaled variable λ = (x − h(t))/ t. The solution of the obtained differential equation is represented in the form of different integral expressions for identification of the absorption coefficient h1 and concentration dependent diffusion D(φ ). In Section 3, testing of the derived integral expressions for estimation of the diffusion and the gas-liquid interface propagation is provided for the volume concentration profiles in the case of known constant and non-constant concentration dependent diffusion coefficient. Sensitivity analysis with respect to the deviations from the true propagation parameter h1 and diffusion coefficient is given. The sensitivity analysis of the diffusion estimations is also conducted with respect to the value of the lowest measured concentration value. In Section 4, eight methods for diffusion estimation are derived for the case when the measured concentration profiles contain noise. The methods are divided into two groups with respect to the concentration interval. The first group of diffusion estimations with finite support includes the concentration profiles matched using functions that are assumed to be vanished outside of some concentration interval. The second group of diffusion estimations with infinite support implies that the concentration are approximated with functions that asymptotically convergent to zero but not zero. Pros and cons of the developed methods are discussed. Section 5 is devoted to the application of the developed methodology for real dissolved volume concentrations obtained from x-ray Computer Tomography (CT). The derived methods are compared with respect to the lowest error for matching the gas-liquid interface propagation and measured concentrations. In Section 6 the results of the presented work are summarized and discussed.

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2 Model Formulation and analysis Let us consider an isothermal binary fluid system containing gas in the upper zone and liquid in the lower zone of a vertical cylindrical reservoir. In the considered system the pressure of the above gas zone is kept constant and higher than the liquid saturation pressure. Therefore, the transport of molecules is assumed to occur only in one direction from the gas zone into the liquid zone. In view of this assumption the concentration of gas in the gas zone is constant. In the liquid zone the transport process can be described using one dimensional diffusion equation based on the Fick’s second law 8 in the form ∂ ∂c ∂c = D , ∂t ∂x ∂x

(1)

where c = c(t, x) is the concentration of the dissolved gas in the liquid phase in kg/m3 at time t and vertical position x, and D is the diffusion coefficient in m2 /s. In the case of a binary mixtures, the self-diffusion coefficients of its components are often different. Therefore, the diffusion coefficient D is naturally expressed as a function of the concentration D = D(c). In the considered dissolved gas-liquid system, the changes of the pressure in the liquid zone can be considered very small and neglectable. In this case the densities of the original liquid and dissolved gas in the liquid phase can be assumed to be constant. The diffusion equation for the dissolved gas concentration in the liquid phase given by (1) can be rewritten in terms of the dissolved gas volume concentration (volume fraction) in the liquid phase, φ (t, x) ∂ ∂φ ∂φ = D(φ ) , ∂t ∂x ∂x

(2)

where c(t, x) = ρgl · φ (t, x) with the density of the dissolved gas in the liquid phase denoted as ρgl . In the case of the binary gas-liquid system, the interface between the liquid and gas zones can be characterized by the discontinuity in the density. The position of the interface of the concentration discontinuity, in general, is movable due to gas swelling and consequent diffusion of the dissolved

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gas into the liquid zone. This position will be referred to as the gas-liquid interface and denoted by h(t). Note that the change in the position of the gas-liquid interface ∆h = h(t) − h(0) defines the gas absorption volume per unit gas-liquid interface area. The bottom position of the gas-liquid column will be assigned to 0, and the total height of the sample is h0 , 0 < h(t) < h0 . Therefore, the diffusion equation given by (1) At the initial time moment we assume that the gas-liquid system has not started interaction, and

h(0) = h0 ,

(3)

0 ≤ x < h0 ,

φ (0, x) = 0,

(4)

with 0 < h0 < H. The diffusion model for dissolved gas concentration in the liquid phase would require boundary conditions at the bottom of the liquid zone and at the gas-liquid interface. At the bottom of the liquid zone zero-flux boundary condition can be assigned

D(c)

∂φ (t, 0) = 0 ∂x

(5)

At the gas interface the first type boundary condition in the form of the Henry’s law 7 can be applied

φ (t, h(t)) = φs =

γ P, ρgl

(6)

where P is the gas pressure under equilibrium conditions, and γ the Henry’s law solubility constant. The constant φs corresponds to the gas solubility volume fraction at the gas-liquid interface. The problem that includes equations (2)–(6) with unknown concentration and position of the gas-liquid interface is underdetermined. It needs to have one more condition to identify both the concentration and interface position. Such condition can be derived from the mass conservation principle for the original liquid, specifying that the amount of the original liquid needs to be con-

6


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served. In the case of the constant pressure this condition can be written in terms of the dissolved gas volume concentration in the form Z h(t)

(1 − φ (t, x))dx = const.

0

Since, at the initial time moment, see equation (4), it was assumed that φ (0, x) = 0, therefore, Z h(t) 0

φ (t, x)dx = h(t) − h0 = ∆h.

(7)

The integral form of this condition may not be very convenient for numerical calculations and analysis. Differentiating equation (7) with respect to t and applying the Leibniz integral rule and the diffusion equation, we obtain the boundary condition in the form

D(φ (t, h(t)))

∂φ (t, h(t)) = (1 − φ (t, h(t)))h0 (t). ∂x

(8)

The free boundary problem in the form (2)–(6) and (8) has been analyzed in a series of works. 37–41

2.1

Short time approximation

Gas dissolves into the liquid starting at the gas-liquid interface. For short time intervals the effect of gas on the bottom of the reservoir is not visible. In such a case the liquid reservoir could be assumed to be infinite acting. The solution to the diffusion problem with the non-flux boundary condition at the bottom of the reservoir can be approximated using the infinite liquid reservoir problem with −∞ < x < h(t), and the boundary condition at x → −∞ specified as

lim φ (t, x) = 0.

x→−∞

7


(9)


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Note that the total amount of gas diffused into liquid also satisfies the relationship Z h(t) −∞

φ (t, x)dx = h(t) − h0 .

(10)

The position and the boundary conditions at the gas-liquid interface remain unchanged. For an unbounded liquid region, the problem can be transformed to the problem with the fixed gas-liquid interface using the transformation y = x − h(t). In this case the equations for the gas concentration with fixed boundary will be changed as follows ∂φ ∂φ ∂ ∂φ − h0 (t) = D(φ ) , ∂t ∂y ∂y ∂y

y < 0,

t > 0,

h(0) = h0 ,

D(φs )

(11) (12)

φ (0, y) = 0,

y < 0,

(13)

φ (t, 0) = φs ,

t >0

(14)

∂φ (t, 0) − (1 − φ (t, 0))h0 (t) = 0, ∂y lim φ (t, y) = 0,

y→−∞

t > 0,

(15)

t > 0.

(16)

It is worth noting the convection term in the diffusion equation (11).

2.2 Constant diffusion case For the case with the constant diffusion coefficient D(φ ) = D0 , the solution of the diffusion problem on the semi-infinite interval given by Equations (2)–(4), (6), (8) and (9) can be easily calculated. The unknown gas-liquid interface and concentration can be found as follows √ h(t) = h0 + h1 t,

√1 x−h √0 2 D0 t h 1 1 + erf 2√D 0

1 + erf φ (t, x) = φs

8


(17)

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where √ h2 h1 h1 π φs = √ (1 − φs ) exp( 1 ) 1 + erf √ . 2 4D0 D0 2 D0

(18)

Note also that the solution for the constant diffusion satisfies the relationship Z √ h1 t =

h(t)

−∞

φ (t, x)dx.

(19)

This relationship illustrates that the amount of gas diffused into the liquid is proportional to the square root of the time. Note that in the case when the constant diffusion is unknown and one concentration profile is measured at some nonzero time t, then h1 could be defined from (19), and the diffusion coefficient D0 could be calculated from (18).

2.3

Heat Integral Method and gas-liquid interface

The idea of the heat integral method is in the application of the polynomial approximation to the solution of the free boundary problem. Such application is used in the case when the boundary condition at the infinity is approximated with ∂ φ (t, δ (t)) = 0 ∂y

φ (t, δ (t)) = 0,

(20)

where δ (t) is located sufficiently far from the gas-liquid interface y = 0, and the true value of the solution φ (δ (t)) is negligibly small. 17,26,27,42,43 The solution of the problem (11)-(15) and (20) is approximated using a polynomial. Here we choose a general form for such approximation as follows y k n y i φ (t, y) = 1 − ∑ ai 1 − δ (t) , δ (t) i=0

9


δ (t) < y < 0

(21)


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with δ (0) = 0, a0 6= 0 and k > 1. Integrating equation (11) between δ (t) and 0, and substituting (21) into it we obtain n

ai . i=0 i + k + 1

h0 (t) = −δ 0 (t) ∑

(22)

Substituting (21) into the boundary conditions (14) and (15), n

n

i=0

i=0

−D(φs ) ∑ (i + k)ai = (1 − φs )h0 (t)δ (t),

∑ ai = φs .

(23)

It follows that s h(t) = h0 +

n 2D(φs ) n ai √ t, (i + k)a i∑ ∑ (1 − φs ) i=0 i=0 i + k + 1

δ (t) =

h0 − h(t) ai . ∑ni=0 i+k+1

(24)

The above expressions suggest that the gas-fluid and zero gas concentration interfaces are moving √ linearly with respect to t. Although in practice the above relationships are also valid (see also the case with constant diffusion), it is worth to remember that these relationships are derived based on the modified boundary conditions in form (20), and these boundary conditions for the diffusion equations can be satisfied only approximately.

2.4

Dimensional analysis and diffusion evaluation

The solution for constant diffusion and the heat integral method reveals that the gas-liquid interface √ position could be described as h(t) = h0 + h1 t. The amount of absorbed gas per unit gas-liquid √ interface area is equal to h1 t. The parameter h1 describes the absorption process and will be referred to as to the gas absorption coefficient. The solution of the absorption-diffusion problem can be obtained as a function of λ = (x − h(t))t −1/2 . After the rescaling in the form (t, x) 7→ (t, λ ), the free boundary problem for (φ (t, x), h(t)) for can be rewritten in terms of the function Φ(t, λ ) =

10


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φ (t, x) as follows

t

∂Φ √ 0 ∂Φ λ ∂Φ ∂ ∂Φ − th (t) − = D(Φ) , ∂t ∂λ 2 ∂λ ∂λ ∂λ h(0) = h0 ,

Φ(t, 0) = φs ,

D(0)

Φ(0, λ ) = 0,

(25) (26)

√ ∂Φ (t, 0) − (1 − φs ) th0 (t) = 0, ∂λ

lim Φ(t, λ ) = 0.

(27) (28)

λ →−∞

√ It is easy to note that for the position of the interface expressed in the form h(t) = h0 + h1 t, it is possible to construct the solution Φ for this problem that is invariant with respect to time. We will attempt to build the solution Φ(λ ) as a monotonic function of λ . Therefore, the diffusion coefficient can be reparametrized as D(λ ) = D(Φ(λ )), and the problem for unknown Φ(λ ) and h1 can be written as follows

−

0 λ + h1 0 Φ (λ ) = D(λ )Φ0 (λ ) , 2 D(0)Φ0 (0) = (1 − φs )

Φ(0) = φs ,

(29) h1 , 2

lim Φ(λ ) = 0.

(30) (31)

λ →−∞

Then the following integral representation is valid 1 D(λ )Φ (λ ) − D(λ0 )Φ (λ0 ) = − 2 0

0

Z λ

(ξ + h1 )Φ0 (ξ )dξ .

(32)

λ0

Practical applications of diffusion for binary fluids naturally suggessts the boundedness of the diffusion coefficients on whole interval Φ ∈ [0, φs ]. The concentration far from the gas-liquid interface is approaching 0, and Φ0 (λ ) → 0 as λ → −∞. Therefore, the left hand side of (32) with the limits of integration λ0 = −∞ and λ = 0 will converge to D(0)Φ0 (0). Taking into account the boundary condition at y = 0 and integrating by parts we obtain simple expression for the gas

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absorption coefficient h1 Z 0

h1 =

−∞

Φ(ξ )dξ .

(33)

Note that the boundary condition given by (30) can be used for estimation of φs from known concentration profiles. Then the diffusion can be estimated from the integral expression (32) by choosing λ0 = −∞ or λ = 0. This will correspond to the forward or backward integration from one or another side of the interval for λ . Specifically, 1 λ 1 λ λ + h1 D(λ )Φ (λ ) = − Φ(λ ), (34) (ξ + h1 )Φ0 (ξ )dξ = Φ(ξ )dξ − 2 −∞ 2 −∞ 2 Z Z h1 1 0 h1 1 0 λ + h1 0 0 D(λ )Φ (λ ) = (1 − φs ) + (ξ + h1 )Φ (ξ )dξ = − Φ(ξ )dξ − Φ(λ ). (35) 2 2 λ 2 2 λ 2 Z

0

Z

Note that the expression (34) is very similar to the Boltzman-Matano expression for the diffusion of two binary liquids. The parameter h1 plays a role similar to the Matano interface value. 44 Nevertheless, this parameter can be excluded from expressions (34) and (35). The resulting equation for the diffusion will become 1 D(λ )Φ (λ ) = (1 − Φ(λ )) 2 0

1 Φ(ξ )dξ + Φ(λ ) 2 −∞

Z λ

Z 0

(1 − Φ(ξ ))dξ .

(36)

λ

It is similar to the expression obtained by Sauer-Freise 45 for diffusion of binary liquids. Integral expressions (33)–(36) can also be rewritten using the inverse function λ = λ (Φ)

h1 = −

Z φs

λ (Φ)dφ ,

(37)

0

Z Φ i λ 0 (Φ) h h1 Φ + λ (Φ)dΦ , 2 0 Z φs h i 0 λ (Φ) D(Φ) = h1 (1 − Φ) + λ (Φ)dΦ , 2 Φ Z φs Z 0 h i 0 λ (Φ) D(Φ) = Φ λ (Φ)dΦ − (1 − Φ) λ (Φ)dΦ . 2 Φ φs

D(Φ) = −

12


(38) (39) (40)

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These integral expressions are convenient because theyprovide the estimation of the diffusion directly in terms of concentration. Further analysis of the integral expressions represented by equations (34) and (35) allow estimation of the diffusion for the values of λ close to −∞ and 0. These diffusion values correspond to the diffusion at zero concentration of the gas in the liquid form and to the concentration of the gas in the liquid form at the gas-liquid interface. For the concentrations Φ that have finite improper integral around λ = −∞, Φλ (Φ)λ 0 (Φ) λ Φ(λ ) = − lim , Φ→0 2 λ →−∞ 2Φ0 (λ )

(41)

h1 (1 − φs ) h1 (1 − φs )λ 0 (φs ) = . 2Φ0 (0) 2

(42)

lim D(λ ) = − lim

λ →−∞

and

D(0) =

3 Application of integral expressions for diffusion calculation In this section we will discuss the application of the integral expressions (33)–(32) and (37)–(40) for diffusion estimation in the case of two diffusion models that allow explicit representation of the diffusion based on the dissolved gas volume concentration. One of the models represents the constant diffusion described by equations (17) and(18) with D = 10−5 and h1 = 0.001 and the other model is based on the case with non-constant diffusion in explicit form given by (47)–(49) with D1 = 10−6 , D1 = 10−5 , A = 0.4 and h1 = 0.001. Figure 1 illustrates the considered models with constant diffusion and diffusion dependent on the volume concentration. The values of φs were obtained as the values of the volume concentration profile at the gas-liquid interface. These values are also defined by expressions (18) and (49) for constant and non-constant models, respectively. In practice, the volume concentration profiles shown in Figure 1 (b) often contain some measurement noise. As result the noise could mask some the concentration values. Mainly the concentration values close to zero would be difficult to distinguish from zeroes. Therefore, limited range 13



11

×10-6

10 9

h1 = 10 -3

8

φs = 0.2527

D

7 6

φs = 0.3501

5

D(Φ) = Const D(Φ) 6= Const

4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Φ (a) 0.4 0.35 0.3

φ s = 0.3501

D(Φ) = Const D(Φ) 6= Const

φs = 0.2527

0.25

Φ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.2 0.15 0.1 0.05 0 -0.015

h1 = 10 -3 -0.01

-0.005

0

λ (b) Figure 1: (a) Constant and non-constant diffusion models, see A, and (b) corresponding volume concentration profiles as functions of λ . of the data with respect to x or λ could be used for diffusion estimation. These limited ranges correspond to the interval of concentrations defined as Φmin < Φ ≤ 0. The value of Φmin will be defined as a precision of the concentration measurement. Since the concentration in general can be between 0 and 1, Φmin is a relative error below which we cannot distinguish the concentration values from 0. Figure 2 shows the error in estimation hˆ 1 for the gas absorption coefficient h1 with respect to the precision in the concentration measurement represented by the number Φmin . It is interesting to note that the method based on the evaluation of the diffusion from Φ(λ ) from equations (33)–(32) provides much better precision in estimation of h1 than the method based on 14


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0.05 0.04

ˆ 1 − h1 |/h1 |h

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ˆ 1 from Φ(λ) h ˆ 1 from λ(Φ) h

0.03 0.02 0.01 0

10-4

10-3

10-2

10-1

min Φ Figure 2: Precision of estimation for the gas absorption coefficient h1 with respect to the lowest available data value of the volume concentration. Solid line shows the estimation of h1 obtained using the discretization of (33), and dashed line shows the estimation from (37). λ (Φ) given by (37)–(40). For example, in order to get 1% error in estimation of h1 , it is enough to have estimation of the concentration with the precision about 0.6% (Φmin = 0.006) using the method based on Φ(λ ) and about 0.06% (Φmin = 0.0006) based on λ (Φ). Note that the results for estimation of h1 are almost identical for constant and non-constant diffusions. Figure 3 presents the sensitivity analysis for diffusion estimation precision with respect to the precision of the volume fracture measurement, Φmin for constant diffusion (panel (a)) and nonconstant diffusion (panel (b)). It was observed that the diffusion estimations obtained using both methods with Φmin < 10−14 are identical. However, when Φmin > 10−14 the diffusion estimation method based on the representation for Φ(λ ) is more precise than based on λ (Φ) for both constant and non-constant diffusion models. The diffusion estimates obtained using the representation for Φ(λ ) for Φmin = 10−14 and 10−12 are identical as shown in the first column of Figure 3 contrary to those obtained using the representation for λ (Φ) in the second column. Further analysis of diffusion estimation reveals that each diffusion estimation regardless of Φmin have much larger estimation error for the values of Φ close to zero than in the middle of the concentration interval. In fact, even for explicit models it is impossible to obtain an accurate estimation of the diffusion at Φ = 0. The estimation error for D(0) is always huge and not acceptable. The reason for this is in the integral expressions for diffusion estimations. These expressions

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ˆ |D(Φ) − D(Φ)|/D(Φ)

10-2

10

10-2

ˆ ˆ D(Φ) as D(λ(Φ)) D(Φ) = 10−5 , φs = 0.2527

-4

= 10−14 = 10−12 = 10−10 = 10−8 = 10−6

Φmin Φmin Φmin Φmin Φmin

10-6 10-8 10-10 10-12

10

ˆ D(Φ) directly D(Φ) = 10−5 , φs = 0.2527

-4

10-6 10-8 10-10

0

0.05

0.1

0.15

0.2

10-12

0.25

0

0.05

0.1

Φ

0.15

0.2

0.25

Φ

(a) 10-2

ˆ |D(Φ) − D(Φ)|/D(Φ)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 51

10-2

Φmin Φmin Φmin Φmin Φmin

ˆ ˆ D(Φ) as D(λ(Φ)) D(Φ) 6= Const., φs = 0.3501

10-4 10-6

= 10 = 10−12 = 10−10 = 10−8 = 10−6

10-8

10

10-4 10-6 10-8

-10

10-12

ˆ D(Φ) directly D(Φ) 6= Const., φs = 0.3501

−14

10

0

0.05

0.1

0.15

0.2

0.25

0.3

-10

10-12

0.35

0

Φ

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Φ

(b) Figure 3: Relative errors in the diffusion estimations based on the volume concentration profiles with different ranges of the data (Φ > Φmin ). (a) Model with constant diffusion, and (b) model with the diffusion as a function of Φ. The first column shows the errors for diffusion estimation using equations (34) , (35) , or (32); the second column using equations (38) , (39) , or (40) . contain improper integration either with respect to λ with the lower limit −∞, or with respect to Φ with the lower limit 0. Since the concentrations are very small for λ → −∞, these improper integrals are estimated using the finite integrals. For intermediate concentration values of Φ, such approximation does not lead to large diffusion estimation errors. However, for very small values of Φ (i.e., Φ < 0.01) this estimation is not valid anymore. Although the absolute error for these integral approximations is extremely small, it does not allow to approximate them with zero for diffusion estimation problem, because the relative errors are increasing as λ → −∞. The problem of the diffusion estimation at the limiting concentration values possesses instability in estimation and is ill-posed. The solution cannot be solved by simple truncation of the concentration data at Φmin which is often necessary for the data with measurement error or for the computer precision

16


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restrictions.

4 Diffusion estimation based on measurement data with noise Direct application of the integration methods for diffusion estimation presented in Section 2 is not possible for measurement data containing noise. This is because the integral methods require estimation of the derivatives for the volume concentration. The presence of the data measurement noise would lead to discontinuities in these derivatives and the diffusion coefficient. To circumvent this problem, smooth approximations to the measured volume concentrations would be needed. In this section, we present a series of methods for diffusion estimation from noisy concentration data based on their polynomial approximations. The methods are divided into two groups with respect to the data location interval. Both finite and infinite data location intervals are considered. The methods derived for the finite data location intervals will be referred to as the methods with finite support. Otherwise, they will be called as the methods with infinite support. The methods are also grouped with respect to the form of approximation of the concentration data. The diffusion coefficient and the gas-liquid interface propagation is estimated based on the explicit form of the concentration approximation Φ = Φ(λ ), or the inverse form λ = λ (Φ). The first approximation type leads to the diffusion estimation directly as a function of the position D = D(λ ), and the second approximation type to the diffusion estimation as a function of the concentration D = D(Φ). The performance of the both types of approximations will be examined and compared in detail. All approximation methods are summarized in Table 1.

4.1

Diffusion estimations with finite support

4.1.1 Methods 1 and 2. Polynomial approximations of (1 − λ /δ ) In practice the heat integral method is sometimes used for estimation the diffusion parameter. 17,26–28 The forward and backward integral expressions for the diffusion given by (34) and (35) can be combined with the heat integral method for estimation of the diffusion coefficient in our case as 17



Page 18 of 51

well. Method 1 can be derived based on the representation of the concentration solution in the form similar to expression (21) λ k n λ i−1 Φ(λ ) = 1 − 1 − a ∑ i δ i=1 δ

(43)

with δ < 0, a1 6= 0 and k > 1. Substitution of (43) into (34) or (35) will lead to the following estimates for h1 and D n

h1 = −δ

ai

∑ k + i,

i=1

h i i δ +(k+i−1)λ λ n a h + 1 − ∑ i 1 k+i δ − λ i=1 δ . D(λ ) = i 2 λ n ∑i=1 ai (k + i − 1) 1 − δ

(44)

Similar expressions for the gas-liquid propagation parameter and diffusion can be obtained based on another polynomial representation of the concentration profile with respect to (1 − λ /δ )k with k > 0 as shown in Table 1, Method 2. Note that both methods for diffusion estimation imply that D(λ ) ∼ (λ − δ ) as λ → δ or Φ → 0. As result the value of the diffusion at zero gas concentration in the liquid phase concentration would approach zero.

4.1.2 Methods 3 and 4. Polynomial approximations of Φ Further examination of expressions (37)–(39) suggests that these expressions could also be used for diffusion approximation for certain form of approximation for the function λ (Φ). Assuming that λ (Φ) can be approximated as polynomials of Φ1/k , k > 0 for Method 3 or of Φ for Method 4, two different forms of estimations for the parameter h1 and diffusion coefficient can be derived as shown in Table 1. Note that for Methods 3 and 4 the estimated diffusion coefficient approaches zero as Φ → 0, moreover, it behaves as D(Φ) ∼ Φ1/k for small concentrations.

18


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4.2

Diffusion estimations with infinite support

4.2.1 Methods 5 and 6. Polynomial approximations of 1/(1 − λ ) The method for diffusion estimation presented above relies on the very restrictive assumption that the concentration profile diverges at the finite location δ . However, the nature of the diffusion contradicts this assumption. The concentration profile is not anymore equal to zero at any time moment t > 0 for the case with the bounded and nonzero diffusion. In order to overcome this assumptions the concentration could be approximated as polynomials of (1 − λ )−k in Method 5 or of (1 − λ )−1 in Method 6. The corresponding estimates of the parameter h1 and diffusion coefficient are shown in Table 1, Methods 5 and 6. In both cases D(λ ) ∼ λ 2 as λ → −∞. Therefore, D(Φ) ∼ Φ−2/k as Φ → 0. In other words, the estimated diffusion is infinite for Φ = 0.

4.2.2 Methods 7 and 8. Polynomial approximations of Φ Similarly to Section 4.1.2, the concentration data can be also approximated using the polynomials of Φ for Method 7 or Φ1/k , k > 1 for Method 8. Table 1 presents the expressions for diffusion and h1 estimations in these cases as well. Again, for both methods D(Φ) ∼ Φ−2/k as Φ → 0, therefore, the diffusion coefficient approaches infinity as Φ → 0.

4.3

Application of polynomial approximations for diffusion calculation for artificial data with noise

The application of the methods presented in Sections 4.1.1–4.2.2 are illustrated for two diffusion models presented in Section 4.2.1 that allows representation in an explicit form. The deterministic volume concentration profiles considered in Section 3 are perturbed with white noise to mimic the realistic measurements. The diffusion estimations are provided for these perturbed profiles. These profiles are first evaluated with respect to their usefulness for analysis. Specifically, it is shown how to eliminate the data values that are contain information completely masked by the noise. These data are specified by the concentrations that are very close to zero and are indistinctive from noise. 19


Finite

Infinite

h1 = −δ

ki λ n Φ(λ ) = ∑i=1 ai 1 − δ ai ∑ni=1 ki+1

ai h1 = ∑ni=1 ki−1 ai h1 = ∑ni=1 k+i−2 a

λ (Φ) = δ + ∑ni=1 ai Φ1/k+i−1

1 Φ(λ ) = ∑ni=1 ai (1−λ )ki

Φ(λ ) = ∑ni=1 ai (1−λ1)k+i−1

λ (Φ) = ∑ni=1 ai Φi−1−1/k

λ (Φ) = ∑ni=1 ai Φ(i−2)/k

4

i/k

5

6

7

8

1/k+i−1


20 φs

(i−2)/k

n i i 1/k ∑i=1 i−1/k φs

1

ai h1 = −φs ∑ni=1 (i−2)/k+1 φs

h1 = −

ai φs h1 = −φs δ + ∑ni=1 1/k+i

λ (Φ) = δ + ∑ni=1 ai Φi/k

ai h1 = −φs δ + ∑ni=1 i/k+1 φs

h1 = −δ

a ∑ni=1 k+ii

h1 - estimation

Approximation form k+i−1 Φ(λ ) = ∑ni=1 ai 1 − λδ

3

2

1

No

ki

i

−1 2Φ1/k

2

∑ni=1 (i − 1 − 1/k)ai Φi−1 ai × h1 + Φ11/k ∑ni=1 i−1/k Φi−1 1 n D(Φ) = − 2k ∑i=1 (i − 2)ai Φ(i−2)/k ai × h1 + ∑ni=1 (i−2)/k+1 Φ(i−2)/k D(Φ) =

D(λ ) =

n (1−λ )2 ∑i=1

i=1 (1−λ )ki ai k+i−1 1+h1 k+i−2 − 1−λ (1−λ )i a ∑ni=1 (k+i−1) (1−λi )i

+kiλ 1− λδ ∑ni=1 ai h1 + δki+1 D(λ ) = δ −λ ki 2 k ∑ni=1 iai 1− λδ 1 n D(Φ) iai Φi/k = − 2k ∑i=1 ai × h1 + δ + ∑ni=1 i/k+1 Φi/k 1/k D(Φ) = − Φ2 ∑ni=1 (1/k + i − 1)ai Φi−1 ai × h1 + δ + Φ1/k ∑ni=1 1/k+i Φi−1 a 1+h1 ki n i − (1−λ )2 ∑i=1 ki−1 1−λ (1−λ )ki D(λ ) = 2k iai ∑n

D(λ ) =

n 1− λδ k+i δ −λ ∑i=1 ai h1 + i 2 ∑ni=1 ai (k+i−1) 1− λδ

D - estimation δ +(k+i−1)λ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Support

Industrial & Engineering Chemistry Research Page 20 of 51

Table 1: Summary of presented models.

Page 21 of 51

0.35 0.3 0.25 0.2

Unperturbed profile Eliminated perturbed data Selected perturbed data Cut-off λ = δ

Φ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.15 0.1 0.05 0 -0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

λ

Figure 4: Perturbed data points with Gaussian noise converted to (λ , Φ) coordinate system. Cut off position λ = δ for elimination of data with the information masked by noise. Eliminated data points are in the left hand side part for λ < δ . Further analysis of the data also includes estimation of such parameters as the gas-liquid interface concentration φs , the exponent k and coefficients ai . 4.3.1 Data filtration. Estimation of δ The measured data contains the dissolved gas volume concentration data points. Figure 4 illustrates these data points in the form of pairs (λ j , Φ j ). From this figure we can see that the data points in the left hand side of the plot contain mainly noise. These values of the real concentration are impossible to distinguish from zeroes. In the analysis these data points would not bring any usable information for diffusion estimation. In general, filtering of the data points can be performed by visual analysis only. The threshold value δ could be identified from the scatted plot containing data, and data points with λ < δ could be eliminated. However, this approach lacks scientific argumentation and is based only on the preferences of an individual conducted data analysis. In order to approach the specified filtering problem, it is suggested to apply the statistic regression analysis. The dissolved gas concentration values are decreasing as the position becomes further from the gas-liquid interface. If the linear regression line is fitted to all concentration data set, its slope would be positive and significantly different than zero. However, for the concentration points located far from the gas-liquid interface, this slope becomes very close to zero. Standard statistical t-test for slope of the linear regression can be used for identification of δ . Specifically, 21



0.36

0.34 0.33

0.05

Data points Selected data points Linear fit φˆs

Mean of |φˆs − φs |/φs

0.35

Φ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 51

0.32 0.31 0.3 0.29 0.28 -1

-0.9

-0.8

-0.7

-0.6

-0.5

λ

-0.4

-0.3

-0.2

-0.1

0 ×10-3

0.04

D(Φ) = Const, φs = 0.2527 D(Φ) 6= Const, φs = 0.3501

0.03

0.02

0.01

0 10-4

(a)

10-3

σ

10-2

(b)

Figure 5: (a) Identification of φs value based on the linear regression approach for points close to the gas-liquid interface at λ = 0. (b) Sensitivity analysis of the estimation φˆs of φs with respect to the standard deviation of the noise added to the unperturbed volume concentration data for the models with constant and non-constant diffusion coefficient described in Appendix A. the concentration data points can be eliminated from right hand side until the p-value characterized by the t-test are smaller than some specified significance level. Figure 4 presents the result of the data filtering using this approach with the significance level of 0.05.

4.3.2 Estimation of the gas solubility concentration φs Since the volume concentration profile is perturbed, the values of Φ at λ = 0 may not necessary be the best estimate for φs , or may not be available. In order to obtain more reliable estimate of the gas solubility concentration φs , the statistical regression approach could be used. Figure 5(a) illustrates the application of the linear regression for estimation of φs . The linear regression is constructed for the data points closely located to the gas-liquid interface, and φs is identifies as an intercept. In general, similar estimation can be obtained using polynomial approximation to the data points in the neighbourhood of the gas-liquid interface. However, in such case the degree of the polynomial should be identified using likelihood-ratio test, Akaike information criteria or other model selection methods. Figure 5(b) shows the sensitivity analysis for the estimation φˆs of the gas-liquid interface concentration φs with respect to the standard deviation of the noise added to the generated data for the unperturbed models described in Appendix A. Naturally, the value of the relative deviation from

22


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the true gas-liquid concentration increases with the magnitude of the standard deviation of the noise. However, for very small noise, the error in the estimation of the parameter φs is not zero. It is because, for very small noise, the perturbation to the typically nonlinear concentration profile is insignificant and the linear regression does not anymore provide the best fit to the data close to the gas-liquid interface. When the standard deviation of the noise is less than approximately 5 · 10−4 , better estimate of the parameter φs can be obtained using polynomial approximation to the data close to gas-liquid interface instead of the linear regression.

4.3.3 Estimation of k The parameter k describes the behavior of the data close to λ = δ for the methods with finite support (Section 4.1), or close to λ = −∞ for the methods with infinite support (Section 4.2). Figures 6(a-f) illustrate selection algorithms for parameter k for different diffusion approximation methods. In the case of the methods with finite support presented in Section 4.1, the parameter k can be obtained from the relationship Φ ∼ (1 − λ /δ )k as λ → δ and λ > δ . It is based on the expressions for the leading order terms in the polynomial approximations for Φ as functions of λ in Section 4.1.1, and for λ as functions of Φ in Section 4.1.2. The parameter k is obtained based on the linear regression model in the form ln Φ = k ln(1 − λ /δ ) + c as shown in Figure 6(a). Specifically, the linear regression can be constructed for the selected data points with λ close to δ and λ > δ . Figure 6(b) shows that the obtained linear regression model can provide a good approximation to the data only for the selected data points. As λ becomes much larger than δ the fit is not anymore satisfactory. For the methods with infinite support, the definition of the parameter k is different for methods presented in Sections 4.2.1 and 4.2.2. The linear regression should be constructed in the form ln Φ = −k ln(1 − λ ) + c for methods from Section 4.2.1, and in the form ln Φ = −k ln(−λ ) + c for methods from Section 4.2.2. Figures 6(c) and (d) show these linear regression fits for selected data points in each case. Figures 6(d) and (f) illustrate how these linear regression line fit the entire set

23



0.35

Filtered data (λ > δ) Selected data ln Φ = k ln(1 − λ/δ) + c

10-1

0.3 0.25

Φ

Φ

0.2 10-2

0.15

Eliminated data (λ < δ) λ=δ Filtered data (λ > δ) Selected data Φ = ec (1 − λ/δ)k

0.1 0.05 10-3

0 10-1

100

-0.015

-0.01

1-λ/δ

-0.005

0

-0.005

0

0.35

All data Selected data ln Φ = −k ln(1 − λ) + c

-1

0

(b)

100

10

-0.005

λ

(a) 0.3 0.25

10-2

All data Selected data Φ = ec /(1 − λ)k

Φ

Φ

0.2 0.15

10-3

0.1

10-4

0.05 0

1

1.005

1.01

1.015

1.02

1.025

-0.015

-0.01

1-λ

λ

(c)

(d) 0.35

10-1

0.3 0.25

10

All data Selected data λ = −ec /Φ1/k

0.2

-2

Φ

Φ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 51

0.15

10-3

All data Selected data ln Φ = −k ln(−λ) + kc

10-4 -3 10

0.1 0.05 0

10-2

-0.015

-0.01

-λ

λ

(e)

(f)

Figure 6: Illustration of the methods for selection of the parameter k based on the selected data points (x). (a) linear regression fit for the selected data points for the methods with finite support and (b) the position of the obtained regression line with respect to selected data points (x), and all data points (.). (c) and (e) linear regression fits for the selected data points for the methods from Sections 4.2.1 and 4.2.2, respectively. (d) and (f) the position of the obtained regression line with respect to data points for the methods from Sections 4.2.1 and 4.2.2, respectively. of data points. Similarly for the methods with the finite support, the fit is satisfactory only for the selected data points. It is also good for the data points with the values of λ smaller than selected data points. However, the fit is not satisfactory for λ larger than the selected data values.

24


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4.3.4 Estimation of ai The final step for the estimation of the gas-liquid propagation front and diffusion coefficient is definition of the coefficients ai . It is easy to note that these coefficients could be defined using simple polynomial fitting approach. However, such approach may lead to overestimation or underestimation of the concentration data when the degree of the polynomial is improperly defined. To overcome this issue it is suggested to apply the stepwise linear regression model with the choice of the predictive variables conducted using the automatic procedure. 46 In this case the problem with definition of the highest degree of the polynomial is eliminated, and only the terms that are the most significant for the fitting of the concentration data are used. Table 2: Definition of the predictive and outcome terms for all estimation methods for application of the stepwise linear regression algorithms. Method 1 2 3 4 5 6 7 8

Xi , i = 1, . . . , n i−1+k 1 − λ /δ ik 1 − λ /δ Φi/k Φi−1+1/k 1/(1 − λ )ik 1/(1 − λ )i−1+k Φi−1−1/k Φ(i−2)/k

Y Φ Φ λ −δ λ −δ Φ Φ λ λ

In order to apply the stepwise linear regression approach, the predictive variables should be specified for the selected estimation method. The forms of the predictive and outcome terms are given in Table 2 for each method. The intercept is taken to be zero. Moreover, the leading term X1 in each approximation has to be present. This term governs the behavior of the concentration profile for either λ → δ in the case of the models with finite support, or λ → −∞ for the models with infinite support. Selection of the predictive variables in the stepwise linear regression approach relies on the forward model selection approach. The variables are added or removed to the model sequentially based on the certain selection criterion. In the current work, selection of variables is performed on the F-test of significance of variables in regression analysis with the significance 25



×10

-3

1.08

1.05

×10

-3

1.06

1.04 1.04

1.03

1.02

1.02

h1

h1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 51

1.01

1

1

0.98

0.99 0.96

0.98 0.97

0.94 M1

M2

M3

M4

M5

M6

M7

M8

M1

M2

Method

M3

M4

M5

M6

M7

M8

Method

(a)

(b)

Figure 7: Comparison of performance of different estimation methods for the gas absorption coefficient h1 in the case of constant diffusion coefficient (a) and variable diffusion coefficient (b). The true value of the parameter h1 is 0.001 (dashed line). The box plots show minimum, first quartile, median, third quartile, and maximum estimations based on 100 realizations. level 0.05. However, different model selection criteria such as R2 , adjusted R2 , Mallow’s C p , AIC, and BIC can also be used.

4.4

Comparisons of methods and uncertainty estimation

The methods for estimation of diffusion and gas-liquid front propagation are applied for constant and variable diffusion cases. The explicit models for these cases were already discussed in Section 3 and Appendix A. The volume concentration profiles for these models were perturbed with random noise sampled from the normal distribution with zero mean and standard deviation of 0.005. Figure 4 illustrates an example of the generated concentration data points for variable diffusion coefficient model. It was considered 100 realizations of each case of diffusion model to assess the uncertainty and compare the methods for diffusion estimation. The calculation of coefficients ai using the stepwise linear regression models was restricted to the first 20 terms (n = 20). Figure 6(b) provides an idea about the uncertainty for the estimation of the concentration φs at the gas-liquid interface. For the standard deviation of 0.005, the relative errors for both cases of diffusion models are about 0.5%. The estimation error of φs is slightly higher for the constant diffusion than for the variable diffusion case. This figure also provides analysis of the uncertainty for estimation of φs for different values of noise. 26


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Figure 7 illustrates the uncertainty in estimation of the gas absorption coefficient h1 describing the dynamics of gas-liquid interface propagation using different estimation methods. It includes the cases with constant and variable diffusion coefficients. It can be concluded from this figure that the methods with finite elements slightly underestimate the parameter h1 . The estimation error is in majority cases less than 2%. The methods with infinite support may either underestimate or overestimate the parameter h1 with the errors up to 5% for Methods 7 and 8. It can also be seen that the best estimation for the gas absorption coefficient h1 is achieved by applying Method 5. The interval between Q25 and Q75 contains the true value of h1 . Figures 8 and 9 gives the uncertainty estimation for the diffusion coefficients in the case of constant diffusion and variable diffusion coefficients, respectively. Overall, as it was discussed in Sections 4.1.1–4.2.2, the methods with the finite support underestimate the diffusion for very low dissolved gas concentrations, and the methods with the infinite support overestimate the diffusion coefficient for these concentrations. Therefore, the estimation errors are larger close to the zero concentration values than in the middle of the concentration interval. It also can be concluded that the diffusion estimation error is larger at the gas-liquid interface concentration value φs . The adequate estimations of the diffusion coefficients are obtained using Methods 1-5. The estimations obtained using Method 6 contain very significant trend in the estimation and are not satisfactory. Although Methods 7 and 8 can be used for diffusion estimation, the obtained results contain significant fluctuations around true diffusion coefficients. Further review of Methods 1-5 reveals that the smallest estimation error and lowest distribution of the diffusion estimates are achieved for Methods 1 and 2. These two methods perform very similarly. Method 5 also performs very well. However, the estimation diffusion errors are slightly larger than in Methods 1 and 2. The largest diffusion estimation errors are obtained by application of Methods 3 and 4. Nevertheless, the interquartile range for the estimations obtained using these methods in most cases contain true diffusion value. It is worth noting that the provided results were obtained for 20 terms in the corresponding approximations of the concentration profiles. Introducing more terms would potentially improve the diffusion estimations. Specifically, better diffusion

27



1.5

×10

-5

1.5

×10

-5

1

D

D

1

Method 2

Method 1 0.5

0.5 0.05

0.1

0.15

0.2

0.05

0.25

0.1

1.5

×10

0.15

0.2

0.15

0.2

0.25

φ

φ -5

1.5

×10

-5

D

1

D

1

Method 3

Method 4

0.5

0.5 0.05

0.1

0.15

0.2

0.25

0.05

0.1

φ 1.5

0.25

φ

×10-5

1.5

×10-5

D

1

D

1

Method 5

Method 6

0.5

0.5 0.05

0.1

0.15

0.2

0.25

0.05

0.1

φ 1.5

0.15

0.2

0.15

0.2

0.25

φ

×10-5

1.5

×10-5

1

D

1

D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 51

Method 7

Method 8

0.5

0.5 0.05

0.1

0.15

0.2

0.25

0.05

0.1

0.25

Figure 8: Case of constant diffusion. Comparison of diffusion estimations using different estimation methods. Box plots illustrate interquartile ranges for different concentration bins. estimations could be achieved for Method 6. On the other hand too many terms in estimation of the concentration profiles would potentially lead to overestimation problems. 28


Page 29 of 51

15

×10 -6

15

×10 -6

Method 2

Method 1

D

10

D

10

5

5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.05

0.1

0.15

φ 15

0.2

0.25

0.3

0.35

φ

×10 -6

15

×10 -6

Method 3

Method 4

D

10

D

10

5

5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.05

0.1

0.15

φ 15

0.2

0.25

0.3

0.35

φ

×10 -6

15

×10 -6

Method 5

Method 6

D

10

D

10

5

5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.05

0.1

0.15

φ 15

0.2

0.25

0.3

0.35

φ

×10 -6

15

×10 -6

Method 7

Method 8

10

10

D

D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5

5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.05

φ

0.1

0.15

0.2

0.25

0.3

0.35

φ

Figure 9: Case of non-constant diffusion. Comparison of diffusion estimations using different estimation methods. Box plots illustrate interquartile ranges for different concentration bins. In summary, Method 5 provides the best estimation of the gas-liquid interface propagation parameter h1 , and Methods 1 and 2 produce the best estimations for the diffusion coefficients. 29



5 Application to bitumen-dimethyl ether binary system In this section we will present a case study for diffusion and absorption estimation for bitumendimethyl ether binary system based on the volume concentration measurement data obtained using x-ray Computer Tomography (CT) methodology. The details for the measurement data collection are given in. 53 The above measurement data set for bitumen and dimethyl ether is used in this section only for parameter estimation and validation. The following steps are conducted: (1) calculation of the pseudo-concentration profiles based on the CT images; (2) estimation of the absorption coefficient h1 based on the gas-liquid front propagation; (3) coordinate translation; (4) estimation of the pseudo-solubility constant φ˜s ; (5) introduction of a new independent variable λ ; (6) pseudo-diffusion estimation based on the pseudo-concentration profiles; (7) estimation of the pseudo-absorption coefficient h˜ 1 ; (8) rescaling to true concentration profiles; (9) diffusion estimation based on the true concentration profiles; and (10) validation of obtained diffusion, absorption and dissolution values using comparison of simulated and measured concentration values.

5.1

Experimental Equipment and Procedure

In order to provide the estimation of the diffusion of dissolved gases into liquids the volume concentration information has to be collected. Such information can be collected using a non destructive method that involves application of CT, 47–53 NMR 54 or MRI. 55 To implement this method, a cylindrical aluminum cylinder with the inner diameter of 5.07 cm was used as a test reservoir. Approximately half of this cylinder was filled with the liquid Peace River bitumen (ρ ≈ 1010kg/m3 , µ ≈ 55, 000mPa · s), and then the selected gas was pumped above the liquid zone. At the initial time moment it is assumed that no interaction between liquid and gas had occurred. During the experiment the gas pressure was kept constant (P = 377kPaa) by regulating gas inflow into the cylinder. The experiment was conducted at 22oC. More details about the experiment can be found elsewhere. 53 In this work x-ray Computed Tomography (CT) technique is used for measurement of the

30


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10 min

130 min

10 min 130 min

-1000

-500

0

CTn

Figure 10: Early and late time two dimensional CT scans and vertical one dimensional CT n profiles taken from the middle of the object. concentration profiles of dissolved gas. Figure 10 presents early and late two dimensional CT scans for the liquid-gas sample. The resolution of one pixel in these scans is 0.195×0.195 mm2 and the measurement error is about 0.6%. The figure also shows one dimensional vertical profiles of CT averaged numbers over horizontal lines within shown rectangular domain in 2D images. This domain is selected from the middle of the scanned object to remove the effect of the boundaries from the consideration. It made it possible because the monitoring of the gas-liquid interaction was performed during relatively short time period. During this period the dissolved gas has not reached the bottom of the liquid solution. The dotted line vertical profile in Figure 10 shows that the dissolved gas concentration is still negligibly low at the bottom of liquid zone. Although the experiment was conducted during relatively short time, it was still possible to identify systematic change in the position h(t) of the gas-liquid interface as well as the change in the concentration of the dissolved gas in the liquid zone. The CT number is linearly related to the density of the scanned material, therefore, the volume concentration of the dissolved gas in the studied dissolved gas-liquid binary system can be

31



-2.816

1 h(t 1 )

h(t 2 )

√ h(t) = h0 + h1 t h0 = −2.8447 h1 = 3.7983 · 10−4

-2.818

0.8 -2.82

h(t k )

h(t)

0.6

φ˜

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 51

0.4

-2.822 -2.824

0.2

Data Fit Confidence bounds

-2.826

0 -2.95

-2.9

-2.85

-2.8

-2.75

-2.7

-2.828 48

50

52

54

56

58 √ 60

62

64

66

68

70

t

x

(a)

(b)

Figure 11: (a) Identification of the front propagation based on the plots for φ˜ . (b) Statistical evaluation and√ identification of coefficients in the relationship for the front propagation in the form h(t) = h0 + h1 t. quantified as follows CT n f −CT n φ˜ = , CT n f −CT ns

(45)

where CT n f and CT ns are the CT number of pure initial fluid and dissolved gas (solvent), respectively. It is important to note that the value CT n f can be easily determined from the CT scans from the initial images; however, the determination of CT ns is not possible directly. It is because the pure dissolved gas is not observed in the experiment. Therefore, as a first iteration the value of CT ns is defined as a CT number for the gas. For constant pressure this number is assumed to be unchanged and can be defined from the CT images in the upper gas zone. The obtained pseudoconcentrations φ˜ will be updated later to the proper dissolved gas concentrations φ with the aid of the conservation principle given in equations (33) and (37), or corresponding relationships for h1 presented in Section 4.1.

5.2

Estimation of the gas-liquid interface propagation and coordinate shift

Let us denote the measurements of the pseudo-volume concentrations of dissolved gas taken at time moments ti and spatial positions xi j as φ˜i j . The values of φ˜i j are constant in the gas zone. Since at the initial initial time moment no dissolved gas is assumed in the liquid zone, the corresponding

32


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dissolved gas concentrations in the liquid zone can be assumed to be equal to zero. Figure 11(a) illustrates the pseudo-volume concentration profiles measured in our experiment at different time moments. The gas and liquid zones are located in the right and left sides of the figure, respectively. The liquid zone increases due to gas absorption and diffusion into it. As result, the interface between gas and liquid moves from left to right. The gas-liquid interface can be characterized by very steep and large jump in the gas concentration. It also can be detected from Figure 11(a) that the concentration profiles at the interface have the same slopes. The intermediate positions of the concentration jumps can be used as gas-liquid interface positions at different time moments as shown in this figure. Although the interface positions identified in the described way might not be the precise positions for the gas-liquid interface, it can be shown that the obtained measurements deviate from the true gas-liquid interface positions by a constant. The described method is sufficient for our further investigations, since the considered diffusion absorption model is invariant with respect to a constant shift of coordinates. As it was noticed before in Section 2, the gas-liquid interface for the diffusion process propagates proportionally to the square root of the time, given that at the initial time moment gas has not propagate into the liquid zone. Therefore, the gas absorption coefficient or the interface propaga√ tion parameter h1 can be determined based on the linear regression in the form h(t) = h0 + h1 t. Figure 11(b) shows the determination of the parameters h1 and h0 with the coefficient of determination R2 equal to 0.995 and the significance of the square root of time relationship p − value = 8.3499 · 10−11 . The estimation of the value of the pseudo-interface gas solubility concentration φ˜s is difficult from the measured concentration profiles given in Figure 11(a). However, after the transformation of the spatial coordinates from the fixed system x to movable system y in the form y = x − h(t), the value of φ˜s can be determined as a branching point for the concentration profiles, see Figure 12(a). Further adjustment allows moving this point to the position y = 0 as shown in Figure 12(a). The corrected value h0 might be obtained from this adjustment, but will not be used for further calculation of the diffusion.

33



0.15

0.08 0.07

Branching point φ˜s = 0.067

0.06

0.1 0.05

φ˜

φ˜

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 51

0.04 0.03

0.05

0.02 0.01 0

0 -0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

y

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

λ

(a)

0

0.5 ×10-3

(b)

Figure 12: (a) Coordinate translation in the form y = x − h(t), and identification of the value for φ˜ ˜ ˜ based on the branching of the plots √ for φ . (b) Representation of the curves for φ at different time moments as a function of λ = y/ t. Figure 12(b) illustrates the data transformation from (t, y, φ˜ ) coordinates to (λ , φ˜ ) coordinates √ √ with λ = (x − h(t))/ t = y/ t. As we can see, the concentration profiles in the new system of coordinates are very close each to other. It is expected as a property of solutions for gas-liquid diffusion process described using the presented free boundary diffusion problem. The negative values of λ correspond to liquid zone, and the positive values of λ describe the gas zone. For diffusion estimation of gas in liquid we will eliminate all positive λ ’s. Note also that not all negative values of λ are useful for estimation of the diffusion. For λ smaller than some critical value, the measurement errors will mask any concentration values. These measurement concentration values should be eliminated because they would not bring any valuable information for the diffusion estimation. Based on the dissolved gas concentrations given in Figure 12(b) as a function of λ , the diffusion can be estimated using the methods presented in Section 4 as a two-step process. Initially, the diffusion estimation is provided for the pseudo-concentration data given in the form of the pairs (λi , φ˜i ). However, such estimation would not satisfy the mass conservation principle. As result the estimation h˜ 1 of the gas absorption coefficient h1 using the corresponding equations given in Table 1 would not match the estimation of h1 based on the concentration profiles, see Figure 11(a). The mismatch in the estimation of h1 occurs because for the pseudo-concentrations φ˜ are used. The second step for the diffusion estimation would require rescaling of the pseudo-concentrations

34


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×10-6

×10

2.5

2.2 2

-6

2

D, cm2 /s

1.8

D, cm2 /s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.6 1.4

M1: M2: M3: M4:

1.2

1 0

0.05

φs φs φs φs

= 0.2238 = 0.2243 = 0.2236 = 0.2239

0.1

0.15

1.5

M5: M6: M7: M8:

1

0

0.2

0.05

0.1

φs φs φs φs

= 0.2193 = 0.2079 = 0.2249 = 0.2188 0.15

0.2

φ

φ

(a)

(b)

Figure 13: Estimated diffusion coefficient using the methods with finite support (a) and the methods with infinite support (b). to proper dissolved gas concentrations based on the mass conservation principle. Such rescaling can be easily performed using the equation

φ=

h1 ˜ φ, h˜ 1

(46)

where h1 is the estimation of the gas absorption coefficient (interface propagation parameter) based on the concentration profiles given in Figure 11(a), and h˜ 1 is the estimation obtained from one of the diffusion estimation algorithms. After rescaling the diffusion estimation is again repeated using the algorithms of Section 4. The new estimations satisfy the mass conservation principle and new estimations of the interface propagation front coincide with the values of h1 obtained from Figure 11(a). Figure 13 presents the diffusion estimations for the bitumen–dimethyl ether binary system using the methods described above. The diffusion estimates using the methods with the finite support are shown in Figure 13(a), and with the infinite support are in Figure 13 (b). Overall all methods provide similar magnitude of the diffusion. Specifically the diffusion estimations obtained using the method with finite support are very similar for intermediate concentration values. The concentrations values at the gas-liquid interface are also very similar. Nevertheless, the estimations of the diffusion differ for zero dissolved gas concentration and for the concentrations close to the gas-liquid interface. On the other hand, the diffusion estimations obtained using the methods with 35



2.6 2.4 2.2 2

1.8 1.6 1.4

×10

-3

8

M1 M2 M3 M4 M5 M6 M7 M8

×10

-3

M1 M2 M3 M4 M5 M6 M7 M8

7

htrue - hsimulated

2.8

RSS (Φ)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 51

6 5 4 3 2 1 0

10

-1

-6

10

D(0)

-6

D(0)

(a)

(b)

Figure 14: Sensitivity analysis with respect to D(0). Estimation of D(0) based on the minimum residual sum of squares for the volume concentrations for different estimation methods (a). Comparison of the measured and simulated values for the gas-liquid interface propagation for different values of D(0) (b). infinite support have different behaviours. As it was noticed in the previous sections, it is impossible to obtain the proper estimate of the diffusion for the dissolved gas concentrations close to zero due to the difficulties of distinguishing of the measurements of the dissolved gas from the noise. Moreover, the diffusion estimates are zeroes at φ = 0 for the methods with the finite support, and converge to infinity for the methods with infinite support. To resolve the problem of diffusion estimation at zero concentration, the sensitivity analysis was performed as follows. It was assumed that the diffusion estimation is known from the estimation methods for the concentration values larger than some small value φmin . The values of D(0) were varied around the value of D(φmin ). The intermediate values of the diffusion between 0 and φmin were linearly interpolated. To select the most appropriate value of D(0), the simulation of the diffusion model (2)–(6) was conducted and the simulated concentration profiles were compared with the measurement data. The value of D(0) was selected as the value that minimizes the residual sum of squares between the measured and simulated concentrations. Figure 14(a) illustrates the results of application of the algorithm for definition of D(0) using each described estimation method. It is clear from this figure that the minimum residual sum of squares is well defined for each estimation method. As a result of the comparison for the performance of different diffusion estimation methods, it was possible to define the best diffusion method.

36


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Specifically, the smallest residual sum of squares was achieved using the estimation method 2. The diffusion value at φ = 0 was identified as D(0) = 1.12 · 10−6 cm2 /s. Further analysis indicates that the diffusion estimations obtained using the other methods with the finite support possess very similar values of the residual sum of squares as well as the diffusion estimates at φ = 0. On the other hand, the diffusion estimates obtained using the methods with infinite support perform significantly worse than the methods with finite support. Figure 13 shows the estimated diffusions that include estimations of D(0). In addition to the above analysis of the residual sum of squares for the concentrations, the propagation of the gas-liquid interface was also evaluated using the measured and simulated data. Figure 14(b) presents the deviation of the simulated and measured positions of the gas-liquid interface. From this figure it is possible to note that simulated and measured gas-liquid interfaces are closest for the diffusion estimation methods 1, 2 and 7. Methods 3-6 and 8 underestimate the position of the gas-liquid interface. Further analysis of the diffusion estimation methods reveals the following conclusions. The diffusion estimation methods with the infinite support are very sensitive to several factors, and obtained diffusions may vary very significantly from each other and from true diffusion. This sensitivity is related to data fitting with the functions described in Table 2. It was observed that the diffusion estimations obtained using these methods are highly affected by the degree of the polynomial n, the thresholds δ and the estimated parameter k. Even small change of these factors often leads to huge difference for estimated diffusions. The methods with finite support are much more stable than the methods with infinite support. The obtained diffusion estimations using the methods with finite support are affected by the uncertainty in various parameters by much lower degree. Finally, Figure 15 evaluates the best diffusion estimate obtained using Method 2 shown in Figure 13 (a) using the bold line. This evaluation includes the comparisons of the measured and simulated gas-liquid interface propagation (Figure 15(a)) and concentrations (Figure 15(b)). To obtain the simulated results, the problem with the gas-liquid interface was converted to the problem

37



0.025

h1 = 3.7983 · 10−4

h(t), cm

0.02 0.015 0.01

Data √ Linear fit: h = h1 t Simulation

0.005 0 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

t, s (a) 0.25

0.2

0.15

φ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 51

0.1

0.05

0 -0.25

-0.2

-0.15

-0.1

-0.05

0

x, cm (b) Figure 15:√ Comparison of the measured (symbols ×), the least squares fit (dotted line) with h(t) = h1 t and simulated (solid line) data for gas-liquid interface propagation (a). Results of simulations (solid lines) for the dissolved gas volume concentrations and comparison with the measured concentrations (symbols ◦, M, O, ., ) at different time moments (b). with the fixed boundary as shown in Section 2 and then solved using the finite difference method.

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6 Discussion and Conclusions An intricate problem of quantification for solvent gas-liquid interaction is considered in this work. This problem involves not only estimation of the diffusion coefficient in the diffusion equation derived based on the mass conservation principle, but also needs evaluation of the absorption of the considered solvent gas into the liquid and specification of the gas-liquid interface propagation. Such multiplex character of the solvent gas-liquid systems is described using a free boundary problem for the diffusion equation with unknown diffusion coefficient. In practical applications the problem of solvent gas-liquid interaction is often solved using simplified approach that neglects the gas swelling into the liquid phase and corresponding gas-liquid interface movement. 3,10,13,14,16–20,22,23,53,56–62 However, this simplification is often not supported by the real measurements of absorption of different gases into liquids. 23,25,28,63,64 In order to solve the problem with gas swelling into liquid with unknown gas-liquid interface, it is necessary to specify an additional condition for the diffusion equation. Such condition was formulated based on the mass conservation of the dissolved gas and represented in the form of the Stefan’s type boundary value condition at the moving interface. 23,26–28,65 An alternative approach for identification of the unknown gas-liquid interface was applied based on the moving grid size refinement. 24 It is also based on the mass conservation of the dissolved gas. The authors of the above works attempted to solve the considered gas-liquid problems with unknown gas-liquid interface and diffusion coefficient using different method under the restrictive assumption about the constant diffusion. Although the obtained results matched the real measurements very well, the assumption about the constant diffusion coefficient is often non-realistic. Specifically such assumption is often not valid when the viscosities of the liquids and dissolved gases are very different. 25,50,51 In this paper we formulated a diffusion model for gas-liquid binary systems that incorporates gas absorption into the liquid and consequent liquid expansion. At the gas-liquid interface the concentration of the dissolved gas is governrned by the Henry’s solubility law and is equal to the gas solubility concentration φs . To identify an equation for unknown moving gas-liquid interface, the additional boundary value condition was derived based on the mass balance of the dissolved gas 39



written in the form of integral equation. The considered problem was examined for relatively short time interval during which no significant changes of the initial liquid was observed at the bottom of the vessel due to gas diffusion. It was shown that in such case the gas-liquid interface, h(t), √ propagates linearly with respect to the square root of the time, h(t) = h0 + h1 t, given that at the initial time moment none dissolved gas is presented in the liquid zone. The parameter h1 describes the change in the position of the gas-liquid interface. It also can be considered as a gas absorption √ coefficient, since the expression h1 t defines the volume of absorbed gas per unit gas-liquid inter√ face area. With the aid of the scaling in the form (x − h(t))/ t, the diffusion model was reduced to the time invariant equation. The obtained equation was used for derivation of the integral equations for diffusion and gas absorption coefficient estimation based on the concentration profiles. Although these integral equations work well in the deterministic case with the concentration profiles not affected by any perturbations, in the stochastic case they are not applicable due to random fluctuations. Therefore, for the real data that contains noise, eight different polynomial methods for diffusion and gas-liquid interface calculation were proposed. These methods were grouped into two categories with respect to the spatial interval for fitting of the concentration data. They were referred in the text to as the methods with finite and infinite supports. The first category included the case when the concentration data was approximated over a finite interval. Outside of this interval the concentration values are very small and are approximated by zeroes. This method is used in many applications modeled using free boundary problems and is referred in literature to as the Heat Balance Integral Methods (HBIM). 17,26,27,42,43,66,67 The second category included the case when the concentration data was assumed to be nonzero for all considered interval. The concentration data were approximated with the functions that are asymptotically convergent to zero far from the moving interface. The proposed approximation methods were evaluated for simulated data as well as for the real measurements obtained from x-ray CT scans. In the case of the simulated data, the evaluation of the methods was performed based on the known constant and variable diffusion coefficients. For real measurement data with unknown diffusion, the comparisons were conducted for measured and simulated dissolved gas concentrations and the gas-liquid interface propagation.

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It was shown that the diffusion approximation methods with the finite support perform better than the methods with the infinite support. The methods with finite support were much more stable with respect to any data fluctuations, than the methods with the infinite support. It is also necessary to understand weak points for the diffusion and gas-liquid interface estimation. The nature of the diffusion estimation as well as free boundary problems is in their ill-posed character. 68 Such problems often experience instabilities of the solutions. This character of the diffusion and free boundary estimation problems is observed in the present studies as well. Specifically, the difficulties in diffusion estimation arose for the concentration values close to zero. In our case such values are present far from the gas-liquid interface. Any fluctuations of the dissolved gas concentration or measurement noise are typically present in any experiment. For concentration values close to zero, it is very difficult or impossible to distinguish the true concentration values from the noise. Therefore, the estimated diffusion at the limiting concentrations is always a challenge. Clear indication of this instability problem for diffusion estimation was observed in the examples presented in Figures 3, 8, 9 and 13. To account for the uncertainty for diffusion estimations close to zero concentration values, we combined the proposed methods with the least square estimation method. It was assumed that the diffusion estimations obtained from Methods 1–8 far from zero concentration values were adequate, and the diffusion at at zero concentration is unknown. This diffusion was estimated using the least square method for comparison of the simulated and measured concentrations. The results are shown in Figures 13 and 14. The other source of uncertainty of diffusion calculation comes from the estimation of the concentration value at the gas-liquid interface. As shown in Figures 8 and 9, small fluctuations in the estimation of this concentration leads to significantly larger diffusion fluctuations than in the middle of the concentration interval. In conclusion, a series of methods for estimation of diffusion and gas absorption coefficient as a gas-liquid interface propagation parameter were developed based on the concentration data. The developed methods were tested using synthetic and real measurement data. Evaluation of

41



Page 42 of 51

the proposed methods showed that the methods developed based on the Heat Balance Integral approach or the methods with finite support perform better than the methods with infinite support. The diffusion estimation methods with finite support provide diffusion estimations with lower uncertainty than the other group of methods. It was pointed out the issues arising for diffusion estimation close to zero concentration values. In the present work this problem was addressed using the least square method for measured and simulated concentrations. Nevertheless, in our future work the problem of diffusion estimations and instability close to zero concentration level will be addressed using different estimation techniques.

Acknowledgment We are grateful for the constructive comments and suggestions by reviewers. The authors also gratefully acknowledge the financial support from NSERC AITF / i-Core Industrial Research Chair in Modeling Fundamentals of Unconventional Resources and sponsoring partners: Alberta Innovates, Athabasca Oil Corporation, Biron Energy, Canada Natural, Devon, Foundation CMG, Husky Energy, Laricina Energy LTD, NSERC, Schulich School of Engineering – University of Calgary and Suncor.

A Example with explicit solution for bounded diffusion In the case when the solution to problem (29)–(31) can be described using the equation √1 −λ1 + (1 − A) 1 + erf λ +h √1 −λ2 A 1 + erf λ +h 2 D1 2 D2 , Φ(λ ) = φs h1√−λ1 h1√−λ2 A 1 + erf 2 D + (1 − A) 1 + erf 2 D 1

42


2

(47)

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the diffusion is evaluated as follows

D(λ ) = √ λ A π 1 − 2

(λ +h1 −λ1 )2 (λ +h1 −λ2 )2 √ √ − − 4D1 4D2 A D1 e + (1 − A) D2 e (λ +h1 −λ1 )2

(λ +h1 −λ2 )2

− 1−A − 4D1 4D2 √A e +√ e D1 D2 √1 −λ1 + λ2 (1 − A) 1 + erf λ +h √1 −λ2 1 + erf λ +h 2 D1 2 D2 − √A e D1

(λ +h1 −λ1 )2 4D1

+

1−A − √ e D2

(λ +h1 −λ2 )2 4D2

(48) .

The relationship between the parameters φs , h1 , λ1 , λ2 , D1 and D2 can be obtained from boundary condition (30) h1 − λ1 h1 − λ2 A(h1 (1 − φs ) + λ1 φs ) 1 + erf √ + (1 − A)(h1 (1 − φs ) + λ2 φs ) 1 + erf √ 2 D1 2 D2 2 2i h (h −λ ) (h −λ ) √ 2φs √ − 1 1 − 1 2 = √ A D1 e 4D1 + (1 − A) D2 e 4D2 π

(49)

.

References (1) Godec, M. L.; Kuuskraa, V. A.; Dipietro, P. Opportunities for Using Anthropogenic CO2 for Enhanced Oil Recovery and CO2 Storage. Energy Fuels 2013, 27, 4183–4189. (2) Zarghami, S.; Boukadi, F.; Al-Wahaibi, Y. Diffusion of carbon dioxide in formation water as a result of CO2 enhanced oil recovery and CO2 sequestration. J. Petrol. Explor. Prod. Technol. 2017, 7, 161–168. (3) Etminan, S. R.; Maini, B. B.; Chen, Z. Determination of mass transfer parameters in solventbased oil recovery techniques using a non-equilibrium boundary condition at the interface. Fuel 2014, 120, 218–232. (4) Bhunia, K.; Sablani, S. S.; J.Tang,; Rasco, B. Hemoglobin diffusion and the dynamics of oxygen capture by red blood cells. Food. Res. Int. 2016, 89, 161–168. (5) Hulst, A. C.; Hens, H. J. H.; Buitelaar, R. M.; Tramper, J. Determination of the effective 43



diffusion coefficient of oxygen in gel materials in relation to gel concentration. Biotechnol. Tech. 1989, 3, 199–204. (6) Longeville, S.; Stingaciu, L.-R. Hemoglobin diffusion and the dynamics of oxygen capture by red blood cells. Sci. Rep. 2017, 7, 10448. (7) Henry, W. Experiments on the quantity of gases absorbed by water, at different temperatures, and under different pressures. Phil. Trans. R. Soc. Lond. 1803, 93, 29–42. (8) Fick, A. Ueber Diffusion. Annalen der Physik 1855, 94, 59–86. (9) Stefan, J. Ueber einige Probleme der Theorie der W´’armeleitung. Wiener Akad. Math. Naturwiss. Abt. 1888, 98, 473–484. (10) Harvey, E. A.; Smith, W. The absorption of carbon dioxide by a quiescent liquid. Chem. Eng. Sci. 1959, 10, 274–280. (11) Dullien, F. A. L.; Shemilt, L. W. Diffusion coefficients for the liquid system: Ethanolwater. Can. J. Chem. Eng. 1961, 39, 242–247. (12) Himmelblau, D. M. Diffusion of Dissolved Gases in Liquids. Chem. Rev. 1964, 64, 527–550. (13) R.K.Gupta,; T.Sridhar, Effect of interfacial resistance on quiescent gas-liquid absorption. Chem. Eng. Sci. 1984, 39, 471–477. (14) Zhang, Y.; Hyndman, C.; Maini, B. Measurement of gas diffusivity in heavy oils. J. Petrol. Sci. Eng. 2000, 25, 37–47. (15) Upreti, S. R.; Mehrotra, A. K. Experimental measurement of gas diffusivity in bitumen: Results for carbon dioxide. Ind. Eng. Chem. Res. 2000, 39, 1080–1087. (16) Etminan, S. R.; Maini, B. B.; Chen, Z.; Hassanzadeh, H. Constant-pressure technique for gas diffusivity and solubility measurements in heavy oil and bitumen. Energy Fuels 2010, 24, 533–549. 44


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(17) Ghaderi, S.; Tabatabaie, S.; Hassanzadeh, H.; Pooladi-Darvish, M. Estimation of concentration-dependent diffusion coefficient in pressure-decay experiment of heavy oils and bitumen. Fluid Ph. Equilibria 2011, 305, 132–144. (18) Pacheco-Roman, F. J. P.; Hejazi, S. H. Graphical estimation of interface mass-transfer coefficient (k) using pressure-decay data. Chem. Eng. Sci. 2016, 146, 35–43. (19) Pacheco-Roman, F. J. P.; Hejazi, S. H. Estimation of Concentration-Dependent Diffusion Coefficients of Gases in Heavy Oils/Bitumen Using Experimental Pressure-Decay Data. Can. J. Chem. Eng. 2016, 94, 2407–2416. (20) Pacheco-Roman, F. J. P.; Hejazi, S. H.; Maini, B. B. Estimation of Low-Temperature MassTransfer Properties of Methane and Carbon Dioxide in n-Decane, Hexadecane, and Bitumen Using the Pressure-Decay Technique. Energy Fuels 2016, 30, 5232–5239. (21) Olander, D. R. Analysis of liquid diffusivity measurements to account for volume changes on mixing – The diaphragm cell. J. Phys. Chem. 1963, 67, 1011–1015. (22) Renner, T. Measurement and Correlation of Diffusion Coefficients for CO2 and Rich-Gas Applications. SPE Reserv. Eng. 1988, 3, 517–523. (23) Riazi, M. R. A new method for experimental measurement of diffusion coefficients in reservoir fluids. J. Petrol. Sci. Eng. 1996, 14, 235–250. (24) Upreti, S. R.; Mehrotra, A. K. Diffusivity of CO2, CH4, C2H6 and N2 in Athabasca bitumen. Can. J. Chem. Eng. 2002, 80, 116–125. (25) Grogan, A. T.; Pinczewski, V. W.; Ruskauff, G. J.; Jr., F. O. Diffusion of CO2 at Reservoir Conditions: Models and Measurements. SPE Reservoir Engineering 1988, 3, 93–102. (26) Do, H. D.; Rice, R. G. Validity of the Parabolic Profile Assumption in Adsorption Studies. AlChE J. 1986, 32, 149–154.

45



(27) Do, H. D.; Pinczewski, W. V. Diffusion controlled swelling of reservoir oil by direct contact with injection gas. Chem. Eng. Sci. 1991, 46, 1259–1270. (28) Jamialahmadi, M.; Emadi, M.; Muller-Steinhagen, H. Diffusion coefficients of methane in liquid hydrocarbons at high pressure and temperature. J. Petrol. Sci. Eng. 2006, 53, 47–60. (29) Ivanchov, M. I. Inverse Problem with Free Boundary for Heat Equation. Ukr. Math. J. 2003, 55, 1086–1098. (30) Hryntsiv, N. M.; Ivanchov, M. I. Inverse problem for the strongly degenerate heat equation in a domain with free boundary. Ukr. Math. J. 2009, 61, 30–49. (31) Ivanchov, M.; Savitska, T. An inverse problem for a parabolic equation in a free-boundary domain degenerating at the initial time moment. J. Math. Sci. 2012, 181, 47–64. (32) Ivanchov, M. I.; Pabyrivska, N. V. Inverse problem for a two-dimensional diffusion equation in a domain with free boundary. Ukr. Math. J. 2013, 65, 1019–1031. (33) Cannon, J. R.; Duchateau, P. An inverse problem for a nonlinear diffusion equation. SIAM J. Appl. Math. 1980, 39, 272–289. (34) Lesnic, D. The determination of the thermal properties of a heat conductor in a nonlinear heat conduction problem. Z. angew. Math. Phys. 2002, 53, 175–196. (35) Fatullayev, A. Determination of unknown coefficient in nonlinear diffusion equation. Nonlin. Anal. 2001, 44, 337–344. (36) Shidfar, A.; Fakhraie, M.; Pourgholi, R.; Ebrahimi, M. A numerical solution technique for a one-dimensional inverse nonlinear parabolic problem. Appl. Math. Comp. 2007, 184, 308– 315. (37) Boley, B. Uniqueness in a melting slab with space- and time-dependent heating. Quart. Appl. Math. 1970, 27, 481–487. 46


Page 46 of 51

Page 47 of 51 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(38) Rogers, C. On a class of moving boundary problems in non-linear heat conduction: Application of a Bäcklund transformation. Int. J. Non-Linear Mechanics 1986, 21, 249–256. (39) Fatullayev, A. Free boundary problems with nonlinear diffusion. Math. Comput. Modelling 1993, 18, 15–34. (40) van de Fliert, B. W.; van der Hout, R. A generalized Stefan problem in a diffusion model with evaporation. SIAM J. Appl. Math. 2000, 60, 1128–1136. (41) Hussein, M. S.; Lesnic, D.; Ivanchov, M. Free Boundary Determination in Nonlinear Diffusion. E. Asian J. Appl. Math. 2013, 3, 295–310. (42) Mitchell, S. L.; Myers, T. G. Application of standard and refined heat balance integral methods to one-dimensional Stefan problems. SIAM Rev. 2010, 52, 57–86. (43) Mitchell, S. L.; O’Brien, S. B. G. Asymptotic and Numerical Solutions of a Free Boundary Problem for the Sorption of a Finite Amount of Solvent into a Glassy Polymer. SIAM J. Appl. Math. 2014, 74, 697–723. (44) Matano, C. On the Relation Between the Diffusion-Coefficients and Compositions of Solid Metals (The Nickel-Copper System). Jpn. J. Phys 1933, 8, 109–113. (45) Sauer, F.; Freise, V. Diffusion in binären Gemischen mit Volumenänderung. Z. Elektrochem. 1962, 66, 353–362. (46) Hastie, T.; Tibshirani, R.; Friedman, J. The Elements of Statistical Learning: Data Mining, Inference, and Prediction; Springer-Verlag, New York, 2009. (47) Wen, Y. W.; Kantzas, A. Monitoring Bitumen-Solvent Interactions with Low-Field Nuclear Magnetic Resonance and X-ray Computer-Assisted Tomography. Energy Fuels 2005, 19, 1319–1326. (48) Luo, H.; Salama, D.; Kryuchkov, S.; Kantzas, A. The Effect of Volume Changes Due to Mixing on Diffusion Coefficient Determination in Heavy Oil and Hydrocarbon Solvent System. 47



SPE Annual Technical Conference and Exhibition, 11-14 November, Anaheim, California, U.S.A 2007, SPE-110522, 1–7. (49) Luo, H.; Kantzas, A. Investigation of Diffusion Coefficients of Heavy Oil and Hydrocarbon Solvent Systems in Porous Media. SPE Symposium on Improved Oil Recovery, 20-23 April, Tulsa, Oklahoma, USA 2008, SPE-113995, 1–10. (50) Aconcha, U. E. G.; Kantzas, A. Diffusion of Hydrocarbon Gases in Heavy Oil and Bitumen. In Latin American and Caribbean Petroleum Engineering Conference, 31 May-3 June, Cartagena de Indias, Colombia 2009, SPE-122783, 1–11. (51) Song, L.; Kantzas, A.; Bryan, J. Investigation of CO2 Diffusivity in Heavy Oil Using X-Ray Computer-Assisted Tomography Under Reservoir Conditions. In SPE International Conference on CO2 Capture, Storage, and Utilization, 10-12 November, New Orleans, Louisiana, USA 2010, SPE-138205, 1–11. (52) Luo, H.; Kantzas, A. Study of Diffusivity of Hydrocarbon Solvent in Heavy-Oil Saturated Sands Using X-Ray Computer Assisted Tomography. J. Can. Pet. Technol. 2011, 50, 24–34. (53) Diedro, F.; Bryan, J.; Kryuchkov, S.; Kantzas, A. Evaluation of Diffusion of Light Hydrocarbon Solvents in Bitumen. In SPE Canada Heavy Oil Technical Conference; Calgary, AB, Canada, 2015 2015, SPE-174424, 1–22. (54) Wen, Y. W.; Bryan, J.; Kantzas, A. Estimation of Diffusion Coefficients in Bitumen Solvent Mixtures as Derived From Low Field NMR Spectra. J. Can. Pet. Technol. 2005, 44, 29–35. (55) Fayazi, A.; Kryuchkov, S.; Kantzas, A. Evaluating Diffusivity of Toluene in Heavy Oil Using Nuclear Magnetic Resonance Imaging. Energy Fuels 2017, 31, 1226–1234. (56) Azizi, S.; Kargari, A.; Kaghazchi, T. Experimental and theoretical investigation of molecular diffusion coefficient of propylene in NMR. Chem. Eng. Res. Des. 2014, 92, 1201–1209.

48


Page 48 of 51

Page 49 of 51 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(57) Civan, F.; Rasmussen, M. Determination of gas diffusion and interface-mass transfer coefficients for quiescent reservoir liquids. SPE J. 2006, 11, 71–79. (58) Etminan, S. R.; Pooladi-Darvish, M.; Maini, B. B.; Chen, Z. Modeling the interface resistance in low soluble gaseous solvents-heavy oil systems. Fuel 2013, 105, 672–687. (59) Rasmussen, M. L.; Civan, F. Parameters of gas dissolution in liquids obtained by isothermal pressure decay. AIChE J. 2009, 55, 9–23. (60) Sheikha, H.; Mehrotra, A. K.; Pooladi-Darvish, M. An inverse solution methodology for estimating the diffusion coefficient of gases in Athabasca bitumen from pressure-decay data. J. Petrol. Sci. Eng. 2006, 53, 189–202. (61) Tharanivasan, A. K.; Yang, C.; Gu, Y. Measurements of molecular diffusion coefficients of carbon dioxide, methane, and propane in heavy oil under reservoir conditions. Energy Fuels 2006, 20, 2509–2517. (62) Unatrakarn, D.; Asghari, K.; Condor, J. Experimental Studies of CO2 and CH4 Diffusion Coefficient in Bulk Oil and Porous Media. Energy Procedia 2011, 4, 2170–2177. (63) Sekerci-Cetin, M.; Emek, O. B.; Yildiz, E.; Unlusu, B. Numerical study of the dissolution of carbon dioxide in an ionic liquid. Chem. Eng. Sci. 2016, 147, 173–179. (64) Zheng, S.; Yang, D. Experimental and Theoretical Determination of Diffusion Coefficients of CO2-Heavy Oil Systems by Coupling Heat and Mass Transfer. J. Energy Resour. Technol. 2017, 139, 1–15. (65) Haugen, K. B.; Firoozabadi, A. Mixing of two binary nonequilibrium phases in one. AIChE J. 2009, 55, 1930–1936. (66) Hristov, J. The heat-balance integral: 1. How to calibrate the parabolic profile? C. R. Mecanique 2012, 340, 485–492.

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(67) Hristov, J. The heat-balance integral: 2. Parabolic profile with a variable exponent: The concept, analysis and numerical experiments. C. R. Mecanique 2012, 340, 493–500. (68) Hadamard, J. Sur les Problèmes aux Dérivées Partielles et Leur Signification Physique. Princeton University Bulletin 1902, 13, 49–52.

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Page 50 of 51

Graphical TOC Entry t1

.....

t1

tn

tn

√ h(t) = h0 + h1 t h(t)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


√ t

h(t)

D(φ)

Page 51 of 51

0

0.1

0.2

φ -1000 -800 -600 -400 -200

0

CTn

51


Estimation of gas absorption and diffusion coefficients for dissolved

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