TRANSIENT PRESSURE AND PRODUCTIVITY ANALYSIS IN CARBONATE GEOTHERMAL RESERVOIRS WITH CHANGING EXTERNAL BOUNDARY FLUX

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1 Wang,., et al.: Transient Pressure and Productiity Analysis in Carbonate Geothermal... THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S77-S84 S77 TRANSIENT PRESSURE AN PROUCTIVITY ANALYSIS IN CARBONATE GEOTHERMAL RESERVOIRS WITH CHANGING EXTERNAL BOUNARY FLUX by ongying WANG, Jun YAO *, Mingyu CAI, and Piyang LIU School o Petroleum Engineering in China, Uniersity o Petroleum, Qingdao, China Original scientiic paper In this paper, a triple-medium low model or carbonate geothermal reseroirs with an exponential external boundary lux is established. The pressure solution under constant production conditions in Laplace space is soled. The geothermal wellbore pressure change considering wellbore storage and skin actor is obtained by Stehest numerical inersion. The well test interpretation charts and Fetkoich production decline chart or carbonate geothermal reseroirs are proposed or the irst time. The proposed Fetkoich production decline cures are applied to analyze the production decline behaior. The results indicate that in carbonate geothermal reseroirs with exponential external boundary lux, the pressure deriatie cure contains a triple dip, which represents the interporosity low between the ugs or matrix and racture system and the inading low o the external boundary lux. The interporosity low o carbonate geothermal reseroirs and changing external boundary lux can both slow down the extent o production decline and the same ariation tendency is obsered in the Fetkoich production decline cure. Key words: carbonate geothermal reseroir, changing external boundary lux, triple-medium, well test interpretation chart, Fetkoich production decline Introduction As a renewable and clean energy, geothermal energy has great potential or deelopment and utilization [, ]. Carbonate heat storage systems or geothermal energy are the most important heat storage systems besides olcanic heat storage systems. Because o the uniue karstiication o carbonate rocks, there are many ractures in carbonate areas and many oids in the ground, which represent ideal reseroirs or storing hot water [3-5]. Well test analysis is an eectie means or the dynamic identiication in geothermal reseroirs. Howeer, because o high heterogeneity and multiscale low in carbonate geothermal reseroirs [6-8], conentional sandstone-based well test methods can not be used directly. Wu et al. [9] proposed an analytical approach or pressure transient test analysis in naturally ractured uggy reseroirs based on the triple-continuum concept. Jia et al. [] study results show that the cures o well test type are dominated by interporosity low actor, external-boundary conditions and luid-storage capacitance coeicient. Xiong et al. [] proposed a laboratory experiments method o well testing or racture-cae carbonate gas reseroirs and the eects o racture and large-scaled cae properties on the behaior o well testing cure * Corresponding author, rcogr_upc@6.com

2 S78 Wang,., et al.: Transient Pressure and Productiity Analysis in Carbonate Geothermal... THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S77-S84 were ealuated. Chen et al. [] treated the large-scale cae as euipotential body and deemed matrix and racture as double porous media. Gao et al. [3] established a radial composite reseroir model including the inner and outer regions with dierent characteristics. A well test model with changing external boundary lux was established by del Angel et al. [4], although the object o that study was homogeneous reseroirs. Because o the high-storability ugs and high-permeability ractures in carbonate geothermal reseroirs, luid inside the ugs or at the external boundary is more likely to inade the reseroir. Thereore, in this study, a mathematical model or carbonate geothermal reseroirs that considers the changing lux at the external boundary was established and studied. Fetkoich cures were also used to analyze production decline characteristics, which are signiicant or obtaining the dynamic parameters o geothermal reseroirs based on pressure and production data. Well test model and solution Model assumption A physical model o a well penetrating a single-layer geothermal reseroir with constant production was deeloped according to the ollowing basic assumptions: a triple-continuum geothermal reseroir or anisotropic ormation is considered, with uniorm thickness layers that are horizontal with closed upper and lower boundaries; both rock and luid in the geothermal reseroir are slightly compressible and the compressibility is constant; the initial ormation pressure is distributed homogeneously and eual to P i ; the inading luid at the external boundary lows into the geothermal reseroir through the racture system and only luid in ractures lows into the wellbore; the interporosity low is at pseudo-steady state [9]; and the luid low is considered to be -. Mathematical model and solution The ollowing euations describe the low o a single-phase luid with slight compressibility in a carbonate geothermal reseroir: 3.6k m P φmcm φc = φc () µ t t t αmkm m ( Pm P ) = φmcm () µ t αk ( P P ) = φc (3) µ t with a series o dimensionless parameters gien: r 3.6k r =, t = t, k = kxky r µ r ( φ C + φ C + φ C ) w w m m φ jc j k j ω j =, j = m,,, λj= α jrw, j= m, φmcm + φc + φc k kh Pj ( r, t ) = [ P (, )] (,, ) 3 i Pj rt j= m.84 µ B

3 Wang,., et al.: Transient Pressure and Productiity Analysis in Carbonate Geothermal... THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S77-S84 S79 where r is radius, C the compressibility, t the time, ω the storability ratio, φ the porosity, λ the interporosity coeicient, k the permeability, α the shape actor, µ the luid iscosity, the production rate (constant), r w the wellbore radius, P i the initial pressure, and h the geothermal reseroir thickness. We assume that kx = ky and the euations can be transormed into dimensionless orm: m r ωm ω = ω (4) r r r t t t Pm λm ( Pm P ) = ω m (5) t P λ ( P P ) = ω (6) t The initial pressure in the geothermal reseroir is assumed to be P i with a uniorm distribution: P ( r,) = P ( r,) = P ( r,) = (7) m For the inner boundary under constant production conditions: r r = = (8) In e. (9), the inasion lux gradually increases to the inal ixed alue. The ixed alue increases with increasing, R. Furthermore, the time o arrial to the ixed alue o inasion lux is determined by τ and a higher alue o τ indicates a later arrial at the inal ixed alue. t τ r = R e (9) r r= R where R dimensionless geothermal reseroir radius and t dimensionless time. Laplace transormation was applied to the preious model. According to the deinition: s τ i i the mathematical model in Laplace space is: ( τ) P = e P r, d τ, i = m,, d P dp + s () s P = () dr r dr For the inner boundary condition: r = For the external boundary condition with changing lux: dp = () dr s dp R τ R r = ext = + dr s sτ + r= R ()

4 S8 Wang,., et al.: Transient Pressure and Productiity Analysis in Carbonate Geothermal... THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S77-S84 where ωmλm ωλ () s = + + ω ωms+ λm ωs+ λ For the general solution o e. (): ( ) ( ) P ( r,) s = AI s () s r + BK s () s r (3) where In ( x ) is the irst category Bessel unction o imaginary argument, n =,, and K ( ) n x is the second category Bessel unction o imaginary argument, n =,,. Combining es. () and (), the solution in Laplace space is: where u = s(s). P w + K( r u) I( R u) + I( r u) K( R u) s u K( u) I( R u) I( u) K( R u) ( s) K( r u) I( u) + I( r u) K( u) ( ) ( ) ( ) ( ) = + ext R u K u I R u I u K R u Solution o Wellbore pressure considering Wellbore storage and skin actor uring the process o drilling and completion, the permeability o the geothermal reseroir around the well could be aected. Furthermore, luid in the well hole could expand because o pressure drop in geothermal deelopment. In order to consider skin actor and wellbore storage, the dimensionless skin actor and wellbore storage are introduced into Laplace space through uhamel s law, e. (5) shows the bottom-hole pressure solution in Laplace space: zpw ( z) + S P ( z) = (5) z{ + Cz zpw ( z) + S } Subseuently, the Stehest numerical inersion method is applied to obtain the pressure solution o geothermal well P considering the wellbore storage and skin actor in real space. Pressure response and sensitiity analysis Pressure characteristic cure A single well at the center o a cylindrical carbonate geothermal reseroir produces at a constant rate and the lux at the external boundary obeys the exponential unction. Concrete parameter alues can be ound: R =, ω =.3, ω =.3, λ = 4, λ m = 5 7, R =.8, τ = 5, C =., and S =. As shown in ig., in the stage o pure wellbore storage, the pressure (P ) and pressure deriatie (dp ) cures coincide to a straight line with a slope o. Then, a hump can be seen in the pressure deriatie cure, the pattern o which is determined by the skin actor. Ater that, the pressure deriatie cure exhibits three dips. With parameters shown in ig., the irst dip represents the interporosity low between ugs and racture systems, the second dip re- (4)

5 Wang,., et al.: Transient Pressure and Productiity Analysis in Carbonate Geothermal... THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S77-S84 S8 lects the interporosity low between matrix and racture systems, and the third dip describes the characteristics o the external boundary luid lowing into the geothermal reseroir, which can be seen as the interporosity low between external lux and geothermal reseroir. For carbonate geothermal reseroirs, this part o the luid is likely to be the luid in the large-scale ugs or the discontinuous water o the external boundary. In the late stage, when the pressure wae reaches the boundary, the pressure and the pressure deriatie cure again coincide to a straight line with a slope o. The inluence o R on pressure response As shown in ig. (a), when R <, the inlux is less than well production, which represents a eature o the closed boundary. When R =, the inlux at the external boundary is eual to the well production, which is euialent to the constant pressure external boundary condition. When R >, the elocity o the boundary luid inasion is greater than that o production, and the pressure o the whole reseroir system rises, while the dimensionless bottom hole pressure gradually decreases. This paper ocuses on the case o R <. As is shown in ig. (b), the external boundary luid inasion has no eect on the early and middle stages o well testing, while on the late stage with increasing R, the time to appearance o the boundary response will be later and the dip in the pressure deriatie cure will be deeper. Finally, = represents a closed external boundary. R P&dP/dln(t/C) 3 imensionless pressure imensionless pressure deriatie t /C Figure. Transient pressure in a carbonate geothermal reseroir with changing external boundary lux (or color image see journal web site) P 4 3 R =.5 R = R =. P&dP/dln(t/C) 3 R =.89 pressure R =.89 pressure deriatie R =.5 pressure R =.5 pressure deriatie R = pressure R = pressure deriatie (a) t (b) t /C Figure. The inluence o R on the pressure and pressure deriatie cures (or color image see journal web site) The inluence o τ on pressure response The duration o the late transition period is aected by the dimensionless parameter τ. With increasing τ, the external boundary luid inasion elocity becomes slower, the dip representing the late transition period in the well test cure appears later, and the caity amplitude becomes smaller. As shown in ig. 3, when τ = 9, the pressure wae irst reaches the boundary o the reseroir, and then relects the nature o the luid inasion, such that the pres-

6 S8 Wang,., et al.: Transient Pressure and Productiity Analysis in Carbonate Geothermal... THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S77-S84 P&dP/dln(t/C) τ = 7 pressure 3 τ = 7 pressure deriatie τ = 8 pressure - τ = 8 pressure deriatie τ = 9 pressure τ = 9 pressure deriatie sure deriatie cure no longer appears as a dip, but instead as a sidestep that represents the inlux at the external boundary. Fetkoich production decline cure analysis The geothermal deelopment process inoles transporting energy to the ground by geothermal reseroir luid, and when the reseroir luid production drops beyond a certain leel, it will be unreliable to ensure economic deelopment. Thereore, it is ery important to analyze the productiity o carbonate geothermal reseroirs during their deelopment. The widely used Arps decline cure can only be applied to analyze the boundary dominated low stage, whereas the Fetkoich method also accounts or the unsteady low stage. In this paper, the Fetkoich dimensionless production decline cure is used to study the production decline o carbonate geothermal reseroirs with ariable external boundary lux. The dimensionless ariable is redeined: where α = ln [ (/)] and R t /C Figure 3. The inluence o τ on the pressure and pressure deriatie cures (or color image see journal web site) β = t d d ( R ) = α (6) = βt (7) ln R ue to the dierent deinitions o the dimensionless time t d, τ d needs to be redeined: τ d = βτ (8) In order to urther inestigate the characteristics o the Fetkoich decline cure in carbonate geothermal reseroirs with changing external boundary lux, the Fetkoich decline cures o our typical geothermal reseroir types were analyzed: (a) homogeneous geothermal reseroirs with a closed boundary, (b) homogeneous geothermal reseroirs with changing external boundary lux, (c) carbonate geothermal reseroirs with a closed boundary, and (d) carbonate geothermal reseroirs with changing external boundary lux. As shown in ig. 4, or geothermal reseroir type (a), the production declines ery slowly in the early transient low stage and then exhibits exponential decline in the subseuent boundary dominated low. For geothermal reseroir type (b), the Fetkoich decline cure behaes the same as that or geothermal reseroir type (a) in the early transient low stage. In the subseuent boundary dominated low, the geothermal wellbore production irst declines exponentially and then slows down because o the inasion o external boundary lux. In this paper, we ocus on the condition o R <, which indicates that the boundary still behaes as closed ater large lux inasion. This explains why the production also declines exponentially in the

7 Wang,., et al.: Transient Pressure and Productiity Analysis in Carbonate Geothermal... THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S77-S84 S83 inal stage. For geothermal reseroir type (c), only the racture system contributes to the production in the early transient low stage, which results in relatiely lower production compared with the corresponding production or homogeneous geothermal reseroirs. Ater the interporosity low period between the ugs and ractures, there occurs the irst sidestep in the Fetkoich chart. The geothermal wellbore production decline slows down and the second sidestep appears ater the ugs and ractures hae both supplied luid or a while, with the interporosity low between the matrix and ractures. For geothermal reseroir type (d), there then appears a third sidestep, which represents the inasion o external boundary luid, namely, the interporosity low between the external boundary luid and the geothermal reseroir. From the preious analysis, it is apparent that the carbonate geothermal reseroir and ariable external boundary lux can both slow down the tendency o production decline. Conclusions A mathematical model or carbonate geothermal reseroirs with exponential external boundary lux has been deeloped in this paper, and a pressure solution with constant production o the geothermal well has been obtained. Furthermore, well test interpretation charts and Fetkoich production decline chart has been constructed. The ollowing conclusions can be drawn. y Changing external boundary lux results in the appearance o triple dips in the pressure deriatie cures or carbonate geothermal reseroirs. y ierent alues o R relect dierent boundary properties when the external boundary lux obeys an exponential law. While τ d determines the time when the external boundary luid inades into the geothermal reseroir. y The interporosity low in carbonate geothermal reseroirs and the changing external boundary lux show consistent patterns in the Fetkoich chart. The production declines rapidly in the initial period and subseuently enters a stable production period relecting the characteristics o interporosity low. Ater reaching the closed external boundary, the production shows exponential decline. Acknowledgment This work is supported by the Program or Changjiang Scholars and Innoatie Research Team in Uniersity (IRT_6R69). d Homogeneous reseroir with closed boundary - - Homogeneous reseroir with changing external boundary lux Carbonate reseroir with closed boundary Carbonate reseroir with changing external boundary lux Figure 4. Fetkoich production decline chart under dierent geothermal reseroir conditions (or color image see journal web site) t d Nomenclature C compressibility, [MPa ] h geothermal reseroir thickness, [m] k permeability, [μm ] P i initial pressure, [MPa] production rate (constant), [m 3 d ] R dimensionless geothermal reseroir radius, [m] r radius, [m] r w Wellbore radius, [m] t time, [h] t dimensionless time, [s]

8 S84 Wang,., et al.: Transient Pressure and Productiity Analysis in Carbonate Geothermal... THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S77-S84 Greek symbols α shape actor, [ ] λ interporosity coeicient, [ ] μ luid iscosity, [mpa s] ϕ porosity, [ ] ω storability ratio, [ ] Reerences [] Sayigh, A., Renewable Energy the Way Forward, Applied Energy, 64 (999), -4, pp. 5-3 [] Shortall, R., et al., Geothermal Energy or Sustainable eelopment: A Reiew o Sustainability Impacts and Assessment Frameworks, Renewable & Sustainable Energy Reiews, 44 (5), C, pp [3] Goldscheider, N., et al., Reiew: Thermal Water Resources in Carbonate Rock Auiers, Hydrogeology Journal, 8 (), 6, pp [4] Montanari,., et al., Geothermal Resources within Carbonate Reseroirs in Western Sicily (Italy): A Reiew, Earth-Science Reiews, 69 (7), June, pp. 8- [5] Teng, M. A., et al., Eolution o Middle-Low Temperature Carbonate Geothermal System in Taiyuan, Northern China, Earth Science, 37 (),, pp [6] Lucia, F. J., Carbonate Reseroir Characterization: An Integrated Approach, Springer, New York, USA, 7 [7] Vik, B., et al., Connectiity in Vuggy Carbonates, New Experimental Methods and Applications, Transport in Porous Media, 93 (), 3, pp [8] Manuel, G. C., et al., Integration o Multiscale Carbonate Reseroir Heterogeneities in Reseroir Simulation, Journal o Petroleum Science & Engineering, 3 (5), July, pp [9] Wu, Y. S., et al., A Triple-Continuum Pressure-Transient Model or a Naturally Fractured Vuggy Reseroir, Lawrence Berkeley National Laboratory, Berkeley, Cal., USA, 7 [] Jia, Y. L., et al., Flow Modeling o Well Test Analysis or Porous-Vuggy Carbonate Reseroirs, Transport in Porous Media, 97 (3),, pp [] Xiong, Y., et al., Laboratory Experiments o Well Testing or Fracture-Cae Carbonate Gas Reseroirs, Petroleum, 3 (7), 3, pp [] Chen, P., et al., A Pressure-Transient Model or a Fractured-Vuggy Carbonate Reseroir with Large-Scale Cae, Geosystem Engineering, 9 (5),, pp [3] Gao, B., et al., Pressure Transient Analysis o a Well Penetrating a Filled Caity in Naturally Fractured Carbonate Reseroirs, Journal o Petroleum Science & Engineering, 45 (6), Sept., pp [4] Angel, Y. N.., et al., Pressure Transient Analysis with Exponential and Power Law Boundary Flux, Journal o Petroleum Science & Engineering, (4),, pp Paper submitted: March, 7 Paper reised: May, 7 Paper accepted: May 8, 7 7 Society o Thermal Engineers o Serbia Published by the Vinča Institute o Nuclear Sciences, Belgrade, Serbia. This is an open access article distributed under the CC BY-NC-N 4. terms and conditions

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