coupling thm

Journal of Earth Science, Vol. 26, No. 1, p. 047–052, February 2015 ISSN 1674-487X Printed in China DOI: 10.1007/s12583-015-0518-y

Karrech, A., Beltaief, O., Vincec, R., et al., 2015. Coupling of Thermal-Hydraulic-Mechanical Processes for Geothermal Reservoir Modelling. Journal of Earth Science, 26(1): 47–52. doi:10.1007/s12583-015-0518-y

Coupling of Thermal-Hydraulic-Mechanical Processes for Geothermal Reservoir Modelling

Ali Karrech*l, Oussama Beltaief2, Ruyan Vincec1, Thomas Poulet3, Klaus Regenauer-Lieb3, 4, 5

1. School of Civil and Resource Engineering, The University of Western Australia, Crawley WA 6009, Australia 2. Tunisia Polytechnic School, Rue Alkhawarizmi, La Marsa, Tunisia

3. Earth Science and Resource Engineering, CSIRO, Kensington WA 6151, Australia 4. School of Petroleum Engineering, University of New South Wales, Sydney NSW 2052, Australia

5. School of Earth and Environment, The University of Western Australia, Perth WA 6000, Australia

ABSTRACT: This paper uses a fully coupled framework of thermal-hydraulic-mechanical processes to investigate how the injection and extraction of fluid within a geothermal reservoir impacts on the dis-tributions of temperature, pore pressure, and deformation within the rock formations. Based on this formulation, a numerical model is developed in light of the thermodynamics of porous materials. The proposed procedure relies on the derivation of dissipative flow rules by postulating proper storage and dissipation functions. This approach opens new horizons for several resource engineering applications. Since it allows for full coupling, this formulation can play a key role in predicting risks when used for reservoir simulation. The results indicate that the injection-extraction process and temperature change have a definite impact on altering the in-situ properties of the reservoir. KEY WORDS: poro-mechanics, resource engineering, fluid injection and extraction, temperature change, pore pressure, stress, deformation, uplift, subsidence.

0 INTRODUCTION As the population of the world is set to double by the end

of this century, sustaining our current lifestyle requires the production of clean energy at affordable prices. While fossil fuels are still expected to be important energy resources for the next decades, the share of renewable energies is expected to increase significantly. The need to increase the supply of re-newable energy sources has led to an increasing amount of research on harvesting geothermal energy. The geothermal energy process involves the utilisation of the Earth’s natural geothermal gradient to extract heat and transform it into a di-rectly useful energy such as electricity. To maintain recover-ability and limit the footprint of geothermal energy, the process involves injecting cold fluids into an underground reservoir of hot permeable rock, allowing the fluid to flow through the rock formation and then extracting the heated fluid. The injection and extraction of fluids and subsequent temperature change throughout the reservoir impacts the physical properties of the permeable formations. The basic mechanism underlying the geo-mechanical response of a reservoir is related to the magni-tude of the pore pressure and temperature variations. Deep underground reservoirs that have low values of porosity, per-meability and compressibility result in greater pressure *Corresponding author: [email protected] © China University of Geosciences and Springer-Verlag Berlin Heidelberg 2015 Manuscript received August 13, 2014. Manuscript accepted October 21, 2014.

changes, generating large surface deformation responses along the top of a reservoir. As fluid is injected, the pore pressure increases above its initial state, decreasing the effective stress in the ground. The expansion that occurs due to fluid injection is primarily caused by the vertical effective stress release due to the pore pressure increase. There is also the need to consider the thermal effect on the deformation in a reservoir. If the tem-perature of the injected fluid is lower than the in situ tempera-ture, the poro-elastic and thermal effects on stresses and defor-mations tend to counteract and cancel each other out.

From the sites investigated by Teatini et al. (2011), two major conclusions where determined: (i) the observed land uplift may vary from a few millimetres to tens of centimetres over a time interval of some months to several years according to the specific location and (ii) the largest uplift depends on a number of factors, including the fluid pore pressure increase, the depth, thickness and areal extent of the pressurized and heated geological formation, and the hydro-geo-thermo- mechanical properties of the porous medium involved in the process. Teatini et al. show that if the fluid injection in a reser-voir occurs at a deep enough level, the uplift and subsidence is expected to be uniform between the injection and extraction wells. If the injection occurs at a shallow level, the surface uplift and subsidence may be irregular due to the influence of the reservoir boundaries. Similarly, Chen (2012) investigated the induced stresses and deformations in hydrocarbon and geo-thermal reservoirs. The purpose of his study was to investigate deformation-based monitoring techniques. This involves using surface deformation, such as tiltmeter monitoring, to control hydraulic fractures, subsurface fluid migration and heat trans-

Ali Karrech, Oussama Beltaief, Ruyan Vincec, Thomas Poulet and Klaus Regenauer-Lieb

48

port as fluid is injected or extracted from a reservoir. Chen used a numerical model of an idealised ellipsoidal poro-elastic res-ervoir embedded in a surrounding rock body at a specified depth to study the effects of fluid injection. The numerical model predicted the expansion and/or con-traction of a reser-voir due to pressure change. The pore pressure, fluid mass con-tent and temperature within the reservoir were allowed to change over time, while the same values in the surrounding rock were fixed. He also included analytical solutions for the induced stresses and deformations obtained using the theory of inclusion and inhomogeneity. Chen concluded that both the shape and mechanical properties of a reservoir can have a sig-nificant effect on any resulting deformation that occurs due to fluid injection given that the reservoir and surrounding media are homogenous. Due to the elliptical reservoir geometry, the surface tilt vectors that occurred due to fluid injection were in a radial pattern, pointing outwards from the centre of the reser-voir and corresponding fluid injection point.

The usefulness of coupled thermal-hydraulic-mechanical processes for poro-materials goes beyond the modelling of geothermal reservoirs as it is relevant for several resource en-gineering applications such as mineral exploration and mining, nuclear waste storage, and carbon dioxide storage. Most of these applications involve large deformation of porous materi-als and/or large temperature variations. Considerable progress has been made in the last decades to develop such a framework based on the pioneering work of Biot (1972) and the compre-hensive formulations of Coussy (2003). The main purpose of this paper is to apply a simplified version of the developments of Karrech et al. (2012) to geothermal reservoir modelling and predict their responses under fluid injection and extraction. 1 GOVERNING EQUATIONS

The idea of describing processes in a coupled manner as-sumes that energy is smooth in the space of state variables (Karrech et al., 2012). The smoothness of energy potentials is a classifying but powerful assumption. Along with Schwartz’ theorem, this assumption replaces many particular hypotheses such as the Onsager and Maxwell principles as will be shown at a later stage. State variables are selected to cover the different processes that govern the behaviour of a mechanical system. These variables should be, a priori, independent and enough to control the system and determine its evolution with respect to time. In the current framework, we express Helmholtz free energy in terms of the common measure of deformation, ε, porosity, ϕ, and temperature, T, as follows

),,(ss (1)

Using the second law of thermodynamics and the inde-pendence of processes, it can be shown that

sf

ss and,, pT

σ (2)

where σ is the Cauchy stress, η is the entropy, pf is the pore pressure, and ρ is the density of the solid-fluid mixture. The subscripts “f” and “s” denote the fluid and solid phases. The above relations can be expressed as follows: ρdψs=σ: dε–ρηdT+pfdϕ. Instead of using the free energy ψs, it is cus-

tomary in poromechanics to consider its dual Ψ which can be obtained by Legendre transformation with regard to ϕ as fol-lows: ρΨ=ρψs–(pf–p0f)ϕ. This transformation shows that

)(

Ψand,

Ψ,

Ψ

f0f pp

σ (3)

The free energy function Ψ is assumed regular with re-spect to its variables. In particular, its mixed derivatives exist and they are continuous. Therefore, differentiation of the dif-ferent terms in equation (3) and application the Schwartz’ theorem results in the following incremental relationships

1d1d3d21d)3

2(d fpbTKGG σ

fd3d

d3d pC (4)

N

pTb fd

d3dd

where ϑ=trε, K is the bulk modulus of the porous material and G is its shear modulus, α is the thermal expansion coefficient of the porous material, b is the Biot coefficient, Cυ is the heat ca-pacity of the porous material at constant volume, αϕ is the latent heat coefficient associated with the change of porosity, and N is the Biot modulus. Note that the isochoric and isobaric heat capacities are duals. The relationship between them involves temperature as follows: Cp=Cυ+9Tα2K. In addition, the follow-ing thermo-elastic relationships can be deduced

spsp

ss

bCC

bb

),(,)1(

,1

11,

)1(

11

(5)

where ϖ=ϕ/J is the porosity in the current configuration, J is the determinant of the gradient of deformation, which is as-sumed to be close to unity. As explained in details by Karrech et al. (2011) the assumption of local equilibrium as well as the second law of thermodynamics deliver the following expres-sion for dissipation

0)( ff f

JρD p gV (6)

where Vf=ϖ(υf–υs) is the infiltration vector, υf is the fluid ve-locity, υs is the solid velocity and J is a heat flux, g is a body force, and ρf is the density of the pore fluid. The right hand terms represent the hydraulic and thermal dissipations, respec-tively. In this study, we maximise the dissipation associated with the hydrothermal behaviour and use the theorem of Schwartz (if h: Rn has continuous second partial derivatives at a point (p1,…,pn), then ,i j∈1…n ,

),,(),( 1

2

,1

2

nij

nji

pph

pph

) to show that

Τ

ΤL)ρp(L

Τ

ΤL)ρ(LJ

fqff

qqpq

gV

g

fff

f

and

,ff

(7)


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In the above equation Lij are symmetry terms which are similar to the Onsager coefficients (Onsager, 1931a, b). For simplification we consider Lij=0 if i=j, in this study. This as-sumption allows us to use the classical Darcy and Fourier laws without unnecessary complexity. When included in the local equations of conservation of mass, momenta, and energy the above fluxes deliver the following partial differential equations that govern the behaviour of the fully coupled system. We de-scribe first the equation of motion

uduun Γat,Γat,0 dΤσgσ (8)

where Td is a specified stress vector and ud is a specified dis-place geo-infiltration problems involve small fluid velocities,

the terms

fff

f 3f VV p can be neglected. In the above

equations Γ denotes the surface boundaries corresponding to the respective Dirichlet and Neumann conditions. Therefore, the partial differential equation of hydraulic flow in porous media can be obtained

pdqn

pff

ffm

ppqp

ttb

t

p

Γat,Γat, where

)(d

d3

d

d

d

d1

fff

ff

f

f

g

(9)

where 1/M=1/N+ϖ/Kf, N is the Biot modulus, Kf is the fluid bulk modulus, αm=αϕ+ϖαf, αf is the fluid thermal expansion,

f ff

ffL , Kff is the intrinsic permeability, ηf is the fluid

dynamic viscosity, qf is a specified pore pressure gradient, and pd is a specified pore pressure. Similarly, the equation of heat transfer reads

Tdqqn

p

TTq

Ct

CC

q Γat,Γat,whered

d)( fff

V (10)

where 0/ qqL is the thermal conductivity, qq is a speci-

fied temperature gradient, and Td is a specified temperature.

2 NUMERICAL MODEL From the equations of conservation of mass, energy and

momenta described previously, we formulate the quasi-static coupled thermo-hydro-mechanical problem. As the governing equations are difficult to solve analytically, we develop a nu-merical model to determine the impact of fluid flow and tem-perature change on the initial conditions of a geothermal reser-voir. The numerical model calculates the changes in tempera-ture, pore pressure, stress and deformations that occur over time. The model idealises a geothermal reservoir containing porous sedimentary rock that is fully saturated. Fluid is injected into one side of the rock and extracted at the same rate on the opposite side of the model. The general purpose commercial Finite Element tool Abaqus (Anonynous, 2011) is used for simulation. The coupling of temperature, fluid flow, and tem-perature is described by dimensional element C3D8PT, which contains 8 degrees of freedom, including pore pressure and temperature, allowing for variations in these areas to be simu-lated, and the resulting stress and deformation to be calculated.

The model has dimensions of 20 m×l 000 m×l 000 m. The width and height of the model were selected large enough as they are the primary directions, which allow for fluid flow and temperature variation to occur throughout the model without the edges of the model affecting the results. The thickness of the model is small in comparison to the other dimension to allow for shorter simulation times. Table l summarises the pa-rameters of simulation: The density of water considered in this study varies with respect to temperature and pressure as:

,1)(1 f

0f

0f

0f

pp

where T0 and ρ0 are refer-

ence temperature and pressure, respectively. In addition the permeability varies with respect to porosity in accordance with

Carman-Kozeny expression: .1

12

0

00

ffff The per-

meability is related to the hydraulic conductivity via the rela-

tionship .f

ffg In this study the permeability is as-

sumed to be orthotropic and the ratio of vertical permeability over horizontal permeability is of 1/3.

Table l Material properties used for simulations

Property Solid Fluid

Density (kg/m3) ρs=2 600 ρf(T,pf)|(0,0)=1 000

Bulk modulus (GPa) Ks=6.89 Kf=2.2

Specific heat (J/(kg·K)) Cps=2 500 Cf=4 193

Thermal conductivity (W/(m·K))

κs=2.5 κf=0.6

Thermal expansion (K-1)

αs=1.5×10-6 αf=6×10-4

Poisson coefficient ν=0.25 _

x

y

z

Injection Extraction

u y· =0

qyqT ��,

u z· =0

u y· =0

Reservoir surface, =σ T·z d

ff, qp

y��

qyqT ��,

ff, qp

y��

Figure l. Fluid flow path and boundary conditions. The 3D block used for simulation is subjected to sliding

boundary conditions restricting its normal motion at all sides except the top. This allows the reservoir to expand and contract vertically as its motion is restricted due to adjacent rock masses delimiting the rest of its periphery. Out of the injection zones, Dirichlet boundary conditions are applied in terms of tempera-ture and pore pressure in accordance with the initial gradients. In other words, temperature and pore pressure at these bounda-


50

ries remain unchanged. The magnitudes of the initial gradients of temperatures and pore pressure are respectively 0.03 ºC·m-1 and ρfg. The centre of the model is initially positioned at 2.17 km, below the Earth’s surface. The initial vertical and horizontal stresses are respectively expressed by

))1)((6671(6671 fss zgσ and σh=0.55συ. The

loading process comprises the injection and extraction of fluids at a rate of 50 L/s across 40 m span at the centre of the left and right hand sides, respectively. The injection area is character-ised by an imposed temperature of 50 ºC.

Figure 2a shows the temperature distribution along the centreline from the injection area to the extraction area, for different time instants. Figure 2b shows the evolution of tem-perature in the vicinity of the extraction point. The initial tem-perature of the injection-extraction path is at 90 ºC. It starts decreasing progressively due to diffusion and advection effects. The profile of temperature represents a front like pattern which progresses towards the extraction point. The front disperses with respect to time due to diffusion. It is worthwhile noticing that temperature decreases slightly at the extraction point due to the heat flux induced by the extraction process. This phenome-non is affected by the gravity, which enhances the cooling process at the extraction point.

Figure 3a shows the spatial distribution of pore pressure along the injection-extraction path at different instants of time. The results show that the problem is not symmetric at least due to the effect of coupling and exchanged feedbacks. In the ab-sence of thermal feedback, pore pressure stabilises within one day. In the vicinity of the injection area the size of the cold zone increases progressively and affects the distribution of pore pressure. This behaviour is explained by the following coupled incremental equation: dpf/M=dmf/ρf–bdε+3αmdT It can be seen that at constant water content and deformation, the pore pres-sure increases with temperature. Figure 3b shows the evolution of pore pressure with respect to time in the vicinity of the injec-tion and extraction points. There are three distinguishable phe-nomena that can be reported. The first is the opposite overall increase of pore pressure at the injection area and decrease at the extraction area by around 4 MPa. This behaviour is mainly attributed to the variation of the fluid content dmf/ρf. The sec-ond component is the small decrease of pore pressure observed at the injection point at around t=5 000 s. This variation is mainly explained by variation of the local volume of the solid skeleton bdε. The third phenomenon acts progressively on the injection and extraction area. It reflects the variation of tem-perature with respect to time 3αmdT.

Figure 2. (a) Spatial distribution of temperature along the centreline between the injection and extraction areas, (b) temporal evolution of temperature at the vicinity of the extraction area.

Figures 4a and 4b show respectively the spatial distribu-

tion of the porosity along the injection-extraction path in dif-ferent instants and the evolution of porosity with respect to time at the vicinities of the injection and extraction points. Initially, the porosity is selected equal to 0.3, but it reduces to 0.292 3 due to geostatic loading. As for pore pressure, porosity varies in

accordance with .d3dd

d sf

t bN

p It increases when

pore pressure increases, the skeleton expands and/or tempera-ture decreases. As shown in Figs. 3 and 4, the behaviours of pore pressure and porosity with respect to time and space are similar since they are thermodynamic duals.

Figure 5a shows the distribution of uplift at the surface of the model with respect to time. After 1 day of injection, there is a positive vertical displacement (uplift) occurring on the injec-tion side of the reservoir and a negative vertical displacement occurring on the extraction side of the reservoir (subsidence). This initial displacement is due to the increase and decrease in pore pressure that occurs after 1 day due to the injection-

extraction process. We can see that the deformation reaches a maximum after at this particular stage and initial deformation is equal and opposite on either side of the reservoir surface, which is the same as the initial change in pore pressure. After 1 year, the effect of temperature change in the reservoir begins to im-pact the vertical displacement. We see that the uplift near the extraction side decreases after 1 year. This is due to the tem-perature decreasing near the injection point, causing the reser-voir to contract. After 10 years, there is a further decrease in the uplift on the injection side of the reservoir, as there is only a significant change in temperature on the left side of the model. There is subsequently no change in subsidence between the 1 and 10 years mark because the temperature does not change significantly within this period. Figure 5b shows the effect of the permeability on the expansion of the rock mass. Changing the permeability of the reservoir model affects the ease to which a fluid can move through the pore spaces. The values of permeability used for this analysis are in the typical range of sedimentary rocks, being 0.25, 0.5 (as seen in the reference set of


51

Figure 3. (a) Spatial distribution of pore pressure along the centreline between the injection and extraction areas, (b) tempo-ral evolution of pore pressure at the vicinity of the injection and extraction areas.

Figure 4. (a) Spatial distribution of porosity along the centreline between the injection and extraction areas, (b) temporal evolution of porosity at the vicinity of the injection and extraction areas.

Figure 5. (a) Uplift along the surface of the model at different time instants, (b) impact of permeability on uplift.

results), 0.75, 1 and 1.25 Darcy. It can be seen that the lower the permeability the higher the induced deformation of the reservoir. 3 CONCLUSION

This paper introduces the effects of coupled thermal- hydraulic-mechanical processes on the response of an idealised geothermal reservoir. The model shows that the temperature change throughout the reservoir is extremely important as its affects the in-situ properties of the reservoir. Through the sensi-tivity analysis we determine that the permeability of the reservoir does not have a significant impact on temperature change, but it affects significantly the pore pressure distribution. The results show that the pore pressure changes to a new equilibrium level immediately after the injection and extraction of fluid begins. As time progresses and the reservoir changes in temperature so does the pore pressure. Decreasing the permeability or increasing the

injection rate in the model results in the initial increase of pore pressure. These changes in temperatures and pressures affect the stress state directly through coupling terms such as the expansion and Biot’s coefficients, respectively.

The deformation in the reservoir is affected by the pore pressure and temperature. On the injection side, the pore pres-sure initially increases and induces a change in stress, which corresponds to uplift. As the temperature decreases over time, the subsequent increase in pore pressure and stress causes ex-pansion in the reservoir. The temperature decrease also results in the reservoir contracting. On the extraction side of the reser-voir, subsidence occurs due to a negative change in pore pres-sure and stress. As temperature decreases, these values increase to a larger negative amount, and its contraction is added to the contraction caused by a temperature decrease. Changing the permeability and injection rate results in the deformation vary-


52

ing drastically over time. This investigation focuses on specific parameters and

outputs to identify how pore pressure changes results in defor-mations occurring. In order to gain a complete understanding of the behaviour of geothermal reservoirs, it would be highly recommended to further research the effect of changing other parameters. These parameters would include those that affect the rate at which the temperature decreases throughout the res-ervoir over time, other parameters affecting the pore pressure and if it is possible to change the in-situ stress without chang-ing the pore pressure. A more comprehensive analysis of the expansion and contraction of the reservoir due to temperature change would also be beneficial. Combining all of these dif-ferent investigations could potentially allow for the surface deformation in reservoirs to be reduced while still enabling the reservoir to produce a fluid that can economically produce energy.

REFERENCES CITED Anonynous, 2011. Abaqus User Manual Version 6.11. Hibbitt,

Karlson and Sorensen Inc., USA

Biot, M., 1972. Theory of Finite Deformations of Porous Solids. Indiana University Mathematics Journal, 21(7): 597–620

Chen, Z. R., 2012. Poroelastic Model for Induced Stresses and Deformations in Hydrocarbon and Geothermal Reservoirs. Journal of Petroleum, Science and Engineering, 80: 41–52

Coussy, O., 2003. Poromechanics. Wiley, Chichester. 312 Karrech, A., Poulet, T., Regenauer-Lieb, K., 2012. Porome-

chanics of Saturated Media Based on the Logarithmic Fi-nite Strain. Mechanics of Materials, 51: 118–136

Karrech, A., Regenauer-Lieb, K., Poulet, T., 2011. Frame In-different Elastoplasticity of Frictional Materials at Finite Strain. International Journal of Solids and Structures, 48(3–4): 397–407

Onsager, L., 1931a. Reciprocal Relations in Irreversible Proc-esses i. Physical Review, 37: 405–426

Onsager, L., 1931b. Reciprocal Relations in Irreversible Proc-esses ii. Physical Review, 38: 2265–2279

Teatini, P., Gambolati, G., Ferronato, M., et al., 2011. Land Uplift due to Subsurface Fluid Injection. Journal of Geo-dynamics, 51: 1–16

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