Solution and Type Curve Analysis of Fluid Flow Model for Fractal Reservoir

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1 Wold Jounal of Mechanics, 0,, 09-6 doi:0.436/wjm Published Online Octobe 0 (htt:// Solution and Tye Cuve Analysis of Fluid Flow Model fo Factal Resevoi Yulong Zhao, Liehui Zhang State Key Laboatoy of Resevoi Geology and Exloitation, Southwest Petoleum Univesity, Chengdu, China @qq.com Received July 6, 0; evised August 4, 0; acceted August 30, 0 Abstact Conventional essue-tansient models have been develoed unde the assumtion of homogeneous esevoi. Howeve, coe, log and outco data indicate this assumtion is not ealistic in most cases. But in many cases, the homogeneous models ae still alied to obtain an effective emeability coesonding to fictitious homogeneous esevois. This aoach seems easonable if the emeability vaiation is sufficiently small. In this ae, factal dimension and factal index ae intoduced into the seeage flow mechanism to establish the fluid flow models in factal esevoi unde thee oute-bounday conditions. Exact dimensionless solutions ae obtained by using the Lalace tansfomation assuming the well is oducing at a constant ate. Combining the Stehfest s invesion with the Vongvuthionchai s method, the new tye cuves ae obtained. The sensitivities of the cuve shae to factal dimension (θ) and factal index (d) ae analyzed; the cuves don t change too much when θ is a constant and d change. Fo a closed esevoi, the u-cuving has little to do with θ when d is a constant; but when θ is a constant, the sloe of the u-cuving section almost emains the same, only the essue at the stating oint deceases with the incease of d; and when d = and θ = 0, the solutions and cuves become those of the conventional esevois, the alication of this solution has also been intoduced at the end of this ae. Keywods: Factal Resevoi, Factal imension, Factal Index, Tye Cuve, Well Test. Intoduction The concet of factal geomety, develoed by Mandelbot [], suggests that stuctues which aea to be comletely andom can be descibed within a geometic mathematical famewok and offes many ossibilities in scientific alications. The class of stuctues teated in this study is limited to those that exhibit self-simila geometical oeties which mean that the stuctue looks the same when obseved unde vaious scales of measuement. He is also one of the fist studies to find that many stuctues in natue exhibit these self-simila geometical oeties. Many geological oeties affecting the flow of fluids in oous media ae known to descibe the factal chaacteistics. Hewett [] eoted that oosity and emeability distibutions in ock fomation demonstated factal chaacteistics and intoduced the use of factal inteolation technique to descibe the heteogeneities in a esevoi. Bakke [3] and oe [4] also develoed simila models fo the inteetation of constant essue oduction/ injection well tests, they both assumed the esevoi oeties to vay accoding to a owe law elationshi with distance fom the souce. Chang and Yotsos [5] esented a well testing model which assumed that the factue netwok could be descibed by a factal distibution. He is also obtained the analytical esults, which showed that a factal esevoi can be identified by a log-log staight line with the slo equals to a function of the factal dimension and the factal index. Beie [6] and Ailian [7] alied factal esevoi model to analyze well test data fo comlex esevois which could not be matched by taditional model, and the esults wee consistent with field actice. Poon [8] extended the concet of factal distibution to study the effect of a comosite esevoi. He et al. [9] established a factal model fo unsteady-state flow in dual-oosity and emeability esevois based on Waen-Root [0] Model, and solved it by Coection Pediction method. Meanwhile, they analyzed essue efomances and thei effect on diffeent factos. Li [] modified a well test model fo infinite dual-oosity esevoi with wellboe stoage and Coyight 0 SciRes.

2 0 Y. L. ZHAO ET AL. skin based on factal theoy, and also esulted a seies of alicable analytic solutions. Kong et al. [] set u equations of flow ate, emeability and oosity fo fluid flow in factal dimensional oous media, and diffeential equations in diffeent coodinate systems. They solved the model with consideation of wellboe stoage and skin, and lotted the solution as tye cuves. On the basis of Waen-Root mode, Zhang et al. [3], set u a model fo defomed dual-oosity factal gas esevois by intoducing factal aametes and comessibility factos. They solved this model by finite element method with consideation of seconday bounday conditions, lotting the esults into tye cuves. To conside theshold essue gadient in low-em esevoi and gadual essue oagation in the fomation, Hou and Tong [4] established non-acy flow model fo defomed dualoosity media. Fo comosite esevoi, thee ae also many scholas [6-] have studied it and they also obtained the chaacteistic cuves. Tye cuves fom models established by above scholas ae mainly lotted the with t on bilogaithmic gah, which ae geatly diffeent fom the taditional ones, and not so alicable in field actice. Theefoe, this ae established the flow model fo factal esevoi based on the mechanics of fluid flow in oous media and factal theoy. Analytical solutions ae obtained by Lalace tansfomation and genealized Bessel equation fo thee kinds of oute-bounday conditions. Accoding to the simila method of Vongvuthionchai [] used to lot tye cuves fo non-newton fluid flow, the coesonding tye cuves is lotted by Stehfest [3] numeical invesion.. Mathematical and efinitions The model was develoed with factued ock in mind, ignoed the oosity and emeability of the matix. The main assumtions of the model include: ) d-dimensional factal flowing gids in two-dimensional Euclidean nonemeable ock, full enetation of the wellboe into esevoi with thickness of h and unde adial flow; ) esevoi fluid is slight-comessive; 3) ignoe the effect of gavity and theshold essue gadient. Accoding to factal theoy [], emeability and oosity of the system is: k d kw w w w Continuity equation fo fluid: d () () acy flow equation: v t k v Substitution of Equations (), () and (4) into (3): whee: d c t w t w kw c l 3. Model Solutions, c, ct cl c. Intoducing the following dimensionless vaiables: πk h q (3) (4) (5) w i (6) t (7) w k t (8) w octw Allows the continuity equation to be witten in dimensionless tems and conside the bounday and initial conditions, then d t The initial essue distibution is assumed unifom, eveywhee,0 0 (0) Inne bounday conditions w C d dt w (9) S () And the oute bounday conditions ae as follows: () lim, t 0 (Infinite esevoi) (3) e 0 (Closed cicle esevoi) (4), z 0 e (Cicle esevoi with constant-essue bounday)(5) Coyight 0 SciRes.

3 Y. L. ZHAO ET AL. To simlify, Equations (9) to (5) ae tansfomed into Lalace sace. whee: 0 d e zt dt Inne bounday conditions: z w C z w Oute bounday conditions: Infinite esevoi Closed cicle esevoi (6) S (7) (8) z lim, z 0 (9) e 0 Cicle esevoi with constant-essue bounday e, (0) z 0 () Equation (3) can be solved accoding to the theoy of genealized Bessel equation. Let, n d to convet Equation (3) into the fomat of genealized Bessel equation R n R m n zr () whee: m = d. Combining m with n, we have m n = θ and θ is a constant. Because of + m n = + θ > 0, and using the method of Bessel function [], a solution of Equation (6) is assumed in the fom: n whee: v. m n efine β by: mn z AI n v m n mn z BK v mn Equation (3) educe to: (3) mn (4) v n v AI z BK z (5) Using the method of modified Bessel function [], the deivative of Equation (5) is: m n AIv z z BKv z Combine Equations (5) and (6) with (7) and (8): v v v w AIv z zsi z BK z zsk z v (6) (7) Czw A ziv z B zk z (8) z 3.. Fo Infinite Resevoi Accoding to the chaacteistics of Modified Bessel Function [], as the dimensionless adius aoaches infinity I v becomes infinity; theefoe to satisfy the oute bounday condition the constant A must be zeo. To detemine the constant B, aly the inne bounday condition (Equations ()-(3)) to Equations (7)-(8). The esulting solution of nomalized bottom-hole essue is the follows: w Kv z S zkv z 3 Cz Kv z S zkv z z Kv z (9) 3.. Fo Closed Cicle Resevoi o Constant-Pessue Bounday Resevoi Alying the simila method of infinite esevoi, we can obtain the following solution: v v v w AIv z zsi z BK z zsk z whee: A, B, A Z A, AA AA Z z 3 z A CzIv Cz Iv S z ziv (30) Coyight 0 SciRes.

4 Y. L. ZHAO ET AL. z 3 z A CzKv Cz Kv S z zkv Fo closed cicle esevoi: z z A Iv e A Kv e, Fo cicle esevoi with constant-essue bounday: z z A I A K, v e v e 4. Tye Cuves and Flow Peiod Analysis Using the method of Vongvuthionchai [9], analytical solutions abstained above can be lotted as tye cuves in bilogaithmic coodinate system with [t /C ] (4-d+θ)/ as x axis and [ /C ] (-d+θ)/, *[t /C ] (4-d+θ)/ as y axis. The tye cuves of infinite factal esevoi ae shown in Figues and, Figues 3 and 4 ae fo closed cicle esevoi and cicle esevoi with constant-essue bounday, esectively. Accoding to the tye cuves, the following flow eiods ae obtained: Stage I is the wellboe stoage flow eiod, which is chaacteized by a unit slo line and not affected by fac- Figue. Tye cuves of infinite esevoi with diffeent factal dimension d (C = 000, S =, θ = 0.0). Figue. Tye cuves of infinite esevoi with diffeent factal facto θ (C = 000, S =, d =.7). Coyight 0 SciRes.

5 Y. L. ZHAO ET AL. 3 Figue 3. Tye cuves of closed and constant-essue bounday esevoi with diffeent factal dimension d (C = 00, S =, θ = 0.0, e = 000). Figue 4. Tye cuves of closed and constant-essue bounday esevoi with diffeent factal facto θ (C = 00, S =, d =.8, e = 000). tal dimension d and facto θ. Stage II is the tansition flow eiod afte wellboe stoage. Based on the tye cuves, length and eak value of this eiod ae contolled by wellboe stoage effect, skin, d and θ. Fo a constant θ value, length and eak value decease with the incease of d; when d is constant, they decease with θ. Accoding to Figues 3 and 4, the tansition eiod becomes shote and shote when d deceases o θ inceases. Stage III stands fo adial flow eiod. Fo factal esevoi, it can be noticed fom the cuves that the adial flow eiod is no longe a 0.5 staight line as taditional tye cuves. They become a gou of uwads staight lines, whose sloes and stating oints ae elated to d and θ. Fo constant d and θ, the sloe is also a constant with diffeent stating oints and essue (essue deivative) values. Stage Ⅳ eesents the bounday effect. When the esevoi is infinite, this eiod will not exit. Fo closed o constant-essue bounday esevoi, aid u- o down-cuving can be seen on the tye cuves (see Figues 3 and 4). Close the bounday is, ealie this change Coyight 0 SciRes.

6 4 Y. L. ZHAO ET AL. haens, vice vesa. Accoding to Figues 3 and 4, fo constant-essue bounday esevoi, when d is fixed, the down-cuving haens ealie with θ becoming smalle; and when θ is a constant, the cuves do not change too much when d changes. Fo closed esevoi, when d is a constant, the u-cuving has little to do with θ; and when θ is fixed, the sloe of the u-cuving section is almost a constant, only the essue at stating oint deceases with incease of d. When d = and θ = 0, the esults become the conventional esevois solutions, and the tye cuves become the coesonding ones, which ae shown in Figue 5. It can conclude that the conventional esevois ae the secial fom of factal esevois. 5. Alication A essue buildu test in an oil well is influenced by skin and wellboe stoage. The measued essue date as a function of time is listed in age 94 of the Boudet (adial flow examle) []. And the othe known esevoi and well data as follows: = 0.5, w = (m), C t = (MPa ), h = (m), B o =.06 (m 3 /sm 3 ), μ 0 =.5 (c). A log-log lot (full test histoy t with essue and essue deivative functions) and final match of this examle is shown in Figue 6. ates ae matched against the conventional esevoi tye cuves fo a well with wellboe stoage and skin in a esevoi with homogeneous behavio. The oety of match tye cuve is C = 800, and the S = 7.6. So, fom the definition of C, we have C = 0.E 3 (m 3 /MPa), and the K h = 359 (m m), and so, the effective emeability is 0.89 (m), which the esults obtained ae consistent with efeence []. 6. Conclusions Based on mechanism of fluid flow in oous media and the factal theoy, a model with factal chaacteistics fo factal esevoi is constucted and solved by Stehfest s [0] invesion method. A new method to lot tye cuves of factal netwoks with factal chaacteistics was develoed. These new tye cuves have the simila chaacteistics as those of the conventional ones. Comaed to the evious factal esevoi cuves, the imovement is valuable. Tye cuves fo factal esevoi ae geatly affected by dimension d and facto θ, esecially duing the adial flow eiod, whee the essue deivative is not a 0.5 staight line any moe, and the slo of the line inceases with the decease of d and incease of θ. uing the wellboe stoage and bounday flow eiod the cuves have the simila chaacteistics as conventional esevoi, and when d = and θ = 0, the solutions and cuves become those of the conventional esevoi, the examle is also sulied duing the alication when d = and θ = 0, which is a secial condition of the factal esevoi. 7. Acknowledgements This wok was suoted by National Pogam on Key Basic Reseach Poject (973 Pogam, Gant No. 0CB0005), Reseach Fund fo the octoal Po- Figue 5. Tye cuves of infinite esevoi with the (d =, θ = 0, S = ). Coyight 0 SciRes.

7 Y. L. ZHAO ET AL. 5 Figue 6. The log-log matching lot. gam of Highe Education of China (Gant No ). 8. Refeences [] B. B. Mandebot, The Factal Geomety of Natue, W. H. Feeman, New Yok, 98. [] T. A. Hewett, Factal istibution of Resevoi Heteogeneity and Thei Influence on Fluid Tansot, Society of Petoleum Enginees, New Olean, 5-8 Octobe 986, ocument I: [3] J. A. Bakke, A Genealized Radial Flow Model fo Hydaulic Tests in Factued Rock, Wate Resouces Reseach, Vol. 0, No. 0, 986, [4] T. W. oe, Factional imension Analysis of Constant Pessue Well Tests, The 66th Annual Technical Confeence and Exhibition of SPE, allas, 6-9 Octobe 99. ocument I: 70-MS. doi:0.8/70-ms [5] J. Chang and Y. C. Yotsos, Pessue Tansient Analysis of Factal Resevoi, SPE Fomation Evaluation, Vol. 5, No., 990, doi:0.8/870-pa [6] R. A. Beie, Pessue Tansient Field ata Showing Factal Resevoi Stuctue, Intenational Technical Meeting of the SPE Calgay, Calgay, 0-3 June 990, ocument I: 553-MS. [7] S. Ailian,. Abdassah, L. Muchaant, et al., Alication of Factal Resevoi Mode fo Intefeence Test Analysis in Kamojiang Geothemal Field (Indonesia), The 68th SPE Annual Technical Confeence and Exhibition, Houston, 3-6 Octobe 993, ocument I: 6465-MS. doi:0.8/6465-ms [8]. Poon, Tansient Pessue Analysis of Factal Resevois. The Petoleum Society of CIM, The 46th Annual Technical Meeting of the Petoleum Society of CIM, Calgay, 7-9 June 995, ocument I: [9] G. L. He and K. L. Xiang, Mathematical Model and Chaacte of Pessue Tansient of Unstable Seeage Flow in efomation ual-poosity Factal Resevois, Jounal of Southwest Petoleum Univesity, Vol. 4, No. 4, 00, (in Chinese) [0] J. E. Waant and P. J. Root, The Behavio of Natually Factued Resevois, SPE Jounal, Vol. 3, No. 3, 963, []. P. Boudet, J. A. Ayoub and Y. M. Piad, Use of Pessue eivative in Well Test Inteetation, SPE Fomation Evaluation, Vol. 4, No., 989, [] S. C. Li, A Solution of Factal ual Poosity Resevoi Model in Well Testing Analysis, Pogess in Exloation Geohysics, Vol. 5, No. 5, 00, (in Chinese) [3] X. Y. Kong, Advanced Mechanics of Fluids in Poous Media, Univesity of Science and Technology of China Pess, Hefei, 999. (in Chinese) [4] L. H. Zhang, J. L. Zhang and B. Q. Xu, A Nonlinea Seeage Flow Model fo efomable ouble Media Factal Gas Resevois, Chinese Jounal of Conutational Physics, Vol. 4, No., 007, (in Chinese) [5] Y. M. Hou and. K. Tong, Non-Steady Flow of Non-Newtonian Powe-Low Pemeability with Moving-Bounday in ouble Poous Media and Factal Resevoi, Engineeing Mechanics, Vol. 6, No. 8, 009, (in Chinese) [6] K. L. Xiang and X. Q. Tu, The Analytical Solutions of Mathematical Model fo a Factal Comosite Resevoi with Non-Newtonian Powe Low Fluids Flow, Chinese Jounal of Comutational Physics, Vol., No. 6, 004, (in Chinese) Coyight 0 SciRes.

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