COMSOL in a New Tensorial Formulation of Non-Isothermal Poroelasticity

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1 Excert rom the Proceedings o the COMSOL Conerence 009 oston COMSOL in a New Tensorial Formulation o Non-Isothermal Poroelasticity Mario-César Suárez A. *, 1 and Fernando Samaniego. 1 Faculty o Sciences Michoacán University, Faculty o Engineering National University o Mexico * Ediicio Cd. Universitaria, Morelia, Mich., Mexico, mcsa50@gmail.com Abstract: The resence o a moving luid in a orous rock modiies its mechanical resonse. Poroelasticity exlains how the luid inside the ores bears a ortion o the total load suorted by the rock. The remaining art o the load is suorted by the elastic skeleton, which contains a laminar luid couled to the ramework by equilibrium and continuity conditions. This work introduces an original tensorial ormulation o iot s theory and the exerimental thermooroelastic arameters that suort the theory. y deining a 4-dimensional total stress tensor and three basic oroelastic coeicients, we deduce a system o dierential equations couling the bulk rock and the luid. The inclusion o the ourth dimension allows extending the theory o solid linear elasticity to thermooroelastic rocks, taking into account the eect o both the luid hase and the temerature. Introducing three volumetric thermal dilation coeicients, one or the luid and two or the skeleton, a comlete set o arameters or geothermal oroelastic rocks are obtained. Keywords: Poroelasticity, thermooroelasticity, iot s theory, inite elements. 1. Introduction Several actors aect the geomechanical behavior o orous crustal rocks containing luids: orosity, ressure, and temerature, characteristics o the luids, issures, and aults. Rocks in underground systems (aquiers, geothermal and hydrocarbon reservoirs) are orous, comressible, and elastic. The resence o a moving luid in the orous rock modiies its mechanical resonse. Its elasticity is evidenced by the comression that results rom the decline o the luid ressure, which can shorten the ore volume. This reduction o the ore volume can be the rincial source o luid released rom storage. A rock mechanics model is a grou o equations caable o redicting the orous medium deormation under dierent internal and external orces. In this aer, we resent an original our-dimensional tensorial ormulation o linear thermo-oroelasticity theory. This ormulation makes more comrehensible the linear iot s theory, rendering the resulting equations more convenient to be solved using the Finite Element Method. To illustrate ractical asects o our model some classic alications are outlined and solved using the Earth Science Module o COMSOL-Multihysics. 1.1 Exerimental background In classic elastic solids only the two Lamé moduli, (l, G) or Young s elastic coeicient and Poisson s ratio (E, n), are suicient to describe the relations between strains and stresses. In oroelasticity, we need ive oroelastic moduli or the same relationshis (undschuh and Suárez, 009), but only three o these arameters are indeendent. The iot s ield variables or an isotroic orous rock are the stress s acting in the rock, the bulk volumetric strain ε, the ore ressure and the variation o luid mass content ζ. The linear relations among these variables are the exerimental oundations o iot s oroelastic theory (iot & Willis, 1957; Wang, 000): ε = +, ζ = + K H H R 1 ε C H = 1 1 ζ H R (1) Where K, H, and R are oroelastic coeicients that are exerimentally measured as ollows (Wang, 000): Δ Δε 1 ε =, C =, K = Δ C 1 Δε Δζ 1 Δζ = =, = H R Δ Δ Δ ()

2 The ollowing igure (1) illustrates all the arts orming a oroelastic medium. iot (1941) and (iot & Willis, 1957) introduced three additional arameters, b, M and C, that are undamental or the tensorial ormulation herein resented. 1/M is called the constrained seciic storage, which is equal to the change o ζ when changes measured at constant strain. oth arameters M and C are exressed in terms o the three undamental ones deined in equation (): 1 Δζ 1 K = = M Δ R H ε RH K M = ; C = M H K R H (4) Figure 1. Skeleton o sandstone showing its ores and solid grains (3 3 3 mm 3 ). (Piri, 003). Here is the bulk volume, consisting o the rock skeleton ormed by the union o the volume o the ores F and the volume o the solid matrix S (Fig. 1). The control volume is D. The drained coeicients K and C are the bulk modulus and the bulk comressibility o the rock, resectively; 1/H is a oroelastic exansion coeicient, which describes how much D changes when changes while keeing the alied stress s constant; 1/H also measures the changes o ζ when s changes and remains constant. Finally 1/R is an unconstrained seciic storage coeicient, which reresents the changes o ζ when changes. Inverting the matrix equation (1) and relacing the value o s in ζ we obtain: K = Kε H (3) K 1 K ζ = ε + H R H The sign conventions are stress s > 0 in tension and s < 0 in comression; the volumetric strain ε > 0 in exansion and ε < 0 in contraction; the luid content ζ > 0 i luid is added to the control volume D and ζ < 0 i luid is extracted rom D ; the ore ressure > 0 i it is larger than the atmosheric ressure. Let C S = 1/K S be the comressibility o the solid matrix. The iot-willis coeicient b is deined as the change o conining ressure k with resect to the luid ressure change when the total volumetric strain remains constant: K C K b= = 1 = = KS M H k ε (5) The coeicient C reresents the couling o deormations between the solid grains and the luid. The coeicient M is the inverse o the constrained seciic storage, measured at constant strain (Wang, 000); this arameter characterizes the elastic roerties o the luid because it measures how the luid ressure changes when ζ changes. These three arameters b, M and C are at the core o the oroelastic artial dierential equations we introduce herein (undschuh and Suárez, 009).. Isothermal Poroelasticity Model Let u s and u be the dislacements o the solid and luid articles; let u = u u s be the dislacement o the luid hase relative to the solid matrix resectively. Let ε s, ε, j s, j, s and be the volumetric dilatations, orosities and volumes o each hase; ε is the volumetric deormation o the luid hase relative to the solid hase. The mathematical exressions o these variables are:

3 Δ S Δ = εs = us; = ε = u S ε = ε ε ; u = u u ε = S S u u x y uz ( us u ) = u = + + x y z (6) iot and Willis (1957) introduced the strain variable ζ (u, t), deined in equation (3), to describe the volumetric deormation o the luid relative to the deormation o the solid with homogeneous orosity: ζ ϕ ϕε ϕ ε ϕ ε ( ) (u,t) = us u = s = (7) The unction ζ reresents the variation o luid content in the ore during a oroelastic deormation. The total alied stresses in the orous rock are similar to the equations o classic elasticity. However, we need to coule the eect o the luid in the ores. The linear comonents o the global stresses, deduced exerimentally by iot, (iot, 1941; iot and Willis, 1957; Wang, 000) are: Where: = λ ε δ + Gε Cζ δ (8) ij U ij ij ij 1 u u j i 1, i i = j εij = +, δij = xj x i 0, i i j λ = λ + Cb; or i, j= x, y, z U The luid ressure is deduced rom equation (3): RH ζ ε K H K = H K R = = Mζ Cε ; i, j = x, y, z (9) We deine a two-order tensor s T = (s ij ) in our dimensions, which includes the bulk stress tensor s acting in the orous rock and the luid stress s F acting in the luid inside the ores, ositive in comression: = + T F ( C ) = λ ε ζ δ + Gε ij U ij ij (10) This tensorial equation becomes identical to the Hookean solids equation, when the rock has zero orosity and b = 0. From equations (8) and (9), we deduce that: = τ b δ (11) ij ij ij τ = λe δ + Gε (1) ij ij ij Tensor t ij is called the Terzaghi (1943) eective stress that acts only in the solid matrix; b is the ore-luid ressure. Since there are no shear tensions in the luid, the ore luid ressure aects only the normal tensions s i (i = x, y, z). The unctions s ij are the alied stresses acting in the orous rock saturated with luid. The solid matrix (t ij ) suorts one ortion o the total alied tensions in the rock and the luid in the ores (b ) suorts the other art. This is a maximum or soils, when b º 1 and is minimum or rocks with very low orosity where b º 0. For this reason, b is called the eective stress coeicient. Inverting the matrices o equations (8) and (9), we arrive to the ollowing tensorial orm o the oroelastic strains: ii 3ν ij εii = M + ; εij = (13a) G E 3H G M C + K ζ = + = H R MK C M U U xx + yy + zz M = = Kε bmζ 3 K = λ + G; KU = K + b M 3 (13b) (14) The coeicient K U is the undrained bulk modulus, which is related to the revious deined coeicients. Note that both tensorial equations (10) and (13) only need our basic oroelastic constants. The resence o luid in the ores adds an extra tension due to the hydrostatic ressure, which is identiied with the ore ressure, because it is suosed that all the ores are interconnected. This linear theory is aroriate or isothermal, homogeneous, and isotroic orous rocks.

4 3. ThermoPoroelasticity Model The equations o non-isothermal oroelastic rocesses are deduced using the Gibbs thermooroelastic otential or available enthaly er unit volume and the energy dissiation unction o the skeleton (Coussy, 1991). Analytic exressions are constructed in terms o the stresses, the orosity, the ore ressure, and the density o entroy er unit volume o orous rock. As we did or the isothermal oroelasticity, we can write in a single ourdimensional tensor the thermooroelastic equations relating stresses and strains. We have or the ore ressure: ( ) = M ζ ζ Cε 0 0 ( ϕ γ )( Mϕ γ T T 0 ) (15) The volumetric thermal dilatation coeicient g [1/K] measures the dilatation o the skeleton and g j [1/K] measures the dilatation o the ores: 1 1 γ = T K 1 ϕ 1 ϕ 1 γϕ = = ϕ T ϕ T K k (16) (17) The luid bulk modulus K and the thermal exansivity o the luid g [1/K] are deined as ollows: 1 1 ρ = C = K ρ (18) T 1 1 ρ γ = = T ρ T (19) The term k is the conining ressure. Exanding the corresonding unctions o the Gibbs otential and equating to zero the energy dissiation we obtain the 4D thermooroelastic equations, which include the thermal tensions in the total stress tensor (undschuh and Suárez, 009): ( b( 0 ) ) Gε K γ ( T T ) 0 ij ij = ij λε δ + + ij 0 (0) In this case, an initial reerence temerature T 0 and an initial ore ressure 0 are necessary because both thermodynamic variables T and are going to change in non-isothermal rocesses occurring in orous rock. The luid stress is deduced in a similar way: ( 0 ) ( ϕ γ )( 0 ) = = M ζ ζ Cε Mϕ γ T T 4. Dynamic Poroelastic Equations (1) The ormulation we introduced herein is very convenient to be solved using the Finite Element Method. The undamental oroelastic dierential equation is the tensorial orm o Newton s second law in continuum orous rock dynamics: u T div T + F = ρ ; div T = T t where : = ε ; ε = u T T T () The terms s T and ε T are the equivalent vectorial orm o tensorial equations (0) and C is the matrix o oroelastic constants. While F is the body orce acting on the rock and the tensor dierential oerator is given by: x 0 0 y z 0 x T = 0 y 0 x 0 z y 0 0 z 0 x y z u x u y= εt = ( εx εy εz εxy εxz εyz er) u z (3) Where u = (u x, u y, u z ) is the dislacement vector o equation (6). Using the oerator in equation (), the dynamic oroelastic equation becomes: T u ( ) u + F = ρ (4) t

5 4.1 Solution o Thermooroelastic Equations: The Finite Element Method Equation (4) includes iot s oroelastic theory. It can be ormulated and numerically solved using the Finite Element Method (FEM). Let Ω be the bulk volume o the orous rock, and let Ω be its boundary, u is the set o admissible dislacements in Eq. (); b is the volumetric orce and s is the orce acting on the surace Ω. Ater doing some algebra we arrive to a FEM undamental equation or every element e in the discretization: e e e e d e d + = F ; e = 1,M (5) t d e is a vector containing the dislacements o the nodes in each e. Equation (5) aroximates the dislacement u o the oroelastic rock. F e is the vector o total nodal orces. K e and M e are the stiness and equivalent mass matrices or the inite element e. The mathematical deinitions o both matrices are: layers o the sequence are each 0 m thick. The irst and third layers are aquiers; the middle layer is relatively imermeable to low. edrock ste Figure. Simliied geometry o the aquier and the imermeable bedrock in the basin. Initial state. e T = d; e= 1, M e e T == = ; = ρ d e (6) Where is the matrix o shae unctions that interolate the dislacements (Liu and Quek, 003). Matrix is called the strain oroelastic matrix. 5. Use o COMSOL Multihysics This section contains two brie illustrations o the deormation o an aquier (Leake & Hsieh, 1997) and the orm that a temerature change can aect its oroelastic deormation. In the irst examle, we assume cold water at 0 C (1000 kg/m 3 ). Ater, we consider a temerature o 50 C (50 bar, kg/m 3 ). Results are shown in igures (4) to (9). To simliy the discussion we use the same model reviously solved by COMSOL-Multihysics and described in the Earth Science Module (COMSOL A, 006): Three sedimentary layers overlay imermeable bedrock in a basin where aulting creates a bedrock ste (S) near the mountain ront (Fig. ). The sediment stack totals 40 m at the deeest oint o the basin (x = 0 m) but thins to 10 m above the ste (x > 4000 m). The to two Figure 3. The mesh o the basin with 967 elements. As given by the roblem statement, the materials here are homogeneous and isotroic within a layer. The low ield is initially at steady state, but uming rom the lower aquier reduces hydraulic head by 6 m er year at the basin center (under isothermal conditions). The head dro moves luid away rom the ste. The luid suly in the uer reservoir is limitless. The eriod o interest is 10 years. The corresonding FE mesh has 967 elements excluding the bedrock ste (Figure 3). The rock is Hookean, oroelastic and isothermal. In the irst examle we use the same data o the ES Module user s guide, excet or the iot- Willis coeicient we assume that b = 0.3.

6 Figure 4. Poroelastic deormation o the basin or the S roblem with cold water (0 C). Streamlines reresent the luid to orous rock couling. Figure 5. Poroelastic deormation o the basin or the S roblem with hot water (50 C). Streamlines reresent the luid to orous rock couling. Figure 6. ertical strain at the basin with a S. Case o cold water (0 C). Figure 7. ertical strain at the basin with a S. Case o hot water (50 C). Figure 8. Horizontal strain at the basin with a S. Case o cold water (0 C). Figure 9. Horizontal strain at the basin with a S. Case o hot water (50 C).

7 6. Discussion o Results The two examles resented herein were solved using COMSOL Multihysics or a wellknown roblem o linked luid low and solid deormation near a bedrock ste in a sedimentary basin described in the Earth Science module. The roblem concerns the imact o uming or a basin illed with sediments draing an imervious ault block. In the irst examle, we considered the water in the aquier to be cold, at 0 C. In the second examle, the water is hot, at 50 C. The basin is comosed o three layers having a total deth o 500 m and is 5000 m long in both cases. Figures (4) and (5) show simulation results or years 1,, 5, and 10, resectively. The dierence here with the results o COMSOL Multihysics is that we have erormed the simulation o a thermooroelastic - couled deormation when the water in the aquier corresonds to geothermal conditions (luid density o kg/m 3, temerature o 50 C, and ressure o 50 bar). Figures (6) and (7) comare the vertical strains and igures (8) and (9) comare the horizontal strains, in both cases resectively. Figures (8) and (9) also illustrate the evolution o lateral deormations that comensate or the changing surace elevation above the bedrock ste. Note that vertical scales are dierent in both examles or clarity. 7. Conclusions All crustal rocks orming geothermal reservoirs are oroelastic and the luid resence inside the ores aects their geomechanical roerties. The elasticity o aquiers and geothermal reservoirs is evidenced by the comression resulting rom the decline o the luid ressure, which can shorten the ore volume. This reduction o the ore volume can be the rincial source o luid released rom storage The immediate hysical exerience shows that the suly or extraction o heat roduces deormations in the rocks. Any variation o temerature induces a thermo-oroelastic behavior that inluences the elastic resonse o orous rocks. We introduced herein a general tensorial thermooroelastic model that takes into account both the luid and the temerature eects in linear orous rock deormations, and resenting two ractical examles solved with COMSOL. The second examle illustrates the inluence o temerature changes on the oroelastic strains. For cold water, the estimated value o ε z is about -1.5ä10-4, while or hot water ε z is -7.5ä10-4. Thereore, the oroelastic deormations are much higher in geothermal reservoirs than in isothermal aquiers. In the irst case the bulk modulus o water K w = 0.45 GPa, corresonding to T = 50 C. For cold aquiers K w =.5 GPa. Water bulk modulus aect other oroelastic coeicients, including the exansivity o rocks, which is relatively small, but its eects can roduce severe structural damages in orous rocks subjected to strong temerature gradients, as haens during the injection o cold luids. 8. Reerences 1. iot, M.A., General Theory o Three- Dimensional Consolidation, Journal o Alied Physics, 1,, , (1941).. iot, M.A. and Willis, D.G., The Elastic Coeicients o the Theory o Consolidation, J. o Alied Mechanics, 4, , (1957). 3. undschuh, J. and Suárez M. C., Introduction to the Modeling o Groundwater and Geothermal Systems: Fundamentals o Mass, Energy and Solute Transort in Poroelastic Rocks. Taylor & Francis CRC Press, (009, in ress). 4. Coussy, O., Mécanique des Milieux Poreux, Ed. Techni, Paris, (1991). 5. Leake, S.A. and Hsieh, P.A., Simulation o Deormation o Sediments rom Decline o Ground-Water Levels in an Aquier Underlain by a edrock Ste. US Geological Survey Oen File Reort 97 47, (1997). 6. Liu, G.R. and Quek, S., The Finite Element Method A Practical course, utterworth Heinemann, ristol, (003). 7. Terzaghi, K., Theoretical Soil Mechanics. John Wiley, New York, (1943). 8. Wang, H.F., Theory o Linear Poroelasticity - with Alications to Geomechanics and Hydrogeology. 87. Princenton University Press, (000).