ray Storage 62 (2028) 106825 Contents lists available at ScienceDirect Ener Storage Journal of Energy Storage ELSEVIER Journal homepage: ww.clsevier.comiocatelest Review article i Recent advances in geothermal energy reservoirs modeling: Challenges and potential of thermo-fluid integrated models for reservoir heat extraction and geothermal energy piles design ‘Mohamed B. Zayed", Bashar Shboul ”, Hongmei Yin‘, Jun Zhao“, Abdelhameed A.A. Zayed ® agrmen of Mechel Bowe gen Fy of Eger Tae Une, eta 31521, pt * Deparment Renee Ey Enger Faay of Een AL AUBay Unie, a, oan «Stance an Tcl Ress ine China Te Goes Caparo, Beli, Ch "Ry Laberay of fo Uaston of Land Mean Gre ry, Ta Uno, Tn 20850, China. * prod Engicrig an Mehl Den Deparment, Fr of gern, Towa Unies, Tans a ‘hermareponse to energy injecton/extraeton. n this context, the optimisation strategies and sing meth- Sling and design of geothermal plants ere presented. Moreover, 4 ctl review of geothermal reservoir ‘modeling fer het extraction purposes is oulined.Dilfreat numerical mods, such 2s the faite element ‘method, fine dference mehod, Sate volune method, ete, which are wed to simulate and descbe the est leagferextaclon of geoteral well, geotal resevois, and geothmal eneay ples are dlcused Furthemore, a general methodologieal pespocive of itgzated anal s presented in an overview ofthe Unite of te reservoir simulation, Moreover, the recent advances In vaious parameters effecting te Best ceracdon performance of geothermal het exchanger including pipe dame, pipe arrangement, pipe pitch, enter t-enter distance, ppe meter, pipe length, and borehole depth are aso discussed. Ths paper can be regarded as 2 decisionmaking too and provide a useful pathway fr sizing parameters selection and usanable eothemal wiston epsom, 1, Introduction ‘lds within its pores and fracture, causing the rock to facture, evolve, and eventually decompose (3)- Fven more, regional and local uerus tions exist due to the non-uniformity of geothermal gradient and hest flow. Most importandy, the average heat flow feom the earth is est [Ax present, the energy crisis is considerably increasing due tothe world population grow, the massive development in industries, the decline inthe curent reserves of fossil fuels, and climate changes Issue 1,2], Wherefore, slentists are being reearched to figure out innovative technologies and concepts to deal with the energy erisis wouble in remote regions [3,"). Geothermal energy is a sustainable energy resource that extracts the thermal energy saved inthe earths erus (5. In principle, geothermal heat is generated when natural radioactive isotopes suchas thorium, uranium, and potassium enclosed in magma ‘decompose during the earth's formation (6,7). Magmas and circulating ‘water aso carry the heat toward the earth's surface. During this process, rocks beneath the earth's surface heat up, which subsequently heats the mated to be approximately 820 GW/m, while the overall worldwide ‘output is about 4 « 108 GW (9) In the meantime, many countries, sich asthe USA. Japan, New Zealand, Germany, Franee, Indonesia, Leeland, tte, consider geothermal energy the primary energy source for elec tricity generation and cooling/heating of buildings 10 In the period veen 1995 and 2022, geothermal technology demonstrated a growth teend, as depicted in Fig. 1. Moreover, it was highlighted that the installed power capacity reached 16 GW by 2022 worldwide; at that point, 8.3 % ofthe world’s power will be generated fom geothermal resources, serving 17 9 of the world’s population (11. Adeitionally, mal eddrese pohmrs aye at-ng anes OE Ze, p//do org/10.1016/: 2022106885, Received 4 November 202; Received in revised form 22 Jamonry 2023; Accepted 4 Febru 2023Me Zot ita iw) 1995 2000 2005 2010 2018 2020 2022 ig- 1. Global insaled geothermal power capacity 1). (C0z emissions can be lowered by billion of tons annually due to the usage of geothermal energy. Further, geothermal resources are fore- ‘asted to provide 5 % ofthe heating load and 3% of power generation in 2080 (121 is worth mentioning that 2800 years of power could be supplied at ‘a constant rate when people consumed about 18 f the expected overall available goothermal energy 13). Discovering the proper location and ‘choosing the suitable extraction technology are the main hindrances to geothermal energy extraction. Typically, areas, where tectonic plate boundaries collide are considered potetial. These locations are located approximately 2km below the earth's surface. Indonesia has some ofthe Ihighese geothermal potentials globally, owing to its proximity to the Pacife Rim. Surprisingly, only 4 % is being used at present, despite the availabilty of around 2.14 GW to be used (14]. In 2010, the USA installed 5.093 GW of geothermal power and was reached co install realy 3.90 GW in 2022 (15) The geothermal resource temperature plays & vitel potential in determining the applications of the geothermal sector. For instance, iret applications such as chemical procesing (16), cooling/eating of buildings [17], agriculture greenhoures (18), hydrogen production 19), saltwater distillation (20,21), recovery of heavy’ fl (22), air en= ‘ergy storage [29], and fish farming (24) hamess the energy stored at shallow depths, whieh have alow temperature between 30°C and 90°C. Curently, low-temperature geothermal resources are being ‘employed for direct applications in around 82 counties, with 70,885 [MW of total installed thermal power and total thermal energy use of 164,635 GWh annually (25). Meanwhile, electricity generation and direct use of geothermal resources at temperatures ranging between 110°C and 160 *G account for around 70 % of the geothermal resources alobelly (26). Technically, geothermal resources. possessing. nearly 150 °C ot higher are used for eleteety production (27). Moreover, resources with a temperature higher than 150 "Care considered high temperature geothermal resources (27), Most of these resources are ound in active voleanic areas and touching the tectonic plate bound aries. Steam vents, hot springs, fumaroles, and geysers form vigorous ‘geothermal activity and are also major high-temperature resources in these locations |27). In addition, electricity could be generated by instaling power plants near areas containing rocks with radioactive ‘materials that constitute high-energy reservoirs (28,29) ‘Another essential heat source is found at depths of 3-10 km beneath the earths surface, formed by high heat production granites or hot dry rock (HDR) Both ofthese reservoirs save lot of hestand are considered ‘8 good source of energy [25]. Compared to fosil energy sources, HDR reservoir energy sources are centralized ata depth of 10 km below the ‘earth’ surface, which contains about two times the amount of energy that fossil energy sources, To pu it another way, the amount of energy ‘contained within @ 1.3 * 10") reservoie is 100-1000 times greater than the amount of energy contained in fossil fuels (30), The main una of Bary Sue 2 (2028) 108585. {feature of the HDR reservoirs is that they are near impenetrable and dy in theie natural state. Many techniques ae used to improve the permeability of the reservoirs. These methods include hydra, éhemical, of thermal simulation [30]. Among these technique, the most prevalent one isthe hydro-fracturing method In hydro-acturng, Dumping & pressurized (above the miaimum insitu principal stress) fluid nto the reservoir’ rock at a given depth is done. As a result, a feature enss and lead othe opening ofthe pre-existing interlocking joins with the possiblity of new factres (The nj Mud ean be ‘moved continuously trough the factures/eracks based on the improvement of the permeability of the reservoir. A heat exchanger exists between the jected hud andl the host rock, represented by the fracture’s surface area, Three physical reservoir measurements can assure superior utilization ofthe resourees in an eshanced geothermal system (BGS) reservoir. Inthe EGS system, two stages are involved: drilling a well int the reservoirs ock and injecting pressurized water to ‘widen natural fractures or open joints Modeling the performance of geothermal reservoirs using numerical methods isan important part of understanding the interaction between new fui injected into the reservoie and the existing ids therein. lis also an effective way of understanding the mutual influences between the reservoir's rocks and the injected uid. Water injetion ata ten perature lower than the reservoirs fig woutd stimulate the reservoirs mechanical, thermal, and chemical equilfrium and thus increase the porosty/permeabiity ofthe reservoir. Indeed, there sa elar corel tion between permeability and porosity of the geothermal field changes in peciitaion/dssoltion kinetes, reservoir mineralogy, and working conditions. However, the mutual influence among these procescs is often intricate and occurs concurrenly while injection/productionoc- curs. For safe and sustainable exploitation of the geothermal resouree, is vital to understand the attzude of these processes inside the reservoir. Furthermore, to improve the efficiency of working conditions based on the chemical and physical eonitions ofthe servo, the prediction ofa spatiltemporal behavior ofthe reservoir, as well asthe seale modeling ae eritieal elements ofthe optimization proces. Many previous review studies have been aimed to demonstrate the developments and challenges of geothermal technologies including: the development of geothermal energy pilings for improved sizing [52], heat transfer improvement of geothermal heat exchangers [33], geothermal electrety generation (34.95), geothermal pants install. thon (25, geothermal use of abandoned gos and oll wells 31, coupled hhydro-thermo-chemical-mechanical approaches (7), alterative mate- rials fr geothermal wells (98, stimulation mechanisms of EGS [59 and economics of EGS [10], However, a few reviews have been attempted on the hydraulic and heat transfer processes of geothermal wells, geothermal reservoirs, and geothermal energy piles utilizing different geothermal modeling for heat extraction. In an earlier review study established by our research team (35), the opportunites, chal lenges, and furare tends of geothermal electricity generation (GEG) technologies in China were presented. The potential of realizing the established GEG strategies in China since 2020 was fest highlighted Moreover, a methodological analysis was performed to determine the essential technical obstacles, inching the immature technology of GEG plants and the pauper energy endowment of the Chinese geothermal resources. was asserted tha various opportune for ameliorating the grade of GEG in China would be based on toning abandoned gas wells into geothermal wells, promoting mult-energy harmonizing systems, oF exploiting HDR resources Herein, the methods of numereal modeling of geothermal systems 3s support tothe sizing and design of geothermal plants are presented “Moreover, a cial review of geothermal reservoir modeling for heat extraction is outlined, Different numerical methods such asthe finite element method (FEM, ite difference method (FDM), finite volume method (FVM), ete. to simulate and describe che hea ransfr extraction af geothermal wells, geothermal reservoirs, and geothermal energy piles ae discussed underlying the recent advances in modeling approachesMr cay eat and the various parameters that are affecting the heat extraction per formance of geothermal systems. The challenges ofthe system's heat ‘extraction performance modeling and the survey gaps in previous studies are discussed and highlighted. Finally, perspectives and limits tions on the measures to boos: heat extraction rate and reduce energy ‘eonsumption are also critically presented for future research recom- mendations. ig. 2 shows a lowchaet describing the overall framework ‘and organization of the literature research reported inthis paper 2. Applications of geothermal energy worldwide ‘The installed capacity of geothermal energy globally is ulized either indireey in generating electricity or using t directly in thermal, in- dustrial, and agricultural applications, as will be presented in this ‘According to 8 recent survey, the world’s installed eapacity for ‘geothermal electricity production is as indicated in (Pg. 3) (#1). Ae ‘cording to recent statistics reported by the Fnergy Information Admin- istration, the USA produces more geothermal electricity than any other ‘county in the world, producing nealy 17 billion KWh from geothermal ‘power plants acrss seven states. Accordingly, it represented 0.4 96 of the total electricity generated at the utlity-scale in the USA (10) ‘Meanwhile, the existence of 200 ative voleanoes in Indonesia means that Indonesia has contsibuted to 40 % ofthe whole world’s geothermal ‘energy potential. Surprisingly, only 5 9% is used for generating elec- ‘wicity. To improve this percentage and develop geothermal use in the county, the government simed to gonerate 7 GW by 2025 (34). In ‘Mexico, geothermal plants are employed to produce 2 % of electricity, 42). In addition, Turkey aims to 600 MW electricity generation by 2023 utilizing geothermal resources with an overall potential of 4.5 GW (43) ‘The HDR resources are being utilized to generate >5.0 GW of power in Ira of rary Surge 52 (2023) 10585 Ethiopia, which has great geothermal energy potential (4) Direct geothermal energy utilization is one of the most prominent and diverse forms of geothermal energy exploitation. In particular, ‘China isthe leader ofthe world inthe direct use of geothermal energy ‘Three areas of special attention are proved by the hot springs (461 Firstly, medical and recovery aspects, as recent medical research ex- Ibis, many diseases, particularly the eure rate of skin diseases, are ‘herapeutiealy fected by hot springs. Afterward, the second place Is represented by agricultural, Industial, and fishery applieations. Atlas, the tie aspects the bathing applications, which utilize around 60 96 of the thermal fields. 1 is noteworthy that China depends on sever ‘geothermal proeets for district heating, while geothermal heat pump projects are located in more than eight regions. Moreover, China has ‘many greenhouses, industrial, agricultural drying, swimming/bathing, and washing projects. Most importantly, China used the annual direct ‘amounts 25 follows: for district heating; 90,650 TJ ané 7011 MW, for heating greenhouses; 4255 TJ and 346 MW, for agricultural drying; 2145 TJ and 179 MW, for fish farming: 5016 TJ and 482 MW, for swimming and bathing: 86,993 TJ and 5747 MW, for industrial process ‘neat; 8221 TJ and 395 MW, and geothermal heat pumps 246,212 TJ and 26,450 MW, thereby, a total of 443,492 TJ and 40,610 MW [47,8 In the meantime, annual dicect uses in the USA sce: 958.3 TJ and 89.60 MW for distriet hesting, 1073.2 TJ and 89.43 MW for individual space heating, 730.2 9 and 79.78 MW for greenhouse heating, 97.5 TJ and ‘6.45 MW for agricultural drying, 2241,9 9 and 122.13 MW for fish arming, 8.0 TJ and 2.34 MW for ania farming (other), 2153.27) and 89.85 MW for swimming and bathing, 18.6 TJ and 2.06 MW for ‘meiting the sow, 17.6 TJ and 0,90 MW for industrial applications, ad 145,460 TJ and 20,280 MW for geothermal heat pumps, giving a total of 152,810 TS and 20,712 MW with a capacity factor of 0.23 [49 In Spain, the numerous annual uses of geothermal energy are as Problem Definition: Literature on Modeling of Geothermal Energy Reservoirs Finite element method (FEM) Finite volume method (FVM) Fig. 2 Howehart of the over amewoek abd contrbutio of he conducted erature researchMr cay eat in MW Indonesia 1,935 Philippines Turkiye New Zealand Mexico Kenya Ite Iceland 754 aly Japan 621 (Other 1.097 ig 3. Geothermal elecrty gener cones in 2022 45) capacity In MW i the highest ten follows: for greenhouse heating; 165.4 TJ and 22.0 MW, for various spaces heating; 133.6 and 5.20 MW, for swimming and bathing; 92.0, ‘TJ and 3.80 MW, for geothermal heat pumps 3542.0 TJ and 513.0 MW, thus aggregating co a net of 3933.0 TJ and 544.0 MW (50). In Germany, ‘eathermal energy has various uses per annum: for district heating: $3634.87 TJ and 346.2 MW, for swimming and bathing 1708.56 and 56.8 MW, for Individual space heating: 35.21 TJ and 3:34 MW, for _geothermal heat pumps; 23,760°T9 and 4400 MW, and ths ora total for Germany of 29,138.6 1) and 4806;3 MW (51). Even though the total ‘number of ground-source heat pumps (GSHP) installed in Japan is stil ‘relatively smal, the numberof GSHPs is growing at an exponential rate For instance, GSHPs are used in 2662 foclities using is 2652, with 327 ‘open-loop systems, 2314 closed-loop systems, and 21 facilities that use both types of systems (92). In tren, industrial applications and green- ‘house heating are expected to utilize 81.3 MW of geothermal energy 52 3, Numerical ulation of geothermal wells "To interpret the coupled heat transfer and fluid flow processes in geothermal well systems (GWS) and their influences on. long-term ‘operation. A GHS simulation that ean concomitantly solve all the complicated nonlinear governing equations deseribing energy, ms, and heat transfer in an integrated coupled manner is ital forthe long term realization of the injected Huds’ interaction with the rocks of the ‘reservoir within GWS, The rexervoir ofthe GWS is embedded ina rock or sol mass tht interchanges the heat wth ts environment. The constit- ‘ent laws are distinguishing for each type of reservoir, while the analytical and numerical analyses are also substantial to fulfill areliable solution forthe GWS. The up-growth of the numerical simulation should ‘be tro main key issues, namely, the physical phase ofthe GW, and the ‘usage scenarios during the geothermal exploitation. elf ry Serge 62 (2028) 106895. 2.1. Numerical model conceptuaiation Miscellaneous phases in the structure ofthe GWS modeling can be recognized. An essential primary step of interpretation and collection of| data must be performed initially for the building of the conceptual model. Then, a model blockstruecure is festly conducted with the related datasess of the hydrogeological parameters which suit what is waited by the developed model, as it should take into aecount the Inydrogeologial structure of the geothermal reservoir, geometric ‘itera of wells and fractures: process coupling conditions such as ‘ermal, mechanical, hydraulic, and chemical conditions. Secondly, a further procedure of refinement and calibration should proceed, [tis an iterative process during which the boundary conditions (BCS) and ‘hydrogeological parameters are adapted based on the physical and ‘conceptual mode! previously described. Various thermo-physical prop- ‘ties (density, thermal conductivity, permeability, capacity of specific heat) vary with depth and temperature. The values ofthese properties should be set, and the mesh can be purified at this step. In ease of high ‘uncertainty about both the heat and energy transport praceses and. hydrogeological datasets, simplified 1D or 20 models canbe firstly run ‘Thereafter, geothermal energy exploitation scenarios should be then processed sarting from the unperturbed state simulation as boundary ‘onditions (32) For sustainability assessment considerations, in Pg. 4, an informa tion flow diagram concerning how the establishment of GWS numerical ‘modeling can asist the susiinability evaluation is effectively consid ‘ered, To recognize that the size and type of the GWS are specified, the ‘geometrical domain should be large enough to understand every mot ‘ating energy/mass transfer boundary, but also small enough to obtain ‘an appropriate computation time (53). Recently, diversified sofware ‘such at TOUGH, COMSOL, GeoFrac, TRNSYS, and mesh generators ike FLUENT employ to solve very complicated GWS geometries, but i i better to start with simple mesh kinds and domain configurations (eg. radial, quadrangular. A refinement of the mesh can be beneficial only for the ares associated with energy/mass transpots, ike fe reinjoe- ‘on, and surface manifestations ofa GWS. Once the mesh and geomery are set up, the equation input parameters ace considered. Mote speci lealy, the constituent laws (e, Darey Laws) are applicable and effec tively describe the energy/mass transport occurring in the coupled _geothermal reservoir-wel system (5). The supposition ofthe values of the well parameters and soll properties is crucial through the modeling definition. One elementary value of a considerable parameter such a5, ‘permeability or porosity is ypecifed toa layer or element (likely with an ieregular shape), Oftentimes some of these parameters have no potential ‘of assigning tensors or directional parameters, as they deduce fom precise interpretations and measurements. And then, the chief role of validation is recognized, a it permit the showing of parameter values ‘ting with eal output data of she GWS when reliable measurements are ‘obtainable [55]. The key factors to be essentially assigned are: + Porosity is the property ofa fraction of the void's volume over the ‘entire volume ofthe GWS. + Permeability isa property of both the injection heat carrier id ICE) and the roek, showing a conception of the productivity of ‘erin rock formations. + Thermal conductivity of oth Maid and rocks. + Density ofboth fluid and rocks. + Specific hea capacity of both fluid and rock. 3.2. Heat transfer extraction in geothermal well and reservoir models and {governing equations formulation GWSs most have 2 to 5 cemented casings and hotter and hotter reservoits, which are required a considerable numberof casings (56). Te ‘elucidate geothermal wells in reservoir simulations, -D finite elements ‘that evenly conser the heat transport between wells and encirclementMr cay eat olf ry Sores 62 (2029) 106895. Sustainability “Assessment Fig 4. Information ow diagram ofthe development af the numerical modeting ofthe coupled reservot-well modeling rocks are deduced based on the single well structure (hy. 5), Herein, the governing equations of the vertically single geothermal well and ‘geothermal reservoir models will be described as well showing. a detailed framework forthe sizing and desig ofthe geothermal system. 3.2.1. Geothermal reservoir made Reservoir rock Darcy's law is employed to formulate the low of uid in the reser- voir rock The conservation of mass and momentum equations describing the flow behavior of the geothermal HCF within the reservoir In the rock macrx isa given (57); o ‘here isthe rock's porosity, pis the éensity ofthe HCF, wis the flow speed, and Sy the mass transport between the rock and fractre ma tei The Darey flow velocity w is computed as; w= 2 (Pep, 899 2 2 ‘where 4 is the permesbiity of the intrinsic rock, sis the dynamical viscosity coeticient of HCF, P Is the static pressure, which ean be ‘computed in terms ofthe water level h and the well depth Hf Pape (Hen) eo “The energy balance forthe fully and grid-saturated rock matrix can ‘be charaecerized by the dtfuson-convection equation (58); Gg Bt nsGas WNT V. (Ra VT) He “w ‘where Gus the heat capacity of HF, and qi the heat sink (or sour). Heat exchanger, errs pomel Inner pipe Casing Cement Fig 5. Schematle of heat wanser and Dud of the vet! sage model ofthe GWS (57Me Zot ita ‘The thermo-physicel properties are averaged to consider the fid and, the reservoir ocks volumetrically as fellows, “The mean volumetric heat capacity of rock matrix (pGeg i (2715 (Gg = AG All 40.6.9 where ps isthe density frocks, Gs the heat capac rocks “The effective thermal conductivity, Keg, canbe calculated in terms of the conductivities of the HCF Ky reservoit rocks K, 7 Key = Bil —@)4+ Kp © Fracture elements ‘The tangential formula of Darcy's lw is utilized to demonstrate ud mass and heat transfer in fractured elements (59); dy (CrP 40, V2) o Here, Vp isthe volumetric low rate through the use of the facture clement, i isthe permeability of fracture, dp i the fracture element aperture, and Vrrefers tothe gradient operato limited tothe tangential plane of the facture clement. “The mass conservation equation incorporated over the cross-section of fracture canbe described as (60), Lie z £ (re 100) Nu 3000 < Re © 58,05 < Pr < ® ca Bde py uot o 40) Ve ldekin VT) He "The high temperatures range in the reservoir neluctaby impacts the [HCE density, thermal conductivity, viscosity, and heat capacity, and. therefore the velocity pattern and heat flow rate. Hence, itis eucial to ‘consider the change in thermal conductivity, fluid density, and viscosity as 2 function of temperature [61] BAST ay FCT Ky = AITI Gy = 17) ao 3.22. Geothermal well model ‘The heat transfer in the GWS comprises convective, conductive slong the axis ofthe well and the heat transfer becween the surrounding rocks ‘columnar and fd co the geothermal wel axis Conservation of energy forthe fluid in a geothermal well For a fluid flow through a 1D element of GWS, the energy equation forthe flow is (62), a Loaw Au VT VAKVT = an 2000 Forres una of Bary Sue 2 (2028) 108585. where Ais the well area, wis the average flow speed along the well axis, Ana Eepresents the interchange heat through the wall ofthe GWS. “The friction factor is formulated as a function of the Reynolds number (Re) and the roughness ofthe well surface (e) as given; | ee a2) ‘Heat transfer between the surrounding rocks andthe fluid “The heat transfer between the rock matrix and the fd Qua can be expressed as (62): 2e( 7) nas = a3) ‘where T; and Tj denote the temperatures of the surrounding rocks fui in GWS, respectively The film convective coefficient can be calculated as; Mak a an [Nusselt numer Nu is expressed by (6315 reo convection os) 43. iil sit and boundary conditions In the cave of fluid flow, the initial conditions are primarily deter- ‘mined by the difference in pressure between the injecting and producing, 1 ¢= 0, the pressure is hydrostatic based Eq (3) ro) , ar the injection wel locaton P (see Sp Soe) Bg = Pea the proton wel neton an ‘able Boundary ad inal condions for geothermal well end reser fincion Plt ntave Fh el” Ty 08) Ty i Sean on trae ° ream Ef ‘Where Ty the rock masts temperature surrounded the well Tithe aner temperatize of he casing, the open hole lug ofthe GS.Irural of rry Surge 52 (2028) 10585 ees Prey er eee 7 ere ig. 6. Computation procedure forthe ‘The primary temperature in the well is defined by the measured ‘thermal gradient AT and the ground surface temperature T, a5; rersarz as) ‘Theo, the BCs can be stratified by applying elther Neumann or Dirichlet BCs on the bottom, lateral and top boundaries, as well as atthe ‘coupled interfaces between GWS and reservoir rocks, as clearly indi= ‘ated in Table 1 ‘Based on the sbove mathematical formulation, the thermal perfor- ‘mance of such GWS in any geothermal field can be simulated by means fof the calculation flowchart for the coupled well-eservoir model ‘demonstrated in Fig. 6. 4. Existing modeling technologies of the reservoir heat ‘extraction and geothermal energy piles design Reservoir modeling, design, and prediation of the Spatiotemporal behavior ofthe reservoir may assis n optimizing the operating condi tions accountng for the chemical and physical tates ofthe reservoir. To investigate the behavior of the reservoir, several analytical models, ‘numerical methods, and software tools were adopted for performing ‘numerical simulations for the prediction of the performances and opti- ‘mization ofthe processes of the geothermal reservoirs, 41. Analytical models ‘Most analytical models for the BGHE apply an infinite line source ‘model in which grouted BGHE i considered asa line hea source within ‘homogeneous ground environment, where thermoptyscal dimensions ‘of & BGHE are unconsidered, andthe constant intial ground tempera- ture taken (35). This model ean instantaneously caleulate the response ‘of temperature ata certain distance orthogonal tothe line heat source axis in terms of the input of heat output per length of the line source, ‘ground difusviy, and thermal conductivity ofthe ground. ‘On the other hand, the eylindical source model where the size ofthe [BGHE is considered can also estimate temperature response atthe wall, ‘ofthe borehole 6). This analytical model needs the same input criteria asthe line source approach, besides the outer diameter ofthe borehole. ‘Both analytical models are mostly used to depict the heat transfer bchavior outside the BGHE via calculating the wall temperature ad ‘thermal resistances ofthe BGHE. combined reservoie-well modell. Eskilson (65) originally introduced the line source model concept, ‘which proposed a finite length of the BGHE. The boundary and initial ‘conditions and the ground temperature were considered constant as wel 4s the effect of borehole heat capacity was neglected. He derived a temperature response dimensionless (y) at the wall of the borehole, ‘which was defined ag function to simulate the performance of BGHE. for different GEP configurations, ané can be described as; =! where Ty and Tyre the borehole wall and initial round temperatures, q isthe heat ux per borehole length, H and 7 are the length and radius of ‘the borehole, and ais the soll dtfsivty. “Zeng et a. (55) developed the analytical solution of Fsklson’s line source approach, which was further defined as two steady-state wall temperatures of the borehole, the center point temperature, and the ‘average temperature, The differences bevween the two maels were compared, and good agreement was obtained. Javad et al. (67) analytically proposed a short-term response modeling for the BGHE, which considered the thermal capacitances of flowing fluid and bore- ‘hole grout, appreciate for determining the average lu temperature of te borehole. In this model, a single equivalen-pipe diameter was ‘considered to simulate the pipe legs of a single U-tube, in which the Circulating fluids temperature equals the mean of return end supply Pipe leg temperatures, Moreover, radial lor heat transfer was modeled ‘and solved through Laplace transformation ‘Zhang et al. (68) deduced of advectionconduction heat transfer ‘equation ina porous medium to describe the advection of groundwater ‘with an infinite line source model It was considered radial groundwater flow at such speed through an iafaitely porous media. Moreover, twas analyze thatthe advection of water in a BGHE fed, indicates thatthe ‘How of groundwater permits to fulfil of a steady-state heat transfer much quicker than those without water advection, It also tabulated the hydraulic and energetic properties of diverse solls and rocks fora fast, ‘evaluation of the advection of the water effect on the energy perfor. ‘mance of GEPs ‘hang etal. (62) proposed finite and infinite solid cylindrical source ‘models to investigate the effect of groundwater advection. It was ind ‘cated thatthe advection of groundwater could signifcandy augment the as) aME Zot tat ‘neat ranser performance of GEPs upto ive compared to that ofthe case without advection of groundwater, Wang eta. (0) further improved the validation accuracy of Zhang etal. model [69] based on ANSYS software. It revealed enhanced analytical modeling which was accurate at velocities of groundwater higher than le" m/s, ‘A coupled analytics! model based on the combination of cylindrical ‘and line models was proposed by Hu etal 71). The features of the proposed model were Its susceptibility 1 shortatime step simulation, which considered the varying heat capacitance of the GEP. The hybrid ‘ylindica/line model was wsed in TRT to obtain the volumetsc heat ‘capacity and soll thermal conductivity. It was shown that the volumetric hheat capacity and soil thermal conductivity were decreased by 20 5 ‘compared tothe line-source madel results, Moreover, the hybrid model ‘was verified against real TRT results and showed a fit verification with ‘TRE results Gordon et al. (72| used a composite cylindrical heat source and TAT procedure to evaluate the ground thermal conductivity and fuiétem- perature fora U-tube and coaxial BGHEs. An RMSE of 0.1 °C was indi- ‘ated after the ealeulation of the ground thermal conductivity, which ‘was decreased from 3.9 W/m"C for the coaxial BGHE to 3.7 W/m" for the U-tube BGHE, ‘A ground modeling addressing phase change influences by handling the effective heat capacity method to assess heat transfer fom the wall ‘ofthe borehole to the ground was developed by Najed et al. (75]- The heat eapacitance of fd and thermal interchanges between U-tube Tegs Jn the BGHE model were considered. Moreover, the BGHE model ‘considered the axal uid temperature change along with the depth of the borehole. Heat transfer within the BGHE and in the soil region was ‘modeled. They showed that the wall temperature of the borehole remained about OC for various days when the ground froze while it ‘decreased to lower values in non-feezing circumstances. ‘Yin etal. (7) developed an integrated distributed geothermal sys- tem powered by shallow and deep geothermal scenarios based on the reverse and forward cycle for heating, cooling, and electricity tri- ‘generation. The analytical analysis indicated that the energetic eff- clency of the multisystem in the power and heating made was 16.10 96 higher than that in the power and cooling mode with high accuracy ‘compared tothe results of the experimental eld, Ma eal. (75) adopted the analytically piecewise technique both on the depth dimensions and time scale, to simulate the heat extraction processes of downhole coaxial heat exchanger (DCHE) The model resus Were verified with reported field experiments, and the efficiency of the heat extraction was defined for detailed assessment of heat extraction ‘characteristics of DCHE, It is asserted that the proposed analytically piecewise approach as a function of the time and heat flux via the borehole wall instead of the borehole wall temperature as the inte- rating interface obtained more accurate fluid temperatures and heat Wei et al. [76] modeled the temperature behavior in fractured ‘geothermal reservoirs considering the Joule-Thomson, adiabatic heat ‘compression expansion, convection heat, and conduction hest effects. ln these models, a thermal interporosity and three heat radial Now re- -gimes with V-geomietry characteristics were identified, The simulations ‘declared that the coefficient of matrix thermal interporosity and facture thermal scoratilty are derived fom the thermal interpoosity zone from the temperature derivative curve, Moreover, the temperature transient analysis estimated the adiabavc expansion heat coefcient, Joule- ‘Thomson coefficient, and fracture intrinsic porosity Galyao etal 77) developed a coupled unsteady analytical modeling of reservoir/wellbore/tubing/easing systems. The density of well hore-lud was modeled in terms of temperature, and the unsteady heat-flow differential equations were solved through the Laplace transformation. The proposed model yielded more accurate tran- sienttemperature-low profiles along the wellbore in comparison with Previous analytical models. una of Bary Seg 62 (2028) 108885. 42. Numerical models ‘Numerical borcholes/energy piles models ean be categorized into ‘three approaches based on finite element, nite volume, and finite dif= ference methods 42.1. Rite dference method Rotumayer etal. (78) developed quas-3-D FDM co simulate the ‘heat transfer behavior within the BGHE region. Axial heat tansfer in the ‘uid was modeled, while axially heat transfer in grout was ignored. In \bismodel, the borehole mesh was splitintaacyindeal grid of nodes 10 consider the radial heat transfer, Lee otal. [79] proposed 3D modeling fora single GWS based om DM. In this study, two various sil regions were considered; one was soil surrounding the GEP inthe radial direction, and the other was a layer of bedrock below the GEP. Morcover, the temperatures and heat capacitances of fluid and pile grout were modeled, whereas the he éapacitance ofthe pipe wall was neglected. It wes applied for diverse U- Pies and various configurations of pipe connections. The model results ‘were compared toa finite line source model results and showed a high degree of validity with the analytical solution wih a maximum devie- tion of 0.11 °C. it was also indicated that 4 U-pipes parallel legs connection had a marginal effect on the GEP performance, causing a temperatute variation of 0.02 °C compared to series legs connection. However, the reduction i the distance between U-pipe legs resulted ina considerable deerement in pile heat rejecton/extraction performance Fare et al [0 developed a cylinder heat source model, which used explicit FDM to investigate the temperature distribution within axial and ‘ail pile/sil schemes fora single GEP, as seen in Fig. 7. The model was accurately conducted at a short-time step, in which heat capac tances of grout, ud, and sol were considered. The model was abe to consider non-homogencous soil/ple material conditions. Heat transfer fluid was simulated ony in the axial direction, and the intial BCs such 13s top boundary, radial boundary, and bottom boundary temperatures were selected as node temperature values. Hence, it was found tha the temperature of the variable surface could be adopted in the model, which was very vital for modeling the thermal interface and energy pile ofthe floor structure, Moreover, a sensitivity analysis showed thet the pile energy output depended significantly on the temperature difference Derween soi and inet Muld temperatures, the circulation pipe site, and ‘he soil thermal conductivity Holmberg et al. (81) numerically used the FDM to predict the per- formance of coaxial BGHE, Parametric analysis forthe coaxial BGHE with different flow rates, borehole depths, and borehole properties was borehole depths. It was illustrated that the deeper ACHE was well efficient in ameliorating the performance of heat extraction. More- over, a performance chat for coatal BGHEs with depths of 200-1000 m ‘was presented, which can be helped as @ guideline when designing BGHES. Song eal (82) analyzed the energetic performance of coaxial BGHE, and the FDM was applied to solve the modeling. It was revealed thatthe outer temperature remarkably deereased at che primary stage and then remained stable relatively. Also there was a eriteal low rate value to fulfil larger heat power with a suitable pressure drop, Moreover, the cement sheaths thermal conductivity ad a geeat effect onthe thermal processes Yang et al (88) integrated the FDM modeling and Monte Carlo ‘method to quanltaively assess the impacts of diferent variables on BBGHE performance. The results pointed out thatthe bottom temperature was reduced with an increase inthe density, circulation time, and heat capacity of the fuid. Moreover, it also increased with the increase in ‘thermal conductivity and inlet temperature. Furthermore, the temper- ature initially dereased and then increased asthe flow rate increase. 42.2, Finite volume method Heer al [5] slmulated the performance of HE using FVM in three