QUATERNIONS TO MODEL NON-ISOTHERMAL POROELASTIC PHENOMENA

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1 PROCEEDINGS, hirty-sixth Worsho on Geothermal Reservoir Engineering Stanord University, Stanord, Caliornia, January - February, SGP-R-9 QUAERNIONS O MODEL NON-ISOHERMAL POROELASIC PHENOMENA Mario-César Suárez Arriaga and Fernando Samaniego V. Faculty o Physics and Mathematical Sciences, Michoacán University Ediicio Cd. Universitaria Morelia, Mich., 586, Mexico mcsa5@gmail.com ASRAC Quaternions are hyercomlex quantities in our dimensions (q, q, q, q ); they are a generalization o the classical two-dimensional comlex numbers (x, y) = x + i y, where i = -. Quaternions satisy all the algebraic roerties o the traditional comlex numbers excet that their multilication law is not commutative. hey have imortant alications in aeronautics, sace light dynamics, robotics, control theory, signal rocessing, and in comuter grahics because o their stability and ability to reresent three-dimensional rotations. In this aer we resent or the irst time, an original alication o quaternions to model non-isothermal oroelasticity. It is well nown that the resence o a moving luid in elastic orous rocs modiies their mechanical resonse. Poroelasticity exlains how the luid inside the ores bears a ortion o the total load suorted by the roc. he other art o the load is suorted by the seleton. he total deormation o geothermal rocs originates rom lithostatic comression, ore ressure and thermal stresses. he introduction o the volumetric Gibbs ree enthaly as a undamental thermodynamic otential, allows to include the thermal stresses directly. We introduce the thermooroelastic theory by deining oroelastic coeicients couling our quaternions: two or the bul roc, one or the luid and one or the total thermal exansion/contraction. he need o the ourth dimension aears naturally and ermits to extend the theory o linear elasticity to geothermal rocs, taing into account the eect o both the luid ressure and the temerature changes. Isothermal oroelasticity contains 6 dierent coeicients, but only our o them are indeendent. Introducing our volumetric thermal dilation coeicients, one or the luid and three or the seleton, a comlete set o arameters or non-isothermal rocs are obtained. We show how the set o quaternions we are introducing is equivalent to a 4D-tensorial oroelastic model. In other words, the thermooroelastic unctions mimic the algebraic structure o a non- commutative "ield". o illustrate the ractical use o this ormulation some alications are outlined: ull deduction o the classical iot s theory couled to thermal stresses. Construction o a general, well osed mathematical model to reresent the hysical behavior o geothermal rocs. Deormations o a non-isothermal reservoir, solved with inite elements, are included. hese results can be alied to geothermal and hydrocarbon reservoirs, and to dee aquiers. INRODUCION All crustal rocs orming natural reservoirs are oroelastic. he luid inside the ores and any temerature change aect their geomechanical roerties. his elasticity is evidenced by the comression resulting rom the ore ressure decline, which can shorten the ore volume. his reduction o the ore volume can be the rincial source o luid released rom storage. On the other hand, any variation o temerature induces a thermo-oroelastic behavior that inluences the elastic resonse o rocs. he exansivity o rocs is relatively small, but its eects can roduce an increase in ermeability and severe structural damages in rocs subjected to strong temerature gradients (undschuh and Suárez, ), as haens during the injection o cold luids. his mechanism is o great imortance in enhanced oil reservoirs and geothermal systems, when the injected cold luid circulates in the underground. his transort can change both the ermeability and the roc thermal conductivity. o develo our roosed our-dimensional model, we need to introduce irst the main roerties o quaternions, which will be a new mathematical tool to reresent thermooroelastic henomena. PROPERIES OF QUAERNIONS Algebraic deinitions We can deine the quaternions as hyercomlex numbers in a our-dimensional sace q 4 (this means that there is an isomorhism between both saces, the irst one is denoted by in honor o Sir William Rowan Hamilton); they satisy

2 two basic oerations, addition and multilication, deined as ollows. Let q and be two quaternions such that: q = (q, q, q, q ) and = (,,, ), where all their individual comonents are real numbers ( n, q n œ, n =,,, ). Eachh individual comlex art o the quaternionn q has a very simle geometric reresentation illustrated by the ollowing two-dimensional grah. Addition: q ( q, q, q, q ) Product: q ( q q q q q q q q q q q q q, q, q, q ) () () wo quaternions and q are equal i and only i each one o their our comonents are equal: q q, n,,, n n o obtain a vectorial reresentation o q, we deinee the ollowing unit vectors o : h = (,,, ), i = (,,, ), j = (,,, ) and = (,,, ). In this way, every quaternion can be written as a linear combination o our-dimensional basis vectors: Figure : Geometric reresentation o each comlex art o q, where ξ = r (Cos θ + i Sin θ ). Matrix reresentations in We can build quaternions into comlex matrices and also into real matrices. he irst one uses the hyercomlex ormm o equation ( ): q q h qiq jq () In this context, q is a thermooroelastic unction, which willl be deined later in this aer. he unit vectors satisy the ollowing multilication table which is built on rom deinition (): q q i q q q i i q qi q qi z q qi q qi z z z i able : * h i j Quaternions roducts o the unit vectors (Hamilton, 84; ater Graves, 88). h i j i i - - j j j - - i - j i - Using this table, we deduce another hyercomlex orm o any quaternion in terms o simle comlex numbers: q qhqi q jq () q q i q q i j z z j he symbols in thee last line are called the comlex conjugates o z and z resectively. he second matrix orm o quaternions uses the ollowing ourdimensional objects: where: z q q i, z q,, q i (5) Where z = q + q i and z = q + q i are traditional comlex quantities, z and z ; and i * j =. hereore, the hyercomlex olar orm o q is: q z z j re re j i n wher e: e Cos isin n, n i n i (4), (6)

3 In terms o these unit and anti-symmetrical matrices, any quaternion can be exressed as a linear combination o arrays, which have the ollowing roerty:, q q q q q q q q q q qqq q q q q q q q q (7) Note that the square matrix roerty (.) = -, is also satisied by the comlex ( ä ) matrices o equation (5). he matrix reresentation (7) allows to deine, in an alternative way, the quaternion multilication () as a our-dimensional, sew-symmetric matrix-vector roduct: q q q q q q q q q q q q q q q q q q q q q qqqq q q q q q q q q q (8) his result comared to equation () demonstrates the non-commutative nature o quaternion multilication. From equation () a third matrix reresentation is deduced in the orm o a diagonal (4 ä 4) array or every q œ : q q q (9) q q I we searate the matrix q rom the global array (7), we observe that each quaternion can be reresented by the sum o a real diagonal matrix lus an anti-symmetrical matrix called the imaginary matrix art o the quaternion: q q q q q q q q q q q q q () q q q It is well nown that any anti-symmetrical matrix reresents a ure rotation where (q, q, q ) are the corresonding comonents o the rotation vector. Hybrid scalar-vector reresentation in he result () suggests the ollowing hybrid scalarvectorial deinition o a quaternion which clearly searates its two arts: q qhqqq Q q iq jq (a) he term q is the real scalar art o q, and Q is its imaginary vector art. Relacing deinition (a) into equation (8) and ater doing some algebra we arrive to the ollowing ractical result which is useul to comute quaternion roducts in terms o classic vector oerations: q q hq, hp q ( Q Q q PQ)hqP P (b) where P ÿ Q reresents the scalar roduct and P ä Q is the vectorial roduct o P and Q resectively. ecause o the act that P ä Q Q ä P we conirm again that the quaternion multilication is a noncommutative oeration. Arithmetic roerties Quaternions satisy all the algebraic and arithmetic roerties o the classical two-dimensional comlex numbers, excet the multilication commutative law (Hamilton, 85; Kuiers, 999). he main roerties o q œ are summarized as ollows:. Additive inverse: - q = - q h - Q, q+(- q) =. Hyercomlex Conjugate: q q h Q. Norm o q: q qq q q q q q 4. Multilicative Inverse: q, qq qq q 5. Division o quaternions: q q q q 6. Normalized quaternion: i qq q q he addition is commutative and associative; the multilication is associative but non-commutative; both oerations obey the distribution law. he set

4 o quaternions, along with the addition, roduct and roerties already deined, has an algebraic structure called a non-commutative division ring, or simly an anti-commutative algebra. he uture useulness o quaternions was clearly envisioned by its creator: "ime is said to have only one dimension, and sace to have three dimensions.... he mathematical quaternion artaes o both these elements; in technical language it may be said to be time lus sace, or sace lus time: and in this sense it has, or at least involves a reerence to, our dimensions. And how the One o ime, o Sace the hree, Might in the Chain o Symbols girdled be.... quaternions would be ound to have a owerul inluence as an instrument o research." (William Rowan Hamilton; quoted in Graves, 88). he trile roduct o quaternions conjugate Let q = q + Q be a non-zero quaternion and v œ be a vector or ure imaginary quaternion (v = ). he alication o equation (b) to the trile roduct o q, its conjugate q - Q and v gives the ollowing result: qvq q QQ v Qv Qq Q v () I q is a normalized quaternion o unit norm: qvq q v Qv Qq Q v () Rotations in three dimensions using q œ We can erorm easily three-dimensional rotations using quaternions and unit vectors to deine an axis. A rotation in about a unit vector u by an angle θ can be comuted using the ollowing unit q and the trile roduct o quaternions: q q Q Cos u Sin wqvq Cos v u vu Sin Cos Sin u v (4) and thereore w œ. A series o two connected successive rotations, irst q and then, is comuted using the ollowing ormula: q Cos u Sin, Cos u Sin w qvq q v q (5) LINEAR POROELASICIY IN ERMS OF QUAERNIONS q œ A ey question that arises when develoing the very basic oroelastic equations is: why do we need a our-dimensional sace in oroelasticity? I we erorm an exeriment in a simle, non-orous solid under an uniaxial stress state, the resulting relationshi between the stress s x and the strain ε x is a one-dimensional equation: E (6a) x Where E is the Young's modulus. ut i we carry out the same one-dimensional exeriment in a orous roc, because o the luid contained in the ores, we obtain two simultaneous linear relationshis that are equivalent to a ( ä ) matrix equation (undschuh and Suárez, ), which corresonds to a twodimensional ormula: K C, and C M x x U x x x KU C x (6b) C M Where K U is the undrained bul modulus (Aendix), C is a coeicient reresenting deormations couling between the solid grains and the luid, M is a arameter characterizing the luid elastic roerties, is the luid-ore ressure, and ζ is a secial strain reresenting the variation o the luid mass content in the ores. Using only the arameter K U we obtain similar D relationshis: K K U U x x KU KU b (6c) Where b = C/M is a quotient called the iot-willis coeicient (iot & Willis, 957); is the Semton coeicient ( = C / K U ). he inverse matrix equation o (6b,c) is: x K H x H R (6d) Where K - = C is the bul comressibility, K is called the drained bul modulus; H - reresents an exansion coeicient and R - is an unconstrained seciic storage modulus (Wang, ). In terms o these moduli, the iot and Semton coeicients are deined as: b = K /H and = R/H, resectively. hese coeicients reresent roerties that are not encountered in standard elasticity o simle solids. he exerimental deinitions o these oroelastic moduli are in the Aendix.

5 Equation (6d) reveals that changes in σ x and roduce changes in ε x, which require luid to be added or removed rom storage because ζ also changes as a unction o σ x and. Ater this observational result, we conclude that the luid contained in the ores o the roc acts inside a second hidden satial dimension, although the exeriment was done in only one dimension. his result is immediately generalized to two- and threedimensional exeriments: an extra third or ourth satial dimension will always aear because o the luid contained in the ores. At this oint, a natural question arises: what are the quaternions roerties that are useul in oroelasticity? he undamental oroelastic relationshi in In order to introduce our model, we deine irst the ollowing oroelastic quaternions or homogeneous, isotroic, orous rocs. q hxi y jz q h xi y j z C q hi j M q h i j b (7) ecause o the hysical symbols involved, we name the quaternions: q σ the stress, q ε the ure strain, q the volumetric strain, and q the luid ressure. hese exressions do not contain shear strains (ε ij ), because changes in the luid ressure or in comressions are suosed not to induce shears (Wang, ). he undamental linear relationshi among the quaternions in equation (7) is very simle: q Gq q bq C M G b, n x,y,z n n (8) he symbols in equations (7) and (8) are o common use: s n (n = x, y, z) holds or the three normal stresses acting in the roc; ε n are the normal strains describing the elastic resonse o the orous roc; ε = ε x + ε y + ε z is the volumetric strain, G and l are oroelastic coeicients. he rigidity modulus G reresents the roc resonse to shear; the Lamé module l = K - G/ measures the roc resonse to comression under drained conditions. In terms o quaternions it is straightorward to invert the stressstrain relationshi (8). For examle: q q q bq (9) G From this equation we deduce the inverted comonents relationshis: b M n b n ; n x,y, z G G G () Ater doing some algebra we deduce other common inverse exressions (iot, 94; Wang, ): M H R n n M ; n x,y, z G E H x y z M K C () Where σ M is the average o the normal stresses, and ν is the Poisson's module. I the roc is anisotroic and inhomogeneous, the corresonding quaternions are deined as ollows: q hxi y jz q hxiy jz q hi j q h i j () he undamental linear relationshi among the nonisotroic quaternions can be written similarly as in equation (8) using the ollowing diagonal matrices (quaternions) o oroelastic coeicients: C Gx x G, Λ Gy y G z z () M bx b by b z he undamental anisotroic stress-strain relationshi in terms o quaternions is a linear combination o matrix - quaternion roducts: q G q Λ q b q (4) Note that we use here the common matrix-vector roduct, and not the quaternion multilication (), its direct use still needs more research. Equation (4) is valid or both isotroic and non-isotroic rocs.

6 Inclusion o the shear stresses s ij and iot's theory Note that there are no shear stresses nor shear strains involved in quaternions (7) and (). hey are going to be introduced in a dierent way. We deine two symmetric matrices s G and ε G. he irst one contains the shear stresses and the second one the shear strains. Equations () and (4) are diagonal matrices o tye (9) containing only rincial stresses and strains. oth inluence the bul roc deormation through tension or comression in the rincial axes. Since there are no shear tensions in the luid, the ore luid ressure aects only the three normal strains ε n (n = x, y, z), as shown in equation (). hereore the shear stresses can only wor on the shear strains: xy xz xy xz xy G yz xy yz xz yz xz yz Gxx Gxy Gxz G is the shear matrix Gxy Gyy G yz Gxz Gyz G zz σ G ε G n,m x,y,z G G nm nm nm (5) Where the ull shear matrix G holds or anisotroic rocs. Note that the shear strains and corresonding stresses cannot be reresented by quaternions. Formula (5) comletes the set o linear oroelastic equations when added to Eq. (4) in matrix orm: q σ G q G ε Λq b q (6) G G I the roc is isotroic and homogeneous the ull set o our-dimensional arrays is simly: x xy xz x xy xz G xy y yz xy y yz xz yz z xz yz z M C b b (7) Equation (6) corresonds to the ormulation o the theory or anisotroic rocs in terms o quaternions and shear matrices. he connections among the individual comonents o equation (7) are nown as iot's linear oroelastic theory or isotroic rocs (iot, 94, 955). he comonents o equation (7) can be written in comact orm (δ nm is the unit tensor): or n,m x, y, z (8) G b nm nm nm nm he term t nm is called the erzaghi eective stress, which acts only in the solid matrix: e G (9) nm nm nm Another imortant act concerning the inclusion o the shear stress-strains is that the shear matrices connected with linear oroelasticity are symmetric arrays. A secial case o the sectral theorem in linear algebra (Lang, 969) establishes that any real symmetric (4 ä 4) matrix M can be transormed into a diagonal matrix D in the ollowing way:, where: () he matrix U is called an orthogonal matrix and is built on the eigenvectors o M. he corresonding basis o eigenvectors or the our-dimensional sace 4 is orthogonal. he matrices reresenting the quaternions in the undamental equations (8), (4) and (6) are all symmetric. hereore, any oroelastic matrix equation, which includes shear stresses and shear strains, can be transormed into a diagonal matrix and, consequently it can be reresented by a quaternion o tye (9) which is valid or every q œ. HERMOPOROELASICIY IN ERMS OF QUAERNIONS q œ he basic thermodynamic otentials in orous rocs are the internal energy and the seciic enthaly or the luid hase. he ree energy describes the solid matrix while or the seleton the aroriate otential is the ree enthaly (undschuh and Suárez, ). In this section we ocus only on isotroic, homogeneous geothermal rocs. he Gibbs ree enthaly or the orous roc he volumetric Gibbs ree internal enthaly g S is deined as the algebraic dierence o the enthaly h S minus the roduct o the absolute temerature and the seleton entroy S S er unit volume o roc: g S i,, hs S S J/m = Pa ()

7 I the rocess is reversible and there is no energy dissiation in the roc, the total dierential o this thermodynamic otential is useul to describe the seleton thermal behavior with shear strains and orosity φ included (Coussy, 99; undschuh and Suárez, ): dg d d S d () S ij ij S hese dierential equations are integrated between an initial state g S (ε i =, =, = ) (i, j = x, y, z) at zero strain, and a inal state g S (ε ij,, ). Note that an initial temerature and an initial ore ressure are needed because both thermodynamic variables and are changing in non-isothermal rocesses: gs i dgs ij ij SS () g S Ater integrating equations (), an exact, analytic exression or the Gibbs otential o the seleton is obtained: K b K K M CV S S ij Initial State: g g =,, ; Final State: or i, j x,y,z and or t : g, S ij M ij M ij ij ij S K M G ij ij S (4) Where K is the bul modulus o the luid. he arameters g and g j [/K] are volumetric thermal dilatation coeicients. he irst one g measures the bul dilatation o the seleton, while g j is related to the variation o the orosity when the temerature changes: V, V (5) C V is the seleton volumetric heat caacity under constant deormations. he thermodynamic behavior o the orous roc is obtained by comuting the ollowing artial derivatives: gs gs gs ij,, SS ij (6) hermodynamics o the luid mass content he symbol ζ reresents the variation o the luid content in the ores deined as ollows (iot, 94): m m (7a) Where m is the luid mass content er unit roc volume, ρ is the luid density, ρ is a reerence density, φ is the eective orosity and Δ reresents a change in the corresonding variable. An exlicit thermodynamic exression or z can be obtained by dierentiating m in equation (7): dm d d d (7b) he luid density is a unction o both, the luid ressure and the temerature r (, ), thus: d d d (8) he luid comressibility C and the luid thermal exansivity g are deined as: C K V V (9) We deine another thermal exansion coeicient g m, which measures the changes in the luid mass content when the average stress s M and are held constant: m m m M, (4) Substituting the coeicients (9) into equation (7b): d d d d (4) K

8 Integrating this exression, between the initial and inal states, and relacing the result into equation (6) or the orosity we obtain: b m (4) M Equation (4) is a useul exression or the variation o the luid content in a linear non-isothermal rocess. he thermooroelastic relationshi in With the revious results, we can deine the thermal stress quaternion q or thermooroelastic rocs: q hij (4) Where Δ = -. he corresonding matrix o thermal exansivity coeicients is the ollowing diagonal array: M m K Γ K K (44a) he linear relationshi among the quaternions in non-isothermal rocesses is: q G q Λ q b q Γ q (44b) In terms o our-dimensional matrices, the equation relating stresses and strains in isotroic rocs is deduced directly: q C x G y z A A b K b b Where: A M, A M, m K (44c) Note that every quaternion is equivalent to a diagonal matrix. he act that linear thermooroelasticity theory can be reresented by quaternions is very imortant because all quaternions roerties, that are useul in this context, can be alied to geothermal rocs deormations. his allows to the develoment and discovery o other unnown oroelastic rocesses. Some numerical values o the oroelastic moduli and the thermal exansivity coeicients o construct the ollowing table o oroelastic coeicients the basic exerimental data set used was {E, G, φ, K S and K } (see Aendix). able : Poroelastic coeicients o dierent rocs. Roc φ E n n U G l l U ye (%) (GPa) (GPa) (GPa) (GPa) [] [] [] [] [4] [5] [6] Roc ye K (GPa) K U (GPa) K S (GPa) b M (GPa) R (GPa) [] [] [] [] [4] [5] [6] he roc tyes are: [].- Clay, (K =.9 GPa). [].- oise sandstone, (K =. GPa). [].- erea sandstone (K =.5 GPa); [].- Indiana limestone, (K =. GPa). [4].- Andesite, (K =. GPa, = 5 bar, = 5 C). [5].- ennessee marble, (K =.5 GPa). [6].- Roc with celestite, (K =. GPa) Reerences used: Rocs [], [], [] and [], Wang (); [4], [5] and [6], undschuh and Suárez (). he igures in italics were estimated using oroelastic ormulas. Rocs o tye [] and [6] are reresentative o the two limit cases o linear oroelastic theory, or b º and b º, resectively. In this section, we also give some exerimental values o the volumetric thermal dilatation coeicients g, g, g j, g m and g U [/K]. As a general trend, thermal exansion increases when temerature

9 rises. he volumetric thermal exansivity g is about -5 K - or solids, -4 K - or liquids and - K - or gases. For the exansivity o the ores in granite we estimated g j º.7 ä -4 K -. For water, at 8 C and bar, g = 6. ä -4 K -. For low orosity hard volcanic rocs, having a orosity j =.8, b =., and =.9, Guéguen & outéca (4) ublished the values g = 5. ä -5 K -, g m = ä -5 K - and g U = 5.75 ä -5 K - and the corresonding bul moduli K = 7 GPa and K U = 8 GPa. Using these data, the corresonding variations o the luid mass content and o the luid ressure or drained and undrained conditions are: 5 m b 58. K, KU m b 8. 4 b, U K 5 m 67. b, Pa units in the last two equations: K hese numerical values conirm that in low orosity and low ermeability rocs, the thermal variation o the luid content is very small because o the short hydraulic arameters. For the same reason, a signiicant luid ressure increase occurs in the resence o thermal stresses, seciically in undrained conditions. his is a common case when injection o cold water taes lace in hot dry roc reservoirs. he values resented in able were obtained under isothermal conditions. ut there are evidences suggesting a great sensitivity o the oroelastic arameters with regard to temerature changes in non-isothermal rocesses (Suárez, ). For examle, the bul modulus o cold water at 5 C is equal to. GPa (able ); whereas the bul modulus o hot water is lower under dee geothermal conditions (K =.45 GPa at 5 C), roducing larger variations o the luid content and inducing higher thermooroelastic deormations. hereore, the oroelastic deormations can be much higher in geothermal reservoirs than in cold isothermal aquiers. his eect is more imortant in high orosity sedimentary rocs. Figure shows the relationshi between b, the iot-willis modulus and the bul modulus o water K (equation 45), which deends on both, the temerature and the luid ressure (undschuh and Suárez, ): KS b KU K (KU K ) 4K S / K KU K KS 5 (45) K GPa Figure : he iot-willis coeicient as a unction o the bul modulus o water. iot modulus b A GENERAL POROELASIC MODEL IN D he undamental law o dynamic oroelasticity in two-dimensions is (undschuh and Suárez, ): u x ux u u x y C G C Fx t x y xy u y uy uy ux C G C Fy t y x x y ux uy u t ux, uy,, 46 L C bcgcm Where (u x, u y ) are the comonents o the dislacement vector; (F x, F y ) are the comonents o the external body orce acting on the orous roc; the let term in equation (46) is the vectorial acceleration and r is the average density, when r and r s are the luid and solid hase densities resectively; L is a reresentative dimension o the orous seleton. his general model is valid only or small, reversible deormations o Hooean oroelastic rocs. We consider only the imortant case o static oroelastic equations when the external body orce is the vertical gravitational orce er unit bul volume and the acceleration in equation (46) is zero. he governing oroelastostatic equations in condensed tensorial notation is equivalent to the ollowing ormula: u i G ui G b Fi i j i or i, j x, y, z (47) Equation (47) is couling the oroelastic mechanism between the ore ressure and the dislacement o the orous roc articles. wo drained coeicients l and G emerge because the couling term is the luid ressure.

10 he mathematical model o a oroelastic roc, ully saturated with a moving luid, is a couled grou o artial dierential equations governing the luid low inside the ores o the deormable seleton. All the coeicients, concets and relationshis described in revious sections are suicient to develo this model. We only need to deine the unnowns and methodically arrange the aroriate equations. here are twelve undamental unnown variables in three dimensional thermooroelasticity: the six stresses (s ij ) o equation (7); the three coordinates (u x, u y, u z ) o the relative luid-solid dislacement; the variation o the luid content z (Eq. ), the ore ressure (Eq. 8), and the temerature (Eq. 4). he six strain comonents (ε ij ) can be comuted using either the quaternion equation (9) or their deinition in terms o the vector dislacement. he corresonding equations o the reviously mentioned unnowns are the seven comonents relating stresses, strains and the ore ressure (Eq. 6) or (Eq. 7), the three artial dierential equations obtained rom the equilibrium conditions in the seleton (Eq. 47), the eleventh equation comes rom the conservation o the luid mass relating z with the Darcy s law through the luid ressure (Eq. ); while the inal twelth comes rom the general heat low equation in geothermal reservoirs, which includes conduction and convection through an advective term related to the transort o energy by the moving luid in the ores: c c v Q t H (48) Where vector v reresents the Darcy s luid velocity, c is the seciic heat coeicient at constant ressure, is the thermal conductivity tensor, and Q H [W/m ] is the volumetric thermal energy roduction. his grou o equations rovides an equilibrated system that can be numerically solved using dierent numerical techniques. SOLUION OF HE POROELASIC MODEL USING FINIE ELEMENS: A D EXAMPLE Equation (47) includes iot s oroelastic theory. It can be ormulated and numerically solved using the Finite Element Method (FEM). Let Ω be the bul volume o the orous roc, and let Ω be its boundary, u is the set o admissible dislacements in equation (46); there is a volumetric orce and a orce acting on the surace Ω. Ater doing some algebra (Liu and Que, ) we arrive to a FEM undamental equation or every element V e in the discretization o Ω comosed o N e inite elements: e e e e d e K d M F ; e,n e (49) t d e is a vector containing the dislacements o the nodes in each V e. Equation (49) aroximates the dislacement u o the oroelastic roc. F e is the vector o total nodal orces. K e and M e are the stiness and equivalent mass matrices or each inite element V e. he mathematical deinitions o both matrices are: e e e V e V K C dv; == LN; M N N dv; e, Ne (5) Where C is an array o oroelastic coeicients, N is the matrix o shae unctions that interolate the dislacements, L is a matrix dierential oerator and array is called the strain oroelastic matrix. o illustrate the model (49), two-dimensional strains o an elastic aquier are resented, including the orm in which a temerature change can aect its oroelastic deormation. In the irst examle, we assume that the aquier contains cold water at C ( g/m ). In a second examle, we consider an uniorm aquier temerature o 5 C. In both cases water obeys Darcy s law or head h. hree sedimentary layers overlay an imermeable bedroc in a 5 m long basin where aulting creates a bedroc ste (S) near the mountain ront (Figure ). he sediment stac totals 4 m at the deeest oint o the basin (x = m) but thins to m above the ste (x > 4 m). he to two layers o the sequence are each m thic. he irst and third layers are aquiers; the middle layer is relatively imermeable to low. bedroc ste Figure : Initial state and aquier's geometry with the imermeable bedroc in the basin. he two examles resented herein were rogrammed and solved using the inite element sotware COMSOL Multihysics (9) or this well-nown roblem o lined luid low and solid

11 deormation near a bedroc ste in a sedimentary basin described in a revious ublication (Leae & Hsieh, 997). he roblem concerns the imact o uming or a basin illed with sediments draing an imervious ault bloc. he 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). he head dro moves luid away rom the ste. he luid suly in the uer reservoir is limitless. he eriod o interest is years. Streamlines reresent the luid to orous roc couling. he second simulation was erormed or a oroelastic couled deormation when the water in the aquier corresonds to geothermal conditions (luid density o 8.4 g/m, temerature o 5 C, and ressure o 5 bar). he roc is oroelastic, homogeneous and obeys quaternion equation (8). Water obeys Darcy s law or head h (K X, K Y are the hydraulic conductivities and S S is the seciic storage): Figure 4: Horizontal strain at the basin containing cold water at C. h h h K X KY qv SS x x y y t (5) For the comutations, data o able were used. In the irst examle or the iot-willis coeicient we assume that b =.; in the second examle b =.9 because it was aected by the higher temerature through the water bul modulus (Fig. ). able : Parameters used in the simulations. Hydraulic conductivity, uer & lower aquiers Hydraulic conductivity conining layer iot-willis coeicient (cold water at C) Young s modulus K X 5 m/day K Y. m/day b. E 8ä 8 Pa Poroelastic storage coe. uer aquier Poroelastic storage coe., lower aquier iot-willis coeicient (hot water, at 5 C) Poisson s ratio S S -6 S S -5 b.9 n.5 he corresonding FE mesh has 967 elements excluding the bedroc ste (Fig. ). Figures (4) and (5) show the horizontal strains in the aquier or cold and hot water resectively. Figures (6) and (7) show the aquier vertical deormation ater years o uming with cold and hot water resectively. Figure 5: Horizontal strain at the basin containing hot water at 5 C. CONCLUSIONS A general thermooroelastic linear model based on quaternions was introduced. his new ormulation is equivalent to a 4D-tensorial model, which contains the classic iot's theory. he model taes into account, in a simle way, both the luid ressure and the temerature eects in orous roc deormations. he examles resented suggest the inluence o temerature changes on the oroelastic strains and on the coeicients. he oroelastic deormations are much higher in geothermal reservoirs than in isothermal aquiers. For cold water, the estimated value o ε z is about -.5-4, while or hot water ε z is about

12 Figure 6: Poroelastic deormation o the basin or the S roblem with cold water ( C). Figure 7: Poroelastic deormation o the basin or the S roblem with hot water (5 C). REFERENCES iot, M. A. (94), "General theory o threedimensional consolidation, Journal o Al. Physics,, iot, M. A. (955), "heory o elasticity and consolidation or a orous anisotroic solid, Journal o Al. Physics, 6, iot, M. A. and Willis, D. G. (957), " he elastic coeicients o the theory o consolidation, Journal o Al. Mechanics, 4, iot, M. A. (96), " Mechanics o deormation and acoustic roagation in orous media Journal o Alied Physics, /4, undschuh,j. and Suárez, M.C. (), "Introduction to the Numerical Modeling o Groundwater and Geothermal Systems: Fundamentals o Mass, Energy and Solute ransort in Poroelastic Rocs", 55., ISN: COMSOL (9), "Couled low laws solved with COMSOL Multihysics.5a", COMSOL A, User's Manual, Stocholm, Sweden. Coussy, O. (99). "Mécanique des Milieux Poreux", Editions echni, Paris. Gueguen, Y. and outeca, M. (editors; 4), "Mechanics o luid - saturated rocs". International Geohysics Series 89, Elsevier- Academic Press, New Yor, NY. Graves, R. P. (88), "Lie o Sir William Rowan Hamilton, ublished in three volumes: Vol., 88, Vol., 885, Vol., 889. Dublin University Press. Hamilton, W. R. (85), "Lectures on Quaternions, a set o lectures delivered by Hamilton between 848 and 85 in the rinity College o Dublin. Hodges and Smith, Graton-Street. he original text can be read at this web site: htt://dlxs.library.cornell.edu/cgi/t/text/agevieweridx?c=math&cc=math&idno=5&q=le ctures+on+quaternions&rm=rameset&view=i mage&seq=9 Hamilton, Edwin (son o Sir William Hamilton) ublished the boo "Elements o Quaternions" in 866, Dublin University Press. C. Jaser reared a nd edition in two volumes; the st one in 899 and the nd in 9, ublished by Longmans, Green & Co. Kuiers, J.. (999), "Quaternions and Rotation Sequences: a Primer with Alications to Orbits, Aerosace, and Virtual Reality, Princeton University Press, 7, ISN: Lang, S. (969), "Linear Algebra, 94., Addison- Wesley Pub. Co., New Yor. Leae, S. A. and Hsieh, P. A. (997), "Simulation o Deormation o Sediments rom Decline o Ground-Water Levels in an Aquier Underlain by a edroc Ste". US Geological Survey Oen File Reort Liu, G. R. and Que, S. (), "he inite element method A ractical course", utterworth- Heinemann, ristol, UK. Suárez, A. M. C. (), "A Four-Dimensional Formulation o Geothermal Poroelasticity", Proceedings o the World Geothermal Congress,. -. ali, Indonesia, 5-9 Aril. Wang, H. F. (), " heory o linear oroelasticity - with alications to geomechanics and hydrogeology, Princenton University Press, NJ.

13 APPENDIX - NOMENCLAURE Poroelastic coeicients: exerimental deinitions In exerimental oroelasticity, there are our dierent tyes o bul moduli and two tyes o deormations, drained and undrained. oth modes reresent limiting resonses o the roc (iot, 94; Wang, ): Drained conditions: he roc is conined and subjected to suort an external hydrostatic ressure s H. he luid in the ores is allowed to escae and the total stress is entirely suorted by the roc seleton. he deormations are achieved at constant ore luid ressure. iot (96) called this state, an oen system. Undrained conditions: Deormations are measured at constant luid mass content m (D z = ). he roc is entirely submerged in a luid in such a way that the external hydrostatic ressure is balanced by the ore ressure (s H = - ). he luid in the ores remains constant; no luid is allowed to move into or out o the control volume. he luid remains traed in the seleton. For iot (96) this state was characteristic o a closed system. ul elastic modulus: he drained bul coeicient K is one o the central concets in oroelasticity; it is exerimentally deined as the ratio o the change in conining lithostatic ressure relative to the volumetric strains when the luid ressure remains constant, under isothermal conditions: C K (A) Undrained bul comressibility: he undrained bul modulus K U measures the resistance o the bul roc against deormations roduced by the conining lithostatic ressure when the luid content z remains constant: KU (A) CU Comressibility o the solid hase: he elastic modulus K S measures the resistance o the solid roc against deormations roduced by changes on hydrostatic comression (s H = - ) when the dierential ressure d = remains constant: C KS S S d (A) Comressibility o the luid hase: he isothermal comressibility o the luid C is deined as the change o luid volume with resect to the eective ressure change er unit volume: V C = K V (A4) Comressibility o the ore volume: he unjaceted comressibility o the ore volume C j, is deined as the change o ore volume with resect to the ore ressure change er unit volume when d remains constant: V C (A5) K V d d he iot-willis Coeicient: his undamental arameter is equal to the change o ressure with resect to the luid ressure change when the total volumetric strain remains constant: K K V b K K V S (A6) he last exression in (A6) gives another deinition o b as the ratio o ore volume change to total bul volume change under drained conditions. his equation also rovides two ways o comuting the drained bul modulus o the ore sace K F. hese ormulae are obtained rom exeriments in which the ore luid can low in or out o the ores to maintain the ore ressure constant. he oroelastic exansion coeicient: H - = /H describes how much DV changes when changes while eeing the alied stress s constant; /H also measures the changes o ζ when s changes and remains constant: H (A7) he unconstrained seciic storage coeicient: /R measures the changes o ζ when changes when the alied stress s remains constant. Its numerical value is determined by the comressibility o the rame, the ores, the luid and the solid hase: R (A8)

14 he constrained seciic storage coeicient: /M is equal to the change o ζ when changes, and it is measured at constant strain. his arameter can be exressed in terms o the three undamental ones reviously deined: K M R H (A9) Deormations couling coeicient: he arameter C reresents the couling o deormations between the solid grains and the luid: K K C H H M C M b (A) he Semton coeicient: reresents the change in ore ressure when the alied stress changes or undrained conditions: R H m (A)