Adv. Geosci., 49, 207–214, 2019
https://doi.org/10.5194/adgeo-49-207-2019
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.
Reservoir-scale transdimensional fracture network inversion
Márk Somogyvári1,2, Michael Kühn3,4, and Sebastian Reich2
1Department of Hydrogeology, TU Berlin, Berlin, 10587, Germany
2Institute of Mathematics, University of Potsdam, Potsdam, 14476, Germany
3Section 3.4 Fluid Systems Modelling, GFZ German Research Centre for Geosciences, Potsdam, 14473, Germany
4Institute of Geosciences, University of Potsdam, Potsdam, 14476, Germany
Correspondence: Márk Somogyvári (mark.somogyv[email protected])
Received: 31 May 2019 – Revised: 2 September 2019 – Accepted: 22 October 2019 – Published: 8 November 2019
Abstract. The Waiwera aquifer hosts a structurally com-
plex geothermal groundwater system, where a localized ther-
mal anomaly feeds the thermal reservoir. The temperature
anomaly is formed by the mixing of waters from three dif-
ferent sources: fresh cold groundwater, cold seawater and
warm geothermal water. The stratified reservoir rock has
been tilted, folded, faulted, and fractured by tectonic move-
ment, providing the pathways for the groundwater. Charac-
terization of such systems is challenging, due to the resulting
complex hydraulic and thermal conditions which cannot be
represented by a continuous porous matrix.
By using discrete fracture network models (DFNs) the dis-
crete aquifer features can be modelled, and the main geolog-
ical structures can be identified. A major limitation of this
modelling approach is that the results are strongly dependent
on the parametrization of the chosen initial solution. Classic
inversion techniques require to define the number of fractures
before any interpretation is done.
In this research we apply the transdimensional DFN in-
version methodology that overcome this limitation by keep-
ing fracture numbers flexible and gives a good estimation on
fracture locations. This stochastic inversion method uses the
reversible-jump Markov chain Monte Carlo algorithm and
was originally developed for tomographic experiments. In
contrast to such applications, this study is limited to the use
of steady-state borehole temperature profiles – with signifi-
cantly less data. This is mitigated by using a strongly sim-
plified DFN model of the reservoir, constructed according to
available geological information.
We present a synthetic example to prove the viability of
the concept, then use the algorithm on field observations for
the first time. The fit of the reconstructed temperature fields
cannot compete yet with complex three-dimensional contin-
uum models, but indicate areas of the aquifer where fractur-
ing plays a big role. This could not be resolved before with
continuum modelling. It is for the first time that the trans-
dimensional DFN inversion was used on field data and on
borehole temperature logs as input.
1 Introduction
Aquifer systems in fractured rocks can be modelled by us-
ing either continuum or discrete models. Continuum models
substitute the fractured rock with an equivalent porous media
(Day-Lewis et al., 2000; Hao et al., 2008; Illman et al., 2009;
Illman and Neuman, 2003; Sahimi, 2011; Vesselinov et al.,
2001; Zha et al., 2015). These methods often fail to model
the exact location of the fractures due to their strong averag-
ing behavior and they are considered most suited to problems
with high fracture density (Long et al., 1982).
Discrete fracture network models (DFNs) are more real-
istic representations of the fractured media, where fractures
and the rock matrix are modelled separately (Dorn et al.,
2013; Jang et al., 2008; Neuman, 2005; Niven and Deutsch,
2012). This separation allows to solve flow and transport
problems effectively, limiting these expensive computations
to the open fracture space. The use of DFN models for in-
version purposes however is not as straightforward as classic
equivalent porous medium models.
DFN models behave in a nonlinear, non-continuous way,
as small changes in fracture locations could open new path-
ways for flow, or completely isolate large parts of the domain.
Fractures are parametrized separately, leading to models with
large numbers of parameters, leading to ill-posed inversion
Published by Copernicus Publications on behalf of the European Geosciences Union.
208 M. Somogyvári et al.: Reservoir-scale transdimensional fracture network inversion
problems. And this number is usually an unknown, making
any classic inversion approach impossible to use directly.
Substituting the complex DFN model with a simplified
model has been proven a good solution to this problem. Sim-
plified DFNs limited to a few main fractures were used to
interpret hydraulic, tracer and temperature observations in
several studies (Le Borgne et al., 2007; Le Goc et al., 2010;
Klepikova et al., 2013, 2014). The simplifications made pos-
sible the identification and characterization of the preferen-
tial flow paths, but could not reconstruct complex fracture
patterns.
The use of complete DFN models is typically limited for
forward modelling purposes, or to the inversion of statistical
DFN parameters instead of exact geometries. The unknown
number of model parameters was a main obstacle, as stan-
dard inversion methods require a predefined parameter set to
adjust.
Hestir et al. (1998) and Niven and Deutsch (2012) used
activation-deactivation of fractures to keep the number of pa-
rameters fixed, while Dorn et al. (2013) stochastically gener-
ated fracture networks with a fixed number of fractures. Hui
et al. (2019) and Pan et al. (2016) used the statistical param-
eters of the fracture network in combination with connectiv-
ity to model the dynamic behavior of fractured reservoirs.
Yao et al. (2018) changed the parametrization of the fracture
network into a set of continuous parameter fields by using
Hough-transformation, making them suitable for inversion
methods requiring Gaussian distributions.
Somogyvári et al. (2017) implemented a transdimensional
DFN inversion approach, that resolved the problem of pa-
rameter numbers by keeping them flexible throughout the in-
version. The method was capable to explore different DFN
geometries and find the preferential transport trajectories.
The method has been used to interpret synthetic tracer to-
mography experiments, and to characterize cross-borehole
aquifer conditions.
The presented study uses this transdimensional DFN in-
version algorithm to characterize the geothermal reservoir of
Waiwera in New Zealand. After introducing the field site,
we show how the methodology needed to be modified for
applicability in a large-scale non-tomographic setting using
temperature borehole data. We demonstrate the applicability
on a synthetic example for proof-of-concept, then present the
results of the interpretation of the field data, to infer the fault
geometry of the Waiwera geothermal reservoir. This is the
first application of this methodology on field observations.
2 Field site
Waiwera is a small east coastal township north of Auck-
land on the northern island of New Zealand. A low temper-
ature (max. 50 ◦C) geothermal reservoir is located under the
city, making the area a popular recreational resort. At this
unique location, artesian flow from bores occurred until the
Figure 1. Geological and thermal conditions of the Waiwera
geothermal site. (a) Structural elements of the aquifer, (b) interpo-
lated temperature distribution.
end of the 1960s and thermal springs were emanating next
to oceanic beaches until the mid of the 1970s. As a result of
overproduction the warm water has to be pumped from the
reservoir since then (Kühn and Stöfen, 2005).
A detailed description of the local geological and hy-
draulic setting can be found in (ARWB, 1980).
In the aquifer the upflowing geothermal water is mixing
with fresh groundwater and seawater. These complex flow
conditions are forming a steady state temperature anomaly at
the site, which is shown in Fig. 1b. The site is well studied,
as well based on borehole information (Kühn and Altmanns-
berger, 2016).
The aquifer body is formed of the Waitemata sandstone
formation, which is a strongly stratified sandstone with a hy-
draulic conductivity of 2.3×10−5m s−1(average of entire
setting determined by a pumping test). The spacing of the
stratification is unknown (Kühn et al., 2016). The stratifica-
tion is not horizontal, because the formation has been folded,
fractured and faulted (Kühn and Schöne, 2017). It is also in-
tersected by two high inclination fault sets (Fig. 1a). One of
these faults provide the pathway for the geothermal water at
the aquifer bottom. From hydraulic tests and well drill ac-
tivities the hydraulic conductivity of faults and fractures is
estimated (in the range of 10−4–10−6m s−1) and used in the
inversion (ARWB, 1987).
Previous studies modelled the aquifer using continuum
approaches, combining the hydraulic properties of the frac-
tures, the stratification and the aquifer matrix (Kühn et al.,
2016; Kühn and Altmannsberger, 2016; Kühn and Stöfen,
2005). These models however do not include the discrete
features and could not describe the thermo-hydraulic behav-
ior of the aquifer. The aim of this study is to interpret the
borehole temperature profiles with DFN models, to infer the
geometry of the faults.
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M. Somogyvári et al.: Reservoir-scale transdimensional fracture network inversion 209
Figure 2. The transdimensional DFN inversion algorithm. Every
iteration of the inversion consists of two phases. In the model update
phase, the DFN is modified randomly, via fracture deletion, addition
or horizontal translation of a fault. In the model evaluation phase,
the updated model is evaluated by simulating the temperature field
in the aquifer.
3 Methodology
3.1 Inversion
In this study we use the transdimensional DFN inversion
introduced by (Somogyvári et al., 2017), based on the
reversible-jump Markov chain Monte Carlo (rjMCMC) algo-
rithm (Green, 1995). This iterative technique uses subsequent
random geometry perturbations to generate and test possible
model configurations (Fig. 2). Every iteration consists of two
phases: an update and an evaluation phase.
In the update phase, the current DFN realization (θn) is
altered by fracture addition, deletion or movement. The type
of the update and its properties are chosen randomly, with
a predefined probability (q(θn→θ∗)). Fractures could only
be inserted to specific discrete points along the existing DFN.
This is an important condition to preserve the reversibility of
the chain (see Somogyvári et al., 2017).
In the evaluation phase the temperature distribution in the
aquifer is modelled on the updated DFN realization (θ∗) us-
ing the forward model (details in Sect. 3.2). The model eval-
uation is done using the Metropolis-Hastings-Green accep-
tance criterion (Green, 1995):
αθ∗|θn=min1,p(θ∗)
p(θn)
L(ξ|θ∗)
L(ξ|θn)
q(θ∗→θn)
q(θn→θ∗)|J|.(1)
The likelihood-ratio (L(ξ|θ∗)/L(ξ|θn)) is calculated from
the improvement of the RMS error of the simulated obser-
vations (ξ|θ). We consider the observations (ξ) normally
distributed, hence the Lfunctions have a gaussian shape.
The proposal ratio (q(θ∗→θn)/q (θn→θ∗)) is calculated
as the ratio of the probabilities of the reverse and the forward
update. Because these updates are discrete, the calculation
of these probabilities are straightforward. For example, the
probability of a fracture deletion is one over the total number
of fractures. A detailed description on the update probabili-
ties can be found in (Somogyvári et al., 2017). The so-called
Jacobian (|J|) is a consequence of the transdimensional up-
dates, and it is required to maintain the stability of the rjM-
CMC. With discrete model updates however, its value is al-
ways one (Denison et al., 2002).
If a proposal gets accepted the new iteration starts with it,
if it gets rejected the chain continues with the previous re-
alization. The final product of rjMCMC inversion is not one
calibrated model, but a set of realizations often referred to as
ensemble. The ensemble is suitable for further statistical in-
vestigations and for uncertainty analysis (Somogyvári et al.,
2017). To eliminate the effect of the arbitrary chosen initial
model, the first part of the chain is discarded and not used in
the ensemble (Somogyvári and Reich, 2019).
3.2 Forward model
To simulate the steady state temperature distribution in the
DFN model, the tracer transport simulator developed by
Jalali (2013) was used. This efficient algorithm could simu-
late temperature distributions in two dimensional DFN mod-
els within a few seconds, making it suitable for MCMC ap-
plications.
The forward model takes the DFN geometry and hydraulic
and temperature boundary conditions as an input and returns
the steady state temperature distribution in the fracture net-
work. The simulator is transient, both the pressure and tem-
perature simulations start from an initial field with perturba-
tions by the boundary conditions, then the simulation is run
until a steady pressure and temperature field is obtained.
First the DFN model is discretized to 1-D segments and
nodes. Pressure distribution is simulated by using mass con-
servation, with an implicit finite difference method. Flow in
the fractures are calculated by Darcy’s law from the pres-
sure gradient. For thermal transport, the advection-dispersion
equation is solved considering a steady-state flow field, with
an implicit upwind finite difference solver (Jalali, 2013;
Ringel et al., 2019).
In this simple model no temperature transport is simulated
within the impermeable rock matrix, only the temperature
loss from the fracture space towards the rock matrix is sim-
ulated. Hence, cross-fracture heat transport (between non-
connected near fractures) is not possible. The simulation is
limited to 2-D and no complex hydraulic and thermohaline
boundary conditions are considered.
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210 M. Somogyvári et al.: Reservoir-scale transdimensional fracture network inversion
The temperature field inside the non-fractured aquifer ma-
trix is calculated via linear interpolation with triangulation.
The interpolated field is then intersected with the boreholes,
and the extracted 1-D temperature profiles were used as the
simulated observations. In theory it would be possible to
compare the interpolated 2-D temperature field, with the in-
terpolated field from the site (see Fig. 1b), but we wanted
to demonstrate the robustness of the method with only using
temperature logs from a limited number of boreholes. The
complete simulation could be done faster than a second on a
laptop, thus this simple DFN transport model is efficient and
fast to be used in an MCMC framework.
3.3 Conceptual model
The methodology introduced in (Somogyvári et al., 2017)
needed to be modified to work on reservoir scale scenar-
ios. The main modification is a simplification to mitigate
the limited amount of observations available. Three fracture
sets are defined: two for the major fault directions and one
for the stratification. In this modified setting, fractures are
completely intersecting the domain. This simplification is
done based on the geological model of the aquifer (Fig. 1a).
Hence, fracture lengths are not a free parameter of the inver-
sion as they are calculated from the fracture position within
the domain. In addition, fracture centers are aligned to the
vertical or horizontal centerline of the domain (depending on
the fracture set) – basically reducing the free parameters to
one per fracture.
Some further modifications were also done. Fracture ad-
dition and deletion is only used for the low-inclination frac-
ture set (stratification), and fracture movement only for the
high inclination sets (the exact location of these faults is not
known).
The hydraulic and thermal fracture properties were chosen
after available borehole information. The two fault sets are
simulated with an aperture of 0.5 mm. The stratification aper-
ture is set to 0.8 mm. These values are chosen after (Kühn
et al., 2016) using the cubic law to convert transmissivities
to apertures. The values were further tuned by preliminary
modelling trying to obtain similar extent of the temperature
anomaly as the observed one. These tests also showed that
the anisotropy has a stronger effect than the exact aperture
values. The hydraulic conductivities of the fractures are cal-
culated from the apertures using the cubic law. Note that both
flow and heat transport is limited to 2-D in the model, and no
3-D effects are considered.
Boundary conditions are defined on the top and bottom
borders (Fig. 3). In the DFN forward models, they appear
when a fracture intersects with these borders. Note that the
used pressure values are not absolute and were selected af-
ter the hydraulic gradient. For the source of the geothermal
water, a 100 m long source region with increased pressure
and temperature is defined in the bottom center of the model.
The observed maximum temperature of 50 ◦C was assigned.
Pressure of the source region was arbitrarily selected after the
parameter tests with inspecting the extent of the temperature
anomaly (Fig. 3).
Note that the used methodology is limited to invert the ge-
ometry, and does not estimate the physical fracture proper-
ties. Introducing the hydraulic parameters of the fractures as
free parameters into the inversion would result in an increase
in model freedom, which without involving additional data
would lead to different equivalent solutions (Somogyvári et
al., 2017).
4 Results
4.1 Synthetic modelling
The method was first tested on a synthetic dataset. The cho-
sen synthetic DFN model is shown in Fig. 4a. The inversion
here is limited to the fractures representing the stratification.
The high inclination faults are considered known, and not es-
timated by the algorithm.
The transdimensional DFN inversion was implemented in
python and was run on a standard office laptop with an In-
tel i5-7267U CPU. The chains were run for 1000 iterations
which took 5 min on average. Because of the stochastic na-
ture of the used inversion technique, we ran several simula-
tions with the same initial parameters to see the stability of
the results. All these simulations led to reconstructed geome-
tries with the same general features, showing the robustness
of the algorithm.
The result of the synthetic example is shown in Fig. 4b
in the form of a fracture probability map. This plot visu-
alizes the realizations of the rjMCMC chain by rasterizing
the DFN models first, then counting the pixels with fractures
along the ensemble. This is equivalent of taking the mean of
the MCMC chain for visualization (Bodin and Sambridge,
2009). The fracture probability map shows perfect fits with
the two fractures in the bottom part of the profile. The three
fractures in the top are not captured exactly, but the exten-
sion of this fractured zone is well resolved. The absence of
fractures in the center part is captured perfectly. This result
is a proof of concept that the methodology is applicable on
the proposed geological problem.
4.2 Field modelling
For field application, the three virtual boreholes defined as
1-D slices of the profile shown in Fig. 1b were chosen. One
borehole is taken from the center of the anomaly, the other
two from the two sides. These virtual boreholes were selected
instead of real wells for practical reasons, as all temperature
profiles from the site were already compiled and integrated
to a 3-D temperature dataset. This removes any local effects
of the actual boreholes and also provide room for sensitivity
analysis of different borehole locations numbers along the
profile, which is to be done in the future.
Adv. Geosci., 49, 207–214, 2019 www.adv-geosci.net/49/207/2019/
M. Somogyvári et al.: Reservoir-scale transdimensional fracture network inversion 211
Figure 3. Boundary conditions of the forward modelling.
Figure 4. Reconstruction of the synthetic DFN example: (a) Synthetic DFN model, (b) Reconstructed fracture probability map.
To improve the estimation, background temperature trends
were removed from the virtual boreholes (using the temper-
ature trends from the two sides of the model). With the re-
moval of these trends, we have also mitigated the effect of
freshwater-seawater mixing, which is not simulated by the
forward model.
Simulations were run for 1000 iterations, and similarly to
the synthetic case the first half of the chain was discarded.
The simulations were completed within 15 min, a little longer
then the synthetic case. The results are presented in Fig. 5.
The reconstructed fracture probability map has the fol-
lowing three distinct features. Strong stratification is present
close to the bedrock with high probability. Fractures in this
area are close to the source and provide pathways for the
heat to spread out horizontally. A non-stratified aquifer body
present in the center of the domain. It could not only mean
that this part is non-fractured, this could be also explained
with the existence of fractures that do not play a role of form-
ing the anomaly. This is supported by the similarity of ad-
ditional synthetic examples where this feature was usually
present. In contrast, there is high probability of stratification
in the top. The role of these fractures is to close the anomaly,
by allowing mixing with the colder waters. The inversion
also placed faults close to the location of two boreholes for
similar reasons.
Figure 5b shows the mean of the reconstructed tempera-
ture fields of the ensemble. Compared to the observed tem-
perature anomaly, the reconstruction shows similar proper-
ties. The two anomalies have a comparable extent horizon-
tally and vertically. The skewedness of the anomaly is recon-
structed. Originally this was explained by the mixing of dif-
ferent waters, but our results show that it could be explained
by the structure of the aquifer.
Figure 5c shows the fit quality of the borehole temperature
profiles. Continuous models from the same site provide bet-
ter fits for most of the temperature observations (Kühn and
Stöfen, 2005) but cannot match some observations which are
supposed to be based on the fractured characteristic of the
reservoir rock. The main limitations here are the 2-D simu-
lation and the lack of complex geometry options, which re-
stricts the shape of the reconstructed anomaly. This could be
addressed by generating and testing more complex fracture
patterns, but can only be executed by involving additional
data into the inversion.
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212 M. Somogyvári et al.: Reservoir-scale transdimensional fracture network inversion
Figure 5. (a) Fracture probability map of the Waiwera geothermal aquifer, (b) mean temperature field of the reconstructions (c) observed
borehole temperature profiles (black) and reconstructed borehole profiles (gray).
5 Conclusions
Temperature observations are usually considered as comple-
mentary information for thecharacterization of aquifer struc-
ture (Bravo, 2002; Klepikova et al., 2014; Leaf et al., 2012).
In this paper we have presented a methodology that treats
steady-state borehole temperature observations as the main
driver of the interpretation process. We have shown that a
handful of these profiles could provide enough information
to resolve the main structural elements of the investigated
aquifer using a DFN based transdimensional inversion.
We used the inversion algorithm introduced by Somo-
gyvári et al. (2017) which provides a data driven approach
by keeping the number of the modelled fractures flexible
throughout the inversion. The DFN model is highly concep-
tualized and suitable for solving the ill-posed inversion prob-
lem, given the limited amount of observations.
Using synthetic models, we have proven that the transdi-
mensional DFN inversion, which was originally developed
for tomographic observations, could be used in this signifi-
cantly different scenario.
We also present the first field application of the method-
ology, for the characterization of the well-studied Waiwera
geothermal reservoir. Transdimensional DFN inversion indi-
cated possible locations of strongly stratified aquifer parts,
and areas which are less likely to be fractured. These find-
ings are to be validated by future explorational campaigns
whose results will be also used to infer DFN models with
more complexity.
Compared to other interpretations from the site using three
dimensional porous medium models, the shape of the re-
constructed anomaly, and the fit of the temperature profiles
are less accurate. The simplified DFN model does not con-
sider flow and transport in the rock matrix, it does not take
into account the complex hydraulic and thermohaline condi-
tions and it is also limited to two dimensions. Still despite
these limitations it indicated discrete aquifer features which
the continuum modelling was not capable to capture. Thus,
for a complete aquifer characterization campaign, we recom-
mend combining the two approaches, by enhancing the three-
dimensional continuous models by the discrete features ob-
tained from DFN modelling.
We did not explore the stochastic behavior of the method-
ology, the resulted ensemble could be the subject of fur-
ther statistical analysis and uncertainty quantification (Som-
ogyvári et al., 2017; Somogyvári and Reich, 2019). This as-
Adv. Geosci., 49, 207–214, 2019 www.adv-geosci.net/49/207/2019/
M. Somogyvári et al.: Reservoir-scale transdimensional fracture network inversion 213
pect will be investigated in a future study. The presented in-
terpretations are to be used to select aquifer sections for more
detailed exploration, to characterize the hydraulic properties
of the fractures and identify the locations of the faults.
Data availability. The data of this study is not openly available, but
it can be provided upon request by the corresponding author.
Author contributions. The analysis of the data and the preparation
of the scripts were performed by MS. The data was provided and
prepared by MK. SR assisted with the methodology development.
The manuscript was prepared by MS.
Competing interests. The authors declare that they have no conflict
of interest.
Special issue statement. This article is part of the special issue “Eu-
ropean Geosciences Union General Assembly 2019, EGU Division
Energy, Resources & Environment (ERE)”. It is a result of the EGU
General Assembly 2019, Vienna, Austria, 7–12 April 2019.
Financial support. This research has been supported by the Geo.X,
the Research Network for Geosciences in Berlin and Potsdam
(grant no. SO_087_GeoX) and the Deutsche Forschungsgemein-
schaft (DFG) (grant no. CRC 1294 “Data Assimilation (Project
B04)).
This open-access publication was funded
by Technische Universität Berlin.
Review statement. This paper was edited by Thomas Nagel and re-
viewed by four anonymous referees.
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