Analytical Solution for Determining Brine Leakage Along a ... · Leakage Along a Salt Wall Submitted by ... gas storage, waste-water disposal, ... for leakage nearby a subsurface

Universitat Stuttgart - Institut fur Wasser- undUmweltsystemmodellierung

Lehrstuhl fur Hydromechanik und HydrosystemmodellierungProf. Dr.-Ing. Rainer Helmig

Bachelor Thesis

Analytical Solution for Determining Brine

Leakage Along a Salt Wall

Submitted by

Simon Scholz

Matriculation Number

2714653

Stuttgart, December 9, 2014

Examiners: apl. Prof. Dr.-Ing. Holger Class

PD Dr. rer. nat. Bernd Flemisch, M.Sc

Supervisor: Dipl.-Ing. Alexander Kissinger

I hereby state that this Bachelor thesis has been written independently. I noted all

sources that were used and marked all statements that originate from other authors.

This work, or essential parts of it, is not part of other examination procedures.

Furthermore, I did not publish this work or parts of it. I also assure that the

electronic copy of this work is identical to other copies.

Hiermit versichere ich, dass ich die vorliegende Arbeit selbststandig verfasst habe. Ich

habe keine anderen als die angegebenen Quellen verwendet und alle Aussagen, die

wortlich oder sinngemaß aus anderen Werken ubernommen wurden, als solche

gekennzeichnet. Diese Arbeit, oder wesentliche Teile davon, ist nicht Gegenstand

eines anderen Prufungsverfahrens. Sie ist außerdem von mir weder vollstandig noch

teilweise veroffentlicht worden. Ebenfalls bestatige ich, dass das elektronisch

eingereichte Exemplar mit anderen Exemplaren ubereinstimmt.

Stuttgart, December 9, 2014

Simon Scholz

Contents

1 Motivation 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Problem Description and Assumptions 5

2.1 Fault structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Salt wall structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Governing Equations and Solution Method 8

3.1 Equation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.1 Darcy’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.2 Diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.3 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Fourier and Laplace transform and inversion . . . . . . . . . . . . . . . 13

3.2.1 Fourier and Laplace transform . . . . . . . . . . . . . . . . . . . 13

3.2.2 Fourier and Laplace inversion . . . . . . . . . . . . . . . . . . . 13

4 Scenarios 15

4.1 Zeidouni two-layer analytical model . . . . . . . . . . . . . . . . . . . . 15

4.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.2 Leakage rate to the upper aquifer for different values of αu . . . 16

4.1.3 Pressure change in the aquifers . . . . . . . . . . . . . . . . . . 18

4.1.4 Leakage rate comparison to numerical model . . . . . . . . . . . 20

4.2 Multiple layer analytical model . . . . . . . . . . . . . . . . . . . . . . 23


4.2.2 Leakage rate comparison for multi-layer or two-layer systems . . 25

4.3 North German Basin multiple layer analytical model . . . . . . . . . . 26


4.3.2 Choice of confining layers . . . . . . . . . . . . . . . . . . . . . 27

4.3.3 Leakage rate comparison . . . . . . . . . . . . . . . . . . . . . . 29

I

CONTENTS II

4.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Summary 31

6 Outlook 33

List of Figures

1.1 Schematic of a typical fault system with a fault core, damage zones and

injection well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Fault components in rock . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Idealized two-layer fault system with boundary conditions modified after

Zeidouni [20] showing a single injection well and the fault zone. . . . . 7

3.1 Schematic overview of the solution method to model leakage through a

fault zone for a two-layer system. . . . . . . . . . . . . . . . . . . . . . 9

4.1 Idealized two-layer fault system with boundary conditions modified after

Zeidouni [20] showing a single injection well and the fault zone. . . . . 15

4.2 Dimensionless leakage rate qlD using different values for αu, considering

zero pressure change in the upper aquifer . . . . . . . . . . . . . . . . . 17

4.3 Pressure changes in the injection aquifer . . . . . . . . . . . . . . . . . 19

4.4 Pressure changes in the upper aquifer . . . . . . . . . . . . . . . . . . . 20

4.5 Dimensionless leakage rate qlD comparison to numerical model for the

two-layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.6 Dimensionless leakage rate qlD for numerical two-layer model . . . . . . 23

4.7 Idealized multi-layer fault system with boundary conditions modified

after Zeidouni [20] showing a single injection well and the fault zone

intersecting all aquifers. . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.8 Dimensionless leakage to upper aquifers 1, 8 and 15 . . . . . . . . . . . 25

4.9 Idealized North German Basin multi-layer fault system with boundary

conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.10 Leakage to upper aquifers - North German Basin multi-layer . . . . . . 28

4.11 Dimensionless leakage rate to upper aquifers of North German Basin

with DuMuX comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 30

III

List of Tables

4.1 Input parameters for the two-layer analytical model. . . . . . . . . . . 16

4.2 Input parameters for the two-layer analytical model compared to the

numerical model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Input parameters for the multi-layer analytical model. . . . . . . . . . . 24

4.4 Layer properties for the North German Basin multiple layer analytical

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

IV

Nomenclature

Subscript “D” stands for a dimensionless term

Subscript “f” stands for fault properties

Subscript “u” stands for upper aquifer

Subscripts “1” (with injection) or “2” (without injection) stand for regions separated

by the fault

α dimensionless transmissibility [-]

δ dirac delta function [-]

η diffusivity coefficient [m2

s]

κ intrinsic permeability [m2]

µ viscosity [Pa·s]

ρ density [ kgm3 ]

φ porosity [-]

a well to fault distance [m]

ct total compressibilty [ 1Pa

]

h aquifer thickness [m]

k aquifer permeability [m2]

L leakage pathway [m]

N number of overlying aquifers [-]

P pressure [Pa]

q injection rate [m3

s]

Q volumetric sink/source term [m3

s]

t time [s]

TD dimensionless transmissivity [-]

V

LIST OF TABLES VI

wf fault width [m]

x, y spatial coordinates [m]

FFT Fast Fourier Transform function in MATLAB [-]

Chapter 1

Motivation

1.1 Introduction

Despite all plans and technologies to reduce the energy demand of modern day society,

the industrialization and the immense population growth world wide still have a large

effect on the climate. According to the Third Assessment Report by the International

Panel on Climate Change (IPCC), “human influences are expected to continue to

change atmospheric composition throughout the 21st century” [6]. To match the

demands of modern day society, the use of the subsurface is more popular than ever.

Considering efforts by various governments and organizations to use subsurface sites

for carbon dioxide storage projects, gas storage, waste-water disposal, enhanced oil

recovery or fracking, it has become more relevant to model and monitor possible

geological formations and sites [8].

Especially geologic sites for sequestration of carbon dioxide have been discussed in

the last 10 to 20 years. Most of the considered geologic sequestration sites are either

depleted hydrocarbon reservoirs, brine-bearing saline formations or unmineable coal

seams [16]. If there is a leaky fault, an abandoned well or any other potential pathway

for leakage nearby a subsurface fluid injection or disposal project, the potential of

contaminating possible underground sources of drinking water or other sensitive

aquifers needs to be taken into account. Oldenburg et al. [11] define migration of

brine or CO2 out of a defined storage region as leakage. In our context leakage is only

possible through faults or wells.

Salt formations, such as salt walls and diapirs are an important part of common

rock formations around the world, especially in the North German Basin. Due to

their structure and the disturbed sediments surrounding them, they may be potential

pathways for displaced brine or trace metals from deep aquifers similar to a regular

fault zone.

Salt wall flanks often intersect multiple geological formations and create faulted and

disturbed sediments surrounding them. In the North German Basin, many intrusive

salt walls that deform sediments of upper formations can be found. The permeability

1.2 Analytical solution 2

or any other parameter of the salt wall and the surrounding sediments are not known.

In this thesis we assume that the salt wall itself is impermeable and the damage zone

is simplified as homogenous. Therefore the leakage along a salt wall is described as

leakage through a fault zone in our context. Given the potential risks and uncertainties

in the subsurface, it is important to model leakage along a salt wall to demonstrate

that, depending on the rock formations, CO2 or other fluids can or can not be stored

safely over a long period of time.

1.2 Analytical solution

There are two different approaches to model leakage, namely analytical simulations

and numerical simulations. The analytical simulation gives a closed form solution for

a given problem, whereas numerical simulations give an approximation. The question

what are the advantages and disadvantages of the analytical solution.

On the one hand, numerical simulations allow more details. Especially details around

the fault zone or salt wall can be portrayed and calculated. But the model is always

bound by a certain grid or cell size and when modeling leakage on a large scale

(10-100 km) the computational demand is very high. The numerical model allows

complex geometries and physics, but the calculation time is much longer than in an

analytical model. On the other hand, a lot of assumptions for the involved physics

are needed for the analytical model. With these assumptions, parts of the geometry

or the heterogeneity might be lost. Nevertheless, analytical models are extremely

efficient and still capture the essential features of the leakage process.

The reason why we still use the analytical solutions is that by comparing the two

different approaches to the same problem, a cross-check is possible. The results of

the complex numerical model can be validated by a simplified analytical model for

the same system. If the results match, the numerical model can be used for further

research. If the results do not match, both models can be checked, adapted and

improved during the process.

Different analytical solutions for determining fault leakage or well leakage are

developed, e.g. by Cihan et al. [4], Nordbotten et al. [10] or Zeidouni [20].

Cihan et al. [4] focus on pressure perturbations and leakage to multiple upper

formations through aquitards or confining layers and old wells. The novelty here is,

that Cihan et al. combine a solution for diffuse leakage with a solution for focused

leakage. Later Cihan et al. [3] also focused on CO2 sequestration.

Kang et al. [7] describe two-phase subsurface flow for a leaky fault with vertical flow

effects.

Nordbotten et al. [10] describe how leakage can migrate through abandoned or old

wells during waste fluid injection. With a single-phase flow and the essential constant

1.3 Aims 3

properties they use the superposition principle to investigate the effect of multiple

leakage pathways.

Zeidouni [20] developed a solution for fault leakage involving multiple overlying

formations. He uses existing analytical solutions to verify his results and gives

potential applications for his analytical model. We will focus on the analytical model

developed by Zeidouni, because of his implementation of the fault zone and the

multiple overlying formations. He also provided his MATLAB code, so that the results

can be reproduced easily.

1.3 Aims

This study focuses on modeling leakage to upper formations through a fault and the

corresponding pressure changes in both the injection zone and upper formations. Since

the characterization of the fault and the aquifers itself is nontrivial, a simplified two-

layer aquifer system of Figure 1.1 is set up to verify the analytical model developed by

Zeidouni [20].

The results are evaluated and the system is evolved to a multi-aquifer system with

a vertical pathway for flow. The test results are compared with the results given by

Shan et al. [14] and Zeidouni [20]. Then the impact of accounting for horizontal

transmissibility on the total leakage is discussed, as well as the corresponding pressure

changes in the target and upper aquifers.

Figure 1.1: Schematic of a typical fault system with a fault core and its surrounding

damage zones as well as an injection/production well by Zeidouni [20, Fig. 1, page 2]

Further a multiple layer aquifer system with a vertical pathway for flow along an

impermeable salt wall in the North German Basin is developed. The leakage is plotted,

evaluated and compared to results from a numerical model set up with the numerical

1.4 Structure of this thesis 4

simulator DuMuX [5]. Salt walls are considered because they are an important part

of the geological composition in the North German Basin and are subject of research

in terms of migration to upper aquifers. Salinity of the fluids and transport processes

could be investigated as well, but are not subject of this thesis.

1.4 Structure of this thesis

Chapter 2 describes the problem in general and explains all necessary assumptions.

The aim of Chapter 3 is to provide fundamental knowledge of the problem as well as a

description of the system setup. Then the basic mathematical equations and methods

are given in Section 3.1. The different scenarios with their corresponding parameters

are discussed in Section 4.1 and 4.2, and the results are compared to our reference

models (by Zeidouni [20] and Shan et al. [14]).

Finally the model is adapted to determine brine leakage along a salt wall in the

North German Basin. Therefore, the model is adapted and the results compared to

numerical simulations run in DuMuX in Chapter 4.3.

Chapter 2

Problem Description and

Assumptions

The simplifications used in this thesis are explained here. Most of the system

properties are described in Chapter 3. We take a look at an injection scenario with

one injection well. Fluid is injected into a deep aquifer. It is assumed there is only

one vertical, linear and impermeable fault [20] intersecting all overlying aquifers in

the system without other perturbations or abandoned wells. The aquifers themselves

are assumed to be homogenous and isotropic. Between each aquifer a confining layer,

that does not allow leakage, is set up so that the aquifers are separated and the fault

is the only connection between the aquifers.

2.1 Fault structure

The fault structure is non-trivial and difficult to model in detail due to the different

components of the fault. Caine et al. [2] and others describe faults as a conduit, barrier

or a combination of both.

Faults generally have a fault core which is surrounded by damage zones, as we can see

in Fig. 2.1. The fault core often consists of a clay-rich, soft and low permeability gouge

zone called fault gouge [20]. Further, there is fault zone breccia, which may be highly

permeable according to Shan et al. [14] and is broken and sheared fault material [12].

With Zeidouni’s analytical model we do not focus on a detailed fault structure, but

simplify geometry and properties to roughly describe the flow through the fault. Zei-

douni [20] assumes that horizontal resistance of the fault does not have an impact on

the leakage. In Appendix A of his paper Zeidouni demonstrates that lateral transmis-

sibility α and the ratio of upperzone thickness to that of the injection zone hD do not

have an effect when the arithmetic mean of the pressure change at the fault plane is

calculated [20, Paragraph 23]. We will discuss later whether these assumptions are

valid or not.

2.2 Salt wall structure 6

Figure 2.1: Fault components in rock formations by Caine et al., 1996 [2] taken from

http://crustal.usgs.gov/projects/rgb/faults_gw.html [18].

2.2 Salt wall structure

Salt walls and salt domes are common rock formations around the world. Usually

they start growing from relatively small anomalies and can be intrusive (always in

the subsurface) or extrusive (reach the surface). Intrusive salt domes often buckle the

formations above them, whereas extrusive salt domes do not deform upper formations

as much. Each salt wall is unique, and no detailed information on salt walls can be

found. We assume, that they consist of an impermeable salt core and a homogenous

and permeable surrounding transition zones. These zones are assumed to be connected

across all intersecting geological layers for our test cases. Therefore, we describe salt

walls as fault zones and use Zeidouni’s model for fault leakage to model leakage along

a salt wall.

2.3 Assumptions

Various assumptions are made, for the analytical model to work.

The fault is considered to be a vertical, linear, semipermeable interface with no vertical

displacement.

Fault storativity as well as reservoir permeabilities on the tangential flow are neglected.

This means that everything that enters the fault either leaks to the upper formation

or to the other side of the fault. For the salt wall test cases, the only way for leakage

is up the fault. A basic setup is given in Figure 2.2

http://crustal.usgs.gov/projects/rgb/faults_gw.html

2.3 Assumptions 7

wf

no-flow boundary

∆P2(x,±∞,t

)=

∆P2(±∞,y,t

)=

0

q

x = 0 x = a

region 2 region 1

k

ku

L

upper zone

injection zone

confining layer

h

hu

Figure 2.2: Idealized two-layer fault system with boundary conditions modified after

Zeidouni [20] showing a single injection well and the fault zone.

No-flow boundary conditions bound the top and bottom of the model for most test

cases and the model is semi-infinite, with negligible pressure changes in region 1 and 2

towards the infinite lateral boundaries. We assume an initial state of equilibrium in all

aquifers.

The formations are assumed to be homogenous and isotropic.

The fault is assumed to be the only potential leakage pathway. This simplifies the

analysis and is valid with respect to focused leakage through fault zones because the

sequestration sites are assumed to be well chosen and are probably not dangerous sites

with unknown field properties. Implied with the above assumption is that no leakage

occurs through the aquitards or the confining layers. We make this assumption, even

though Cihan et al. [3] and Birkholzer et al. [1] state that diffuse leakage should be

considered for large scale pressure evaluations because of their long term effects for

large scale projects.

Chapter 3

Governing Equations and Solution

Method

In this Chapter, the equations for the model described in Chapter 2 are explained. Us-

ing the code obtained from Zeidouni [20], we model leakage through a fault to multiple

overlying formations. The system setup is based on Zeidouni [20] and Shan et al. [14].

The model is an idealization of the complex system shown in Figure 1.1. They both

assume that different aquifers are separated by aquitards or confining layers that do

not allow flow. The aquifers are assumed to be homogenous and isotropic and are

intersected by a single fault.

For the derivation of the governing equations and the solution method we take a look at

two aquifers with one confining layer in between. The system has a no-flow boundary

condition at the top and the bottom, as well as negligible pressure changes towards

the infinite lateral boundaries. There is a single fault splitting the system in regions 1

(with injection) and 2 (without injection). The Cartesian coordinate is set in a way

that x = 0 at the fault. The injection is at a constant flow rate and no other faults or

old wells are considered.

A detailed description of the general assumptions is given in Chapter 2.3. The pa-

rameters and input data for each test case are given in the corresponding sections of

Chapter 4.

3.1 Equation System

In this thesis an analytical solution for single-phase flow is applied to a hypothetical

two-layer system. In Section 4.2 the system is evolved to a hypothetical multi-layer

system and finally a real multi-layer system is modeled. Because we are only interested

in brine leakage and do not consider local two-phase flow, variable density can be

neglected here [3]. Leakage and pressure buildup in the far field for CO2 storage can

then be reasonably described by a single-phase flow model according to Nicot [9].

3.1 Equation System 9

The solution is based on the diffusion equation and Figure 3.1 gives an overview of the

solution method. First of all new dimensionless terms are introduced. Then the set of

partial differential equations is Laplace transformed in the time domain and Fourier

transformed in the space domain. Combining all the equations, we get a system

of coupled ordinary differential equations. This is evaluated in the Laplace-Fourier

domain and then retransformed to calculate the leakage either using the pressure

changes or the derivatives of the pressure changes.

Figure 3.1: Schematic overview of the solution method to model leakage through a

fault zone for a two-layer system.

3.1.1 Darcy’s Law

Darcy’s law is a relationship that was determined experimentally by Henry Darcy.

It provides an accurate description for the ability of a fluid to flow through porous

media. In a simple discrete form it shows that the flux q is a function of the intrinsic

permeability κ and the pressure gradient ∇p divided by the fluid viscosity µ

q = −κµ∇p . (3.1)


3.1.2 Diffusion equation

Diffusion is a fundamental transport process in engineering science, especially in envi-

ronmental fluid mechanics. With the mass balance defined as

∇ · (ρ q) + ρQ =∂(ρ φ)

∂t, (3.2)

q can be substituted with Equation 3.1. With the Dupuit assumption for an isotropic

and homogenous aquifer, as well as constant density ρ we get the diffusion equation

∂2∆P

∂x2+∂2∆P

∂y2+��>

Dupuit assumption∂2∆P

∂z2+Q =

φµ ctk

∂∆P

∂t. (3.3)

The diffusivity coefficient η is defined as

η =k

φµ ct, (3.4)

for further simplifications with φ being porosity, µ fluid viscosity and ct total compress-

ibility.

3.1.3 Governing equations

This Section is an explanation of the equations used in this thesis to model flow through

the fault zone and the corresponding pressure perturbations. We follow Zeidouni here.

A more detailed explanation can be found in Appendix A of his paper [20].

Equation 3.3 is used in a special form, with the Dirac delta function to add a point

source term, is used here for region 1 of the injection zone (as previously defined in

Fig. 2.2) to get∂2∆P1

∂x2+∂2∆P1

∂y2+qµ

khδ(x− a)δ(y) =

1

η

∂∆P1

∂t(3.5)

and respectively

∂2∆P2

∂x2+∂2∆P2

∂y2=

1

η

∂∆P2

∂t(3.6)

for region 2, where ∆P1 and ∆P2 stand for pressure changes in the corresponding re-

gion.

q is the volumetric injection rate, µ fluid viscosity, k injection zone permeability, h in-

jection zone thickness, δ the Dirac delta function and a the distance from the injection

well to the fault.

The upper zone equations are similar to Equation 3.6.


The boundary conditions are given by

∆P1(x, y, 0) = ∆P2(x, y, 0) = 0, (3.7)

∆P1(x,±∞, t) = ∆P2(x,±∞, t) = 0 (3.8)

and

∆P1(±∞, y, t) = ∆P2(±∞, y, t) = 0. (3.9)

With the corresponding boundary conditions and the assumption of an initial state

of equilibrium in the aquifers, Equation 3.5 is used to describe the fault outflow of

region 1 as the sum of the flow toward region 2 and the vertical leakage to the upper

aquifer of region 1. This is demonstrated in Equation 3.10.

outflow of region 1︷︸︸︷kh

µ

∂∆P1(0, y, t)

∂x=kfhh

µwf

(∆P1(0, y, t)−∆P2(0, y, t))︸︷︷︸flow to region 2

+kfvwf

2µL(∆P1(0, y, t)−∆Pu1(0, y, t))︸︷︷︸

vertical flow to region 1 of upper aquifer

(3.10)

For region 2 we get

kh

µ

∂∆P2(0, y, t)

∂x+kfvwf

2µL(∆P2(0, y, t)−∆Pu2(0, y, t))

=kfhh

µwf

(∆P1(0, y, t)−∆P2(0, y, t)) , (3.11)

and for the upper zones we get

kfvwf

2µL(∆P1(0, y, t)−∆Pu1(0, y, t))

=kfhhuµwf

(∆Pu1(0, y, t)−∆Pu2(0, y, t))−kuhuµ

∂∆Pu1(0, y, t)

∂x(3.12)

and

kfvwf

2µL(∆P2(0, y, t)−∆Pu2(0, y, t)) +

kfhh

µwf

(∆Pu1(0, y, t)−∆Pu2(0, y, t))

=kuhuµ

∂∆Pu2(0, y, t)

∂x. (3.13)

Then the equations are Laplace transformed in time and exponential Fourier trans-

formed in the y-coordinate to obtain a system of coupled ordinary differential equations,

defined by

∆P (x, y, s) = L{∆P (x, y, t)} =

∫ ∞0

∆P (x, y, t)e−stdt (3.14)


and

∆P (x, ω, s) = F{∆P (x, y, s)} =

∫ ∞−∞

∆P (x, y, s)eiωydy , (3.15)

where ∆P stands for the Laplace transformed pressure and ∆P for the Laplace and

Fourier transformed pressure.

The total leakage rate to the upper zone is calculated by integrating the pressure

differences [14].

ql =kfvwf

2µL

(∫ ∞y=−∞

(∆P1(0, y, t)−∆Pu1(0, y, t))dy

+

∫ ∞y=−∞

(∆P2(0, y, t)−∆Pu2(0, y, t))dy

). (3.16)

We now want to get our solution in dimensionless form, therefore the following param-

eters are defined as:

PD1 =kh

qµ∆P1 , PD2 =

kh

qµ∆P2 ,

PDu1 =kh

qµ∆Pu1 , PDu2 =

kh

qµ∆Pu2 , (3.17)

where PDi stand for the dimensionless pressure changes in the corresponding aquifer.

By evaluating Equation 3.15, new dimensionless terms are introduced to further sim-

plify the problem.

TD =kuhukh

, α =kfhwf

/k

a, αu =

kfvwfv

2L

/kh

a, tD =

ηt

a2. (3.18)

TD is the dimensionless transmissivity, which is calculated by dividing upper aquifer

permeability ku and thickness hu by injection zone permeability k and thickness h.

α is a measure for the horizontal transmissibility of the fault, dividing effective

horizontal fault permeability kfh by the fault width wf , which is divided by the

injection zone permeability k and the distance from the injection to the fault a.

αu is a measure for the vertical transmissibility of the fault, dividing effective vertical

fault permeability kfh by the fault width wf and two times the leakage path way L,

all divided by the injection zone permeability k and the injection thickness h divided

by the distance from the injection to the fault a.

tD is the dimensionless time, defined by the diffusivity coefficient η and the actual

time t divided by the distance from the injection to the fault a squared. tD is only

used for calculation of the leakage and for some reference test cases.

3.2 Fourier and Laplace transform and inversion 13

With these equations there are two ways to calculate dimensionless leakage and the

dimensionless pressure changes. The first one is with the actual pressure differences

qlD = αu

∫ ∞yD=−∞

(PD1(0, yD, tD)− PDu1(0, yD, tD)) dyD

+

∫ ∞yD=−∞

(PD2(0, yD, tD)− PDu2(0, yD, tD)) dyD . (3.19)

The other way is by combining Equation 3.19 with 3.12 and 3.13. So dimensionless

leakage can be calculated using the pressure derivatives

qlD = TD

∫ ∞yD=−∞

(∂PDu2(0, yD, tD)

∂xD− ∂PDu1(0, yD, tD)

∂xD

)dyD . (3.20)

To obtain the pressure changes in the aquifers, Equations 3.17 are evaluated. In a final

step the pressure changes in a dimensional form are calculated using

∆Pi =qµPDi

kh. (3.21)

3.2 Fourier and Laplace transform and inversion

3.2.1 Fourier and Laplace transform

The Fourier and Laplace transform are used to reduce our system of initial and bound-

ary conditions to a system of coupled ordinary differential equations. Therefore, the

sequential Laplace transform is used to transform the time domain and the exponential

Fourier transform is used to transform the space domain. This is for example obtained

for the two-layer system, by solving a system of four equations and four unknowns.

3.2.2 Fourier and Laplace inversion

To invert the Fourier transform, a numerical inversion technique developed by Var-

doulakis and Harnpattanapanich [19] in 1986 is considered.

The inverse Laplace transform is performed using the Stehfest [15] algorithm. Both

these inversion techniques are computationally expensive, which leads to a long run-

time if large regions are modeled in detail. So even though this is an analytical solution,

we have to use numerical inversion techniques and therefore decrease computational

efficiency.

To increase computational efficiency the Fast Fourier Transform function (FFT), which

is implemented in MATLAB, is chosen. According to Zeidouni [20], it is exactly

2π times the inverse Fourier transform of the method described here. The results

are divided by the sampling frequency because FFT uses a sampling rate of dyD = 1.

3.2 Fourier and Laplace transform and inversion 14

Finally, our results are shifted using fftshift in MATLAB to ensure that the zero-

frequency component is at the center.

For a more detailed explanation of the inversion techniques, please take a look into

Appendix B of Zeidouni [20].

Chapter 4

Scenarios

Based on by Zeidouni’s [20] multi-layer code we rebuilt a two-layer system. The results

are compared to his original explicit calculations in Chapter 4.1. Next, the multilayer

analytical model is used and compared in Chapter 4.2 and finally in Chapter 4.3 the

working code is used to determine brine leakage along a salt wall in the North German

Basin. For all systems the fault or salt wall is the only communication to the upper

formations. No diffuse leakage or any other pathway is possible.

4.1 Zeidouni two-layer analytical model

wf

no-flow boundary

∆P2(x,±∞,t

)=

∆P2(±∞,y,t

)=

0 q

x = 0 x = a

region 2 region 1

k

ku

L

upper zone

injection zone

confining layer

h

hu

Figure 4.1: Idealized two-layer fault system with boundary conditions modified after

Zeidouni [20] showing a single injection well and the fault zone.

4.1 Zeidouni two-layer analytical model 16

4.1.1 Boundary conditions

The system has a no-flow boundary condition at the top and the bottom of the system,

as well as insignificant pressure changes towards the infinite lateral boundaries. There

is a single fault splitting the system in regions 1 (with injection) and 2.

Figure 4.1 shows the system and Table 4.1 gives all the input parameters for this test

case. The exact parameters are chosen to match the test cases of Zeidouni [20] and

Shan et al. [14]. The fault width is assumed to be 1.68 meters. This is the value

given by Zeidouni and is supported by Porse [13] who assumes an arbitrary value of

1.68 meters for Hastings field, Texas.

Parameter Value

Injection rate q 0.005 m3

s

Injection well - fault distance a 100 m

Fluid viscosity µ 0.001 kgms

Injection aquifer diffusivity coefficient η 10 m2

s

Injection aquifer permeability k 1 · 10−11 m2

Injection aquifer thickness h 20 m

Upper aquifer diffusivity coefficient ηu 10 m2

s

Upper aquifer permeability ku 1 · 10−11 m2

Upper aquifer thickness hu 20 m

Time (logarithmic & dimensionless) t 10−1 − 106

Table 4.1: Input parameters for the two-layer analytical model.

4.1.2 Leakage rate to the upper aquifer for different values

of αu

We want to compare different values for αu, considering zero pressure change in the

upper aquifer. As previously defined in Equation 3.18, αu is a measure for resistance

that stands for the fault properties divided by the injection zone properties.

In this specific test case, Zeidouni’s code [20] is compared to Shan et al.’s [14] solution.

To get the same results we have to consider a Dirichlet boundary condition of insignif-

icant resistance to flow in the upper aquifer. TD, defined in Equation 3.18, has to be

set to TD → ∞ for the Dirichlet boundary condition. We achieve this by setting the

dimensionless transmissivity TD to a value of 109 in the code.


The results of the dimensionless leakage, defined by the leakage to the upper formation

divided by the injection rate, are plotted over dimensionless time tD and compared to

Shan et al.’s [14] solutions in Figure 4.2.

10−1

100

101

102

103

104

105

106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time tD [−]

lea

ka

ge

ql /

in

jectio

n r

ate

qIn

j = q

lD [

−]

Dimensionless leakage depending on αu

αu = 0.01

αu = 0.1

αu = 1

αu = 10

Shan reference

Figure 4.2: Dimensionless leakage rate qlD using different values for αu (measure of

fault transmissibility), considering zero pressure change in the upper aquifer.

It is clearly visible, that there is almost perfect agreement to Shan et al.’s [14] solution

for all values of αu. Only for the first couple of time steps a minor difference is visible.

As we can see, increasing αu results in an increased dimensionless leakage rate qlD.

With increased qlD, the steady state of qlD = 1 is reached faster. This implies that, if

we wait long enough, 100 % of the injected fluid flows through the fault. It is obvious

that this does not reflect most real systems. This is why reaching qlD = 1 only makes

sense in this theoretical test case with the dimensionless transmissivity TD set to 109.

The minor differences, especially in the beginning and towards the end of the injection

period can be neglected here. They can probably be traced back to a much higher

resolution used by Shan et al. [14] for his calculations.


4.1.3 Pressure change in the aquifers

In this section, pressure changes ∆P due to injection and leakage in the respective

aquifers are evaluated after 10 hours of injection. Zeidouni [20] once again uses

Shan et al.’s [14] solution to validate his results. Shan et al. calculate the drawdown

of a pumped aquifer and plot the depletion curves for a constant-head fault, a leaky

fault, and no fault at all, which results in a solution developed by Theis [17].

It is interesting to see that the drawdown of the leaky fault scenario is between the

other two curves. The Theis solution is achieved by not allowing vertical flow up

the fault zone, which reduces the system to a simple pumping test. Usually the

Theis solution is used during a aquifer or pumping test with a fully penetrating well.

Therefore, it is reasonable that for a system with no fault, Shan et al. get the Theis

solution. The Theis solution is often used for different type of tests today, namely to

create a Theis type curve. The aquifer properties are evaluated for the unsteady flow

phase of a pumping test to determine the hydraulic properties of a nonleaky aquifer.

The effect of horizontal resistance between the aquifers or regions might be of interest

for the pressure analysis. Zeidouni does account for horizontal resistance in some of his

results at first, but later shows that “for mere calculation of leakage rate, the lateral

resistance of the fault plane can be neglected” [20, Paragraph 23]. Since our focus is

on modeling the pressure changes with the already existing code for modeling leakage,

we do not account for horizontal resistance in our code. Zeidouni does give the explicit

equations for PDiin Paragraph 17 - 20 of his paper [20], which could be used for

future two-layer test cases. Since they are only given for the two-layer system and do

not present any new information, we refrain from using or comparing to them in this

thesis.

Pressure change in the injection aquifer not accounting for horizontal resis-

tance

Not accounting for horizontal resistance leads to a pressure plot with no discontinu-

ities at the fault. The system setup is the same as before, where αu = α = 1 and

the resistance to flow in the upper aquifer is TD = 1. Taking a look at the results in

Figure 4.3, we see that the pressure in the respective aquifer systems matches perfectly

with Zeidouni’s solution [20]. A minimal difference is visible at the fault. This might

be a result of different fault parameters or inaccuracies trying to get the data out of

Zeidouni’s plot with multiple scenarios plotted in just one figure.

The largest pressure change is at the injection site with more than 21 kPa. It decreases

towards the outer boundaries. The left side towards the fault decreases a little faster

due to the leakage through the fault. The faster decrease can also be measured by the

pressure changes in the upper aquifer. The increase in the upper aquifer is proportional

to the faster decrease on the left side of the injection site. The pressure at the right side

drops slower and reaches a steady state for the outer boundaries. There are no disconti-


nuities at the fault plane because we assume perfect communication and no horizontal

resistance. If horizontal resistance would be introduced here, the arithmetic mean of

the pressure change would still be the same as we can see in Zeidouni’s [20] paper.

−100 −50 0 50 100 150 2002

4

6

8

10

12

14

16

18

20

22

Distance x from the fault [m]

Pre

ssu

re d

iffe

ren

ce

∆ P

[kP

a]

Pressure change in the injection aquifer

Scholz P

1

Scholz P2

Zeidouni reference

Figure 4.3: Pressure changes ∆P in kPa for the injection aquifer not accounting for

horizontal resistance compared to the results of Zeidouni [20].

Pressure change in the upper aquifer

As Figure 4.4 shows, our solution also matches Zeidouni’s [20] results for the upper

aquifer. The largest pressure change is at the fault with 3.65 kPa and decreases sym-

metrically towards 0 for x→ ±∞.

Shan et al. did not give a solution or a reference for the upper aquifer, so we can not

verify the results. Since the fault represents the only communication between the two

aquifers, pressure increases at the fault and decreases symmetrically towards the outer

boundaries.

At first we had small differences calculating ∆P , because the size of the modeled area

was not given explicitly. It is interesting that our pressure changes for all evaluated x in

this section of the system were 0.4 kPa larger than Zeidouni showed in his paper for the

upper aquifer. The pressure change in the injection aquifer did not show any noticeable

differences. While searching for the reason we realized that the upper aquifer reacts


more sensitive to small changes of the domain size that is used to calculate leakage.

Therefore when modeling pressure changes with this code, the size of the aquifer has

to be well chosen. For mere calculation of leakage this does not have an effect, because

leakage is calculated using the arithmetic mean.

−100 −50 0 50 100 150 2001.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

Distance x from the fault [m]

Pre

ssu

re d

iffe

ren

ce

∆ P

[kP

a]

Pressure change in the upper aquifer

Scholz P

u

Zeidouni reference

Figure 4.4: Pressure changes ∆P in kPa for the upper aquifer not accounting for

horizontal resistance compared to the results of Zeidouni [20].

4.1.4 Leakage rate comparison to numerical model

To get a better understanding of how the analytical model reacts to more realistic

cases, a new test case is set up for the analytical model and a comparison is run in the

numerical simulator DuMuX [5]. The models are set up with similar parameters, given

in Table 4.2. The DuMuX model only models 50 km in y-direction, so we reduce our

analytical model to the size of the numerical model, even though it only has minimal

effects on the leakage.

We take a look at a scenario with a 50 year injection period and use the same

parameters for the injection and the upper zone. There is a single confining layer

separating the two zones with a thickness of 50 m and all leakage is assumed to flow

along a salt wall. The salt wall is stretched out across the entire region and is assumed

to act like a fault that does not allow horizontal leakage in the numerical model.


Therefore, we double the initial fault width of 50 m and consider only one region in

the analytical model. As previously mentioned in Section 2.1 and 4.1.3, the horizontal

resistance does not have an effect on the leakage.

Parameter Value


s



Aquifer diffusivity coefficients η 0.55556 m2

s

Aquifer permeabilities k 1 · 10−13 m2

Aquifer thicknesses h 50 m

Salt Wall / Fault permeability kf 1 · 10−12 m2

Injection period t 50 years

Table 4.2: Input parameters for the two-layer analytical model compared to the nu-

merical model.

We run the simulations for two different boundary conditions. The first one with

a Neumann no-flow boundary condition at the top. The other one with a Dirichlet

boundary condition at the top for the numerical model.

In our analytical model we implement the Dirichlet boundary condition by setting

the dimensionless transmissivity of the upper aquifer to TD = 109. The Neumann

boundary is implemented by setting the parameters given in Table 4.2.

Figure 4.5 shows that good agreement is achieved for both boundary conditions. It is

visible that for the analytical solution there is more leakage into the upper aquifer. The

difference between the numerical and the analytical solution is noticeable, especially for

the Dirichlet boundary condition where the difference at the end of the 50 year injection

period is a total of 2.83 %. The difference for the Neumann boundary condition test

case is only 1.41 %. These differences might be the result of slightly different model

setups and the interpretation of MATLAB or DuMuX.

Another reason might be the modeled aquifer size in the DuMuX model. The size has

been adapted here, but could play a role for future test cases.

The curves show similar behavior, but the numerical solution is still bound to a certain

grid, while the analytical solution is not. Therefore, the numerical solution will reach

a steady state earlier. Overall, the results show a similar behavior for both models

and therefore are proof that, for this specific case, the domain size of the numerical

model is well chosen. Furthermore the analytical solution can act as a validation of


0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Flux into shallow aquifers along salt wall

Ma

ss F

low

Ra

te /

In

jectio

n R

ate

[−

]

Time [years]

DuMuX Neumann

DuMuX Dirichlet

Analytical Neumann

Analytical Dirichlet

Figure 4.5: Dimensionless leakage rate qlD comparison to numerical model for an equiv-

alent two-layer reference system set up in DuMuX.

the numerical solution. Both describe the problem quite well and the small differences

can be subject of future research.

As described earlier, the size of the modeled area does matter. While the analytical

model is only sensitive to the size when it comes to pressure evaluations, the DuMuX

model shows that the size does have an effect on the leakage in Figure 4.6. A steady

state is reached faster for smaller areas due to the domain size and the Dirichlet bound-

ary conditions set there. Consequently, when modeling large systems, numerical models

would be computationally expensive and might still reach a steady state too early.

4.2 Multiple layer analytical model 23

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Ma

ss F

low

Ra

te /

In

jectio

n R

ate

[−

]

Time [years]

Neumann 25 kmDirichlet 25 kmNeumann 50 kmDirichlet 50 kmNeumann 100 kmDirichlet 100 km

Figure 4.6: Dimensionless leakage rate qlD for numerical two-layer model with different

domain sizes set up in DuMuX.

4.2 Multiple layer analytical model

In this Section the two-layer system by Zeidouni is evolved to a multiple layer system.

The code is built to consider N overlying formations, where each formation is separated

by a confining layer. Zeidouni [20, Paragraph 24] states, that “obtaining the leakage

rate is more relevant than accessing the pressure attenuation caused by leak-off into

the upper formations.” By focusing on leakage only, the model is further simplified.


The multiple layer system shown in Figure 4.7 has a no-flow-boundary condition at the

top and the bottom, as well as insignificant pressure changes towards the infinite lateral

boundaries just like the two-layer model in Section 4.1. There is a single fault splitting

the system in regions 1 (with injection) and 2. Each upper aquifer is 40 m thick and the

injection zone has a thickness of 60 m. There is a total of 16 overlying aquifers (injection

zone and 15 upper aquifers) with confining layers in between. Diffusivity coefficient and

permeability are the same for all overlying aquifers. The confining zones are assumed

to block leakage completely. All the parameters are given in Table 4.3.


wf

no-flow boundary∆P2(x,±∞,t

)=

∆P2(±∞,y,t

)=

0

upper zone 2

L1

q

x = 0 x = a

region 2 region 1

k

kuN

upper zone 1

injection zoneh

huN upper zone N

upper zone N-1huN−1kuN−1

Figure 4.7: Idealized multi-layer fault system with boundary conditions modified after

Zeidouni [20] showing a single injection well and the fault zone intersecting all aquifers.

Parameter Value


s



Injection aquifer diffusivity coefficient η 10 m2

s

Injection aquifer permeability k 1 · 10−12 m2

Injection aquifer thickness h 60 m

Upper aquifers diffusivity coefficient ηu 10 m2

s

Upper aquifers permeability ku 1 · 10−12 m2

Upper aquifers thickness hu 40 m

Time (logarithmic) t 10−3− 102 years

Table 4.3: Input parameters for the multi-layer analytical model.


4.2.2 Leakage rate comparison for multi-layer or two-layer

systems

As we can see in Figure 4.8 the leakage rate to the bottom aquifer is significantly

higher than to the middle or top aquifer. All 16 aquifers are evaluated, but only three

aquifers are plotted here to get a better understanding and to simplify the evaluation.

After 25 years of injection the total leakage to the bottom aquifer is 20 % of the total

injection volume. Only 0.13 % reaches the middle aquifer and the top aquifer is only

reached by 0.0001 %.

10−3

10−2

10−1

100

101

102

0

0.05

0.1

0.15

0.2

0.25

0.3

Time [Years]

Le

aka

ge

ql /

In

jectio

n R

ate

qin

j = q

lD [

−]

Faultleakage multiple layer leakage to upper aquifers

Aquifer 1 (lower)

Aquifer 8 (middle)

Aquifer 15 (upper)

Two−layer reference

Figure 4.8: Dimensionless leakage to upper aquifers 1 (lower aquifer), 8 (middle aquifer)

and 15 (upper aquifer) plotted versus time in years. The aquifers are listed from the

lowest to the uppermost.

It is interesting to see that the leakage for the bottom aquifer increases at first, and

decreases again after approximately 25 years of injection. An explanation for this

behavior can be found in the increased pressure due to the injection. The increased

pressure reduces the pressure gradient between the aquifers. Zeidouni states that the

leakage drops after 25 years and does not mention longer periods. Our results show

that the leakage drops slightly after 25 years, but for a constant injection rate increase

again after approximately 50 years. This can also be traced back to the pressure

gradient, which is still impacted by an increased pressure, but the leakage starts to

4.3 North German Basin multiple layer analytical model 26

reach all aquifers by then and is therefore causing changes in the entire model again.

Given these small increases after the initial decrease, the leakage rate is going to reach

a steady state in all aquifers at some point.

The leakage to the middle or uppermost aquifer is relatively small compared to the

leakage into the lowest aquifer. This shows that multiple layers with confining layers

should always be considered. The approach is supported when the equivalent two-layer

system is compared to the multi-layer solution in Figure 4.8.

For the two-layer system the thickness of the upper aquifer is assumed to be 600 m (sum

of all overlying aquifers in the multi-layer system) and the confining layer is assumed

to be 450 m (sum of all confining layers in the multi-layer system). By the end of a

25 year injection period the bottom layer alone has more leakage than the equivalent

two-layer system. This demonstrates how important multiple layers are in hindering

up-fault leakage. Other test scenarios with leakage through abandoned wells e.g. run

by Nordbotten et. al [10] show similar results when it comes to multiple layer systems

with confining layers.

4.3 North German Basin multiple layer analytical

model

In this Section, leakage along a salt wall for a real case scenario is modeled. Several

simplifications and assumptions are made to adapt the real case parameters to our

analytical solution.

All layers are assumed to be homogenous and of constant thickness. The leakage occurs

only through the fault zone, i.e. along the salt wall, which does not reflect the real

scenario, where diffuse leakage through the confining layers takes place.

The results of the analytical solution are compared to a numerical simulation with a

similar setting.

Figure 4.9 shows the idealized system. It illustrates how the boundary conditions are

interpreted. Since the code of the analytical solution does not consider a horizontally

sealing fault and Zeidouni [20] shows in his paper that a horizontally sealing fault does

not have an impact on the system, we still model flow through the fault to both re-

gions but are only interested in leakage to region 1. To achieve a similar setting to the

numerical model, the fault width is doubled and we focus on just one region when it

comes to the evaluation. By doubling the fault width, our leakage along the salt wall

should be the same as in the numerical model.

The real system also has another aquifer on top of the Miozan aquifer, called Quar-

ternary which is considered by the numerical model. For the analytical solution this

is not possible. That is why the two aquifers are summed together, which could be

performed due to their similar properties. However, this would also result in a much

longer fault. Another solution is to only consider the Miozan aquifer and neglect the

Quarternary aquifer.


no-flow boundary

∆P1(x,±∞, t) = 0∆P1(±∞, y, t) = 0

q

x = 0 x = 13.5 km

Solling (injection zone)

Rupelian-clay

Miozan

Oligozan

Kreide

Oberer BuntsandsteinOberer Mittl. Buntss 2

wf

salt

wal

lTD →∞

Figure 4.9: Idealized North German Basin multi-layer fault system with boundary

conditions. Permeabilities range from 1 ·10−13 m2 for the Miozan aquifer to 1 ·10−18 m2

for the Rupelian-clay aquifer. Leakage only occurs along the salt wall.


For the first test case, a no-flow boundary is set at the bottom of the system, as

well as insignificant pressure changes towards the lateral infinite boundaries. The

numerical simulation has a Dirichlet boundary at the top and we try to adapt that for

the analytical solution by setting TD → ∞, with TD = 109 being used in the code.

The parameters for the different layers are given in Table 4.4 where the diffusivity

coefficient η is calculated using Equation 3.4. The fault is assumed to be horizontally

sealing because no leakage is supposed to go through the salt wall. This is achieved

by doubling the fault width and focusing on just one side of the aquifer, in this case,

region 1. The injection rate is constant at 0.0235711 m3

sand the injection period is

50 years.

4.3.2 Choice of confining layers

In this Section the focus is on confining layers. As previously described in Section 4.2,

confining layers need to be taken into account. For the real test case we will now

take a look at layers that are not confining, but are small in thickness and have low

permeabilities. For the first test we decide to consider permeabilities of 1 ·10−17m2 and

lower as confining. The aim is to investigate whether these layers can be neglected for

leakage calculations and can be considered as confining layers instead. This does not

only increase computational efficiency, but also simplifies the problem even further.

The impact on the results is dependent on how accurate the initial assumptions are.


Layer Thickness h Diffusivity

coefficient η

Permeability k

Tertiary Post-Rupelian

(Miozan)

400 m 1.481 m2

s1 · 10−13 m2

Rupelian-clay 80 m 2.222 · 10−5 m2

s1 · 10−18 m2

Oligozan, Eozan, Palaozan 350 m 2.222 m2

s1 · 10−13 m2

Kreide 900 m 0.317 m2

s1 · 10−14 m2

Oberer Buntsandstein 50 m 5.556 · 10−4 m2

s1 · 10−17 m2

Oberer Mittl. Buntss 2 20 m 5.556 · 10−3 m2

s1 · 10−16 m2

Solling (injection zone) 20 m 1.222 m2

s1.1 · 10−13 m2

Table 4.4: Layer properties for the North German Basin multiple layer analytical

model.

Therefore these further assumptions should be handled with care and should only

have a minimal impact on the results.

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

North German Basin without confining layers

Time [years]

Mass F

low

R

ate

ql /

Inje

ction R

ate

qin

j = q

lD [−

]

Oberer Mittl Buntss 2

Oberer Buntsandstein

Kreide

Oligozän

Rupelian−clay

Miozän

(a) North German Basin multi-layer all layers

with real parameters.

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

North German Basin with confining layers

Time [years]

Mass F

low

R

ate

ql /

Inje

ction R

ate

qin

j = q

lD [−

]

Oberer Mittl Buntss 2

Kreide

Oligozän

Miozän

(b) North German Basin multi-layer with some

layers defined as confining.

Figure 4.10: Leakage to upper aquifers in the North German Basin multi-layer system.

Comparison of idealizing layers (Oberer Buntsandstein and Rupelian-clay) as confining

layers (b) or not (a).


Figure 4.10 a) and b) show that the leakage to the relatively thick aquifers is almost the

same for calculations which take the aquifers into account and for calculations which

consider them as confining layers. For both test cases 36 % of the total leakage are in

the top aquifer “Miozan” by the end of the 50 year injection period. The “Kreide” and

“Oligozan” aquifer also match perfectly. Only 0.019 % of the leakage migrates into the

“Rupelian-clay” aquifer by the end of the 50 year injection period. The assumption

of considering the above mentioned aquifers as confining in this scenario seems to be

valid.

The bottom aquifer “Oberer Mittl. Buntss 2” could probably be considered as confining

as well for future scenarios. In this study, we decided on keeping the properties, because

it is the first aquifer above the injection zone, and for small and short injections the

aquifer might become more important.

4.3.3 Leakage rate comparison

The analytical solution is plotted and compared to a numerical simulation run in

DuMuX [5]. We now use the results of the confining layers from Section 4.3.2 to

simplify the calculations. A new compressibility ct = 9 · 10−10 1Pa

is introduced for the

comparison to the numerical model. Therefore, the diffusivity coefficients in Table 4.4

need to be recalculated for this case with Equation 3.4. Since the initial compressibility

was set to ct = 4.5 · 10−10 1Pa

, the diffusivity coefficients need to be divided by 2. For

this case only the leakage to the uppermost aquifer is plotted.

Figure 4.11 shows the difference between the two simulations. The analytical simu-

lation was done twice with different boundary conditions at the top. One is with a

Dirichlet boundary and the other one with a Neumann boundary on top. The nu-

merical simulation uses a constant head boundary at the top of Quarternary aquifer,

which is located on top of the Miozan aquifer. As we can see, the simulations with

the Neumann boundary match for the first years, but a significant difference is visible

after just 7 years of injection. The analytical model does not allow as much leakage to

the uppermost aquifer for no flow boundary conditions on top.

For the Dirichlet boundary condition, the analytical solution shows a similar behavior

like the numerical solution. The behavior is the same for the first 40 years. How-

ever, we get a higher dimensionless leakage rate for the analytical solution, but this

matches the results of the two-layer solution in Section 4.1.4. The numerical simulation

is close to reaching a steady state after 40 years, whereas the analytical solution has

not reached a steady state by the end of the 50 year injection period. This is due to the

boundary conditions set in the numerical simulation. The domain size might just be

too small. The difference in leakage might come from a slightly different interpretation

of the system. In addition, the effect of the confining layers is not the same in the two

different approaches, as diffuse leakage and storage in the confining layers is considered

in the numerical model .

4.4 Limitations 30

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Ma

ss F

low

Ra

te q

l / I

nje

ctio

n R

ate

qin

j = q

lD

[−]

Time [years]

DuMuX constant head

Analytical Dirichlet

Analytical Neumann

Figure 4.11: Dimensionless leakage rate to the uppermost aquifer for the North German

Basin Multiple layer with a DuMuX comparison.

The agreement between the two solutions is not very good, but still acceptable for

the first real test scenario. It is interesting to see the range of results we get for the

analytical solution depending on the boundary conditions. This shows how important

it is to set the proper boundary conditions. A validation for the real system cannot

be achieved since both, the analytical and the numerical model, are based on different

simplifications.

4.4 Limitations

The model itself may be valid and works as expected, but it will always be subject to

the input data and simplifications we assumed. It only considers straight vertical faults

or salt walls. In reality, they might have an angle and the actual flow path between

two aquifers might be longer.

Diffuse leakage or multiple pathways for leakage are not taken into account here at all.

Especially for the North German Basin multiple layer scenario, other pathways and

diffuse leakage have a large impact in reality. Our analytical solution cannot portray

anything to that level of complexity. We also assume an initial state of equilibrium

and existing groundwater flow with an existin gradientis not considered.

Chapter 5

Summary

Over the last 10 to 20 years, there has been rising interest in ways to use the subsurface.

Most of these projects are motivated by the changing atmospheric composition and

the demand to use or store energy. Multiple possible scenarios of injection into the

subsurface are subject of current research in Germany. Especially the North German

Basin, which consists of special rock formations like salt diapirs and salt walls, is

of interest. These salt walls could act like a fault and be a possible pathway for

migration of injection induced leakage to shallow aquifers. If a fluid is injected into

a deep aquifer, brine or other fluids could migrate to shallow aquifers and possibly

contaminate ground water resources. The aim of this thesis is to model the leakage

along a salt wall to upper formations analytically to compare and validate existing

numerical models.

Since there is no explicit analytical model for leakage along a salt wall, a model for

leakage through a fault, developed by Zeidouni [20], is used. This basic model is

checked and compared to other simple two-layer reference cases and numerical models.

Then multiple layers are introduced and finally, a more realistic multiple layer system

of the North German Basin is modeled and compared to the numerical simulations

run with DuMuX [5].

For the first step, the aim was to validate a simple two-layer solution. This was done by

using Zeidouni’s approach and Shan et al.’s solution as well as a numerical solution. The

different models were set up similarly and all started with an initial state of equilibrium.

The analytical solution by Zeidouni shows good agreement to all the reference models.

The effect of different parameters was investigated and the simplifications introduced

by Zeidouni were found to be accurate for this specific test case. It is interesting to

see that the leakage to upper aquifers is independent of the horizontal fault properties.

We use this solution to model leakage along a salt wall, which is described as a fault

that only allows leakage to one side of the fault in our context. Pressure evaluation of

the the injection zone and the upper aquifer for the two-layer model showed that the

pressure changes in the aquifers are extremely sensitive to the domain size.

32

The next step was done by introducing multiple layers with multiple confining zones

that do not allow any leakage. The results were compared to an equivalent two-layer

system. Like Zeidouni, we found the difference between single and multiple layer

models were significant. After a short injection period of 20 years, the model already

shows that the confining layers hinder upfault leakage extremely. When modeling real

systems with confining layers or strongly varying parameters, multiple layers need to

be considered for future simulations.

Finally, the real parameters of the North German Basin were used with the analytical

model for multiple layers. Many simplifications and assumptions of the physics were

needed to describe the real system. The analytical model showed quite close agreement

to the numerical simulation, but differences, depending on the top boundary condition,

were identified. The boundary conditions for the analytical and the numerical model

cannot be set exactly the same due to the different approaches. The numerical model

also considers diffuse leakage and storage in the confining layers and the analytical

model does not. The results are good enough to show that there is some agreement, but

the models need further investigation, especially concerning the boundary conditions.

A big advantage of the analytical model can be seen when modeling large areas. The

numerical model is always bound to a certain domain size, whereas the analytical model

theoretically stretches out towards infinity. This can be seen for the last years of the

injection period when the numerical model is about to reach a steady state, because

the model boundaries set by the domain size are reached faster.

All in all, the analytical model developed by Zeidouni, which is presented in this thesis,

validates the results of numerical models developed in DuMuX, but the results do not

match perfectly yet, which needs to be further investigated. The effect of the confining

layers for multiple layer system or real scenarios is found to be important and the

boundary conditions need to be well chosen.

Chapter 6

Outlook

This thesis showed how powerful analytical solutions can be for modeling leakage. Some

of the disadvantages concerning the analytical model where also shown, but since com-

putational efficiency is very important, analytical models should be subject of further

research. This is especially true for regions with leaky faults or needing a risk analysis.

Analytical models have a lot of advantages considering efficiency, but they are based

on various assumptions and simplifications.

Analytical solution for fault leakage should be further investigated. Based on the lim-

itations of analytical solutions presented here, a new approach with multiple faults or

old wells is of interest. Real systems often show multiple possible pathways for leakage

to upper formations. Therefore, when modeling real scenarios, these pathways should

be portrayed in the analytical solution. Inhomogeneous fault zones and a more detailed

fault structure, as well as fault zones with an angle, are of interest too. The fault zones

are assumed to be homogenous and vertical. For the real system, they are everything

but homogenous and most of the time they do have an angle. Therefore, separating

the fault zone into different sections could help improve the results.

As one of the most important subjects of future research, diffuse leakage needs to be

investigated. The effect of the confining layers was shown in this thesis, but we know

that for the real system diffuse leakage can be important. Where numerical models can

easily take diffuse leakage into account, the analytical model needs a new approach.

Cihan et al. [4] developed an analytical solution that takes diffuse leakage into account.

Due to the fact that some layers are not separated perfectly by a confining layer, diffuse

leakage, especially for large scenarios, is extremely important.

Finally, the analytical solution can not only validate existing numerical models, but

could also be coupled with an existing model. For large scale calculations, the advan-

tages of the analytical solution are obvious. But a complex fault structure or inhomo-

geneous sediments surrounding the salt wall are best portrayed by a detailed numerical

solution. Taking the advantages of both could result in a coupled model which uses

the analytical solution for the regions far from the salt wall and the numerical model

for the complex structures around the salt wall.

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