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The influence of reservoir heterogeneities on geothermal doublet performanceDoddema, Leon

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CIO, Center for Isotope Research

IVEM, Center for Energy and Environmental Studies

Master Programme Energy and Environmental Sciences

The influence of reservoir heterogeneities

on geothermal doublet performance

Leon Doddema

EES 2012-149 T

University of Groningen

Training report of Leon Doddema

Supervised by: Prof. M.A. Herber (Geo-Energy, ESRIG)

Prof.dr.ir. J. van der Plicht (CIO, ESRIG)

University of Groningen

CIO, Center for Isotope Research

IVEM, Center for Energy and Environmental Studies

Nijenborgh 4

9747 AG Groningen

The Netherlands

http://www.rug.nl/fmns-research/cio

http://www.rug.nl/fmns-research/ivem

TABLE OF CONTENTS

SUMMARY ....................................................................................................................................................... 3

1. INTRODUCTION ...................................................................................................................................... 5

1.1 WHAT IS GEOTHERMAL ENERGY .............................................................................................................. 5 1.2 HISTORY AND CURRENT USE .................................................................................................................... 5 1.3 BENEFITS AND RISKS ................................................................................................................................ 6 1.4 RESEARCH AIM AND SCOPE ...................................................................................................................... 6 1.5 STRUCTURE OF THE THESIS ...................................................................................................................... 7

2. ANALYTICAL BACKGROUND ............................................................................................................. 9

2.1 GEOTHERMAL POWER .............................................................................................................................. 9 2.2 DARCY’S LAW .......................................................................................................................................... 9 2.3 PRESSURE DISTRIBUTION ....................................................................................................................... 10 2.4 REQUIRED PUMPING POWER ................................................................................................................... 10 2.5 DOUBLET LIFE TIME ............................................................................................................................... 10

3. NUMERICAL METHOD ........................................................................................................................ 13

3.1 PROBLEM DESCRIPTION AND COMPUTER PROGRAMS .............................................................................. 13 3.2 NUMERICAL SCHEME OF TOUGH2 ........................................................................................................ 13 3.3 FINITE DIFFERENCE METHOD .................................................................................................................. 14 3.4 NEWTON-RAPHSON ITERATION .............................................................................................................. 16

4. DEFINITION OF THE BASE MODEL ................................................................................................. 17

4.1 LITHOSTRATIGRAPHY ............................................................................................................................ 17 4.2 ROCK PROPERTIES ................................................................................................................................. 19 4.3 BOUNDARY CONDITIONS, INITIAL CONDITIONS AND NATURAL STATE .................................................... 22 4.4 PROCESS PARAMETERS ........................................................................................................................... 23 4.5 SIMULATION PARAMETERS .................................................................................................................... 25

5. RESULTS OF THE BASE MODEL....................................................................................................... 29

5.1 NATURAL STATE .................................................................................................................................... 29 5.2 PRODUCTION PHASE ............................................................................................................................... 30

6. EXPERIMENTS AND RESULTS .......................................................................................................... 33

6.1 AMELAND .............................................................................................................................................. 33 6.2 WEDGE .................................................................................................................................................. 37 6.3 WEDGE-AMELAND ................................................................................................................................. 41 6.4 RESERVOIR BOUNDARY .......................................................................................................................... 45 6.5 FAULT .................................................................................................................................................... 47 6.6 COMPARISON BETWEEN EXPERIMENTS ................................................................................................... 49

7. DISCUSSION ............................................................................................................................................ 53

7.1 APPROACH AND METHODS ..................................................................................................................... 53 7.2 RESULTS ................................................................................................................................................ 54 7.3 RESEARCH OVERALL .............................................................................................................................. 55

8. CONCLUSION ......................................................................................................................................... 57

REFERENCES ................................................................................................................................................ 59

3

SUMMARY

The current main problem with deep geothermal energy in the Netherlands is the uncertainty in terms of attainable flow rate and life time. The goal of this research is therefore modeling a geothermal doublet in a heterogeneous reservoir, using a numerical 3D model created in simulator TOUGH2. A base model is defined by choosing stratigraphy, rock properties and boundary conditions, approximately modeling the Slochteren sandstone in the North of the Netherlands, as well as process and simulation parameters. This model is used for two subsequent simulations; the first model creates the naturally occurring steady state situation while the second calculates 200 years of geothermal production. This model then serves as a base and reference for a number of experiments modeling different heterogeneities.

In practice all reservoir properties are heterogeneous to some extent, but in this research only reservoir permeability is chosen as such. Five experiments are conducted, each investigating a particular low permeability structure placed as an obstacle in the reservoir: a horizontal layer, a wedge occupying a part of the reservoir, a slanted layer cutting through the reservoir, a fault partially penetrating the reservoir and a reservoir boundary in the neighborhood of the doublet.

Overall the pressure drop over the doublet takes the same value as that of the base model or is higher; up to 7 times higher pressure drops are observed. The breakthrough time of the cold front becomes both higher and lower than in the base model, representing an improvement or regression in geothermal life time of the doublet. In each experiment several factors that describe the low permeability obstacle are varied. The main conclusion is that the distance between the obstacle and the wells has the largest effect on doublet performance. Surprisingly very little gain in geothermal lifetime can be expected by varying well perforations.

5

1. INTRODUCTION

This chapter provides an introduction firstly to geothermal energy in general and secondly to the aim and scope of the research.

1.1 WHAT IS GEOTHERMAL ENERGY

Geothermal energy is energy present in the form of heat inside the Earth. The dominant source of geothermal heat is the decay of radioactive isotopes, mainly 40K, 232Th, 235U and 238U, in the crust and mantle of the Earth. Geothermal energy is considered as a sustainable energy source because the amount of heat stored below the surface is enormous: According to a rough estimate “the Earth’s energy to a depth of 10 kilometers could theoretically supply all of mankind’s energy needs for six million years” (Lund, 2007). Nevertheless, a local depletion, at least to some extent, is observed in the majority of geothermal projects.

Geothermal energy can be exploited in various ways. Firstly one could distinguish between directly using the thermal energy for heating purposes or converting it to a carrier like electricity. The latter is more flexible and better suited for transport over large distances, but the portion of the geothermal heat that can be converted is limited by the efficiency of the conversion processes. Another division is based on the depth of where energy is extracted. According to the Dutch mining law, wells deeper than 500 m are classified as deep, but in practice depths are generally between 1500 and 4000 m (Actieplan Aardwarmte, 2011). Geothermal energy from depths below 4000 m is often designated as ultra deep. Also temperature (or enthalpy) is a fundamental means for classifying geothermal resources. Different classes have been proposed (Williams, Reed & Anderson, 2011; Dickson & Fanelli, 2005). Often a temperature of 150 °C is considered the minimum for conventional electricity production (Lund, 2007), but for lower temperatures a binary cycle system can be used. The binary cycle is a steam cycle, in which the hot water (temperatures below 100 °C) from the well heats and vaporizes a secondary fluid in order to run it through a turbine and power a generator. Finally, there is a number of ways in which geothermal energy can manifest itself in the subsurface. For example there are convective reservoirs, which can be vapor or liquid dominated, sedimentary basins containing hot water or steam, hot dry rock and magma resources (Lund, 2007).

1.2 HISTORY AND CURRENT USE

A short history of geothermal energy is given in (Barbier, 2002) and (Lund, 2007). The first use for bathing, washing and cooking originated already in prehistory. The world's first geothermal district heating systems was constructed in Chaude Aigues, France in the 14th century and as for the production of electricity, the first experiments and commercial plant originated in Tuscany, Italy in the beginning of the 20th century.

Figure 1: Development of direct use and electricity generation geothermal capacity (Goldstein et al., 2011).

In the year 2009 electricity was being produced in 24 countries with an installed capacity of 10.7 GWe of which around 11% can be attributed to binary plants. In the same year the direct use capacity was between 50 to 60 GWTH (Goldstein et al., 2011). For the development through time of these numbers see Figure 1. In the Netherlands geothermal energy is still in early development, with only nine projects producing or under construction as of August 2012. In the year 2010 in the Netherlands, the total heat delivered by geothermal wells

0

10

20

30

40

50

60

0

2

4

6

8

10

12

1970 1975 1980 1985 1990 1995 2000 2005 2010

Dir

ect

use

ca

pa

city

(G

Wth

)

Ele

ctr

ic c

ap

aci

ty (

GW

e)

6

was a little more than 1 PetaJoule (Actieplan Aardwarmte, 2011). For the distribution of world-wide direct use of thermal energy by type of consumer see Figure 2.

Figure 2: Distribution (percentages) of world-wide direct use of thermal energy by type of consumer in the year 2000

(Barbier, 2002).

1.3 BENEFITS AND RISKS

Geothermal energy can be considered ‘green’ because there are little to no emissions involved. In case of the direct use of geothermal energy for heating, the geothermal system often forms a closed loop and the only emissions result from powering the necessary pumps (and from the construction, decommissioning and maintenance if the whole life cycle is considered). Electric power plants that run on geothermal steam do produce some emissions, due to the release of contaminants present in the steam to the atmosphere, but considerably less than fossil fuel powered equivalents (Lund, 2007). Moreover, there is no noise or landscape pollution (Heekeren, 2011), like it is the case with wind turbines for example. Compared to other sustainable energy sources, geothermal energy strikes a good balance between monetary costs and benefits and is becoming increasingly favorable as gas prices rise (Actieplan Aardwarmte, 2011).

The Netherlands is committed to increase the share of renewable energy to 14% by the year 2020 (Actieplan Aardwarmte, 2011). Geothermal energy is considered a good option in achieving this goal, especially since it is suitable to operate as a base load energy supply (Lund, 2007), unlike most of the other sustainable alternatives.

Induced seismicity and land subsidence are potential problems associated with geothermal energy (Barbier, 2002). Reinjection of waste fluids may control or prevent altogether the subsidence of land, but there is a chance low-magnitude seismic events will occur more frequently in regions with already existing seismic instabilities. However, Barbier argues that for some cases the reinjection of fluids might have a positive effect, by releasing geologic stress gradually instead of instantaneously and in this way reducing the number of higher magnitude events.

1.4 RESEARCH AIM AND SCOPE

A completely different problem is financing a geothermal project. The biggest issue is the large uncertainty during the planning phase, in output power (which is a combination of temperature and flow rate) and life time that may be expected. This uncertainty makes it particularly difficult to justify the large investment associated with drilling a production well and slows down the development of geothermal fields in the Netherlands. One solution could be a guarantee scheme, currently experimented with and under development by the Dutch government (Actieplan Aardwarmte, 2011). Another possible route could be pursuing cheaper production or testing wells, by researching innovative drilling technology for example.

This research tries to tackle the primary problem, which is the uncertainty in geothermal output. On the one hand, this uncertainty originates from site specific geological uncertainties. A solution could be performing more or more detailed measurements of the subsurface or the reprocessing of earlier acquired data (Kramers et al., 2010). The other part of the uncertainty results from the lack of understanding of how aquifer parameters, like permeability, porosity or thickness, will influence the geothermal output, in particular with regard to the heterogeneity of these parameters.

42%

23%

12%

9%

14%

Bathing and swimming pool heating

Space heating

Groundwater heat pumps

Greenhouse heating

Other

7

The aim of this research is further investigating these dependencies, in order to link a feasible geothermal performance to certain geologic conditions. Formulated explicitly as a research question:

• How will reservoir parameters, e.g. permeability, influence the performance of a standard geothermal doublet configuration, in particular with regard to the heterogeneity of these parameters; can different geologic settings be classified based on their quality as a geothermal reservoir?

To answer this question, a geothermal doublet configuration is numerically simulated in a three dimensional model, using the TOUGH2 simulator (Pruess, Oldenburg & Moridis, 1999). This model is based on the geologic characteristics of the Rotliegend sediments in the North of the Netherlands, but not specifically for a single location. The Rotliegend is chosen for its sandstones with good reservoir properties at a large depth, containing brine at around 100 °C. Unlike those of the Triassic and Lower Cretaceous in the West of the Netherlands, the Rotliegend sandstones have not yet been targeted in commercial geothermal projects.

It is expected that the development of geothermal projects in the Netherlands will benefit from this kind of research. This research will also become a basis for further work at the Geo-Energy department of the University of Groningen and may contribute to geothermal science in general.

1.5 STRUCTURE OF THE THESIS

First the analytical background and some numerical mathematics theory will be discussed. Next the definition and the results of the base model will be discussed, followed by the setup and the results of the experiments. In the final chapters a discussion and a conclusion are given.

9

2. ANALYTICAL BACKGROUND

In this chapter some of the analytical background regarding the performance of a geothermal doublet is discussed.

2.1 GEOTHERMAL POWER

The most important performance characteristic of a geothermal doublet is its power output, which is defined as follows:

�������� �� ������������� � ����������� (1)

In other words, the power output is a function of volume flow rate Q in m3/s, temperature difference between injector and producer (Tproduction - Tinjection) and heat capacity of the working fluid (or water) ρwcw in J m-3 K-1. In this research only the aquifer properties are investigated and the influence of the process equipment, e.g. piping and heat exchangers, are disregarded where possible. This means that in this calculation the bottomhole temperatures are used.

2.2 DARCY’S LAW

Fluid flow through porous media is governed by Darcy’s law. Darcy empirically established that the volume flux of water Q trough a sandy formation can be found by (Marsily, 1986):

� ���� (2)

In which A is the area of the cross-section of the porous formation in square meters, ∆h the difference in hydraulic head in meters, L the thickness of the formation in meters and K, a proportionality constant named the hydraulic conductivity (in m/s), see Figure 3. The hydraulic head, which is expressed in units of distance and

sometimes also referred to as “piezometric head”, is given by h = p/ρg + z, where p is pressure, ρ is density, g the gravitational acceleration and z the elevation from a certain datum plane.

Figure 3: Darcy's experiment, adapted from (Marsily, 1986).

The original law of Darcy is valid for one-dimensional flow in an isotropic medium only. Over time it has been extended in such a way that it is applicable to three dimensions and anisotropic media. This generalized form of the law of Darcy can be written as (Marsily, 1986):

� �� !"# $ %"z' (3)

10

In which U is the three dimensional filtration velocity, k the specific permeability tensor and µ the dynamic viscosity (in Pascal-seconds). Just as hydraulic conductivity, permeability is a measure for how easily a fluid can flow through a porous medium, but permeability is only a property of the rock matrix and independent of the fluid properties. The unit of permeability is one of area; for example square meter, but more often the 'darcy' is used (one darcy is equal to 9.87e-13 m2). The term filtration velocity stands for the macro-scale velocity, i.e. the mean bulk velocity over a certain length of porous medium, as opposed to the actual particle velocities in the pores. The filtration velocity is the average fluid velocity over the whole volume including the rock matrix, instead of only the pore volumes (Marsily, 1986).

2.3 PRESSURE DISTRIBUTION

According to Darcy’s law fluid flow in a porous medium is the result of pressure gradients and gravity effects. For the case of a geothermal doublet, gravity induced fluid flow is mainly caused by density variations as a result of the injection of a colder fluid into the reservoir. This kind of coupled heat and mass transfer is problematic to analyze analytically. For an isothermal pressure driven flow however analytical solutions do exist.

For a doublet pair of injector and producer wells located in a horizontal plane at (x,y) = (C,0) and (-C,0) respectively, both flowing with a constant absolute volume rate Q, the steady state pressure distribution in the plane is given by (Muskat & Wyckoff, 1946; Brigham, 2000):

4πkh� !#!,, .' � #�' ln!!1 $ ,'2 $ .2' � ln!!1 � ,'2 $ .2' (4)

This equation is valid for a completely saturated, infinite, horizontal, homogeneous and isotropic aquifer at initial pressure pi for a fluid with constant properties and wells penetrating the full aquifer thickness.

2.4 REQUIRED PUMPING POWER

Because the viscosity of water decreases quite strongly with temperature (Lide, 2007), the application of this equation to geothermal doublets is very limited. However, it is still useful, for example to make a first approximation of the flow rates that can be attained in a certain aquifer. The pressure distribution from equation (4) can be written as a function of coordinate x only as follows:

#!,' #� $ � 434� ln 5!� � ,'2,2 6 (5)

With the injection well located at x = 0 and the producer at x = L. When both well bores have a radius of Rwb, the pressure difference over the doublet ∆p for a known volume flow rate Q is now given by:

Δ# #!7 �' � #!� � 7 �' � 34� ln 8 �7 � � 1: (6)

The power P that is required by a pump with efficiency η to increase the potential by ∆p of a volume flow with flow rate Q is simply given by (Esch & Kemenade, 2007):

� �Δ#; (7)

Without paying attention to process equipment like tubing or heat exchangers, it is now clear from equations (6) and (7) that overall the required pumping power increases quadratically with flow rate and linear with viscosity, permeability and aquifer height. The influence of well spacing and borehole diameter is relatively small.

2.5 DOUBLET LIFE TIME

Cold front breakthrough

For sustainable and economic operation it is important that a geothermal doublet can be used for a sufficiently long time. The moment when the temperature of the produced water starts to decline as a result of the cooling

11

effect of the injector is called “cold front breakthrough”. In the literature, the most referenced analytical approximation for the time at which cold front breakthrough occurs seems to be that of Gringarten and Sauty (Gringarten, 1978):

<���=>�?���@? 3��23� =�= � (8)

In which L is the distance between producer and injector, h is the height of the aquifer, Q is the volume flow rate and ρaca and ρwcw are the volumetric heat capacities of the aquifer and water respectively. However, a slightly different notation established by Tsang, Lippmann and Witherspoon was also found (Lippmann & Tsang, 1980):

<���=>�?���@? 3��23� =�= $ B! � � =�=' � (9)

In this relation the cold front breakthrough time (tbr) is also dependent on porosity φ, contrary to the solution of Gringarten and Sauty, which seems to make sense. On the other hand, it could be that Gringarten and Sauty found that the effect of porosity is minor and may be neglected, but sadly, this could not be verified since for both cases only articles referring to these equations were available and not the original work. However not entirely certain, it seems the following assumptions were made to derive these equations: The aquifer is confined, horizontal, homogeneous and isotropic, no conduction of heat takes place in the horizontal direction or to the confining rocks, heat exchange between the aquifer matrix and fluid is instantaneous, no regional ground water flow, both wells penetrate the total aquifer thickness and dispersion effects are omitted (hydrodynamic dispersion and conductive thermal diffusion).

Decline after breakthrough

The lifetime of a doublet is not limited to the cold front breakthrough time. The demands in terms of production temperature depend on the specifics of the application that the geothermal heat is used for. Next to that, the geothermal plant may operate in conjunction with for example a natural gas fueled boiler to deliver the final amount of heat to the application at hand. In this case the eventual life time of the geothermal doublet depends on economic considerations as well.

In any case, it is convenient to have an approximation for the time dependent temperature decline after cold front breakthrough has occurred. One such approximation is given in (Gringarten, 1978) and depicted in Figure 4. In this graph the dimensionless temperature change at the producer is plotted versus a dimensionless time, for different values of Λ, a coefficient representing the relative strength of heat exchange between the aquifer and the confining rocks (or rather the absence of heat exchange since Λ equal to infinity represents no exchange at all).

Figure 4: Dimensionless temperature change versus dimensionless time, modified from (Gringarten, 1978). Earliest

cold front breakthrough occurs at π/3.

13

3. NUMERICAL METHOD

In order to understand the workings of the simulator and evaluate the results later on some background of the numerical method is needed. In this chapter the used software and main numerical techniques are discussed.

3.1 PROBLEM DESCRIPTION AND COMPUTER PROGRAMS

In a geothermal aquifer two main physical processes take please: fluid flow and heat transfer. To sufficiently describe these processes six equations are necessary, which determine the density ρ, the pressure p, the temperature T and Ux, Uy, Uz, the components of the fluid filtration velocity vector U. These unknowns, functions of both time t and the point in space x, can be solved using the continuity equation, the equations of motion for the fluid (Darcy’s law), the heat equation (or energy balance) and the equation of state of the fluid (Marsily, 1986).

Often a number of assumptions can be made to reduce the number of unknowns or simplify the equations, but in the case of a heterogeneous geothermal reservoir this is possible only to a limited extent. The result is that these equations are strongly coupled and cannot be solved analytically. Numerical methods are necessary to solve, or to give a numerical approximation of the solution, of this initial-boundary value problem.

The goal of this research is not to program our own computer code, but instead the widely used (O’Sullivan, 2001) TOUGH2 computer program is used. Quoting the manual: “TOUGH2 is a general-purpose numerical simulation program for multi-dimensional fluid and heat flows of multiphase, multicomponent fluid mixtures in porous and fractured media” (Pruess et al., 1999). The configuration and control of TOUGH2 relies entirely on quite cumbersome fixed-format input files and no interaction is possible while a simulation is running. Therefor pre- and postprocessor PetraSim is used, which is a graphical user interface for preparing the input data, calling the TOUGH2 executable and visualizing the results (Thunderhead, 2010).

Another pre- and post-processer, but of a different kind, that is used extensively in this research is the program PyTOUGH (Croucher, 2011). This is a very flexible processor based on the scripting language Python, which can create or load existing TOUGH2 input files as well as process the simulation results. During this research PyTOUGH is used to adapt a base TOUGH2 model created with PetraSim for the automation of experiments. An example script which compares different doublet flow rates can be found in Appendix A.

3.2 NUMERICAL SCHEME OF TOUGH2

TOUGH2 is part of the TOUGH / MULKOM family of codes, which has been in development since 19831. It has a modular “MULKOM” architecture, see Figure 5, which allows the application to a variety of multicomponent and multiphase flow problems, by using different equation of state modules. In general, the main purpose of TOUGH2 is solving the following mass and energy balance equation (Pruess et al., 1999):

CC< D EFCG� D HF ⋅ JCK�LMNM

$D OFCG�NM (10)

With M the mass or energy per volume, F the mass or heat flux vector and q a sink or source term, defined for an arbitrary volume Vn with closed surface Γn. Superscript κ = 1, .., NK denotes the mass component (water, CO2 etc.) and κ = NK + 1 is the heat component and symbol n stands for the normal vector on surface element dΓn, pointing inward into Vn. For the exact forms of these accumulation and flux terms for mass and heat the reader is referred to appendix A of the TOUGH2 manual.

1 See http://esd.lbl.gov/research/projects/tough/history.html

14

Figure 5: The modules in the architecture of TOUGH2.

The first step in calculating the numerical solution to eq. (10) is discretizing time and space. In TOUGH2 this is accomplished by means of the finite difference method, which will be explained further in the coming sections. Material properties and initial conditions then have to be specified for each volume element. Next to that, for each element at the border of the simulation domain boundary conditions are needed. Combining all of these together results in a system of strongly coupled nonlinear equations, which is solved simultaneously for every cell in the domain, for every consecutive time step.

The solution scheme of TOUGH2 basically consists of two nested steps. The first step is a Newton-Raphson iteration, which essentially converts the system of nonlinear equations to a series of linear systems. Subsequently, for each Newton-Raphson iteration this linear system is solved using one of several solvers that are supported by TOUGH2. Direct solvers are available, but only the iterative solvers are really suitable to handle large models efficiently (Pruess et al., 1999). While solving the linear systems is the most computationally intensive of the whole simulation (Pruess et al., 1999), the numerical background to this will not be explained, because the underlying theory is quite complex and at the same time contributes little to the understanding of the overall model.

3.3 FINITE DIFFERENCE METHOD

In TOUGH2 the finite difference method (FDM) is used to discretize time and space. The specific implementation for time and space however differs; for the discretization of the spatial domain the so-called integrated finite difference method (IFDM) is applied while for the temporal discretization the backward Euler method is used. Although these methods can both be classified as a finite difference method, they are discussed separately in the following sections.

The reason for using two different methods is understood by realizing the differences between boundary value problems (BVP) and initial value problems (IVP). According to (Heath, 2002), for initial value problems the availability of complete state information about the solution at a single point guarantees the existence and uniqueness of a global solution in most practical cases. Numerical methods for solving IVPs start at this known initial value of the solution and extend it to the desired point by “marching” in small increments. For BVPs however, existence and uniqueness of a global solution is much less certain, because state information about the solution is not concentrated but rather spread across multiple points in the domain. For this same reason the numerical solution to a BVP has to be computed for the whole domain simultaneously.

Backward Euler Method

FDM basically entails replacing the derivatives in a differential equation by finite difference approximations. Finite difference formulations can be formally derived by omitting the higher order terms from a Taylor series expansion or from polynomial interpolation (Heath, 2002). Because here only a first order finite difference is considered it is derived from the formal definition of the derivative (Adams, 2006):

C.C< limR�→T.!< $ Δ<' � .!<'Δ< (11)

Data inputand

Initialization

Solution ofLinearSystems

PrintedOutput

¨EOS¨- Module

Equationof

State

Assembling andIterative Solutionof Flow Equations

Primary

Variables

Secondary

Parameters

15

The first order finite difference formulation is obtained by not taking the limit, but simply replacing the derivative by a finite difference quotient, so that:

C.C< U Δ.Δ< .!< $ Δ<' � .!<'Δ< (12)

This is known as a forward difference formula (Heath, 2002). Using this approximation for the derivative an IVP can be numerically solved. Take for example the following ordinary differential equation:

C.C< V!<, .' (13)

When the solution yk at time tk is known, the solution yk+1 at time tk+1 = tk + ∆t can be obtained by substituting the finite difference formula for the derivative, resulting in:

.>WX .> $ Δ<V!<>, .>' (14)

This approach is known as Euler’s Method. It is called an explicit method, because the function value of the next time step can be computed directly from the known value at the current time. The Backward Euler Method that is used in TOUGH2 however is based on the first order backward finite difference formulation:

C.C< U .!<' � .!< � Δ<'Δ< (15)

With which the numerical approximation for the example ODE of equation (13) is given by:

.>WX .> $ Δ<V!<>WX, .>WX' (16)

Now the function f must be evaluated for the solution value at the new time step which is still unknown; in other words it is impossible to directly compute the solution. Instead, this equation should be solved for yk+1, making Backward Euler an implicit method. Making use of an implicit method may seem overly complicated compared to being able to evaluate the function directly, but in practice an explicit method puts unworkable restrictions on the time step size that can be used before the numerical scheme becomes unstable, making the solution of transient multi-phase flow problems extremely inefficient (Pruess et al., 1999).

Integrated Finite Difference Method

In general, several methods for the discretization of a spatial domain exist. Examples of methods widely used in computational fluid dynamics are the Finite Element Method, Finite Difference Method, Finite Volume Method and Spectral Methods (Zikanov, 2010). The method used in TOUGH2 is called the Integrated Finite Difference Method (IFDM). This is different from regular FDM in that it is derived directly from the integral form of the balance equation. However, for regular, square grids referred to a global coordinate system IFDM is completely equivalent to conventional finite differences (Pruess et al., 1999). On the other hand, IFDM can also be considered as a simplification of the Finite Volume Method (Rozos & Koutsoyiannis, 2010).

An in-depth mathematical background of IFDM can be found (e.g. in Marsily, 1986), but here only the implementation as in TOUGH2 will be discussed. The basic idea is that the integral over an arbitrary domain of a continuous variable (the accumulation of mass or energy) can be written for a discrete volume element, or cell, as the average of the quantity in that cell times the volume of the cell. In the same manner the integral over an arbitrary surface of a continuous flux can be replaced by a sum of the fluxes across the surface segments of the cell, so that the generic balance equation in eq. (10) can be written as the following set of time dependent ordinary differential equations:

CE�FC< 1G� Z��[\�[F

[$ O�F (17)

16

With subscripts n denoting the cell and m the respective neighboring cell. When applying the Backward Euler Method of eq. (16) to the time derivative in this result the following system of fully discretized balance equations is obtained:

E�F,>WX E�F,> $ Δ<G� ]Z��[\�[F,>WX $ G�O�F,>WX[

^ (18)

This set of coupled nonlinear algebraic equations can now be cast in residual form (i.e. {left hand side} – {right hand side} = 0) and solved by Newton-Raphson iteration.

As for the demands on the solution mesh, Pruess (2004) states that the general definition of IFDM does not pose any restrictions on how the discrete subdomains (grid blocks) should be defined. On the other hand however, Rozos and Koutsoyiannis (2010) claim that formally the cell boundaries should coincide with equipotentials and therefore the solution grid should be dynamic for transient flow problems. For the particular IFDM implementation in TOUGH2 Pruess (2004) explains that the interface between two cells should be perpendicular to the nodal line connecting the two cell centers, because the calculated flow through this interface is a function of the nodal values of only the two adjacent cells.

3.4 NEWTON-RAPHSON ITERATION

One way of numerically approximating the root of a nonlinear function f(x) is by means of Newton-Raphson iteration. Starting at some initial guess x0 the approximation of the root is improved by iterating the following formula until the desired accuracy is achieved (Heath, 2002):

,>WX ,> � V!,>'V′!,>' (19)

With f' denoting the first order derivative of f with respect to x. Basically this means the root is determined for the tangent of the function at xk instead of for the nonlinear function itself, see Figure 6. Evidently, this method only works for functions that are sufficiently smooth between the initial guess and the actual root.

Figure 6: Newton-Raphson iteration to approximate the root of a nonlinear function, modified from (Heath, 2002)

For a system of nonlinear equations Newton-Raphson iteration can be applied as follows (Heath, 2002):

`>WX `> $ a> (20)

With sk to be solved from the following system of linear equations:

bc!`>'a> �d!`>' (21)

The Jacobian matrix Jf(xk) contains all first-order partial derivatives of the function f with respect to xk and is computed by numerical differentiation in TOUGH2 (Pruess et al., 1999). As mentioned earlier, the solution of linear systems is not covered in this report.

xkxk+1

17

4. DEFINITION OF THE BASE MODEL

In this study several types of heterogeneities are examined. These heterogeneities are all applied to the same base model, to allow comparison between different types. In this chapter this reference model will be outlined by discussing the lithostratigraphy, rock properties, boundary and initial conditions, process parameters and finally the simulator settings.

4.1 LITHOSTRATIGRAPHY

The target of this research is the sandstone reservoir of the Slochteren Formation, but the formations above and below play a role as well, as they interact with the Slochteren through flow of fluid and heat conduction. In the following section the adjacent lithological units will be discussed, with special attention to the translation from the measured data to the model.

Geological setting

Below the Slochteren lies an unconformity. As a result of uplift and erosion, different lithological layers from the Carboniferous reside directly below the Slochteren. Depending on the location this can be from the Stephanian, Westphalian, Namurian or Dinantian and older periods (Buggenum & Hartog Jager, 2007). In the north of the Netherlands mainly Westphalian A-C will be encountered.

Above the Carboniferous lies the Rotliegend group. The Slochteren formation, part of the Upper Rotliegend Group, consists of Aeolian and fluvial sandstone deposits from the Middle Permian (Geluk, 2007). The lateral equivalent of the Slochteren is the Silverpit formation, of which several tongues penetrate the Slochteren: the Ameland and Ten Boer members.

The Rotliegend is buried under the Zechstein group, from the Late Permian. This group consists of several formations composed of primarily evaporites. Directly above the Rotliegend the Coppershale Member and a succession of anhydrite and limestone layers can be found, followed by rock salt from the Zechstein Z2 evaporite cycle. The thickness of this salt layer varies a lot, because although it was deposited as a flat surface, it was deformed under stress from the overburden and formed diapirs.

Stratigraphy in the model

When cold water is injected in the reservoir, heat from the cap and base rock flows towards the aquifer. This effect is slightly limited, because on the one hand the aquifer is confined, so limited flow of fluid can take place and secondly transient heat conduction has a limited penetration depth. Still, this heat flow adds to the amount of thermal energy that can be extracted and therefore the lithology above and below the reservoir needs to be modeled, at least to a certain extent.

To choose which layers should be included in the model lithostratigraphy data from several wells2 was consulted. The average thickness of the lithological units above and below the Slochteren Fm. was calculated, only taking into account the wells where these lithological formations are actually present, i.e. if the layer was completely absent this would not lower the average value. When the Ameland Member is present, the Upper and Lower Slochteren thicknesses are combined. Results can be seen in Table 1.

Also in Table 1 are the strata selected and used in the base model, together with their respective thickness. Note that thicknesses were rounded for convenience and the Ameland Member was omitted from the base model, because it is only present in a small area and to allow for comparison with a wider range of geometries in the actual research. Next to that, the formations in the lower part of the Zechstein were truncated into one single lithological unit in the model, in order to be more flexible in choosing a solution mesh.

2 AKM-04, GRK-01-S1, GRK-03, KHM-01, MGT-01-S1, MGT-02, NOR-04, NOR-23, SLO-01, SLO-09 and UHM-01-S1 from “NL Olie- en Gasportaal”: www.nlog.nl

18

Table 1: Lithostratigraphy of the base model.

In place lithostratigraphy Average thickness (m) Strata in the model Thickness (m) ZEZ2H Zechstein Salt 446.5 ZEZ2H 500 ZEZ2A Anhydrite 6.8

ZEZ2X 50 ZEZ2C Carbonate 9.2 ZEZ1W Anhydrite 27.9 ZEZ1C Carbonate 12.0 ZEZ1K Coppershale Member 1.0 ROCLT Ten Boer Member 50.2 ROCLT 50 ROSLN Slochteren Fm. 195 ROSLN 150 ROSLA Ameland Member 49.2 - - Carboniferous base - CARBF 400

The thickness of the Slochteren was chosen somewhat smaller, because the calculated average doesn’t correlate very well with the area of interest based on the isopach map of the Upper Rotliegend (which comprises the Slochteren Formation and Ten Boer and Ameland members) in Figure 7. Although the Upper Rotliegend increases in thickness further to the North, the reservoir quality of the Slochteren sandstone diminishes in the same direction, because of the gradual transition into the Silverpit Formation. Together with the permeability calculated in section 4.2, a thickness of 200 meters for the Slochteren Fm. would overestimate the transmissivity of the reservoir.

Figure 7: Isopach map of the Upper Rotliegend in the Netherlands (TNO-NITG , 2004)

The Zechstein Salt and Carboniferous base are picked to be the top and bottom strata of the model. For these layers the correlation with the thickness from the well data is less important. What matters is that they are thick enough to avoid thermal interaction with the model boundaries. This minimum thickness is estimated from basic heat transfer theory. For a semi-infinite solid, initially at a uniform temperature Ti and with a fixed surface temperature Ts, the time dependent one dimensional temperature distribution T(x,t) is given as (Incropera & DeWitt, 2007):

�!,, <' � ���� � �e fgV 8 ,

2√j<: (22)

N

Projection RD 1918, Ellipsiod Bessel 1841

0 10 20 30 40 50km

Thickness in meters

0 - 50

50 - 100

100 - 150

150 - 200

200 - 250

250 - 300

300 - 350

> 350

Not Present

Fault

19

With erf() denoting the Gaussian error function and α the thermal diffusivity. Thermal diffusivity can be expressed as α = (λ/ρc), with λ the thermal conductivity, ρ the density and c the specific heat capacity of the solid. With this function a “penetration depth” can be defined, for example as the depth where the relative temperature change is only 1% or less.

Assuming ZEZ2H and CARBF can be treated as semi-infinite solids with a homogenous bulk temperature and a constant surface temperature (e.g. because of cold water injection in to the aquifer), the minimum thickness of these layers to avoid temperature boundary effects can be estimated. Using the rock properties in section 4.2 and a time span of 200 years, the required thicknesses are calculated as 457 and 271 meters for ZEZ2H and CARBF respectively. For convenience and extra margin these numbers were then rounded to respectively 500 and 400 meters.

Depth of the reservoir

The lithostratigraphy is completed by defining the depth relative to the surface. The reservoir’s depth is important, because it, among other things, as will be explained in the next section, determines its temperature and pressure. Because in reality this depth varies a lot, it will be interesting to use different values in the research, but for the base model a reference level needs to be chosen. Because the Slochteren Formation lies on average about 3 km below the surface, according to data from several wells, the middle of ROSLN in the model chosen a depth of 3000 meters. For an overview of the model stratigraphy see Figure 8.

Figure 8: Stratigraphy of the model.

4.2 ROCK PROPERTIES

Rock properties are important for reservoir heat content and fluid / heat flow behavior. Sources are literature, well logs and drill core measurements. There are five types of rock used in the model, see Table 2. TOUGH2 needs density, porosity, x-, y- and z-permeabilities, wet heat conductivity and specific heat capacity for each rock type.

Table 2: Rock types used in the model.

Name Description ZEZ2H Rock salt of the Zechstein Formation ZEZ2X Zechstein anhydrite, limestone and Coppershale members ROCLT Ten Boer Member, consisting of silty and sandy claystones ROSLN Slochteren sandstone CARBF Carboniferous base, mainly sandstones and claystones

Zechstein salt

Zechstein carbonate & limestoneTen Boer Member

Slochteren

Formation

Carboniferous base

500m

50m50m

150m

400m

z (m)

-3000

-3475

-2325

20

Density

Density data for ROCLT and ROSLN was retrieved from core measurements from wells AKM-04, GRK-01-S1, MGT-01-S1, NOR-23, SLO-01, SLO-09 and UHM-01-S1. Density data for ZEZ2H, ZEZ2X and CARBF was retrieved from density logs from wells NOR-33, BLS-01-S1, GRK-01-S1, KHM-01, MGT-02 and GRK-13. The readout of these logs has a relatively high error. The values across different wells were averaged. For ZEZ2X, the values of lithological units ZEZ2A, ZEZ2C, ZEZ1W, ZEZ1C and ZEZ1K were averaged across the wells and then weighted with respect to their thickness.

TOUGH2 uses the grain densities of the rock, however well density logs provide only the bulk densities, sometimes together with a neutron porosity log. This difference was neglected, because on the one hand these rocks have very small porosity and on the other hand the porosity cannot be determined reliably from the neutron porosity logs.

Porosity and permeability

Porosities and permeabilities of ZEZ2H and ZEZ2X are assumed to be zero, which is reasonable because these are evaporites. Also the porosity and permeability of CARBF was assumed to be zero, because this rock is very old, compressed and therefore tight. Besides that it is also simply a very convenient assumption, because there will be no flow in the bottom of the simulation domain. This may not be a totally accurate assumption because of the unconformity present between the Permian and Carboniferous.

Porosities and permeabilities for ROCLT and ROSLN were taken from core measurements, from the same wells as used for the density determination. Per well the arithmetic mean of the measured data is taken for the depth range corresponding to the lithological unit under investigation. The top and bottom depth of lithological units is provided by NLOG3 with the other well data. The permeabilities of ROCLT and ROSLN were chosen somewhat lower (50 and 250 mD) compared to the average of the measurements (91 and 297 mD respectively), because of the likely presence of zones with lower reservoir quality, which will cause some resistance to flow over the whole length of the aquifer.

TOUGH2 uses separate values for x-, y- and z-permeabilities. For the sake of simplicity and because the Slochteren sandstone was deposited in eolian and fluvial settings, the assumption is made that vertical permeabilities of all rock types are equal to the horizontal permeabilities form the drill core measurements. However this assumption is somewhat questionable, because the few z-permeability measurements that are available show a significantly lower value. The depositional mechanism for ROCLT and the uplift of the Carboniferous (CARBF) promote anisotropy as well. A more detailed analysis is needed to assess the influence of permeability anisotropy, but this is not part of this research.

Thermal conductivity and specific heat

Thermal conductivities and specific heat capacities for all rock types were dug up from various literature sources. The results of this research can be found in Figure 9 and Figure 10. A more detailed overview of these literature data can be found in Appendix B.

Next to these specific values a number of more general dependencies were also found in the literature. For instance, specific heat increases with temperature (Waples, 2004). As for thermal conductivity, it increases with water saturation and pressure and decreases with temperature and porosity (Bär et al., 2011; Clauser & Huenges, 2011; Emirov & Ramazanova, 2005; Šafanda et al., 1995; Schütz, Norden & Förster, 2011).

Since the rock types in the model do not correspond exactly with the ones found in literature, some assumptions have to be made to choose the right values. For ZEZ2H the same values were used as rock salt in (Leeuwen, 2010). For ZEZ2X averaged values for anhydrite and limestone were used, weighted by their average thickness form wells UHM-01-S1, SLO-01, SLO-09, NOR-04, NOR-23, MGT-01-S1, MGT-02 and KHM-01. The Coppershale layer was neglected from this calculation, because it represents on average only 1.8% of the entire thickness of ZEZ2X. For the Slochteren sandstone (ROSLN) the average of literature values for sandstone were

3 “NL Olie- en Gasportaal”: www.nlog.nl

21

used. For ROCLT, ROSLA and CARBF, which consist mainly of sandy and silty claystones or a succession of sand- and claystones, the average values for claystone were used. Claystone thermal properties were the least encountered in literature, but those that were found lie close to that of sandstone and siltstone. Values for thermal conductivity were rounded to the nearest decade to reflect the uncertainty in these figures (since temperature, pressure, porosity and saturation dependencies could not be exactly accounted for).

Figure 9: Thermal conductivity for various rock types.

Figure 10: Specific heat for various rock types.

0

1

2

3

4

5

6

Thermal conductiv

ity (W

m K )

-1-1

Anhydrite Claystone Limestone Rock salt Sandstone Shale Siltstone Carboniferous

Bär et al., 2011

Ondrak et al., 1998

Botor et al., 2002

Schön, 2011

Clauser et al., 1995

Schütz et al, 2012

Muntendam-Bos et al., 2008

van Leeuwen., 2010

0

200

400

600

800

1000

1200

1400

1600

1800

Specific heat (J Kg K )

-1-1

Anhydrite Claystone Limestone Rock salt Sandstone Shale Siltstone Carboniferous

Botor et al., 2002

Schön, 2011

Dezayes et al., 2008

Smoltczyk, 2003

Muntendam-bos et al., 2008

van Leeuwen, 2010

Ondrak et al., 1998

Waples et al., 2004

22

Summary

As a summary of this chapter Table 3 is presented with all the required rock properties.

Table 3: Summary of rock properties.

ZEZ2H ZEZ2X ROCLT ROSLN CARBF Density kg/m3 2.11 2.83 2.72 2.67 2.63 Porosity % 0 0 11 19 0 XY-Permeability mD 0 0 50 250 0 Z-Permeability mD 0 0 50 250 0 Wet heat conductivity W/m/K 5.5 4.0 2.0 3.0 2.0 Specific heat KJ/m3/K 1050 830 870 980 870

4.3 BOUNDARY CONDITIONS, INITIAL CONDITIONS AND NATURAL STATE

The numerical model needs to have boundary conditions and initial conditions. The first reason for this is that the mathematical equations behind the model can only be solved if a unique solution exists. The boundary conditions determine whether this is the case. Secondly, the boundary and initial conditions define the numerical value of that solution, which means they have to be chosen carefully in order to make the model simulate realistic conditions.

The initial conditions for the production model should represent an equilibrium state, so that when the model is run, only the effects of the geothermal exploitation are observed and no secondary behavior. For this reason the simulation is carried out in two phases. The first phase consists of modeling an equilibrium state that matches the naturally existing state as good as possible. The second phase will model the actual geothermal production, with injection and extraction of water in to and out of the reservoir.

Boundary conditions

The boundary conditions constrain the solution of the model at the boundaries of the simulation domain, at every time instant. Either the state of the boundary cell or the flow to or from that cell will be prescribed, which is called a Dirichlet or Neumann boundary condition respectively. A combination of both is also possible and called a Robin condition. Furthermore, a boundary condition can be defined as constant and uniform across the whole boundary, but also time dependent and location specific.

From the equation of state modules present in TOUGH2 "EOS1" is selected, which is the most basic module modeling only pure water in liquid, vapor or two-phase state. In our model only liquid water occurs, as a result of the pressure and temperature ranges used in the model, in which case the primary variables are pressure and temperature. The top and bottom boundary conditions are used to obtain suitable temperature and pressure gradients, while the primary variables can vary freely at the sides of the domain by using no-flow Neumann conditions. Because cells modeling the reservoir are separated from the top and bottom cells by zero permeability layers, the boundary conditions effectively only affect temperature and not pressure.

For the natural state of the base model the temperature gradient is described by steady state “composite wall” heat conduction. This means the vertical heat flux q is related to the overall temperature delta (or gradient) by the heat conduction coefficients λi and layer thicknesses hi according to (Incropera & DeWitt, 2007):

O ������[k����∑�� m�⁄ (23)

To uniquely describe the temperature gradient as well as absolute temperature either two temperatures or one temperature and the heat flux need to be defined. For the sake of simplicity the temperatures at the top and bottom of the domain were chosen constant as boundary conditions for both the natural state and production model.

23

Such a Dirichlet condition is easily implemented to a model boundary by appending a layer of “inactive” or “fixed” cells (TOUGH2 vs. PetraSim terminology). This method of defining the boundary condition is very efficient, because no balance equations need to be solved for such cells. Despite that their thermodynamic variables cannot change, the flows to and from the boundary condition cells are still taken into account each time step (Pruess et al., 1999), which means the boundary will act as a source/sink for fluid and heat flow in the model (Thunderhead, 2010). Because effectively the thermodynamic state is defined in the center of a cell, it is convenient to make the cells in this layer very thin (0.1 meters), so that the boundary condition is placed in the correct location. The numerical values of temperature and pressure at these boundaries are defined by the chosen initial conditions, which will be discussed in the next section.

Initial conditions

As mentioned before, the initial conditions for the producing model are acquired by performing a natural state simulation. However, the latter needs to be initialized in a certain state as well. Using the boundary conditions as described above, the equilibrium temperature gradient that the natural state model will develop depends only on the top and bottom boundary conditions and not on the initial conditions in the rest of the model. Conversely, the equilibrium pressure that will develop in the reservoir is dependent on the temperature change relative to the initial state. For this reason it is convenient to initiate the natural state model as close to the eventual equilibrium state as possible, which has the additional benefit of reducing simulation time.

For the base model this is accomplished by computing the analytical solution for the temperature gradient with eq. (23). On average, the continental geothermal gradient is about 30 °C per km and the continental heat flux about 65 mW/m2. However, these values are incompatible with each other because the lithological column in the model contains a relatively large share of highly conductive material. Instead an overall temperature gradient of 25 °C per km across the model is chosen, which results in a heat flux of 76 mW/m2.

Next to the gradient, the absolute value of temperature needs to be chosen. From measurement data from various wells it became clear that the initial temperature in the Slochteren sandstone should be around 90 °C. To reach this temperature in the middle of the Slochteren sandstone in the model, temperatures of 78 and 107 °C were defined at the top and base of the domain respectively. Also the initial conditions for the whole model can now be determined easily, because according to Fourier’s law the temperature gradient is piecewise linear for each lithological unit.

Analytically determining the initial pressure conditions in the reservoir is very straightforward as well, because, assuming a hydrostatic pressure regime and constant density, steady state pressure p increases with depth h according to:

# %� (24)

With ρ the density of water and g the gravitational acceleration. It is important to understand that TOUGH2 only takes into account the pore pressure, i.e. the pressure of the water inside the pores, and not the lithological pressure.

4.4 PROCESS PARAMETERS

Injection temperature

The key aspect of a geothermal doublet is that the 'spent' water, i.e. the produced water after the heat is extracted and became energetically useless, is reinjected into the aquifer, with the main advantage being that the pressure in the reservoir is maintained. Contrary to the analytical estimates of equations (8) and (9), in reality the temperature of the reinjected water influences the life time of the doublet (larger temperature gradients mean more heat conduction). Next to that it determines the geothermal power output according to equation (1).

It is assumed that the geothermal fluid is exposed to ambient temperature before being reinjected and consequently its temperature is basically reduced to that of the surroundings, assuming 20 °C. Intermediate temperatures e.g. immediately after the heat exchanger are not considered.

24

Well settings

TOUGH2 has several types of generators (sources and sinks) that can be used to model the injector and producer well on a cell level. PetraSim extends this functionality by defining the well as a whole. The geometry of both wells is chosen as simple as possible (no deviation) and both are perforated over the whole thickness of the Slochteren sandstone.

The injection well is represented by a mass flow rate and an enthalpy. This means the injection temperature cannot be set directly, but depends on the value of enthalpy and the pressure in the reservoir. To calculate which enthalpy corresponds to the desired injection temperature, a tool based on the 1967 ASME steam tables is used4. The drawback of this generator is that the injection temperature varies slightly with changing reservoir pressure.

For the production well the so-called deliverability model of TOUGH2 is used (Pruess et al., 1999). This model makes the well operate as a function of the actual bottomhole pressure as calculated during the simulation. The mass production rate of a cell with a generator of this type is given by:

O 4 �o!# � # �' (25)

With p the actual pore pressure of the cell, pwb the prescribed bottomhole pressure and PI the “productivity index”. The prescribed bottomhole pressure is chosen slightly higher than the pressure at the bottom of the reservoir for all the cells containing the deliverability generator, by choosing the “well model” gradient option in PetraSim. Combined with the boundary conditions and injection generator type in use, this means the pressure in the reservoir will rise until the flow rate of the production well matches the prescribed flow rate of the injection well. The value for productivity index is calculated using the appropriate expression in the TOUGH2 manual as 4.86E-12 m3. Note that the definition of PI in TOUGH2 is different from the one commonly used in the oil and gas industry, which is expressed in units of volume delivered per drawdown pressure (e.g. bbl/d/psi).

Well spacing and flow rate

The distance between the injector and producer together with flow rate determine for a large part the life time of the geothermal resource, next to the pressure drop over the doublet. Using the analytical estimates from chapter 2 one can get an idea of how these performance characteristics relate to flow rate and well spacing. For better overview the breakthrough time and the coefficient of performance (COP) were contoured for a range of flow rates and spacings in Figure 11 and Figure 12. The coefficient of performance is defined here as the quotient of thermal power output divided by the pumping power necessary to overcome the pressure drop over the reservoir. In reality however pressure drop over the piping and heat exchanger as well as the efficiency of the pump should be taken into account as well. The values for permeability and density etc. that are used to construct these plots are the same as those mentioned in section 4.2.

Figure 11: Coefficient of performance as a function of well spacing and mass flow rate.

4 Provided by Thunderhead Engineering at www.thunderheadeng.com/petrasim/petrasim-documentation/

25

Figure 12: Breakthrough time as a function of well spacing and mass flow rate.

The required flow rate basically follows directly from the desired power output and the temperature of the reservoir. Well spacing is often dimensioned with a large margin to prevent early breakthrough, especially since a deviated doublet with 1500 to 2000 meters well distance at a depth of about 3 km can be realized with relative ease. In reality the choice of process parameters is probably mostly governed by economic considerations, but these are not investigated further in this research.

For the base model a desired geothermal power of 30 MW is assumed. For reservoir and injection temperatures of 90 and 20 degrees Celsius respectively the required flow rate is calculated at about 104 kg per second or 374 m3 per hour. The well spacing is chosen relatively small, 1500 meters, to reduce the size of domain necessary to accommodate the doublet. By doing so also the life time is shortened, reducing the required simulated timeframe. Both the wells are perforated over the full height of the Slochteren Formation.

4.5 SIMULATION PARAMETERS

Run time and time step

For the modeling of the natural state a constant time step of 1000 seconds is used and the end time criterion is set to infinite, because the goal is to halt the simulation by the steady state criterion built in TOUGH2. This criterion is reached when 10 consecutive time steps converge within a single Newton-Raphson iteration. Indeed, when the boundary and initial conditions are chosen carefully to accurately match the analytical temperature and pressure distributions, steady state is reached after the first 10 time steps.

Contrary to the natural state model the production model is a transient model, which means the time settings of the solver are very different. The simulation should run for at least the cold front breakthrough time, which is about 70 years according to eq. (8), but the simulation end time is chosen at 200 years to also investigate the temperature decline after breakthrough.

On the one hand the time step size should be small to limit the discretization error, but large on the other hand to allow an efficient calculation of the solution. For this purpose TOUGH2 can change the time step size automatically, depending on the convergence rate of the iteration process. TOUGH2 is configured to double the time step when convergence occurs within 3 Newton-Raphson iterations, starting at an initial size of 100 seconds.

Symmetry

For a homogeneous reservoir the mathematical problem to be solved is symmetrical with respect to the center line connecting the two wells and for a number of conceivable models with lateral heterogeneity this applies as well. But is it possible to simulate the reservoir as such and how convenient is this approach?

26

In fact it is quite simple to just model half of the reservoir geometry and apply the appropriate no-flow boundary condition at the interface. This effectively cuts the computation time in half and produces an error in the order of 1 ‰ (when comparing power output after 100 years). However the only benefit is the reduction in calculation time, since the mesh resolution cannot be increased if full models and models utilizing symmetry are to be compared. Considering simulating only half of the model does result in a slight error and computation time is not really a constraint, symmetry was not utilized to produce the final results.

Domain and cell size

Just as with time the discretization of space results in an error, which is proportional to the discretization step size. So again, on the one hand very small cells are preferred for accuracy, but on the other hand large cells reduce calculation time. Next to this, the size of the domain could be chosen either smaller or larger for the same reason. Because the cell count is chosen as close as possible to the TOUGH2 limit of 99,999 cells, cell and domain size are linked together and a compromise between the two needs to be found.

Ideally one would choose domain and cell size in such a way that produces the smallest error. Next to that, an irregular mesh would be used that puts the resolution in the place where it is needed the most. However in practice it is difficult to carry such an optimization through. Also a mesh optimized for the reference model would likely not be the optimal mesh for carrying out the actual experiments. For this reason, the mesh was constructed based on simple guidelines and some trial and error.

The domain size in the vertical direction is defined by the height of the stratigraphical column in the model, see section 4.1. An irregular cell size is chosen, allowing a higher resolution in the reservoir itself than in the confining rocks. The chosen distribution of cells can be found in Table 4. The specified factor F defines how PetraSim distributes a specified number of cells n over a total distance L according to:

� �T\T $ �T\X$. . . $�T\�kX �0Z ��

�k1

�q0 (26)

From this relation L0 can be solved considering L and F are both known. When the factor has value 1 all cells are equal in size. For factors larger than 1 grid cells become smaller with depth and for factors smaller than 1 the opposite applies.

Table 4: Irregular mesh properties in z-direction

Layer Thickness (m) Number of cells Factor

ZEZ2H boundary condition 0.1 1 1.0 ZEZ2H 500 3 1.6 ZEZ2X 50 1 1.0 ROCLT 50 2 1.0 ROSLN 150 8 1.0 CARBF 400 4 0.65 CARBF boundary condition 0.1 1 1.0

In the lateral direction a rectangular solution grid is used. The other option provided by PetraSim is to use a polygonal mesh, which can be automatically refined around the wells. Drawbacks of such a mesh however are that interpolation between the polygons yields somewhat choppy plots of the results and lateral homogeneity cannot be implemented very neatly. Cell spacing is chosen constant and equal for the x- and y-directions, because this allows the implementation of lateral homogeneities with as little dependency on the mesh as possible.

Running a number of simulations with different domain sizes showed that the areal extent should be around 4km in both directions to satisfactory limit the influence of the domain boundaries on doublet performance. Because 70 cells can be placed along both lateral axes before the cell number limit of TOUGH2 is encountered (with 20

27

cells in the vertical direction) the cell size becomes about 58 meters (rounded off). Perpendicular to the center line of the doublet an uneven number of cells is chosen so that the wells are placed in the middle of the cell and not on an intersection. See Table 5 for the summary description of the mesh. Given these domain and cell sizes, the injection and production wells are placed at x = 1479 and 2987 meters respectively (in the middle of the domain in y-direction).

Table 5: Number of cells and cell and domain size

x y z

Cell size (m) 58 58 irregular

Number of cells 70 71 20

Domain size (m) 4060 4118 1150.2

Solver

Finally the linear system solver and solver settings have to be chosen. TOUGH2 contains multiple solvers of which two are recommended: a preconditioned bi-conjugate gradient solver and a stabilized bi-conjugate gradient solver. The latter is chosen because it is the most recent solver added to TOUGH2 and it supposedly converges better when iteration is started close to the solution (Pruess et al., 1999).

Other than the simulation time configuration discussed above only the weighting settings were changed from the default values. Weighting is necessary to determine material properties and thermodynamic variables for flow between cells. Permeability at the interface is harmonic weighted and the density at the interface is determined as the arithmetic mean of adjacent elements, as suggested by the TOUGH2 manual for single phase flow (Pruess et al., 1999).

29

5. RESULTS OF THE BASE MODEL

The results from the base model and also the methods used to extract these result are the subject of this chapter.

5.1 NATURAL STATE

First the outcome of the natural state simulation will be discussed. Following the approach outlined in section 4.3, the natural state simulation was initiated close the analytical solution, in terms of temperature and pressure gradient. Figure 13 and Figure 14 display these analytical solutions compared to the steady state solution subsequently calculated by TOUGH2.

The maximum relative errors are only 0.0019% and 0.0018% for temperature and pressure respectively, which is negligible. Because the initial state is defined close to the analytical solution only the minimum number of 10 time steps are needed to reach the steady state criterion within TOUGH2.

Figure 13: Comparison of numerical and analytical natural state temperature gradients.

Figure 14: Comparison of numerical and analytical natural state pressure gradients.

-0.0025

-0.0020

-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

75

80

85

90

95

100

105

110

2200 2400 2600 2800 3000 3200 3400 3600

Tem

per

atu

re d

iffe

ren

ec (°C

)

Tem

per

atu

re (°C

)

Depth (m)

numerical

analytical

difference (n-a)

-600

-400

-200

0

200

400

600

22

24

26

28

30

32

34

2200 2400 2600 2800 3000 3200 3400 3600

Pre

ssu

re d

iffe

ren

ce (

Pa)

Pre

suu

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5.2 PRODUCTION PHASE

Next the results of the production phase simulation in terms of temperature, flow rate and pressures will be discussed.

To obtain the average production temperature the temperatures of the grid cells containing the production well are weighted to the respective cell production flow rates and summed (the heat loss from the well to the surrounding rock between the reservoir and the surface is neglected). This temperature is plotted versus time in Figure 15. Initially water of 90°C is produced, as this is the average initial temperature of the reservoir. After around 50 years in the simulation the temperature starts to decrease, reducing to around 75°C at the end of the simulation. These production temperatures correspond to thermal power outputs (relative to the 20 °C injection temperature) of 30.4 and 23.9 MW respectively.

Additionally a breakthrough time is defined in a more strict sense, to compare the cold front progression between different simulation runs more easily. To this end, the time when the temperature decreases to 95% of the initial value is determined by interpolating the temperature output data. The 95% breakthrough time is 96.7 years for the base model.

Figure 15: Average temperature versus time for the production well.

The temperature distribution after 200 years can be seen in Figure 16. It is evident that the cold front has reached the production well.

Figure 16: Temperature contours after 200 years for the base model. Lateral cross-section in the plane defined by z = -

3000 and vertical cross-section at y = 0 meters. Injection and production wells are located at x = 1479 and 2987 meters

respectively.

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Interestingly the cold front does not really continue past the production well, although the cold front has reached the production well much earlier already (see Figure 15), which can probably be attributed to heat delivery from the opposite side of the well. Also can be seen that the domain is large enough to accommodate the entire cold front, i.e. the temperature distribution is not affected by domain boundaries. The temperature distribution is not symmetrical around the injection well because the water flow has a preference towards the injection well: for each layer of grid blocks making up the reservoir the pore pressure is lowest in the cell containing the production well.

Flow rate at the injection well is set to 104 kg/s or 374 m3/h. At the production well the flow rate is governed by the deliverability model discussed in section 4.4. The calculated production mass flow rate versus time is given in Figure 17. After a short startup period the flow rate increases from about 103.0 kg/s to 103.6 kg/s at the end of the simulation. This means that a small share of the injected water accumulates in the reservoir, which is probably due to the fact that the water density increases due to decreasing temperature, meaning more water can be stored in the same pore volume.

Figure 17: Production well mass flow rate versus time.

Pressure behavior for both wells is shown in Figure 18. Pressure was determined also as a flow rate weighted average over the generator cells, just as with temperature. The initial pressure in the aquifer is about 29 MPa. Right after the simulation starts, pressure rapidly builds up in the reservoir by the injection of water. As soon as the production well starts flowing the production well pressure remains nearly constant. The pressure at the injection well continues to increase over time, because the aquifer cools down and viscosity increases. The pressure difference between the two wells is 2.84 MPa at the end of the simulation.

Figure 18: Average pressure versus time for both wells.

Pressure correction

The pressure values from TOUGH2 are averaged across the well and weighted to flow rate, but the resulting number is still not very realistic. This becomes apparent when looking at the output of an isothermal doublet simulation, see Figure 19. The analytical pressure distribution shown in this figure, calculated using eq. (5), is limited to a wellbore outer radius rwb of 0.1 meter.

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The correlation between the analytical and numerical solution is quite good, except for two “errors”. Firstly, the numerical pressure distribution seems to suffer a little from boundary effects and does not develop entirely as expected near the edges of the domain. Secondly, a large error occurs in the cells representing the wells. Apparently the resolution of the solution grid is not high enough to resolve the entire pressure gradient close to the well. The result is that the pressure difference across the wells calculated using TOUGH2 is significantly smaller than what it should be.

Figure 19: Analytical and numerical pressure distribution in the top layer of the reservoir for an isothermal doublet.

Injection and production wells are located at x equal to 1479 and 2987 meters respectively and have a wellbore radius

of 0.1 meter.

For the base model but even more so for the experiments it is convenient to have a means of correcting the numerical values of well pressure to a more realistic value. However the pressure distribution in the case of a geothermal doublet is not symmetrical and eq. (5) is difficult to apply. Instead, the equation is used that describes the pressure distribution around a single well in a constant pressure circle (Muskat & Wyckoff, 1946):

#!g' # � $ � 234� ln 8 gg �: (27)

In this equation p is the pressure at radius r around the wellbore with pressure pwb and radius rwb. By choosing rwb as 0.1 meter, r as half the cell width and p the pressure calculated in the cells representing the well, eq. (27) can be solved for pwb. The results of the various pressure calculations can be found in Table 6. From the isothermal model can be seen that applying the aforementioned correction pressure drop is slightly overestimated. This is not such a big issue however, because on the other hand pressure drop from e.g. skin and the well piping is still neglected.

Table 6: Pressure differences between injector and producer wells for base and isothermal model.

Analytical ∆p Average numerical ∆p Corrected numerical ∆p

Isothermal model 2.84 MPa 1.42 MPa 3.04 MPa

Base model - 2.84 MPa 6.32 MPa

The coefficient of performance of the doublet, i.e. the thermal output power divided by the required pumping power, can now be calculated. An “initial” COP is calculated by using the temperature and corrected pressure difference after 30 years of simulation, resulting in a value of 48.6. Combining the corrected pressure drop of 6.32 MPA with the output power after 200 years of 23.9 MW, the final COP is calculated as 35.6 at the end of the simulation, which is 27% lower than the initial COP.

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6. EXPERIMENTS AND RESULTS

This chapter contains the discussion of the experiments together with their results. Some elements apply to all of the experiments and will be discussed first.

The experiments are based on a “one thing at a time” approach as much as possible. This means that in each experiment only one aspect is different compared to the base model. The difference between the experiments is always in terms of permeability, because the assumption is made that, out of the heterogeneities that can occur in reality, changes in permeability will have the largest influence on thermal lifetime and pressure drop.

Generally with a different permeability the porosity will be affected as well, according to the porosity-permeability relation for that specific type of rock. However, for these experiments the choice was made to keep porosity always the same as the base model, to retain the “one thing at a time” philosophy as much as possible. Although perhaps unrealistic in most cases, this way the results will be independent of a specific porosity-permeability relation and applicable to a wider range of reservoirs (within the Slochteren Fm. at least). As a side note, in reality it is possible to have large variations of permeability in a reservoir without significant change in porosity, e.g. through illite growth in the pores.

Each of the five experiments is characterized by a certain “permeability geometry”, i.e. the permeability of the reservoir is adapted in a zone that has a distinctive shape. To obtain a better understanding of the geometry under study, several factors (like permeability value or size of the geometry) are varied. Depending on the experiment these variables can take a number of different values. With one experiment as an exception, every possible combination of variable values is simulated, resulting in a so-called “full-factorial” experiment. Because this amounts to a large number of experiment “treatments” that have to be run, the process of adding the variations to the model and running the simulation was automated with the use of PyTOUGH, see section 3.1.

The base injection volume flow of 374 m3/h is distributed over the cells containing the well perforations, weighted linearly with permeability values of each cell. This is done because in reality a larger flow is expected from the well to the high permeable layers and a lower one to the low permeable layers. Secondly, for a uniform flow rate distribution, the high injection rate in low permeable zones would cause an enormous increase in cell pressure, which disturbs the model and may affect the effective injection temperature. This approach was used for all experiments.

In addition, a production phase test run was done for each simulation, in order to inject water of the preferred temperature of 20 °C. As in TOUGH2 the injection enthalpy and not temperature is specified, the actual injection temperature depends on the pressure in the perforated cells, which is a priori unknown. After a test run of 2 years of production the pressure and temperature of the injection cells are recorded and the “correct” injection enthalpy is estimated, by using the IFC-67 thermodynamic routines built in PyTOUGH (the same thermodynamic formulation is used in TOUGH2).

For each experiment we will discuss what the goal of the experiment is and what geometry will be investigated. Secondly the specifics of the experiment implementation are discussed and finally followed by the results.

6.1 AMELAND

The first experiment is called “Ameland” because the low permeability layer in this model resembles the Ameland member (‘ROSLA’) which is present in the Slochteren Fm. in the most northern parts of the Netherlands. The 37.5 meters thick (2 grid cells) low permeability layer lies in the middle of the Slochteren Fm. and laterally extends through the whole model, see Figure 20. Note that only permeability is different, all other material properties are kept constant. The question is what is the effect of the reservoir dividing layer on the doublet performance, but also what happens when the wells are only partly perforated, instead of over the full thickness of ROSLN.

34

Figure 20: Cross-section at y = 0 showing the low permeability zone resembling the Ameland Member. Note that the

scale of the z-axis is 5 times that of the horizontal axis. The Upper and Lower Slochteren Formations are denoted by

ROSLU and ROSLL respectively.

A number of simulations were run, each with a different set of variables. The permeability of the “Ameland” layer obtained values of 50, 25, 10 and 2.5 mD, the rest of the Slochteren Fm. keeping the base model permeability of 250 mD. Besides permeability also the well perforations were varied. The well perforations are applied on a cell level, meaning the respective cell receives a source/sink term depending on whether it is the injection or production well and depending on the choice of perforation. For both wells the perforation schemes “top”, “bottom”, “both” and “whole” are used, see Figure 21. Because every possible combination of permeability and well perforations is run, this experiment consists of 43 = 64 simulations.

Figure 21: Different perforations used in the Ameland model for both the injection as the production well.

Results

With the explanation of the experiment out of the way it is time to look at the results.

Pressure (at the end of the simulation, after 200 years) was determined for both wells by taking the flow rate weighted average over the generator cells, as an attempt to properly represent the pressure inside the well casing. Exclusively to the Ameland experiment, pressure values were then normalized to a depth of 3000 meters to account for well perforations at different depths. This was done by applying a hydrostatic pressure gradient to the difference between the average perforation depth and the reference depth. The pressure differences between the injection and production well for an Ameland permeability of k = 2.5 mD and different perforation schemes can be found in Figure 22. Pressure values represented here are without the correction mentioned in section 5.2. By applying the correction the pressure deltas become 2-2.5 times higher, but the order remains practically the same.

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Figure 22: Average final pressure drop for every combination of perforation for k = 2.5 mD.

Pressure drops are highest when only the part above or below the “Ameland” is perforated, especially for the injection well. This makes sense, because with the smaller opening in the well casing flow rates are higher and pressure gradients larger. Choosing either “both” or “whole” at the production well produces very similar results, but interestingly the perforation of “whole” always performs slightly better than “both”. However, this could be attributed to the somewhat unrealistically high production rates in the “Ameland” layer due to the well on deliverability model.

As mentioned above also the permeability of the middle layer was varied. The effect of decreasing permeability is twofold: On the one hand pressure drop increases across the board and secondly the spread between the differential pressures increases. Next to the end time values also the pressures after 30 years of production were evaluated. Just as with the base model, see Figure 18, pressure drops after 30 years are slightly smaller than at the end of the simulation, but the difference is not very large.

Figure 23: 95% breakthrough time for every combination of well perforation for k = 2.5 mD.

Figure 23 shows the 95% breakthrough times, as explained in section 5.2, for the Ameland model with an intermediate layer of permeability 2.5 mD. There is a difference of approximately 35 years between the best and worst breakthrough times. In comparison with the base model cold front breakthrough always occurs earlier, except for when the production well is perforated at the top only. This is because producing from the top of the reservoir results in the highest final production temperatures while it has the lowest initial production temperatures, as the reservoir is relatively cold at the top due to geothermal gradient. Since the breakthrough

36

time is calculated relative to the initial production temperature this results in the longest breakthrough time. However for a more reasonable 30 years of running the highest temperatures are still achieved by production in the bottom of the reservoir.

One would expect the breakthrough times to be especially good when injecting and producing on opposite sides of the reservoir splitting layer, however this is not the case: These configurations rank only 3rd and 11th out of the total of 16 experiment (with k = 2.5 mD). This can be explained by the fact that the temperature distribution is for a large part indifferent of the type of perforation, see for example Figure 24. Apparently, despite the low flow of fluid, heat can still pass the middle layer quite easily. On the one hand this can be attributed to heat conduction and a small amount of convection, but perhaps also (to some extend) numerical errors play a role here. For one, the reservoir splitting layer is only two cells thick, so only one single vertical flow connection lies completely in the low permeability zone, separating the top and bottom of the reservoir. Next to that, the temperature distribution may be affected by a phenomenon called numerical dispersion, which is the excessive smearing of spatial gradients due to the discrete nature of the simulation volume (Coats, 1982).

Figure 24: Comparison between perforations on opposite sides of the “Ameland” layer versus across the whole height

of the Slochteren Formation. The cross-section is taken at y = 0 after 200 years of simulation and permeability of the

middle layer is 2.5 mD. Note that both the length as the color of the vectors indicate the amount of fluid flow.

Another thing to note is that in this case the calculated breakthrough times are a bad indicator for the production temperature development over time. Although the 95% breakthrough times exhibit a large spread, production temperatures never differ by more than 1-2 °C at any given time between the models with different perforations, see Figure 25. The large differences shown in breakthrough time are mainly caused by two things. Firstly, the initial production temperature in the bottom of the reservoir is higher than the average reservoir temperature (90 °C), causing the temperature-time curve to dip more rapidly in the first say 50 years, resulting in an earlier relative breakthrough time. Secondly, the cold injection water has a higher density than the water present in the aquifer and therefor a tendency to move towards the bottom of the reservoir. Due to these two factors, producing only in the bottom of the reservoir shows bad performance in terms of the relative breakthrough times shown in Figure 23, while producing in the top shows relatively positive results.

37

Figure 25: Production temperature over time for all 64 experiments in the Ameland model.

The effect of permeability on reservoir lifetime is rather small. The effect of the middle layer does clearly become more pronounced with a lower permeability. However, from Figure 25 can be seen that production temperatures are always within 2 degrees Celsius or less of each other, irrespective of permeability value. The differences in initial production temperature are caused by the different perforation depths at the producer and are independent of injector peforation choices.

Overall it seems perforating the producer only at the top and the injector “whole” or “both” has the best performance, because breakthrough time is the highest and pressure drop one of the lowest. But because absolute temperature differences are so small perforating the “whole” or “both” sides of both wells is also a good strategy since this minimizes pressure drop.

6.2 WEDGE

The geometry in the next experiment is a wedge-shaped formation of lower permeability in the Slochteren sandstone. The wedge is defined to the left in the xz-plane and uniform in y-direction, see Figure 26. The question is of course what the presence of such a shape does with the pressure drop and life time of the doublet compared to the base model. For a better understanding of this geometry a number of factors are varied: The permeability of the wedge is set to 50 and 25 mD (compared to 250 mD for the rest of the Slochteren), the position in x-direction of the wedge inside the simulation domain varies in steps of 290 meters (5 grid blocks) and the slope of the wedge (17.9°, 9.2° and -17.9°). In addition, this sequence is repeated with the low permeable formation to the right of the inclined plane. Well perforations are the same as for the base model, meaning the whole thickness of the Slochteren is perforated.

Figure 26: Cross-section at y = 0 showing the low permeability geometry in the “Wedge” model. The location of the

wedge frontier xwedge is defined at a depth of 3000 meters. Note that the scale of the z-axis is 5 times that of the

horizontal axis and in actuality the slope is one fifth of the one in this drawing.

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Results

In this section we will look at the results of the Wedge model, first at the breakthrough behavior and later the pressure drop will be discussed.

Figure 27: 95% breakthrough times versus x-location for different wedge slopes and permeabilities. The volume

occupied by the wedge is given by 0 ≤ x ≤ xwedge, i.e. the wedge is in the left side of the domain, as shown in Figure 26.

The 95% breakthrough time for different factor combinations and the low permeability wedge in the left side of the domain can be seen in Figure 27. Evidently the largest dependency is on the x-position of the wedge, while other factors influence the breakthrough time significantly less. With the inclined plane of the wedge located at x = 290, at the very side of the domain, the presence of the wedge is insignificant and breakthrough time reduces to that of the base model (96 years). The lowest times occur when the wedge’s edge is close to the injection well, where the cold front is directed away from the wedge into the direction of the production well, see Figure 28 and Figure 29. In the same figures we can see that, with the wedge directly to the left of the production well, the fluid produced comes primarily from the part of the reservoir that has not cooled down yet, resulting in a longer life time.

As noted before, the dependency of breakthrough time on slope and permeability of the wedge is much less. With lower permeability for the wedge the breakthrough time is impacted slightly more but the differences between the simulated values of 50 and 25 mD are relatively small. The minor differences in results between the various wedge slope angles can be attributed to gravity effects.

For the case when the wedge is to the right of xwedge, the breakthrough times are roughly identical, but mirrored in x. Now the maximum breakthrough time occurs with the edge of the wedge at the injection well and the minimum with the edge at the production well. Next to that the maximum breakthrough time is 160 years for the reversed wedge compared to 150 years shown in Figure 27. It is unclear whether this should be addressed to the discrete nature of the results (the real maximum lies somewhere between the result data points) or to the smaller viscosity of water around the injector.

Unlike with the Ameland model, breakthrough time is a sign of faster cooling of the reservoir, because in final temperature (after 200 years) the lowest vs. highest temperature is 69 vs. 82 °C, a difference of 13 degrees Celsius. The dependency of final temperature on xwedge, slope and permeability is very much like (almost a copy of) that of the 95% breakthrough time.

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Figure 28: Flow of fluid vectors and temperature contours at z = -3000 meters for simulations with wedge located

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Figure 29: Flow of fluid vectors and temperature contours cross-section at y = 0 meters for simulations with wedge

located between x = 0 and x = 1451 (bottom) and x = 2901 meters (top). Wedge permeability is 25 mD and slope is

17.9°.

Figure 30: TOUGH2 average pressure difference between the wells versus xwedge for different wedge slopes and

permeabilities. The volume occupied by the wedge is given by 0 ≤ x ≤ xwedge, i.e. the wedge is in the left side of the

domain, as shown in Figure 26.

Pressure drop for the Wedge model is displayed in Figure 30, again for the case with the low permeability geometry to the left of xwedge. Pressure drop is equally dependent on xwedge and permeability, but not so much on slope.

When the wedge is small and clear of the wells (xwedge < 1161) pressure drop remains at the level of the base model. Clearly when the wedge extends beyond one of the wells at x = 1479 and x = 2987 pressure drop increases. With the wedge boundary in between the wells ∆p is proportional to x, which makes sense because the water has to traverse a longer distance in the low permeability zone. With nearly the whole Slochteren taken by

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the low permeability wedge the pressure drop increases to about 12 and 21 MPa averaged TOUGH2 values for k = 50 and 25 mD respectively. When corrected as explained in section 5.2 these values increase to 29 and 55 MPa.

The permeability of the wedge shape has a large influence on pressure drop, as expected. When permeability is cut in half, pressure drop increase (relative to the base model) is approximately doubled. The slope of the wedge plays a minor role, except that a smaller slope produces a smaller pressure drop with the edge of the wedge right adjacent to a well. Simulating either a positive or a negative slope produces the same results in terms of pressure drop.

When the low permeability zone is to the right of the cutting plane the resulting plot looks almost like an exact mirrored copy to the one shown in Figure 30. However, for 1741 ≤ xwedge ≤ 2611 the pressure drop is substantially lower for the setup with the wedge to the right of the domain. With the wedge on the left side pressure deltas are about 9 and 15 MPa for k = 50 and 25 respectively, see Figure 30, but with the wedge on the right these values are only around 6 and 8 MPa respectively. In other words whether or not the injection well is inside the low permeability zone has a larger impact than whether this is the case for the production well. This difference is explained by the fact that water viscosity is lower on the injection side of the doublet.

6.3 WEDGE-AMELAND

This experiment can be regarded as a variation on the Wedge model. Instead of the low permeability zone bounded by a single plane, now two planes are used to define a low permeability “obstacle”, see Figure 31. Because the resulting geometry resembles the one in the “Ameland” model but slanted, this experiment shall be known as “Wedge-Ameland”. To investigate this geometry the following factors were varied:

• Placement in x-direction from 290 to 3370 meters in steps of 290 m.

• Permeability (25, 10 and 5 mD).

• Width: 2 and 4 grid blocks (116 and 232 meters respectively).

• Slope: 17.9°, 9.2° and -17.9°. These angles are chosen such that their tangents are 1, 0.5 and -1 block height divided by one block width.

In order to save computation time not every combination of variables is simulated, making this a partial factorial experiment.

Figure 31: Cross-section at y = 0 showing the low permeability geometry in the “Wedge-Ameland” model. The

location of the wedge frontier xwedge is defined at a depth of 3000 meters. Note that the scale of the z-axis is 5 times that

of the horizontal axis and in actuality the slope is one fifth of the one in this drawing.

Width is expressed in number of blocks and not in units of length, because the effective length is not very well defined. This has two reasons. Firstly, permeability is defined in the cell centers only. For the flow from one cell (center) to another, the value of permeability for that flow connection is a harmonic weighted average of that of the two cells. In other words, for the flow connections at the boundary of the low permeability wedge intermediate values for permeability are used, making the transition somewhat gradual and not entirely discrete.

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Secondly, although a skewed rectangle (in cross-section) is desired, in the discrete model this becomes a staircase shape, see Figure 32. Especially when the slope is 0.5 blocks per column (9.2°) the effective thickness that the water flows through might be less than the width in number of blocks suggests.

Figure 32: Comparison of desired wedge geometry and actual geometry in the discrete model: a) slope is 17.9° and b)

slope is 9.2°. The effect of permeability weighting for flow connections is not shown in this figure.

Results

Because this experiment consists of many factors and is only a partial factorial experiment, the influence of the factors will be discussed one by one. First the pressure drop results will be discussed, followed by the cold front breakthrough times.

Figure 33: TOUGH2 average pressure difference versus xwedge for the Wedge-Ameland model, for different wedge

permeability. The wedge has a slope of 17.9° and a width of 4 cells.

Figure 34: TOUGH2 average pressure difference versus xwedge for the Wedge-Ameland model, for different wedge

width. The wedge has a permeability of 5 mD and a slope of 17.9°.

Pressure drop versus xwedge for different permeability and width can be seen in Figure 33 and Figure 34. Overall pressure drop is the highest when the wedge is located near and at one of the wells (at x equal to 1479 and 2987 meters). With the wedge geometry outside of the doublet pressure drop is about equal to that of the base model and with the wedge in between the wells a small increase in ∆p is recorded. The maximum with xwedge near the injector is higher than at the producer because of lower water viscosity of the injected water. It is questionable

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however if the displayed peaks are really the maximum pressure drops that can occur for this geometry or that the actual maxima lie somewhere between the data points.

Permeability does not influence the pressure drop very much, but especially with the wedge in the area between the wells a difference occurs. This relatively low influence is explained by the fact that the Ten Boer Member is still unaffected and becomes relatively more transmissive as permeability of the wedge decreases, hence it will act as a “by-pass”, see Figure 35. In the same figure can be seen that the water does not flow through the barrier horizontally but takes the path of least resistance. If the width of the wedge is doubled the increase in pressure drop over the base model also roughly doubles.

Figure 35: Vertical cross-section at y = 0 showing fluid flow vectors and temperature contours in ROSLN and

ROCLT in between the two wells. The slope is 17.9°, width is 4 cells, permeability is 5 mD and wedge is located at

xwedge = 2230 meters.

Figure 36: TOUGH2 average pressure difference versus xwedge for the Wedge-Ameland model, for different wedge

slopes. The wedge has a permeability of 5 mD and a width of 4 cells.

When comparing different slopes for the wedge, in Figure 36, it can be noticed that a change of slope sign does not change the outcomes much, only the peaks from the wedge close the wells seem slightly shifted. Again the discrete nature of the results make it difficult to compare maxima, because the “real” maximum can lie somewhere in between the data points. For the reduced slope of 9.2° the peaks around the wells are wider and less tall, which makes sense: because the wedge geometry is wider, the wells penetrate the low permeability cells for a wider range of xwedge. On the other hand a lower number of low permeability cells are penetrated by the well per xwedge, which results in lower maximum pressure drops.

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Figure 37: 95% breakthrough time versus xwedge for the Wedge-Ameland model, for different wedge permeability. The

wedge has a slope of 17.9° and a width of 4 cells.

Figure 38: 95% breakthrough time versus xwedge for the Wedge-Ameland model, for different wedge width. The wedge

has a permeability of 5 mD and a slope of 17.9°.

Breakthrough time versus xwedge for different permeability and width are plotted in Figure 37 and Figure 38. Notably the breakthrough time is lowered with the wedge on the outside of the doublet, where the direction of injected water towards the production well is increased. If the wedge is placed in between the wells as a barrier the breakthrough time is strongly increased. A lower permeability or larger width of the wedge amplifies these effects on tbr.

Breakthrough time vs. xwedge for 3 different slopes is displayed in Figure 39. Here a change in slope sign impacts the results more than is the case with pressure drop. The differences are partly explained by a small shifting and the discrete nature of the result, but the fact that cold water has a tendency to sink to the bottom of the reservoir also may have some influence. Again the reduced slope shows a smaller deviation from the base model results and the presence of a well impacts a larger range of xwedge.

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Figure 39: 95% breakthrough time versus xwedge for the Wedge-Ameland model, for different wedge slopes. The wedge

has a permeability of 5 mD and a width of 4 cells.

Also large differences occur in terms of final temperature. Unsurprisingly, the distribution of low vs. high production temperature after 200 years largely matches that of breakthrough times. The minimum and maximum final production temperatures recorded are 71.1 and 81.0 °C respectively.

6.4 RESERVOIR BOUNDARY

In this experiment the effect of geothermal production near the reservoir boundary is investigated, by adding a low permeability zone on one side of the doublet as displayed in Figure 40. Mainly the distance from the doublet center line to the reservoir boundary yboundary is varied, from 29 to 1885 meters, in 116 meter steps. Not that the domain boundary lies at y = 2059. Next to that the permeability is varied. Not so much to mimic reality but more to check how the model copes with different permeability values. The reservoir boundary will restrict fluid flow; the question is how much will this affect pressure drop and thermal life time of the doublet.

Figure 40: Cross-section at z = -3000 showing the low permeability geometry in the “Reservoir Boundary” model.

Results

An example TOUGH2 output with the reservoir boundary located at yboundary = 145 meters is displayed in Figure 41. A small but negligible flow develops in the boundary itself, because in this case the boundary permeability is 0.1 mD and not zero. As a result of the fact that almost half of the simulation domain is occupied by the reservoir boundary, an increased interaction (relative to the base model) of the cold front with the domain boundary is observed.

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Figure 41: Flow of fluid vectors and temperature contours at z = -3000 meters for a simulation with the reservoir

boundary located at yboundary = 145 meters. Boundary permeability is 0.1 mD.

The main results of the Wedge-Ameland model can be found in Figure 42 and Figure 43. Both the breakthrough times as well as the pressure drop development are completely in agreement with what one expects. With the doublet close to the reservoir boundary the breakthrough time decreases, which is because the injected water has less space to occupy and flow rates will be higher and therefore the cold front will move faster towards the production well. With the reservoir boundary very close to the model domain, breakthrough time and pressure drop become equal to that of the base model. The closer the reservoir edge is to the doublet the higher the pressure drop, increasing to almost 5 MPa.

Figure 42: 95% breakthrough time versus position of the reservoir boundary in y-direction. Three curves for different

values of permeability are shown.

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Figure 43: TOUGH2 average pressure difference versus position of the reservoir boundary in y-direction. Three

curves for different values of permeability are shown.

Interesting to note in the graphs above is the fact that both curves flatten towards maximum yboundary, meaning the presence of the reservoir boundary has less and less influence on the doublet's performance. From these results one can argue that the domain boundary, which is situated at y = 2059 is far away enough from the doublet to have negligible effect. However when looking at final production temperature in Figure 44, the absence of boundary effects is less certain. The final production temperature in the base model is 74.6°C, which indicates that the size of the potentially present boundary effect is limited.

Figure 44: Production temperature in the Boundary model after 200 years of simulation for different values of

yboundary.

6.5 FAULT

In this last experiment a fault that partially cuts through the reservoir is modeled. A very simple representation of a fault is used: only the permeability of a wall-like section in the reservoir (both Slochteren and Ten Boer) is lowered. The fault lies perpendicular to the edge of the simulation domain at y = 2059, see Figure 45. Both the position of the fault in x-direction as the length in y-direction is varied: xfault from 754 to 3538 meters and yfault from -435 to 725 meters. Note that yfault is measured from the doublet center line. The spacing between xfault values is chosen in such a way to avoid placing the fault right on top of a well and is therefore somewhat irregular.

The permeability and width of the fault can also be varied, but this has been omitted because otherwise the experiment becomes too large. The fault has a permeability of 1 mD and a width of 2 grid blocks i.e. 116 meters.

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Figure 45: Top view of the reservoir showing the low permeability obstacle in the “Fault” model.

Results

Pressure drop values (TOUGH2 values averaged over the wells) can be seen in Figure 46. Always when the fault lies closer (in y-direction) to the doublet center line pressure increases. When the fault is furthest away from the doublet, pressure drop values become equal to that of the base model. Looking at the dependency on xfault, pressure drop reaches a maximum when the fault lies in between of the two wells. For the cases where the fault penetrates past the doublet center line (yfault < 0) clear peaks are visible for when the fault is placed near one of the wells (xfault = 1566 or 2900).

The recorded maximum pressure drop depends strongly on the distance from the fault to the wells. In earlier runs of this experiment the fault was occasionally placed in the same cells as the wells, which resulted in pressure drops of up 100 MPa and pore pressures of up to 160 MPa. In reality it is not possible for this kind of stress to occur in a formation like the Slochteren Sandstone. Additionally, the simulation runs with the wells inside the fault had other problems, like a very long computation time and wrong injection temperatures.

Figure 46: TOUGH2 average pressure difference for different x- and y-placements of the fault.

Looking at the 95% breakthrough time, the base model result again appears when the fault is far away from the doublet, see Figure 47. For yfault > 0 the tbr vs. xfault plots have a bowl-shape, with the minimum breakthrough time occurring if the fault lies approximately in the middle of the two wells. For the two curves with yfault < 0 this same shape can be recognized, with an additional response in the area of the wells (xy-coordinates of the wells are (1479, 0) and (2987, 0)): if the fault lies directly on the outside of the doublet the breakthrough time is reduced, while directly next to a well on the inside of the doublet tbr dramatically increases. This is the same behavior as we saw in the Wedge-Ameland model, where water flow towards the production well is either obstructed or stimulated.

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For xfault in the area in between the wells breakthrough time improves or worsens depending on yfault, because two phenomena occur at the same time: on the one hand the fault effectively narrows the reservoir which increases flow velocity, but on the other hand the obstacle makes the cold water travel a longer distance. Which of these two effects is stronger is hard to assess with pen and paper, but it does show very clearly in the results of the numerical experiment.

Figure 47: 95% breakthrough time for different x- and y-placements of the fault.

An example fluid flow vector field and temperature distribution is given in Figure 48. Both fluid flow and temperature distribution are far from symmetrical in the y-axis, as was the case for the base model. Although the fault is still somewhat permeable it appears that only a negligible portion of fluid flows directly through the barrier. The largest part of injected water flows directly around the fault into the direction of the production well, while another significant share first flows around the reservoir along the fault and domain boundaries before reaching the producer.

6.6 COMPARISON BETWEEN EXPERIMENTS

As a summary to all the results, a comparison is made between the 5 experiments. Overall a total of 609 simulations were run to obtain the results of the experiments and a large number of test runs as well. Compared to the base model only the permeability in certain places in the reservoir was adapted, but this was done in a multitude of ways, whereby factors such as location, width and slope are varied, to create a better understanding of the geometry under study. Four of the experiments were full-factorial and only in the Wedge experiment only a selection of all possible variable combinations was simulated. All these simulations produce many result data, in which a few things stand out that will be discussed below.

Are there certain outcomes that are particularly improving the performance of the doublet? Remarkably in none of the simulations an improvement in pressure drop over the base model was recorded. However this is not really a surprise because compared to the base model permeability was only lowered and never higher. For the lowest pressure drop (in these experiments) the wells should be as far away from any lower permeability obstacles as possible. If a choice has to be made the obstacle should preferably be on the outside of the doublet rather than on the inside, in between the two wells.

The base model showed an average pressure difference over the two wells of 2.8 MPa in TOUGH2 and 6.3 after the correction explained in section 5.2. The three experiments that were discussed last all exhibit maximum pressure drops of around 5 MPa, the Ameland model around 7 MPa and Wedge even higher than 20 Mpa. The corrected ∆p values for these maxima are calculated as 16, 56, 8, 9 and 8 MPa for Ameland, Wedge, Wedge-Ameland, Boundary and Fault respectively. The strength of the correction varies greatly because it depends on mass flux and cell permeability, see eq. (27). For the Ameland model the mass fluxes per cell are higher because

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of the smaller number of perforations and the correction for the Wedge model is impacted by the fact that each well-representing cell only has a tenth of the permeability compared to the base model.

Figure 48: Flow of fluid vectors and temperature contours at z = -3000 meters for a simulation with the fault located

at xfault = 1740 and yfault = -203 meters.

51

In terms of cold front breakthrough time both performance improvement and impairment were observed in most of the experiments. For Wedge, Wedge-Ameland and Fault maximum tbr per experiment are around 150 years and minimum around 70 years, compared to the base model breakthrough time of 96.7 years. In the boundary model no improvement was possible: only reductions of tbr to as low as about 50 years. In the Ameland model improvements were not very large: the maximum tbr is 107 years. Remarkable about the Ameland model is that despite some differences in tbr, production temperature after a few years is always almost the same. For the other models final production temperature varied over a larger range with an overall minimum of 62.5 and a maximum of 83.0 °C.

Because production flow rates are almost completely the same in every simulation, see Figure 49, the remarks made on final production temperature above also hold for final geothermal power output. The minimum and maximum outputs after 200 years respectively are 18.5 MW in the Boundary experiment and 27.4 MW in the Wedge experiment. More interesting is to look at the efficiency of the various configurations by calculating the COP of the doublet, i.e. the thermal power output divided by the required pumping power. The final COP (after 200 years) attains the largest value in the Wig-Ameland experiment (36.7), while the lowest is recorded in the Wedge experiment (4.2). Also an “initial COP” is calculated, which is based on temperature and (corrected) pressure drop values after 30 years of simulation. With 50.0 this values is highest in a simulation from the Fault experiment and with 5.4 lowest in the Wedge experiment. As the power output at the 30 year mark is roughly the same in every simulation, the large differences in initial COP can be fully attributed to the differences in pressure drop.

Figure 49: Final mass flow rates for all simulations in the experiments.

Also in most of the experiments there is a choice of variables that reduce the outcome of the simulation to that of the base model, e.g. when the permeability of the heterogeneity is not very different compared to the rest of the reservoir and the distance to the doublet is large. Only in the Ameland model the permanent presence of the reservoir splitting layer ensures a deviation from base model performance in all cases.

Overall, it is established that the location of the obstacle placed in the simulation (in reality the location of the doublet relative to the obstacle) has the largest effect on performance. Next in line are the value of permeability and the size of the obstacle. The depth over which wells are perforated also has a large impact on pressure drop, but production temperature over time is not really affected, at least for the Ameland model. The least influential is the slope in the experiments incorporating a wedge shape. The absolute value of slope has some effect, but slope sign practically does not show any difference, meaning it does not matter on which side of the wedge the injection well or the production well is.

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7. DISCUSSION

This chapter gives an overview of the assumptions and choices made, as well as the methods used in this research. Also the validity of and the methods to acquire the results are discussed. If there is room for significant improvement a recommendation is given.

7.1 APPROACH AND METHODS

Approach

This research entails the investigation of reservoir heterogeneities and the effect of these on geothermal doublet performance. To be able to quantify this effect and to distinguish between small variations in the geometry a certain precision is required. This accuracy cannot be delivered by analytical approaches and physical experiments are practically impossible, which means a numerical simulation is the best choice. From the nature of the doublet setup itself but also from the various heterogeneities it is necessary to use a 3-dimensional model. The choice for TOUGH2 / PetraSim was based on availability and experience with this simulation package and in hindsight this was a good decision. Especially because of the availability of the extension PyTOUGH that was discovered during the research, because the automation drawn from PyTOUGH made it possible to examine each experiment extensively, in an efficient and reproducible way.

From the equation of state modules present in TOUGH2 "EOS1" was selected, which is the most basic module modeling only pure water in liquid, vapor or two-phase state. The choice for this module was very important, because it determines for a large part the complexity and the assumptions used for the simulation. TOUGH2 is a very versatile simulator and capable of including e.g. CO2, NaCL or “dead” oil in the model. In this research the goal is to investigate reservoir properties in conjunction with a geothermal doublet. The emphasis was therefore on the flow of liquid water and heat rather than gases or phase changes and chemical reactions. The choice for EOS1 proved good in terms of practicality and general applicability of the experiments, but unfortunately it is unclear how the results would change with the presence of e.g. natural gas or dissolved salt.

Base model

A lot of effort has been put into a realistic and practical base model, in terms of geological setting, boundary conditions, rock properties, process parameters (e.g. injection temperature and flow rate) and simulation parameters (e.g. solution mesh). The stratigraphy and natural state were modeled after the Slochteren Sandstone reservoir as present in the North of the Netherlands because this region enjoys a special interest, but the results are thought to be more broadly applicable.

Process parameters must also be somewhat realistic, but additionally this was also a matter of matching the different choices to achieve a balanced simulation. For instance well spacing and flow rate have to be in balance with the size of the domain in order to avoid boundary effects. In this process the creation of an ideal solution mesh was attempted, with the maximum number of cells allowed by the TOUGH2 code and high resolution in the places where it is needed the most. Although difficult, the mesh can be optimized for a single model by improving it in a trial and error procedure. However since the mesh of the base model is also used in every experiment, choosing the best mesh, i.e. the one that provides the most accurate results overall, is more a matter of a scientific guess. Two choices were made to make the mesh robust and usable: some safety margin was applied to the size of the domain in order to try to avoid boundary effects under all circumstances and secondly the mesh consists of rectangular cells with constant dimensions in the horizontal plane to be able to introduce heterogeneities in a regular fashion.

Looking back the mesh had a number of disadvantages: First of all the mesh is relatively coarse in the area around the wells, creating a large difference between the cell's pressure and the well pressure. Secondly in the Ameland model the number of cells in z-direction in the reservoir appears somewhat low, resulting in the middle layer consisting of only 2 cells. Lastly the lateral domain size was too small in the Boundary model, giving rise to stronger domain boundary effects. In this model maybe the approach of moving the doublet through the

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domain instead of moving the reservoir boundary might have been a good idea. Still the tradeoffs that were made seem to have resulted in an adequate mesh and there is no reason to reject the results because of its limitations.

However, if improvements to the mesh are desired the following is recommended: Firstly a proper error analysis should be carried out, in order to evaluate and compare different meshes quantitatively. Such an analysis is unfortunately not part of this research because of time constraints. Secondly the number of cells in the model may be pushed beyond the TOUGH2 imposed limits, e.g. by using the TOUGH2-MP code instead of regular TOUGH2.

Modeling of the wells

One of the larger limitations in this research is the way wells are modeled in TOUGH2. The main problems are the fact that injection enthalpy is defined instead of injection temperature and secondly the injection flow rates per cell are predetermined by user input and not dependent on time dependent local pressure values. The first problem was satisfactory overcome by doing a test run prior to each simulation to assess pressure at the injection location and estimate the right value of enthalpy to inject with the desired temperature. Still, this should be regarded as a workaround and impractical because it increases the computation time for each simulation and still introduces small deviations in injection temperature.

For injection flow rates the total mass flow was divided over the injection cells by weighting to permeability, i.e. less water is injected in cells that are less permeable. A linear weighting scheme was used but it is questionable how realistic this is. The distribution of flow rates at the production well was very different as a result of the deliverability model, but it unclear how well this matches reality. To improve the model in terms of well representation first of all a better understanding of well theory is needed, as this subject was underexposed in this research. Next coupling a well simulator to the reservoir model as suggested by the TOUGH2 manual may be considered.

In addition, a downside of discretization is that the flow in the nearest vicinity of the well is modeled rather poorly: In our model the solution mesh resolution was too low to capture the very rapid changes of filtration velocity and pressure gradient directly on the outside of the well. This makes it difficult to assess the pressure drop over the doublet, as will be discussed below. To prevent the issue smaller grid cells need to be used in the area around the well. However, if equally sized cells remain to be used the cell count has to extend beyond the TOUGH2 limit or the domain size needs to be reduced, both of which are undesirable. A better solution is to only refine the grid locally, but this is not possible with rectangular cells in the PetraSim model. It is possible e.g. by directly editing the simulation input file with a text editor or building a TOUGH2 model from the ground up with PyTOUGH, but this was considered too much effort compared to the benefits it brings to this research.

7.2 RESULTS

Base model validation

The base model was validated to analytical solutions in two ways. The natural state simulation was initiated with an analytically determined temperature gradient. The subsequent simulation showed that this analytical solution is also numerically a steady state. This demonstrates that the quality of the used natural state is only dependent of choices like stratigraphy, material properties and boundary conditions and not at all on the numerical model.

The validation of the production phase simulation was done for an isothermal model only, as no analytical solution is available for a geothermal doublet. Overall the numerical solution agrees reasonable well with the analytical one, only near the domain boundary significant differences occur. This boundary effect did not appear to affect the production temperature and was therefore neglected for the remainder of the research. However now there is some uncertainty whether or not some of the results of the experiments are affected by the limited size of the domain. For further research the effect of boundary effects should be investigated into more detail.

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Method to determine pressure drop over the doublet

Normally the pressure gradient in the well is hydrostatic and only one measurement point is needed to know "the pressure" in the well. It was difficult to determine the pressure drop over the doublet, because actual well pressures are unknown; effectively only the pressures on the generator cell's outer edge are calculated, meaning pressure is only known at a distance of 29 meters from the well. For this reason the pressure values of the cells containing the well were averaged. Still, this leaves an error for the pressure gradient over the last 29 meters between the cells outer edge and its center where the well is positioned. To take this into account an analytical solution was used to calculate a correction. Still this solution is not completely satisfactory, because even in the case of an isothermal doublet in a homogeneous reservoir a significant error is produced. In the simulations of the experiments around the wells flow can be far from radial and large temperature gradients exist, so this error increases. Other problems are that "skin" of the well as well as pressure drop in the well tubing are neglected. Nevertheless, it is assumed that the averaged TOUGH2 pressure values are adequately suited for comparing different experiment configurations.

Cold front breakthrough time

In order to detect cold front breakthrough in the simulation the breakthrough temperature was chosen at 95% of the initial production temperature, as a 5% change is large enough to prevent faulty detection due to round off errors or other numerical phenomena. However this 95% threshold temperature was calculated separately for each experiment, which in hindsight was not such a good idea. The problem lies in the post processing of the Ameland model, where a large variety tbr is recorded but the production temperatures over time are almost exactly the same. As part of this discussion it is verified that if an absolute value (85.5°C) as a threshold is used the differences in recorded breakthrough time are much smaller and as such a better representation of the production temperature differences.

7.3 RESEARCH OVERALL

Choice of experiments

Within the topic of reservoir heterogeneities there are numerous things that can be varied and investigated. In this research the choice was made to only vary reservoir permeability, for a couple of reasons. First of al the permeability is known to vary a lot from the various core measurements consulted for setting up the base model. Secondly it is assumed that permeability variations have a large effect on water flow in the reservoir and therefor affect the doublet performance very significantly. However it seems to be a better practice to do an actual screening research to assess which parameters are the most important. In the current situation the effect of for instance varying thermal conduction coefficients or heterogeneous boundary temperatures are unknown and the choice for permeability cannot be motivated by figures.

The choice for varying only permeability does have some practical advantages. For example a single natural state can be used for every simulation in the experiments and this steady state can be directly calculated analytically, unlike when for example thermal conduction coefficients are varied laterally. Also when e.g. the thickness or slope of the entire reservoir is made heterogeneous and variable this puts impractical demands on the solution mesh, whereas with changing permeability only a different rock type has to be assigned to a range of cells.

In terms of permeability heterogeneities the main structures that were deemed interesting are in fact part of the research. Still within the existing experiments some additional factors might be interesting to vary, for instance the slope in y-direction. Additional experiments can be thought of as well, for instance one with multiple horizontal layers or even a stochastically assigned permeability per cell. The Fault model can be improved by making the representation of the fault more realistic, however additional parameters need to be changed aside from permeability.

The one thing at a time approach was desired but in the Ameland experiment not fully implemented. Since the middle layer was implemented together with a change in perforations, the effect of the different perforations

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without the presence of the Ameland layer is unknown. It would have been better to also run experiments for the different perforations while keeping the base model permeability for the whole reservoir.

Applicability of results

The question is how well do the results comply with reality, i.e.: "Can they be trusted?" Off course the model produces an error compared to reality resulting from discretization and boundary effects, but also due to physical assumptions and choices for material properties and boundary conditions. As mentioned earlier a proper error analysis was not carried out. Nevertheless it is assumed that although some kind of error is present comparisons between simulations and experiments are still valid. The quantitative results are not directly applicable to real world situations anyway and it is the qualitative relation between experiments simulations that really matters.

According to the experiments, in many cases it is possible to obtain a significant improvement in geothermal life time if a limited increase in pressure drop is allowed. Now if a geothermal doublet is to be situated in a reservoir relative to some particular low permeability geometry one can try to choose the best location based on this research. In reality however the reservoir layout is probably more complex and more detailed simulations are necessary to optimize a doublet placement for a particular location. Still when orientating on a larger area it is nice to be able to globally estimate what are the things that must be avoided at all cost and which are structures that can be beneficial in terms of cold front breakthrough time.

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8. CONCLUSION

Having gone through the introduction, explaining the base model, describing every experiment together with the results and a small comparison and discussion, the research is concluded by recognizing the following:

• A numerical 3-dimensional model is created for the Permian Rotliegend reservoir, based on literature, well logs and core measurements, which is balanced in terms of process and simulation parameters. This model acts as a strong basis that has fulfilled its purpose for this research, but has more possibilities for future research.

• Experience was gathered in working with PyTOUGH to automate the process of adapting a base model and applying variations to a several experiment parameters. This method is efficient and reproducible and therefor very valuable for coming research.

• Five experiments were conducted, changing the permeability of the reservoir in a number of ways that are modeled after real world structures. These experiments are Ameland, Wedge, Wedge-Ameland, Boundary and Fault. Maybe in reality the porosity is also somewhat lower or e.g. local temperature is slightly different, but the main differentiator is low permeability. For these structures we can say the following:

• Pressure drop over the doublet increases by the presence of low permeability structures in the reservoir, which is of course not a surprise. Striking though is the dependence of distance from the wells to the obstacle: Because flow rates are higher in the vicinity of the wells, the presence of a low permeability zone there has more influence on doublet performance, in terms of both pressure drop and cold front breakthrough time.

• Cold front breakthrough times are both positively and negatively influenced by the modeled heterogeneities. Depending on the corresponding value of pressure drop, advantageous doublet configurations are identified.

• According to our Ameland model, it is beneficial to perforate the well casing over the full height of the reservoir, in order to minimize pressure drop. Perforating the production well only in the lower part of the reservoir improves production temperature in the first couple of years, but contrary to what is expected no significant improvements in geothermal life time are achieved.

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APPENDIX A: PYTOUGH EXAMPLE SCRIPT

In this appendix an example script is given with which a TOUGH2 model (e.g. the base model discussed in chapter 4 of the report) is run a couple of times with a number of different flow rates for the injection well, see below. The scripting language is Python.

In the first lines a module named “os” and several functions from “t2data” and “multiprocessing” are imported. The “t2data” module comes from the program PyTOUGH, see section 3.1. In the succeeding lines a number of constants are defined, like the list of flow rates we are interested in and the number of cells that represent the injection well. Following is a function “flowrate(q)”, which defines the actual experiment. Within this function first the base model is opened, after which the flow rates of all of the injection generators present in the model are adapted. Finally the model is saved in a new directory and the simulation is run by calling the “EOS1” executable from TOUGH2.

The last 3 lines take care of actually doing the experiment. For each entry in the list of flow rates defined earlier the function “flowrate(q)” is carried out. With the help of the function “Pool” from the multiprocessing module the simulations are run simultaneously in order to utilize all 6 processor cores of the workstation.

The script reads as follows:

import os

from t2data import *

from multiprocessing import Pool

basedir = 'C:/sim/Base4'

expdir = 'C:/sim/Base4/flowrate'

qlist = [40., 60., 80., 100., 120., 140.]

layers = 8

def flowrate(q):

os.chdir(basedir)

dat = t2data('base4.dat')

os.chdir(expdir + '/q%s' % q)

for gen in dat.generatorlist:

if gen.type == 'COM1':

gen.gx = q/layers

dat.write('q%s.dat' % q)

dat.run(simulator='eos1')

if __name__ == '__main__':

pool = Pool(processes=6)

pool.map(flowrate, qlist)

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APPENDIX B: ROCK PROPERTIES

Thermal conductivities and thermal capacities for a number of rock types collected from various literature sources can be found in Table 7 and Table 8 respectively.

Table 7: Thermal conductivities for a number of rock types.

Rock λ in W m-1 K-1

Source* Remarks

Mean Range

Anhydrite 5.36 5.09 – 5.63 C 25-35 °C

4.76

C

4.00 1.00 – 6.05 F 77 samples

5.43 4.89 – 5.73 F

4.90 – 5.80 F No mean value given

5.00

G

Claystone 2.04 0.60 – 4.00 F 242 samples

1.90 1.6 – 2.1 G

Limestone 2.90

B

3.03

C Empirical relation, T = 20 °C

2.50

C Empirical relation, T = 100 °C

2.29 0.62 – 4.40 F 487 samples

2.97 2.00 – 4.41 F 26 samples

3.44 1.30 – 6.26 F 445 samples

2.70 2.0 – 3.2 G

Rock salt 5.94 5.11 – 6.77 C T = 27 °C

4.00

E

5.50 5.00 – 7.20 H

Sandstone 1.40

A Porosity of 20%

3.60

A Porosity of 0%

2.90

D T = 89 °C

3.00

B

2.47 0.90 – 6.50 F 1262 samples

2.57 1.56 – 3.86 F 8 samples

3.71 1.88 – 4.98 F 11 samples

5.60 2.3 – 6.6 G

Shale 1.70

D T = 89 °C

1.85

B

2.07 0.55 – 4.25 F 377 samples

1.90 1.6 – 2.2 G

Siltstone 2.00

B

2.67 2.56 – 2.78 F 1 sample

2.68 2.47 – 2.84 F 3 samples

2.90 2.4 – 3.4 G 18 samples

Carboniferous 4.10

D "Base rock (Carboniferous)"

2.65

E "Prepermian, highly compacted"

*A: Bär et al., 2011. B: Botor, Kotorba & Kosakowski, 2002. C: Clauser & Huenges, 1995. D: Muntendam-Bos et al., 2008.

E: Ondrak et al., 2008. F: Schön, 2011. G: Schütz, Norden & Förster, 2012. H: Leeuwen, 2010.

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Table 8: Specific heat capacities for a number of rock types.

Rock Specific Heat

Source* Remarks

J kg-1 K-1 J m-3 K-1

Anhydrite 590

H

940

H T = 20 °C

880

E

Claystone 869 2345 F Calculated from volumetric Cp with ρ = 2.7

Limestone 680

H T = 20 °C

880

H T = 20 °C

939 2600 A Calculated from volumetric Cp with ρ = 2.77

930

E

Rock salt 880

H T = 20 °C

840

D

1050

G

Sandstone 775

H T = 20 °C, ρ = 2640 kg m-3

833 2200 C Calculated from volumetric Cp with ρ = 2.64

1061 2800 A ρ = 2640 kg m-3

710

B ρ = 2200 kg m-3

960

D

1640

E

910

E Range 820–1000

Shale 778 2100 A

1111 3000 C

1180

E

Siltstone 910

H T = 20 °C

989 2650 A Calculated from volumetric Cp with ρ = 2.68

Carboniferous 815 2200 C "Base rock Carboniferous", calculated from volumetric Cp with ρ = 2.7

*A: Botor, Kotorba & Kosakowski, 2002. B: Dezayes et al., 2008. C: Muntendam-Bos et al., 2008. D: Ondrak et al., 2008.

E: Schön, 2011. F:Smoltczyk, 2003. G: Leeuwen, 2010. H: Waples & Waples, 2004.