![(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 c l and c r , are shown as a composition path in the hodograph plane, (a), and as concentration profiles, (b) and (c).](https://www.researchgate.net/profile/Marc-Hesse-2/publication/258121651/figure/fig2/AS:614192579104777@1523446330523/a-Comparison-of-the-integral-curves-for-pH-dependent-adsorption-shown-in-solid-gray_Q320.jpg)
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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
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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
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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
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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)
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1956 V. PRIGIOBBE, M. A. HESSE, AND S. L. BRYANT
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