Caserta, A.

Kanivetsky, R.

Salusti, E.

English

AbstractIn this paper we analyze the presence of “Burgers solitons” in geologic porous rocks. These are quick, strong transients of combined fluid pressure and solute density that are generated from an adjacent matrix, characterized by an intense pressure and contaminant density. The effect that produces such transients is a nonlinear advection process that is not often taken into account in hydrologic science. A brief analysis of solitons in Pierre Shale is also shown in the paper

Elsevier - Publisher Connector 

 Procedia Environmental Sciences  25 ( 2015 )  44 – 49 
1878-0296 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the Scientific Committee of the IAHR Groundwater Symposium 2014
doi: 10.1016/j.proenv.2015.04.007 
Available online at www.sciencedirect.com
ScienceDirect
7th Groundwater Symposium of the  
International Association for Hydro-Environment Engineering and Research (IAHR)   
Towards a theory of solitary shock waves of fluid pressure 
and solute density in geologic porous media 
A. Casertaa*, R. Kanivetskyb and  E. Salustic 
aIstituto Nazionale Geofisica Vulcanologia, V. di Vigna Murata, 605, 00143, Rome, Italy 
bDepartment of Bioproducts and Biosystems Engineering, University of Minnesota, 1390 Eckles Ave, St. Paul, MN 55108, USA 
cCNR-ISAC, via Fosso del Cavaliere 100, Rome 00100,Italy. 
Abstract 
In this paper we analyze the presence of “Burgers solitons” in geologic porous rocks. These are quick, strong 
transients of combined fluid pressure and solute density that are generated from an adjacent matrix, characterized by 
an intense pressure and contaminant density. The effect that produces such transients is a nonlinear advection process 
that is not often taken into account in hydrologic science. A brief analysis of solitons in Pierre Shale is also shown in 
the paper. 
 
© 2015 The Authors. Published by Elsevier B.V. 
Peer-review under responsibility of the Scientific Committee of the IAHR Groundwater Symposium 2014. 
 
Keywords: Porous media; Burgers solitons; Solitary shock waves. 
1. Introduction 
Theoretical knowledge gaps in hydrologic research persist even today, in particular about non linear 
effects as advection in low permeability media. To confront this gap we propose a novel model of solitary 
shock waves of fluid pressure and solute density in geologic porous media. An example of its application 
to Pierre Shale, USA, is also shown. 
 
* Corresponding author: Tel,: +39-06-51860494; Fax: +39-06-51860605 
E-mail address: arrigo.caserta@ingv.it 
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the Scientific Committee of the IAHR Groundwater Symposium 2014
45 A. Caserta et al. /  Procedia Environmental Sciences  25 ( 2015 )  44 – 49 
2. The model 
A novel theoretical model on mechanic and chemo-poroelastic coupling of fluids and solute in porous 
media is here presented. The focus of this model is on advection and the related non linear processes. 
Analyzing the evolution of fluid pressure p and density of solute ρ in a rock under the action of strong 
external stress we obtain two coupled non linear equations [1]. Their solutions thus unveil a novel 
phenomenon: the presence of quick and strong transients of coupled fluid pressure and solute density, 
known in physics as Burger solitons.  
     Merlani, Salusti, Violini ([2], MSV thereafter) obtained the model equations 
 ∂ p
∂ t
+E
∂ρ
∂t
+F
∂2p
∂x2
+ H
∂2ρ
∂x2
= 0                                                                                     (1a)              
∂ρ
∂t
+ M
∂ p
∂x
∂ρ
∂x
+N
∂ρ
∂x
⎛ 
⎝ ⎜ 
⎞ 
⎠ ⎟ 
2
+ S
∂2 ρ
∂x2
+U
∂2 p
∂x2
= 0                                                                               (1b) 
 
(Table 1 and II) where one can see the intrinsic relation F/H = M/N.  
   MSV [2] analysed the non-linear solutions of equations (1), considering a first fluid saturated porous-
permeable rock for x <0 (contaminant ρ0 + ρI  and pressure p0 + pI  at t = 0) and for x > 0 another 
homogeneous patch of fluid saturated porous-permeable rock (contaminant  ρ0  and  pressure p0  at  t = 0) 
(Fig 1). In this way MSV obtained a Burgers-like equation 
     0
2
)(
2
2 =
∂
∂−
∂
∂
+
∂
∂
x
Z
x
A
t
ρρρ                                                                                                                (2) 
where A =  N - EM  and  Z = UE - S.  In addition the Darcy fluid velocity is (Fig. 2) 
    
 uD = - [k/(φη)] ∂p/∂x =  Ek/(φη) ∂ρ/∂x                                                                                                      (3) 
  
 For an initial solute density ρ* the solution of (3) is ruled by a Reynolds number [3; 2]  
  
R = (2Aρ*)/Z                                                                                                                                              (4) 
In more detail if R > 8- 10 as for a strong initial jump ρ*, one has        
    
tttAxx
At
x
B <=<<+= **42
2
0 ρψρρ
                                                        (5) 
and   ρ  =  ρ0  otherwise . The corresponding  p is 
 
tttAxx
t
t
A
HEF
At
Expp B <=<<
−
+−= **4)
*
(ln
22
2
0 ρψ                         (6) 
 
46   A. Caserta et al. /  Procedia Environmental Sciences  25 ( 2015 )  44 – 49 
          
Fig. 1. A sketch of the two rocks system.                                            Fig. 2. The fluid velocity at different times 
 
3. Pierre Shale 
The Pierre Shale, also called claystone, is a geologic formation, or a series, in the Upper Cretaceous. It 
is mostly studied in North America, mainly east of the Rocky Mountains in the Great Plains, from North 
Dakota to New Mexico. This dark-gray shale is remarkably fossiliferous and can reach ~ 210 m in 
thickness over the Sandstone aquifer systems [4]. It is of marine origin and was deposited in the Western 
Interior Seaway. Pierre Shale stratigraphy is correlated with other marine shales that occur farther west, 
such as the Bearpaw Shale in United States and Canada. 
       The data discussed in Table I were obtained in central South Dakota, USA [5; 6]. Its mineralogy is 
70-80% of clay, of which about 80% is a mixed-layer of smectite-illite. The shale, at this site, is saturated 
at the depth of ~ 75 m.  
An estimate of the swelling coefficient ω 0  is questionable. Here we compare our equation on the relation 
among pressure, contaminant, stress and strain, with the recent study of Sarout and Detournay [6]. We 
call here α* their parameter α and χ the ratio of moles of solute over that of fluid, while their v0 is about 
1.8 x 10-5 m3. By comparing all this with data for Pierre shales one has   
     
0
0
**)11(
v
TR
D
αα
ρρ
ω =+
                                                                                                            (7)
 
Sarout and Detournay [6] explained that their parameter α* is difficult to measure even in a laboratory. 
 
 
 
 
 
47 A. Caserta et al. /  Procedia Environmental Sciences  25 ( 2015 )  44 – 49 
Table 1.  Estimated values of relevant parameters of Pierre Shale, in SI. 
 
Parameter 
Value 
in SI 
References 
φ  Rock porosity 0.3  
k  Intrinsic permeability 10−18   
η  Fluid viscosity 3×10−4   
Θ  Solute reflection coefficient 0.25   [6] 
D Solute diffusion coefficient 10−8  [7] 
Ms Solute molar mass (Na Cl) 0.06  
α   Biot coefficient 0.7  
ω 0  Swelling coefficient 105   
K   Bulk modulus 4 ×106   
Ks Bulk modulus of the solid matrix 1.5 ×107  [8] 
Kf  Fluid bulk modulus 2x109  
ρ   Estimated solute density 1  
p Estimated fluid pressure ρE≈  105   
Formation period Cretaceous  
They assume that their α* = 0.9.  Since   ρf ≈ ρD is about 103 and α * ~ 1,  all this gives an estimate value 
of ω 0 ~ 105.  These quantities, from which we compute our Table II, may have uncertainties of about 30-
50 % and in the following we will assume that all the considered rocks have comparable uncertainties. 
The above data confirm that we have A = N - M E = 10-7 in the SI and   Z = E U - S = 3x10-8.  This 
gives R ≈ 7 ρ* and therefore the presence of solitons can occur if initially the external solute density is 
ρ* >1.5,  or equivalently the pressure p* ≈ 105-106. 
4. Conclusions 
We here describe a novel model that provides a simple analytic technique for estimation of coupled 
solute and water transports in porous media. The importance of our nonlinear model is that it shows that 
these transients are much sharper than those predicted by linear models, perturbation theories, or scale 
analyses presently available in hydrologic literature. Thus, under a strong enough initial impulse the 
resulting sharp transients can be the origin of many previously unexplained phenomena such as 
instabilities, or even small earthquakes.  
 
 
48   A. Caserta et al. /  Procedia Environmental Sciences  25 ( 2015 )  44 – 49 
Table 2. Coefficients of equation (1) for Pierre Shale, in SI. 
 
Coefficient                                                                      Numerical value in SI 
E = )11(
2
0
DSKV ρρα
ω −
+
−                                                  -1.5x10
5 
 
F = )(
D
fKkKk
ρη
ρ
η
Θ
+−  /(α2+VK)                                        -2,5x10-8 
 
H= )11(
*
SD
S
f
M
TKRk
ρρη
ρ
+
Θ / (α2+KV)                                  
 
0.2 
 
  M = - f
f D
k
ρ
ρρφη
]1[ Θ+                                                           -10-14 
  N = 
fS
SDM
TRk ρ
ρρηφ
]11[* +Θ                                                      10−7  
  S = -
φ
D                                                                               -3x10-8 
U =
f
S
TR
DM S
ρφ
ρ
*
                                                                    
 7.5x10-16 
D
S
fs TR
M
KKK
V
ρ
ωφφα
*
0−+−=    2.5 x10-8 
 
 
Acknowledgements 
 
The authors would like to thank prof C. Doglioni and dr. D. Bernagozzi, R. Droghei, F. Falcini, M. 
Sommacal for help and providing essential information.  
 
References 
 
[1] Ghassemi A. and Diek A. Linear chemo - poroelasticity for swelling shales: theory and application. Journal of Petroleum 
Science and Engineering; 2003; 38; 199-212.  
[2] Merlani A., E.Salusti and G.Violini. Non-linear waves of fluid pressure and contaminant density in swelling shales. 
Jour.Petroleum.Science.Engeneering 2011; 79; (1-2); 1-9. 
[3] Whitham G.B. Linear and nonlinear waves . Pure and Applied Mathematics; Wiley Interscience Publication, New York, 
London; 1974;  
[4] Bredehoeft, J. D., Neuzil C. E. and Milly P. C. D. Regional Flow in the Dakota Aquifer: A Study of the Role of 
Confining Layers.   U.S. Geological Survey Water-Supply Paper 2237; 1983; 45. 
[5] Neuzil, C. E., and A. M. Provost A. M. Recent experimental data may point to a greater role for osmotic pressures in the 
subsurface. Water Resour. Res.; 2009; 45; W03410, doi: 10.1029/2007WR006450. 
49 A. Caserta et al. /  Procedia Environmental Sciences  25 ( 2015 )  44 – 49 
[6] Sarout J.,Detournay E. Chemoporoelastic analysis and experimental validation of the pore pressure transmission test for 
reactive shales. Int. Jour. Rock. Mec.; 2011; 48; 759-772. 
[7] Barbour, S.L., Fredlund, D.G. Mechanisms of osmotic flow and volume change in clay soils. Canadian Geotechnical 
Journal; 1989; 26; 551-562. 
[8] Simm, R. Practical Gassmann substitution in sand/shale sequences. First Break; 2007; 25; 61-68. 
 


Towards a Theory of Solitary Shock Waves of Fluid Pressure and Solute Density in Geologic Porous Media 

Open Access Repository

 Procedia Environmental Sciences  25 ( 2015 )  44 – 49 1878-0296 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of the Scientific Committee of the IAHR Groundwater Symposium 2014doi: 10.1016/j.proenv.2015.04.007 Available online at www.sciencedirect.comScienceDirect7th Groundwater Symposium of the  International Association for Hydro-Environment Engineering and Research (IAHR)   Towards a theory of solitary shock waves of fluid pressure and solute density in geologic porous media A. Casertaa*, R. Kanivetskyb and  E. Salustic aIstituto Nazionale Geofisica Vulcanologia, V. di Vigna Murata, 605, 00143, Rome, Italy bDepartment of Bioproducts and Biosystems Engineering, University of Minnesota, 1390 Eckles Ave, St. Paul, MN 55108, USA cCNR-ISAC, via Fosso del Cavaliere 100, Rome 00100,Italy. Abstract In this paper we analyze the presence of “Burgers solitons” in geologic porous rocks. These are quick, strong transients of combined fluid pressure and solute density that are generated from an adjacent matrix, characterized by an intense pressure and contaminant density. The effect that produces such transients is a nonlinear advection process that is not often taken into account in hydrologic science. A brief analysis of solitons in Pierre Shale is also shown in the paper.  © 2015 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of the IAHR Groundwater Symposium 2014.  Keywords: Porous media; Burgers solitons; Solitary shock waves. 1. Introduction Theoretical knowledge gaps in hydrologic research persist even today, in particular about non linear effects as advection in low permeability media. To confront this gap we propose a novel model of solitary shock waves of fluid pressure and solute density in geologic porous media. An example of its application to Pierre Shale, USA, is also shown.  * Corresponding author: Tel,: +39-06-51860494; Fax: +39-06-51860605 E-mail address: arrigo.caserta@ingv.it © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of the Scientific Committee of the IAHR Groundwater Symposium 201445 A. Caserta et al. /  Procedia Environmental Sciences  25 ( 2015 )  44 – 49 2. The model A novel theoretical model on mechanic and chemo-poroelastic coupling of fluids and solute in porous media is here presented. The focus of this model is on advection and the related non linear processes. Analyzing the evolution of fluid pressure p and density of solute ρ in a rock under the action of strong external stress we obtain two coupled non linear equations [1]. Their solutions thus unveil a novel phenomenon: the presence of quick and strong transients of coupled fluid pressure and solute density, known in physics as Burger solitons.       Merlani, Salusti, Violini ([2], MSV thereafter) obtained the model equations  ∂ p∂ t+E∂ρ∂t+F∂2p∂x2+ H∂2ρ∂x2= 0                                                                                     (1a)              ∂ρ∂t+ M∂ p∂x∂ρ∂x+N∂ρ∂x⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2+ S∂2 ρ∂x2+U∂2 p∂x2= 0                                                                               (1b)  (Table 1 and II) where one can see the intrinsic relation F/H = M/N.     MSV [2] analysed the non-linear solutions of equations (1), considering a first fluid saturated porous-permeable rock for x <0 (contaminant ρ0 + ρI  and pressure p0 + pI  at t = 0) and for x > 0 another homogeneous patch of fluid saturated porous-permeable rock (contaminant  ρ0  and  pressure p0  at  t = 0) (Fig 1). In this way MSV obtained a Burgers-like equation      02)(22 =∂∂−∂∂+∂∂xZxAtρρρ                                                                                                                (2) where A =  N - EM  and  Z = UE - S.  In addition the Darcy fluid velocity is (Fig. 2)      uD = - [k/(φη)] ∂p/∂x =  Ek/(φη) ∂ρ/∂x                                                                                                      (3)    For an initial solute density ρ* the solution of (3) is ruled by a Reynolds number [3; 2]    R = (2Aρ*)/Z                                                                                                                                              (4) In more detail if R > 8- 10 as for a strong initial jump ρ*, one has            tttAxxAtxB <=<<+= **4220 ρψρρ                                                        (5) and   ρ  =  ρ0  otherwise . The corresponding  p is  tttAxxttAHEFAtExpp B <=<<−+−= **4)*(ln2220 ρψ                         (6)  46   A. Caserta et al. /  Procedia Environmental Sciences  25 ( 2015 )  44 – 49           Fig. 1. A sketch of the two rocks system.                                            Fig. 2. The fluid velocity at different times  3. Pierre Shale The Pierre Shale, also called claystone, is a geologic formation, or a series, in the Upper Cretaceous. It is mostly studied in North America, mainly east of the Rocky Mountains in the Great Plains, from North Dakota to New Mexico. This dark-gray shale is remarkably fossiliferous and can reach ~ 210 m in thickness over the Sandstone aquifer systems [4]. It is of marine origin and was deposited in the Western Interior Seaway. Pierre Shale stratigraphy is correlated with other marine shales that occur farther west, such as the Bearpaw Shale in United States and Canada.        The data discussed in Table I were obtained in central South Dakota, USA [5; 6]. Its mineralogy is 70-80% of clay, of which about 80% is a mixed-layer of smectite-illite. The shale, at this site, is saturated at the depth of ~ 75 m.  An estimate of the swelling coefficient ω 0  is questionable. Here we compare our equation on the relation among pressure, contaminant, stress and strain, with the recent study of Sarout and Detournay [6]. We call here α* their parameter α and χ the ratio of moles of solute over that of fluid, while their v0 is about 1.8 x 10-5 m3. By comparing all this with data for Pierre shales one has        00**)11(vTRDααρρω =+                                                                                                            (7) Sarout and Detournay [6] explained that their parameter α* is difficult to measure even in a laboratory.      47 A. Caserta et al. /  Procedia Environmental Sciences  25 ( 2015 )  44 – 49 Table 1.  Estimated values of relevant parameters of Pierre Shale, in SI.  Parameter Value in SI References φ  Rock porosity 0.3  k  Intrinsic permeability 10−18   η  Fluid viscosity 3×10−4   Θ  Solute reflection coefficient 0.25   [6] D Solute diffusion coefficient 10−8  [7] Ms Solute molar mass (Na Cl) 0.06  α   Biot coefficient 0.7  ω 0  Swelling coefficient 105   K   Bulk modulus 4 ×106   Ks Bulk modulus of the solid matrix 1.5 ×107  [8] Kf  Fluid bulk modulus 2x109  ρ   Estimated solute density 1  p Estimated fluid pressure ρE≈  105   Formation period Cretaceous  They assume that their α* = 0.9.  Since   ρf ≈ ρD is about 103 and α * ~ 1,  all this gives an estimate value of ω 0 ~ 105.  These quantities, from which we compute our Table II, may have uncertainties of about 30-50 % and in the following we will assume that all the considered rocks have comparable uncertainties. The above data confirm that we have A = N - M E = 10-7 in the SI and   Z = E U - S = 3x10-8.  This gives R ≈ 7 ρ* and therefore the presence of solitons can occur if initially the external solute density is ρ* >1.5,  or equivalently the pressure p* ≈ 105-106. 4. Conclusions We here describe a novel model that provides a simple analytic technique for estimation of coupled solute and water transports in porous media. The importance of our nonlinear model is that it shows that these transients are much sharper than those predicted by linear models, perturbation theories, or scale analyses presently available in hydrologic literature. Thus, under a strong enough initial impulse the resulting sharp transients can be the origin of many previously unexplained phenomena such as instabilities, or even small earthquakes.    48   A. Caserta et al. /  Procedia Environmental Sciences  25 ( 2015 )  44 – 49 Table 2. Coefficients of equation (1) for Pierre Shale, in SI.  Coefficient                                                                      Numerical value in SI E = )11(20DSKV ρραω−+−                                                  -1.5x105  F = )(DfKkKkρηρηΘ+−  /(α2+VK)                                        -2,5x10-8  H= )11(*SDSfMTKRkρρηρ+Θ / (α2+KV)                                   0.2    M = - ff Dkρρρφη]1[ Θ+                                                           -10-14   N = fSSDMTRkρρρηφ]11[* +Θ                                                      10−7    S = -φD                                                                               -3x10-8 U =fSTRDM Sρφρ*                                                                     7.5x10-16 DSfs TRMKKKVρωφφα*0−+−=    2.5 x10-8   Acknowledgements  The authors would like to thank prof C. Doglioni and dr. D. Bernagozzi, R. Droghei, F. Falcini, M. Sommacal for help and providing essential information.   References  [1] Ghassemi A. and Diek A. Linear chemo - poroelasticity for swelling shales: theory and application. Journal of Petroleum Science and Engineering; 2003; 38; 199-212.  [2] Merlani A., E.Salusti and G.Violini. Non-linear waves of fluid pressure and contaminant density in swelling shales. Jour.Petroleum.Science.Engeneering 2011; 79; (1-2); 1-9. [3] Whitham G.B. Linear and nonlinear waves . Pure and Applied Mathematics; Wiley Interscience Publication, New York, London; 1974;  [4] Bredehoeft, J. D., Neuzil C. E. and Milly P. C. D. Regional Flow in the Dakota Aquifer: A Study of the Role of Confining Layers.   U.S. Geological Survey Water-Supply Paper 2237; 1983; 45. [5] Neuzil, C. E., and A. M. Provost A. M. Recent experimental data may point to a greater role for osmotic pressures in the subsurface. Water Resour. Res.; 2009; 45; W03410, doi: 10.1029/2007WR006450. 49 A. Caserta et al. /  Procedia Environmental Sciences  25 ( 2015 )  44 – 49 [6] Sarout J.,Detournay E. Chemoporoelastic analysis and experimental validation of the pore pressure transmission test for reactive shales. Int. Jour. Rock. Mec.; 2011; 48; 759-772. [7] Barbour, S.L., Fredlund, D.G. Mechanisms of osmotic flow and volume change in clay soils. Canadian Geotechnical Journal; 1989; 26; 551-562. [8] Simm, R. Practical Gassmann substitution in sand/shale sequences. First Break; 2007; 25; 61-68.  

Towards a Theory of Solitary Shock Waves of Fluid Pressure and Solute Density in Geologic Porous Media

A. Caserta

R. Kanivetsky

E. Salusti

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Towards a Theory of Solitary Shock Waves of Fluid Pressure and Solute Density in Geologic Porous Media

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