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SIAM J. APPL. MATH .c

2013 Society for Industrial and Applied Mathematics

Vol. 73, No. 5, pp. 1941–1957

HYPERBOLIC THEORY FOR FLOW IN PERMEABLE MEDIA

WITH PH-DEPENDENT ADSORPTION∗

VALENTINA PRIGIOBBE†,MARCA.HESSE

‡,AND STEVEN L. BRYANT§

Abstract. A theory for the solution of the Riemann problem for a one-dimensional, quasi-

linear, 2×2 system of conservation laws describing reactive transport in a permeable medium with

pH-dependent adsorption is developed. The system is strictly hyperbolic and nongenuinely nonlinear

because the adsorption isotherms are not convex functions. The solution comprises nine fundamental

structures, which are a concatenation of elementary and composed waves. In the limit of low pH,

the isotherms reduce to convex two-component Langmuir isotherms considered in chromatography,

and the solution comprises only four fundamental structures, as in classical theory. Semianalytical

solutions and highly resolved numerical simulations show good agreement in all cases.

Key words. chromatography, conservation laws, hyperbolic partial diﬀerential equations, non-

genuinely nonlinear, pH-dependent adsorption, p orous media, reactive transport

AMS subject classiﬁcations. 35L02, 35L40, 35L60, 35L65, 35L67, 35Q99

DOI. 10.1137/130907185

1. Introduction. In this paper, a theory for the solution of the Riemann prob-

lem of a 2×2 system of conservation laws describing reactive transport in a permeable

medium with pH-dependent adsorption is discussed. The suitably nondimensionalized

conservation laws for the total concentration of protons, cht, and the concentration

of a charged solute, cs, in an incompressible and isothermal ﬂuid in local chemical

equilibrium are given by

a(c)t+cx=0 on −∞<x<∞,t>0,(1.1)

where c=(cht,cs), and the piecewise constant initial data

c(x, 0) = clfor x<0,

crfor x>0

(1.2)

are deﬁned by the left and right states. The nonlinear coupling between the equations

arises in the accumulation term,

(1.3) a(c)=c+z(c),

∗Received by the editors January 24, 2013; accepted for publication (in revised form) July 15,

2013; published electronically October 24, 2013. This material is based upon work supported as part

of the Center for Frontiers of Subsurface Energy Security (CFSES), an Energy Frontier Research

Center funded by the U.S. Department of Energy, Oﬃce of Science, Oﬃce of Basic Energy Sciences

under Award DE-SC0001114. Acknowledgment is made to the Donors of the American Chemical

Society Petroleum Research Fund, for partial support of this research.

http://www.siam.org/journals/siap/73-5/90718.html

†Department of Petroleum and Geosystems Engineering, University of Texas at Austin, Austin,

TX 78712 (valentina.prigiobbe@austin.utexas.edu).

‡Department of Geosciences and Institute for Computational Engineering and Sciences, University

of Texas at Austin, Austin, TX 78712 (mhesse@jsg.utexas.edu).

§Department of Petroleum and Geosystems Engineering and Institute for Computational Engi-

neering and Sciences, University of Texas at Austin, Austin, TX 78712 (steven bryant@mail.utexas.

edu).

1941

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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1942 V. PRIGIOBBE, M. A. HESSE, AND S. L. BRYANT

which contains the absorbed concentrations, z(c)=(zh(c),z

s(c)), also referred to as

adsorption isotherms, given by

zh=

φztkh1

2cht +c2

ht +4kw

1+kh1

2cht +c2

ht +4kw+kscs

,(1.4)

zs=φztkscs

1+kh1

2cht +c2

ht +4kw+kscs

,(1.5)

where khand ksare the equilibrium constants for the adsorption of protons and

the solute, kwis the dissociation constant of water, ztis the total concentration of

adsorption sites, and φis the dimensionless ratio of solid to ﬂuid volume. Without

loss of generality, throughout the paper we assume the following values: kh=10

8,

ks=10

5,kw=10

−14,zt=10

−2,andφ=0.4. The derivation of the governing

equations and isotherms is outlined in Appendix B. Figure 1 shows the adsorption

isotherms in composition space, also called the hodograph plane, which is spanned

by cht/k and cs/k,wherek=(kh−ks)/(khks) [29]. The isotherms show the rapid

changes in the absorbed concentrations near the cs-axis, the so-called sorption-edges.

Note that cht can attain both positive and negative values while csis nonnegative

(Appendix A).

The analysis presented here builds on the analytic solutions for competitive ad-

sorption and ion-exchange in chemical engineering processes [11, 32, 13, 29, 23] and

its applications to ﬂuid ﬂow in the subsurface [26, 33, 7, 2, 3, 8, 18]. The pH of

the solution has a strong eﬀect on the adsorption of solutes and, therefore, on solute

transport in reactive permeable media. Previous work on analytical solutions for so-

lute transport with pH-dependent adsorption has used charge balance to eliminate

the proton concentration [10, 27]. During adsorption, charges can build up at the

solid-liquid interface and strict charge balance cannot be assumed so that the con-

servation of protons has to be modeled explicitly [12, 22]. The conservation equation

for protons contains a term accounting for the dissociation of water, which intro-

duces additional nonlinearity in the accumulation term and renders the hyperbolic

system nongenuinely nonlinear. This additional nonlinearity fundamentally changes

the structure of the theoretical development and gives rise to new types of solutions

that are not present in previous work on adsorption. Here, semianalytical solutions

to the Riemann problem are developed and compared with highly resolved numerical

simulations.

2. Self-similar solution. The weak solution of the Riemann problem for hyper-

bolic systems is invariant under a uniform stretching transformation of the indepen-

dent variables (¯x=eαx,¯

t=eαt) [6], so a self-similar solution in the similarity variable

η:= x/t is sought. The similarity transformation reduces the system of partial dif-

ferential equations (1.1) to the following system of ordinary diﬀerential equations:

(2.1) ∇ca(c)−1

ηIdc

dη=0,

which is in the form of a nonlinear eigenvalue problem, with eigenvalue σp(c)=1/η =

t/x and the eigenvectors rp(c)=dc/dη. The similarity transformation asserts that c

is constant along trajectories which propagate at the constant velocity, vp=1/σp=η.

The interpretation of the eigenvalue, σp(c), is therefore the retardation of c.The

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pH-DEPENDENT REACTIVE TRANSPORT 1943

0 2 4 6 8 10

0

0.002

0.004

0.006

0.008

0.01

cs/k

zh

0 2 4 6 8 10

0

0.002

0.004

0.006

0.008

0.01

cs/k

zs

−5 −4 −3 −2 −1 0 1 2 3 4 5

0

0.002

0.004

0.006

0.008

0.01

cht/k

zh

−5 −4 −3 −2 −1 0 1 2 3 4 5

0

0.002

0.004

0.006

0.008

0.01

cht/k

zs

ef

ab

cd

Fig. 1.Variation of the adsorption isotherms in composition space and two-dimensional sec-

tions along cs/k =5and cht /k =−1in solid and dashed lines, respectively.

retardation arises instead of the velocity, because the nonlinearity in (1.1) is in the

accumulation term instead of in the ﬂux term [29, 20]. In chromatography [23], it

is common to express (2.1) in terms of the gradient of the absorbed concentrations,

∇cz, and its eigenvalues, θp=σp−1, so that

(2.2) (∇cz−θpI)rp=0.

The eigenvalues of (2.1) are given by

(2.3) θp=1

2zs,s +zh,ht ±√Δand σp=1+θp,

where Δ = (zs,s −zh,ht)2+4zh,szs,ht is the discriminant and zi,j is the partial deriva-

tive of the isotherm ziwith respect to cj. The eigenvalues as a function of the

normalized compositions cht and csare shown in Figure 2.

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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1944 V. PRIGIOBBE, M. A. HESSE, AND S. L. BRYANT

For competitive adsorption, the cross-derivatives of the isotherms, zh,s and zs,ht,

are generally negative [29, 23]. The sign of the cross-derivatives is clear when the

concentration of protons,

ch=1

2cht +c2

ht +4kw>0,(2.4)

is identiﬁed in the isotherms (1.4) and (1.5), and zh(ch,c

s)andzs(ch,c

s) are recog-

nized as standard Langmuir isotherms, with negative cross-derivatives, zh,s and zs,h

[29]. For the pH-dependent case considered here, zs,ht =zs,hch,ht <0, because

ch,ht =1

2+ch

c2

ht +4kw

>0.(2.5)

Therefore, the discriminant, Δ, is positive, and the two eigenvalues θpare real, dis-

tinct, and positive. Because the eigenvalues represent retardations, they are sorted in

decreasing order, θ1>θ

2, to maintain standard ordering of the waves from the slow-

est to the fastest [20]. This implies that the 2×2 quasi-linear system (2.1) is strictly

hyperbolic [19] and the isotherms are physically admissible [23]. The theory of hy-

perbolic systems with genuinely nonlinear ﬁelds was developed by Lax [19] and then

extended by Liu [21] to systems with nongenuinely nonlinear ﬁelds. This extended

theory will be used here to describe the admissible wave structure and the complete

set of analytical solutions of the Riemann problem. To deﬁne the physically cor-

rect, unique, weak solution of the hyperbolic problem given by (1.1), an appropriate

entropy condition must be satisﬁed [20].

3. Wave structure and hodograph plane. The weak solution of the Riemann

problem for a strictly hyperbolic 2×2 system consists of a concatenation of two waves,

W1(slow wave) and W2(fast wave), connecting three constant states, cl(left state),

cm(middle state), and cr(right state)

(3.1) clW1

−−→ cmW2

−−→ cr.

The construction of the solution requires the determination of two waves and the

location of the middle state in the hodograph plane, the space of dependent variables.

In reactive transport problems, the dependent variables are the compositions and,

therefore, the hodograph plane is referred to as composition space and the waves are

said to follow composition paths [13, 26, 18, 25].

3.1. Integral curves and rarefaction waves. Rarefactions are elementary

waves connecting two constant states with a smooth variation in concentration. The

composition paths of p-rarefactions, Rp, are the integral curves of the pth eigenvector

of (2.1), given by

(3.2) cp(c0,η)=c0+η

0

rp(c)dη,

where the eigenvectors are

rp=dc

dηp

=zs,ht

σp−1−zs,s .(3.3)

The two families of integral curves of (2.1) depicted in Figure 2 show two distinct

patterns for negative and positive values of cht, which are separated by a transition

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

pH-DEPENDENT REACTIVE TRANSPORT 1945

−5 −4 −3 −2 −1 0 1 2 3 4 5

0

1

2

3

4

5

cs/k

cht/k

−5 −4 −3 −2 −1 0 1 2 3 4 5

0

1

2

3

4

5

cs/k

cht/k

ab

cd

9.79.6 9.5 9.3 9.0 7.0 5.0 4.74.5 4.4 4.3

pH

9.79.6 9.5 9.3 9.0 7.0 5.0 4.74.5 4.4 4.3

pH

Fig. 2.The eigenvalues, the integral curves, and the inﬂection loci in the hodograph plane. (a)

Eigenvalue σ1;(b) eigenvalue σ2;(c) slow paths (gray) and inﬂection locus I1(black dashed line);

(d) fast paths (gray) and inﬂection locus I2(black dash-and-dot line).

region centered near cht = 0. Far from the cs-axis the integral curves form nets of

intersecting straight lines, but in the transition region they become strongly curved

and approach vertical asymptotes. The complexity of the integral curves in the tran-

sition region is a reﬂection of the rapid changes in the sorbed concentrations that is

a characteristic of pH-dependent adsorption.

The rarefaction waves are physically admissible if the retardation, σp, decreases

monotonically along the composition path from the left to the right. Figure 2 shows

that the σp’s are not monotonic functions near the cs-axis, which leads to the occur-

rence of composite waves discussed in section 3.3.

3.2. Hugoniot locus and shock waves. Shocks are elementary waves con-

necting two constant states, c−and c+, with a discontinuous variation in composi-

tion. Mass conservation requires that the retardation of a shock, ˜σp, must satisfy the

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1946 V. PRIGIOBBE, M. A. HESSE, AND S. L. BRYANT

Rankine–Hugoniot jump condition

(3.4) ˜σp(c−,c+)= [a(c)]

[c]=1+[z(c)]

[c],

where the brackets indicate the diﬀerence between the two states across the disconti-

nuity [29, 20]. Again, the retardation arises naturally, instead of the velocity, because

the nonlinearity is in the accumulation term instead of in the ﬂux term.

For a given state c−, the jump condition deﬁnes two lines in composition space,

the Hugoniot locus H(c−), that give the set of all c+that can be connected to c−

through a mass-conserving shock, Sp. The two lines comprising H(c−)aretangentto

the integral curves at c−, but they deviate signiﬁcantly from the integral curves in the

transition region near the cs-axis, as shown in Figures 3a and 3b. This is important

for the construction of the solution, if cland crlie on opposite sides of the cs-axis

and their integral curves do not intersect.

Ap-shock is physically admissible if it satisﬁes the Lax entropy condition [19]

(3.5) σp(c−)>˜σp(c−,c+)>σ

p(c+),

which ensures that the shock is self-sharpening and therefore stable. Figure 3 shows

the portions of the H(c−) that satisﬁes (3.5) as solid lines. Due to the nonmonotonic

variation of σp, a single branch of H(c−) can satisfy (3.5) only along segments, which

leads to the occurrence of composite waves discussed in section 3.3.

Figure 3c shows that H(c−) can have three branches, two attached to c−and

a detached branch. The presence of the detached branch is an interesting feature

of systems with pH-dependent adsorption, which is essential for the construction of

the solution in some cases where c−and c+lie on opposite sides of the transition

region near the cs-axis. Detached branches have only been reported for relatively few

physical systems modeled by hyperbolic conservation laws [17, 4, 9, 15], and have

not been previously reported for reactive transport. Figure 3d shows the area on the

hodograph plane, where detached branches of H(c−) that satisfy the entropy condition

exist. The extent of this area will vary with ksand kh, but its existence for negative

value s of cht is a general feature of reactive transport with pH-dependent adsorption.

Thus, this type of behavior constitutes a nontrivial generalization of classical theories

of transport with competitive adsorption.

3.3. Inﬂection locus and shock-rarefaction waves. In systems with pH-

dependent adsorption both characteristic ﬁelds are nongenuinely nonlinear, because

the σp’s are not monotonic functions of composition, as shown in Figure 2. Therefore,

each p-wave may consist of a combination of rarefactions and shocks [21].

The inﬂection locus Ipfor the pth characteristic ﬁeld gives the locations where σp

attains a maximum value when moving along integral curves of the p-family and it is

deﬁned as

(3.6) ∇σp·rp=0.

The inﬂection loci for both characteristic ﬁelds are single connected curves indicated

with black dash-and-dot lines on the hodograph plane shown in parts (c) and (d) of

Figure 2.

Acomposedp-wave arises when it joins two states on opposite sides of the cor-

responding inﬂection locus, Ip. The order of the shock and the rarefaction in the

composed wave is determined by the nature of the local extremum of σp[1]. The

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pH-DEPENDENT REACTIVE TRANSPORT 1947

−5 −4 −3 −2 −1 0 1 2 3 4 5

0

1

2

3

4

5

cs/k

cht/k

−5 −4 −3 −2 −1 0 1 2 3 4 5

0

1

2

3

4

5

cs/k

cht/k

−5 −4 −3 −2 −1 0 1 2 3 4 5

0

cs/k

cht/k

1

2

3

4

5

c

ab

d

S2

S1

S1

S2

S1

−5 −4 −3 −2 −1 0 1 2 3 4 5

0

1

2

3

4

5

cs/k

cht/k

9.79.6 9.5 9.3 9.0 7.0 5.0 4.74.5 4.4 4.3

pH

9.79.6 9.5 9.3 9.0 7.0 5.0 4.74.5 4.4 4.3

pH

9.79.6 9.5 9.3 9.0 7.0 5.0 4.74.5 4.4 4.3

pH

9.79.6 9.5 9.3 9.0 7.0 5.0 4.74.5 4.4 4.3

pH

cl

Entropy-satisfying

Entropy-violating

Inflection loci

Entropy-satisfying

Entropy-violating

Inflection loci

no detached branch

detached branch

Inflection locus I2

cl

cl

Fig. 3.Hugoniot loci. (a) and (b) Hugoniot loci emanating from a left state, cl,intheleft

and right halves of the hodograph plane, respectively. (c) The attached and detached branches of the

hugoniot locus H1for left states with increasing pH. (d) The region in the hodograph plane where

the H1of a left state has a detached branch.

inﬂection loci shown in Figure 2 correspond to local maxima of σp, which correspond

to minima in the velocity, so that with the rarefaction the inﬂection loci is always

slower than the shock. The shock-rarefaction of the pth family connecting two con-

stant states, SRp, is a curve consisting of a p-shock emanating from c−, connected at

an intermediate point c∗toap-rarefaction, which ends at c+. Any discontinuity Sp

connecting the states c−and c∗must satisfy the Liu entropy condition [21] given by

(3.7) σp(c−)≥σp(c−,c∗)>σ

p(c+).

4. Fundamental structure of the solutions of the Riemann problem.

The nine fundamental structures of the solutions of the Riemann problem are illus-

trated in this section in the same fashion as in [16]. The solutions are a concatenation

of the entropy-satisfying waves of rarefaction, shock, and shock-rarefaction as listed

in Table 1. The nine fundamental entropy-satisfying weak solutions of the Riemann

problem are shown together with the numerical simulations for a representative set of

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1948 V. PRIGIOBBE, M. A. HESSE, AND S. L. BRYANT

Tabl e 1

Complete set of the fundamental structures of the solutions of the Riemann problem comprising

four concatenations of elementary waves, four concatenations of composed waves and elementary

waves, and one concatenation of two composed waves.

Solution W1W2Illustration

1S1S2Figure 4(a–c)

2R1R2Figure 4(d–f)

3R1S2Figure 4(g–i)

4S1R2Figure 5(a–c)

5SR1R2Figure 5(d–f)

6SR1S2Figure 5(g–i)

7S1SR

2Figure 6(a–c)

8R1SR

2Figure 6(d–f)

9SR1SR

2Figure 6(g–h)

left and right states in Figures 4 through 6. They are illustrated as composition paths

on the hodograph plane and as concentration proﬁles as a function of the similarity

var iable η.

The numerical solutions were calculated using a ﬁnite volume scheme for the

discretization of the nonlinear system given in (1.1). The accumulation term was

not expanded, but diﬀerentiated directly to treat the nonlinearity implicitly, and the

advection term was integrated explicitly and approximated with an upwind ﬂux [27].

All numerical solutions presented in this paper use a dimensionless domain of length

0.2 divided into 3000 uniform grid cells. The Courant number for the initial time step

was unity. To ensure convergence of the Newton iteration, the time step was reduced

adaptively.

5. Discussion.

5.1. Comparison with the theory of chromatography. For positive values

of cht, i.e., pH lower than 7, the diﬀerence between cht and chis negligible and the

conservation laws given in (1.1) simplify to

∂

∂t (ch+zh)+ ∂ch

∂x =0,(5.1)

∂

∂t(cs+zs)+ ∂cs

∂x =0,(5.2)

and the adsorption isotherms reduce to two-component Langmuir adsorption isotherms,

zh=φchkhzt

1+chkh+csks

,(5.3)

zs=φcskszt

1+chkh+csks

,(5.4)

used in the theory of chromatography [11, 32, 13, 29, 18, 23]. The two-component

Langmuir isotherms are concave functions and systems (5.1)–(5.4) are genuinely non-

linear, so that the fundamental structures of the analytical solutions of the Riemann

problem are four concatenations of elementary waves corresponding to the ﬁrst four

types given in Table 1. The nongenuine nonlinearity of pH-dependent adsorption,

therefore, introduces two inﬂection loci and adds ﬁve additional solution structures

involving composed waves.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

pH-DEPENDENT REACTIVE TRANSPORT 1949

−2 −1 0 1 2

0

1

2

3

4

5

cht/k

cs/k

0 0.02 0.04 0.06 0.08

0

1

2

3

4

5

cs/k

Analytic

Numeric

0 0.02 0.04 0.06 0.08

−1.5

−0.5

0.5

1.5

cht/k

−2 −1 0 1 2

0

1

2

3

4

5

cht/k

cs/k

0 0.02 0.04 0.06 0.08

0

1

2

3

4

5

cs/k

Analytic

Numeric

0 0.02 0.04 0.06 0.08

−1.5

−0.5

0.5

1.5

cht/k

ab

c

S2

S1

de

f

gh

i

R2

R1

cl

cr

cm

S2

S1

S2

S1

cl

cr

cm

cl

cl

cl

cl

R1

R1

R2

R2

−2 −1 0 1 2

0

1

2

3

4

5

cht/k

cs/k

0 0.02 0.04 0.06 0.08

0

1

2

3

4

5

cs/k

Analytic

Numeric

0 0.02 0.04 0.06 0.08

−1.5

−0.5

0.5

1.5

cht/k

cl

cr

cm

cl

cl

S2

R1

S2

R1

S2

R1

9.3 9.0 7.0 5.0 4.7

pH

9.3 9.0 7.0 5.0 4.7

pH

9.3 9.0 7.0 5.0 4.7

pH

gh

i

Fig. 4.Fundamental structures of the solutions 1–3as in Table 1.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1950 V. PRIGIOBBE, M. A. HESSE, AND S. L. BRYANT

−2 −1 0 1 2

0

1

2

3

4

5

cht/k

cs/k

0 0.02 0.04 0.06 0.08

0

1

2

3

4

5

cs/k

Analytic

Numeric

0 0.02 0.04 0.06 0.08

−1.5

−0.5

0.5

1.5

cht/k

η

−2 −1 0 1 2

0

1

2

3

4

5

cht/k

cs/k

0 0.02 0.04 0.06 0.08

0

1

2

3

4

5

cs/k

Analytic

Numeric

0 0.02 0.04 0.06 0.08

−1.5

−0.5

0.5

1.5

cht/k

η

−2 −1 0 1 2

0

1

2

3

4

5

cht/k

cs/k

0 0.02 0.04 0.06 0.08

0

1

2

3

4

5

cs/k

Analytic

Numeric

0 0.02 0.04 0.06 0.08

−1.5

−0.5

0.5

1.5

cht/k

η

ab

c

de

f

gh

i

R2

S1

cl

cr

cm

c*

R1

S2

S1cl

cr

cm

c*

R1

S2

S2

SR1

SR1

R2

S1

cl

crcm

cl

cl

S1

S1R2

R2

R2

R2

cl

cl

SR1

SR1

9.3 9.0 7.0 5.0 4.7

pH

9.3 9.0 7.0 5.0 4.7

pH

9.3 9.0 7.0 5.0 4.7

pH

Fig. 5.Fundamental structures of the solutions 4–6as in Table 1.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

pH-DEPENDENT REACTIVE TRANSPORT 1951

−2 −1 0 1 2

0

1

2

3

4

5

cht/k

cs/k

0 0.02 0.04 0.06 0.08

0

1

2

3

4

5

cs/k

Analytic

Numeric

0 0.02 0.04 0.06 0.08

−1.5

−0.5

0.5

1.5

cht/k

η

−2 −1 0 1 2

0

1

2

3

4

5

cht/k

cs/k

0 0.02 0.04 0.06 0.08

0

1

2

3

4

5

cs/k

Analytic

Numeric

0 0.02 0.04 0.06 0.08

−1.5

−0.5

0.5

1.5

cht/k

η

−2 −1 0 1 2

0

1

2

3

4

5

cht/k

cs/k

0 0.05 0.1 0.15 0.2

0

1

2

3

4

5

cs/k

Analytic

Numeric

0 0.05 0.1 0.15 0.2

−1.5

−0.5

0.5

1.5

cht/k

η

de

f

R2

S2

R1

cl

cr

cm

R1

R1

SR2

SR2

ab

c

gh

i

R2

S2

S1

cl

cr

cm

S1

S1

SR2

SR2

R2

S2

R1

cl

cr

cm

SR

2

SR1

S1

S1

c*

cl

cl

cl

cl

c*

c*

SR

2

SR1

cl

cl

9.3 9.0 7.0 5.0 4.7

pH

9.3 9.0 7.0 5.0 4.7

pH

9.3 9.0 7.0 5.0 4.7

pH

Fig. 6.Fundamental structures of the solutions 7–9as in Table 1.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1952 V. PRIGIOBBE, M. A. HESSE, AND S. L. BRYANT

−5 −4 −3 −2 −1 0 1 2 3 4 5

0

1

2

3

4

5

cht/k

cs/k

0 0.2 0.4 0.6 0.8 1

1.5

2

2.5

3

3.5

4

η

cht/k

0 0.2 0.4 0.6 0.8 1

1

1.5

2

2.5

3

3.5

4

4.5

cs/k

Analytic

Numeric

cl

cr

cm

R1

S2

cl

cr

cl

cr

R1

R1

S2

S2

ab

c

9.79.6 9.5 9.3 9.0 7.0 5.0 4.74.5 4.4 4.3

pH

pH dependent

classical

Fig. 7.(a)Comparison of the integral curves for pH-dependent adsorption shown in solid gray

lines and for classical chromatography shown as dashed black lines [29]. The analytical and numerical

solutions, for a particular set of cland cr, are shown as a composition path in the hodograph plane,

(a), and as concentration proﬁles, (b) and (c).

Figure 7 shows that the integral curves for pH-dependent adsorption coincide with

the integral curves from the theory of chromatography for cht >0 [29]. Here, the in-

tegral curves have been shown to be straight lines that are tangent to a parabolic

envelope curve in the fourth quadrant of the hodograph plane. Therefore, system

(5.1)–(5.4) are of Temple-class [30] and the Hugoniot loci are identical to the corre-

sponding integral curves. In pH-dependent adsorption, this simpliﬁcation is not gen-

erally valid, and the Hugoniot loci can deviate signiﬁcantly from the integral curves

and even have detached branches as illustrated in section 3.2.

5.2. The eﬀect of pH on the solution structure. To illustrate the strong

eﬀect of pH variations on the structure of the concentration fronts, we consider a

sequence of solutions with a ﬁxed right state, cr,atpH=5.0 and a sequence of left

states, cl, with ﬁxed cl

sbut decreasing cl

ht, corresponding to a pH increasing from 4.9

to 9.6. The gradual change of the structure of the analytical solution of the Riemann

problem is shown in Figure 8. As the pH of the left state increases, the fundamental

structure of the solution changes as follows:

1. clR1

−−−→cmS2

−−→ cr,

2. clSR1

−−−→cmS2

−−→ cr,

3. clSR1

−−−→cmR2

−−−→cr,

4. clS1

−−−→cmSR2

−−−→cr.

Figures 8b and 8c show the smooth variation of the analytical solution with the

change in the left state. This illustrates the continuous dependence of the solution

on the initial condition (1.2). In particular, it shows that the occurrence of shocks

to the detached branch of the Hugoniot locus, in structures 2 and 3, does not cause

an abrupt change in the shape of the concentration proﬁles. This provides evidence

for the uniqueness of the solution and indicates the robustness of the mathematical

model for pH-dependent adsorption. In practice, this is of importance for applications

such as laboratory experiments that aim at reproducing the Riemann problem to test

the reactive transport model.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

pH-DEPENDENT REACTIVE TRANSPORT 1953

0 0.2 0.4 0.6 0.8 1

0

1

2

3

cs/k

0 0.2 0.4 0.6 0.8 1

−4.1

−3.1

−2.1

−1.1

−0.1

0.9

1.9

cht/k

η

b

c

−4.5 −3.5 −2.5 −1.5 −0.5 0.5 1.5

0

1

2

3

cht/k

cs/k

cl

cr

S1

SR2

SR1

S2

SR1R1R1

a

SR1

SR1

R2

SR1

9.79.5 9.4 9.2 8.75.3 4.8

pH

11222334

3

2

1

3

1

2

2

4

11

2

3

4

Fig. 8.Gradual change of the fundamental structure of the analytical solutions of the Riemann

problem with the change of the left state. Solutions with the same structure have the same line

thickness and the numbers correspond to the four structures discussed in section 5.2.

5.3. Dispersion-induced waves in pH-dependent reactive transport. Fo r

some sets of cland cr, the combined eﬀect of hydrodynamic dispersion, which adds

a diﬀusive ﬂux, and pH-dependent adsorption leads to the formation of an additional

pulse traveling without retardation. This dispersion-induced wave is not present in the

analytical solution in the hyperbolic limit [31, 5, 27, 28]. The appearance of this wave

in pH-dependent reactive transport is interesting because hydrodynamic dispersion

generally only smooths the concentration proﬁles. This behavior has recently been

analyzed under the simplifying assumption of solution charge balance [27] and it is also

present in numerical solutions of the more complete model of pH-dependent sorption

discussed here.

The dispersion-induced wave arises when cland crare on opposite sides of the

transition zone and cl

ht <c

r

ht, so that the pH decreases strongly from clto cr. Figure

9 shows the analytical and the numerical solutions for a case with the fundamental

structure

(5.5) clS1

−−→ cmSR2

−−−−→cr,

where cr

s= 0. The numerical solution, which includes a diﬀusive ﬂux, shows an addi-

tional wave at the leading edge of SR2at η= 1. Previous simulations of dispersion-

induced waves have used a diﬀerent geochemical model, but Figure 9 shows that this

phenomenon also occurs in this formulation. A detailed discussion of this dispersion-

induced phenomenon, however, is beyond the scope of this work, which is focused on

the hyperbolic structure.

6. Conclusions. In this paper, a theory for the solution of the Riemann problem

for a one-dimensional, quasi-linear, 2×2 system of conservation laws describing reac-

tive transport in a permeable medium with pH-dependent adsorption is developed.

The conservation equation for protons contains a term accounting for the dissociation

of water which introduces additional nonlinearity in the accumulation term. The sys-

tem is strictly hyperbolic and nongenuinely nonlinear, because the pH-dependent

adsorption isotherms are not convex functions. This leads to nine fundamental

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1954 V. PRIGIOBBE, M. A. HESSE, AND S. L. BRYANT

−8 −7−6 −5 −4 −3 −2 −1 0 1 2

0

0.5

1

1.5

2

cht/k

cs/k

0 0.25 0.5 0.75 1

0

0.5

1

1.5

2

2.5

cs/k

0 0.25 0.5 0.75 1

−5

−3

−1

1

cht/k

η

Analytic

Numeric

ab

c

cl

cm

S1

SR2c* cr

Dispersion-induced

wave

cl

S1

Dispersion-induced

wave

cl

S1

SR2

cr

cr

0.95 0.97 0.99 1.011.01

0.8

1

1.2

cht/k

η

cr

Dispersion-induced

wave

9.99.8 9.77 9.6 9.5

pH

9.79.3 8.97.0 5.0 4.7

Fig. 9.Nonclassical reactive transport behavior under the condition of cr

s=0and for kh=

108.73,ks=10

4.47 as in [27].

solution structures, which are a concatenation of elementary and composed waves.

Reactive transport with pH-dependent adsorption can, therefore, develop more com-

plex reaction fronts than competitive adsorption of solutes with negligible complexa-

tion reactions. When both left and right states are at low pH (less than 7), however,

the abundance of protons reduces the inﬂuence of the dissociation of water and the

isotherms reduce to convex two-component Langmuir isotherms considered in chro-

matography. The classic solutions of chromatography are, therefore, a subset of the

solutions discussed here. Numerical simulations show excellent agreement with the

analytic solutions in the hyperbolic limit, but in certain cases the interaction of hy-

drodynamic dispersion and pH-dependent adsorption can lead to an additional wave

traveling without retardation, as previously reported.

Appendix A. Total proton concentration. In an aqueous system, the three

chemical species H2O, H+,andOH

−are linearly dependent and only two can be

chosen as a basis to span the compositional space. In aqueous chemistry [24], H2O

and H+are usually chosen as basis and OH−=H

2O-H

+. Under the assumption of

local chemical equilibrium, only the conservation equations for the basis species have

to be considered. In addition, the concentration of water is assumed to be constant

because the aqueous solution is assumed to be dilute, so that only the conservation

equation for protons is required,

(A.1) ∂

∂t (cht +zh)+ ∂cht

∂x =0,

where zhis the adsorbed proton concentration derived in Appendix B and the total

concentration of the H+basis species in the aqueous phase is given by

(A.2) cht =cH+−cOH−,

where the minus sign arises because the OH−species is generated by subtracting the

H+basis species from the H2O basis species.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

pH-DEPENDENT REACTIVE TRANSPORT 1955

Appendix B. Geochemical model. We considered an aqueous system con-

taining protons (H+) and a general solute (Sn+) of positive charge, which can be

adsorbed onto a reactive surface as

X−1/2+H

+←→ XH+1/2,(B.1)

X−1/2+S

n+ ←→ XSn−1/2,(B.2)

where X−1/2corresponds to a reactive surface site [14]. Neglecting the electrostatic

term of the eﬀective equilibrium constant that accounts for the development of the

surface charge upon adsorption, the eﬀective equilibrium constants equal the intrinsic

equilibrium constants (kg mol−1)

kh={XH+1/2}

chX−1/2=10

8,(B.3)

ks={XSn−1/2}

cs{X−1/2}=10

5,(B.4)

where cicorresponds to the concentrations of the subscripted species (mol kg−1)and

the equilibrium constants resemble those appropriate for the adsorption of strontium

on iron oxide at 25◦C in [34]. The surface site balance is given by

zt={X−1/2}+{XH+1/2}+{XSn−1/2},(B.5)

where ztis the total concentration of the reactive surface sites (mol kg−1).

Combining (B.3), (B.4), and (B.5), the adsorbed concentration (mol kg−1)of

protons, zh, and of the solute, zs, on the reactive surface are

zh=ztkhch

1+khch+kscs

,(B.6)

zs=ztkscs

1+khch+kscs

.(B.7)

These functions resemble the two-component Langmuir isotherm employed in previous

chromatographic studies [29, 26, 7, 33, 8, 2, 3, 23]. In pH-dependent sorption the

isotherms have to be expressed in terms of the total concentration of protons in the

ﬂuid, cht, introduced in Appendix A, instead of the proton concentration, ch.

Combining (A.2) with the law of mass action for the dissociation of water

H2O←→ OH−+H

+,kw=coh ch=10

−14,(B.8)

we derive the expression

ch=1

2cht +c2

ht +4kw,(B.9)

which can be substituted into (B.6) and (B.7), to obtain the pH-dependent adsorption

isotherms

zh=

φztkh1

2cht +c2

ht +4kw

1+kh1

2cht +c2

ht +4kw+kscs

,(B.10)

zs=φztkscs

1+kh1

2cht +c2

ht +4kw+kscs

.(B.11)

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1956 V. PRIGIOBBE, M. A. HESSE, AND S. L. BRYANT

REFERENCES

[1] F. Ancona and A. Marson,A note on the Riemann problem for general n×n conservation

laws, J. Math. Anal. Appl., 260 (2001), pp. 279–293.

[2] C. Appelo, J. Hendriks, and M. Vanveldhuizen,Flushing factors and a sharp front solution

for solute transport with multicomponent ion-exchange, J. Hydrol., 146 (1993), pp. 89–113.

[3] C. Appelo,Multicomponent ion exchange and chromatography in natural systems, Reviews in

Mineralogy, 34 (1996), pp. 193–227.

[4] T. Barkve,The Riemann problem for nonstrictly hyperbolic system modeling nonisothermal,

two-phase ﬂow in a porous medium, SIAM J. Appl. Math., 49 (1989), pp. 784–798.

[5] S. L. Bryant, C. Dawson, and C. van Duijn,Dispersion-induced chromatographic waves,

Ind. Eng. Chem. Res., 39 (2000), pp. 2682–2691.

[6] B. J. Cantwell,Introduction to Symmetry Analysis, Cambridge University Press, Cambridge,

UK, 2002.

[7] R. Charbeneau,Groundwater contaminant transport with adsorption and ion-exchange

chemistry—method of characteristics for the case without dispersion, Water Resour. Res.,

17 (1981), pp. 705–713.

[8] R. Charbeneau,Multicomponent exchange and subsurface solute transport—characteristics,

coherence, and the Riemann problem, Water Resour. Res., 24 (1988), pp. 57–64.

[9] A. H. Falls and W. M.Schulte,Features of three-component, three-phase displacement in

poro us medi a, SPE Reserv. Eng., 7 (1992), pp. 426–432.

[10] E. Glueckauf,Theory of chromatography; chromatograms of a single solute,J.Chem.Soc.,

(1947), pp. 1302–1308.

[11] E. Glueckauf,Theory of chromatography: VII. The general theory of two solutes fol lowing

non-linear isotherms, Discuss. of Faraday Soc., 7 (1949), pp. 12–25.

[12] J. Gruber,Waves in a two-component system: The oxide surface as a variable charge adsor-

bent, Ind. Eng. Chem. Res., 34 (1995), pp. 2769–2781.

[13] F. Helfferich and G. Klein,Multicomponent Chromatography, Marcel Dekker, New York,

1970.

[14] T. Hiemstra, J. Dewit, and W. Vanriemsdijk,Multisite proton adsorption modeling at

the solid-solution interface of (hydr)oxides—a new approach. II. Application to various

important (hydr)oxides, J. Colloid Interf. Sci., 133 (1989), pp. 105–117.

[15] E. Isaacson, D. Marchesin, B. Plohr, and J. B. Temple,Multiphase ﬂow models with

singular Riemann problems, Mat. Apl. Comput., 11 (1992), pp. 147–166.

[16] R. Juanes and T. Patzek,Analytical solution to the Riemann problem of three-phase ﬂow in

poro us medi a, Transp. Porous Media, 55 (2004), pp. 47–70.

[17] R. Juanes and M. J. Blunt,Analytical solutions to multiphase ﬁrst-contact miscible models

with viscous ﬁngering, Transp. Porous Media, 64 (2004), pp. 339–373.

[18] L. Lake, S. L. Bryant, and A. Araque-Martinez,Geochemistry and Fluid Flow, Elsevier,

Amsterdam, The Netherlands, 2002.

[19] P. Lax,Hyperbolic systems of conservation laws, II., Comm. Pure Appl. Math., 10 (1957),

pp. 537–566.

[20] R. J. LeVeque,Numerical Methods for Conservation Laws, 2nd ed., Birkh¨auser, Berlin, 2008.

[21] T. Liu,The Riemann problem for general 2×2conservation laws, Trans. Amer. Math. Soc.,

199 (1974), pp. 89–112.

[22] P. C. Lichtner,Continuum formulation of multicomponent-multiphase reactive transport,

Reviews in Mineralogy, 34 (1996), pp. 1–81.

[23] M. Mazzotti,Local equilibrium theory for the binary chromatography of species subject to a

generalized Langmuir isotherm, Ind. Eng. Chem. Res., 45 (2006), pp. 5332–5350.

[24] F. M. M. Morel and J. G. Hering,Principles and Applications of Aquatic Chemistry,John

Wiley and Sons, New York, 1993.

[25] F. J. Orr,Th eory of Gas Injecti on P rocesses, Tie-Line Publications, Holte, Denmark, 2007.

[26] G. Pope, L. Lake, and F. Helfferich,Cation-exchange in chemical ﬂooding. Part 1.Basic

theory without dispersion, Soc. Petrol. Eng. J., 18 (1978), pp. 418–434.

[27] V.Prigiobbe,M.A.Hesse, andS.L.Bryant,Anomalous reactive transport in the framework

of the theory of chromatography, Transp. Porous Med., 93 (2012), pp. 127–145.

[28] V. Prigiobbe, M. A. Hesse, and S. L. Bryant,Fast strontium transport induced by hydro-

dynamic dispersion and pH-dependent sorption, Geophys. Res. Lett., 39 (2012), L18401.

[29] H. K. Rhee, A. Aris, and N. Amudson,First-Order Partial Diﬀerential Equations, Vol. II,

Theory and Application of Hyperbolic Systems of Quasilinear Equations, Prentice-Hall,

Englewood Cliﬀs, NJ, 1989.

[30] B. Temple,Systems of conservation laws with coinciding shock and rarefaction curves,Con-

temp. Math., 17 (1983), pp. 143–151.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

pH-DEPENDENT REACTIVE TRANSPORT 1957

[31] L. Toran, S. L. Bryant, J. Saunders, and M. Wheeler,A two-tiered approach to reactive

transport: Application to Sr mobility under variable pH, Ground Water, 36 (1998), pp. 404–

408.

[32] D. Tandeur and G. Klein,Multicomponent ion exchange in ﬁxed beds, Eng. Chem. Fund., 6

(1967), pp. 351–361.

[33] A. Valocchi, R. Street, and P. Roberts,Transport of ion-exchanging solutes in

groundwater—chromatographic theory and ﬁeld simulation, Water Resour. Res., 17 (1981),

pp. 1517–1527.

[34] W. van Beinum, A. Hofmann, J. Meeussen, and R. Kretzschmar,Sorption kinetics of

strontium in porous hydrous ferric oxide aggregates I. The Donnan diﬀusion model,J.Col-

loid Interf. Sci., 283 (2005), pp. 18–28.