Dilation occurrence analysis in gas storage based on the different constitutive models response

Journal of Applied Mathematics and Physics, 014, *, ** Published Online **** 014 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.436/jamp.014.***** Dilation occurrence analysis in gas storage based on the different constitutive models response Z.Shahmorad/H.Salarirad/H.Moladavudi (Affiliation): Department of Mining and Metallurgical engineering, Amirkabir university of technology, Tehran, Iran Email: Zahrashahmorad@gmail.com Received **** 014 Copyright 014 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract Caverns constructed by solution mining in salt domes are reservoirs with large dimensions and low working costs. Because of special properties of salt, salt domes are considered as one of the best geo formations for storage of energy resources like oil, natural gas, compressed air and nuclear waste for decades in industrial countries. Among all, storage of natural gas as the second most demanded fuel has a special importance. In fact countries' ability in storage of natural gas has an important role in their energy managements. Predicting the behavior of storage reservoirs which are constructed in salt structures is so important. Until now researchers have used different constitutive models to estimate the behavior of underground gas storages. Burger analytic model, Power Law and WIPP empirical models are the most common models. In this research the behavior of an underground gas storage has been predicted by utilization of Burger, Power Law and WIPP constitutive models. Differences resulted from these different constitutive models utilization in dilation occurrence around the caverns have been discussed. The results have shown that the Power Law and WIPP models are more capable of predicting dilation occurrence around gas storage reservoirs. Keywords storage reservoirs, salt domes, predicting the behavior, Dilation occurrence, Burger, Power Law, WIPP constitutive models 1. Introduction How to cite this paper: Z.Shahmorad, H.Salarirad and H.Moladavudi (014) Study the effect of different constitutive model utilization in dilation occurrence around an underground gas storage made in salt structures. http://dx.doi.org/10.436/jamp.014.*****

Increasing demand of natural gas and also it's fluctuated consumption in different seasons and months of a year, are important reasons to develop the underground storage caverns in industrial countries. The essential role of these storage caverns is to regularize high and low seasonal consumption rate in gas network. In addition to this important function, also in terms of strategic issues, gas storage is so important. In fact, the industrial countries by means of gas storage cavern's extent, have maintained their authority and decisive role in the world energy market. Due to special properties of salt rock geologic formations, caverns made in salt structures have been used as storage places from several decades ago. These properties include, impermeability, suitable mechanical properties, self-healing and solubility by water injection. Time dependent and complicated behavior of salt rock, has made salt cavern prediction behavior to a challenging issue for rock mechanics researchers. Most of researchers have studied the storage cavern's behavior by utilization of different constitutive models and have analyzed salt cavern's stability in different conditions. One of the key parameters in stability analysis of gas storage caverns is dilation occurrence. Because rock dilation due to growth of micro-cracks, increases the permeability and leakage which are not desirable for storage caverns. Therefore dilation occurrence issue has always had a great importance in behavior prediction of storage caverns. In this research a gas storage cavern constructed in a salt structure, was modeled and by use of finite difference method, the differences in dilation occurrence resulted from different constitutive models utilization have been discussed.. Constitutive Models In physics and engineering a constitutive model is a relation between two physical quantities. It approximates the response of the material to external factors, usually as applied fields of forces. Constitutive models are combined with other equations governing physical laws to solve physical problems. For example in a mechanical analysis a constitutive model relates applied stresses or forces to strains or deformations [1]. As the salt rock has a time dependent behavior, the constitutive models used for describing salt rock behavior should be time dependent. In the following some of time dependent constitutive models which are highly used for salt rock are presented..1. Burger Analytic Model A Kelvin element connected in series with a Maxwell element, constitute a model that exhibits instantaneous strain, transient and steady state creep evolution stages, known as Burger model. The initial response of this model to a suddenly applied stress is governed by the spring constant of the Maxwell element. Transient creep is due to the Kelvin element, while secondary steady state creep is due to the dashpot of the Maxwell element []. A schematic sketch of Burger model and its strain variation versus time is shown in "figure1". Figure 1.Schematic sketch of Burger model and its strain versus time curve [3].

Zahra Shahmorad In this model, total strain equals to summation of strains created in Kelvin and Maxwell models. "Equation (1)" shows Burger constitutive equation. G t k k t t e. (1) 9k 3 3G 3G 3G m m k k In "equation (1)", is the applied stress, t is total strain, k is the bulk modulus, m is the Maxwell viscosity, G m is the Maxwell shear modulus, G is the Kelvin shear modulus and k k is the Kelvin viscosity. These parameters can be estimated by using a creep test [3]... Power Law Model Utilization of the empirical Power Law is common in modeling the time dependent behavior of salt rock. "Equation ()" shows the standard form of this model. Q n RT D e. () In "equation ()", is strain rate of the secondary creep, T is temperature in Kelvin, D, n and Q / R are rock properties which can be estimated by using creep tests at different levels of temperatures and stresses. is the von mises stress which is estimated by "equation (3)". 1 1 3 3. (3) In "equation (3)", 1, and 3 are principle stresses[4]..3. WIPP Model WIPP Model is an empirical model for describing time and temperature dependent behavior of salt rocks. At the beginning it was developed for studying nuclear waste sites in New Mexico. But Later, WIPP model was also used as a tool for studying time dependent behavior of underground salt caverns used for hydrocarbon storage. In this model secondary creep strain rate ) s ( and primary creep strain rate ) p ) are estimated by "equation (4)" and "equation (5)", respectively. * A B, if Q n RT s D e p p s s ss. (4) *. (5) ss * p A B p s, if s ss s * In "equation (5)", ss is the critical strain rate and A and B are material constants that can be estimated empirically. As it seems, secondary creep equation in this model is the same as Power Law secondary creep equation. So the only difference between Power Law model and WIPP model is that the primary creep stage is not considered in Power Law model [4]. Burger, Power Law and WIPP constitutive models, are viscoelastic models and are not able to model the plastic behavior and failure of materials. Therefore in this research their corresponding models in which failure occurrence is considered by implementation of a plastic criteria, are used. In these models plastic parameters are defined, so the failure possibility can be discussed. By considering failure possibility, one may have a more comprehensive sight about salt rock behavior. These models are Cvisc, Cpower and PWIPP which corresponds to Burger, Power Law and WIPP constitutive models respectively. 3

3. Dilation Damage of salt rocks due to dilation which is increased by increasing deviatoric stress, is one of the important instability factors of underground rock structures. Salt rock dilation due to growth of microcracks, increases the permeability and leakage which are not desirable for storage caverns. When a hydrostatic loading is applied to a rock specimen, the volume of the specimen decreases due to closure of microcracks. As the deviatoric stress increases, the specimen volume decreases somewhat further until the so called dilatancy boundary is reached. Beyond the dilatancy boundary the volume increases till the occurrence of the failure due to formation of additional microcracks [5]. Researchers have developed equations for dilatancy boundary of salt rock by using triaxial experiments on salt rock specimens. Among these equations, equation developed by Ratigan (1991) have been used extensively by researchers as a safety factor for underground storage caverns made in salt structures. By using this equation, dilation beginning can be estimated and cavern safety versus dilation occurrence can be assessed [6]. "Equation (6)" is dilation safety factor equation developed by Ratigan. SF 0.7I 1. (6) J In "equation (6)", I 1 is the first invariant of stress tensor and J is the second invariant of deviatoric stress tensor. When Sf<1, shear octahedral stress is greater than mean stress and rock dilates. In the following, an underground gas storage is modeled and differences in rock dilation occurrence caused by different constitutive model utilization is discussed. As was mentioned before, theses constitutive models are Cvisc, Cpower and PWIPP. 4. Storage Cavern Modeling 4.1. Cavern Geometry In this research, a capsule shaped cavern was modeled. The height and diameter of the capsule are 140 and 40 meters respectively. It was considered that the cavern is located 900 meters below the ground surface. As the cavern geometric conditions are completely symmetric, only one fourth of the cavern was modeled. The made model and its dimensions are shown in "figure ". Figure.Geometry and dimensions of the model 4

Zahra Shahmorad 4.. Input Parameters for Constitutive Models One of the major obstacles which are encountered in the field of rock mechanics, is the problem of data input for rock mass properties. The usefulness of elaborate constitutive models and powerful numerical analysis programs, is greatly limited, if the analyst does not have the reliable input data for rock mass properties [7]. Values of constitutive model's parameters used in this research, were extracted from a paper which is about a gas storage site in salt structure, located in Poland. In this paper parameters values of Cvisc, Cpower and PWIPP constitutive models are presented. These values are shown in "table 1". Table 1 Input parameters values of constitutive models [8]. Model Parameters Value Unit Elastic G GPa 3.07 K GPa 6.6. gr 3 cm m 9 0.3 10 e GPa. s Burger k 6 0.15 7 10 e GPa. s G 0.15 8.091e GPa Power Law and WIPP WIPP Plastic parameters G k D n m Q R A B * ss C 0.8091e 0.15 1. 10 37 5 5750 4.56 17 4.6 10 8 1 40 0 Pa GPa s n 1 - K - - s 1 MPa - - 4.3. Cavern Excavation and Cyclic Pressure These salt caverns are excavated by solution mining method. For simulation of this method, the in situ stress at cavern walls is reduced to brine pressure which is produced by solution mining. In this research it was considered that the solution mining process is done within 8 months and deburring, which means replacing the brine with gas, is done within 4 months. After deburring, cavern internal pressure has a cyclic change between 7MPa to 18 Mpa to meet demand changes. Stress changes at cavern wall due to solution mining and also cavern cyclic internal pressure is shown in "figure 3". Figure 3.Cavern wall stress changes due to solution mining and cavern internal pressure changes (depth=900m) 5

5. Results of Numerical Analysis As the direction of one of principle stresses is horizontal and the other is vertical, the most critical points of cavern can be considered as the points shown in "figure 4". Figure 4.Position of cavern's critical points Hereinafter, the history of dilation safety factor ("equation(6)"), in function of cavern internal pressure variations with time, is the base of analysis of the three constitutive model's effect on the modeled cavern response. Theses history curves correspond to the mentioned critical points on the cavern wall, with the same color assigned point in "figure 4". Dilation safety factor changes curves, by utilization of different constitutive models are presented in "figure 5" to "figure 7". In these figures, dilation safety factor variations are shown versus time and cyclic cavern internal pressure changes. Figure 5.Critical point's dilation safety factor changes versus time and cyclic cavern internal pressure changes with Cpower utilization As it is shown in "figure 5", by cyclic cavern internal pressure changes, the dilation safety factor changes also have a cyclic trend. If the CPower model is used, dilation safety factors reach their minimum values at the max- 6

Zahra Shahmorad imum and minimum cavern internal pressures. Safety factor changes for all critical points are almost similar. But the red and blue curves which correspond to roof and bottom respectively, show lower minimal values in comparison with other points curves, in maximum cavern internal pressures. This means that cavern's roof and bottom are more unstable than cavern's wall at maximum cavern pressures. However, the safety factor values for all critical points are nearly the same in minimum cavern pressures. Safety factor values' variations, by utilization of PWIPP model are shown in "figure 6". Figure 6.Critical point's dilation safety factor changes versus time and cyclic cavern internal pressure changes with PWIPP utilization Safety factor changes by utilization of PWIPP are similar with safety factor changes when Cpower is used. This similarity is probably due to similar stress distribution around the cavern when Cpower and PWIPP are utilized. Safety factors changes, by using Cvisc model are indicated in "figure 7". Figure 7.Critical point's dilation safety factor changes versus time and cyclic cavern internal pressure changes with Cvisc utilization 7

If the Cvisc model is used the safety factor variations will be different from when Cpower and PWIPP models are used. Which is due to different stress distribution around the cavern when Cvisc model is used. As it is shown in "figure 7", by using Cvisc model, safety factors reach their minimum value only when the cavern pressure is minimum. Incresing the cavern pressure to in situ stresses causes creation and growth of microcracks, which leads to gas leakage[9]. This microcrack creation can be a result of rock dilation. So reduction of dilation safety factor in maximum gas pressure, when Cpower and PWIPP models are used, can be logical. Therefore these two constitutive models are more capable than Cvisc model, for predicting dilation occurrence around gas storage caverns. In fact by using Cvisc constitutive model, the maximum allowable pressure can't be estimated. It's worth to add that dilation safety factor reduction at minimumm cavern pressures, by using all three constitutive models, is due to principle stress differences which is increased by cavern pressure reduction. 6. Numeric results verification For verifying the accuracy of numeric analysis, model's results obtained by using numeric software, are compared with model's results obtained by using closed form solutions, based on the different assumptions, proposed by different researchers. In the following, some analytic methods which are developed for time dependent problems, are presented and their results are compared with numeric analysis's results. 6.1. Analytic method for Burger constitutive model In plain strain condition, if the Burger constitutive model is considered for surrounding environment of a tunnel, then "equation (7)" can be used for wall displacement determination of the tunnel[10]. U t G p G tgk pr 0 pi tgm G M k ( ) (1 ) 1 e1 M 0 M k. (7) In "equation (7)", p0 is the in situ stress, pi is the tunnel support pressure, G M, M, G k, K are Burger model parameters and t is time. As stress condition in the middle of the cavern's wall can be considered as plain strain, so "equation (7)" can be used for verifying radial displacement of this point (middle of cavern's wall), obtained from numeric analysis. In "figure 8", radial displacements of the middle of cavern's wall obtained by numeric and analytic methods are compared. It should be noted that Burger parameters shown in "table 1", were used in the numeric and analytic methods. 8

Displacement (m) Zahra Shahmorad 0.4 Burger Model 0.35 0.3 0.5 Analytic Numeric 0. 0.15 0.1 0.05 0 0 4 6 8 10 Time (year) Figure 8. Values of cavern's wall radial displacements versus time by using Burger model in the case of pi 0 and r 0. As it seems in "figure 8", Values of radial displacements resulted from using both methods are so similar. 6.. Analytic method for Power Law constitutive model In 1986, van sambeek have developed an analytic method for determination of displacements and stresses in a thick walled cylinder which is subjected to a pressure on the outer surface. If creep behavior of this thick walled cylinder is defined by Power Law then "equation (8)" to "equation (10)" can be used for determination of stresses and displacements [4]. n b 1 r r p b p. (8) b b n 1 a n n b 1 n r pb p b. (9) b n 1 a 9

-Radial stress (MPa) n n 1 3 b u r A ( ) p n b t. (10) 4 b r n 1 a In these equations, is the pressure on the outer surface of the cylinder, a and b are the inner and outer radiuses of cylinder. A and n are parameters of Power Law which their values are presented in "table 1". As loading condition in the modeled cavern is not similar to condition defined by van sambeek, a thick walled cylinder with inner radius of (a=0 m) and outer radius of (b=160 m) was modeled in the numeric software. This cylinder was loaded on the outer edge by ( pb 19.8MPa ). In the following, stresses and displacements resulted from numeric and analytic solutions are compared. Radial stress values, obtained by numeric and analytic methods are compared in "figure 9". 5 Power Law Model 0 Analytic Numeric 15 10 5 0 0 4 r (m) 6 8 10 1 Figure 9. Radial stress variations versus distance from inner wall of cylinder Tangentional stress values, obtained by numeric and analytic methods are compared in "figure 10". 10

Displacement (m) -Tangentional stress (MPa) Zahra Shahmorad 30 Power Law Model 5 0 Analytic Numeric 15 10 5 0 0 4 6 8 10 1 r (m) Figure 10. Tangentional stress variations versus distance from inner wall of cylinder Radial displacement values, obtained by numeric and analytic methods are compared in "figure 11". analytic numeric 1.8 1.6 1.4 1. 1 0.8 0.6 0.4 0. 0 Power Law Model 0 4 6 8 10 Time (year) Figure 11. Radial displacement variations versus time using Power Law model By seeing good accommodation of numeric and analytic results, numeric method can be used more confidently 11

7. Summary Prediction the behavior of Salt storage caverns within operation time (several decades) is so important. Because salt rock has a time dependent and complicated behavior even at low deviatoric stresses. This behavior will be more complicated in complicated loading conditions like mechanical conditions rulling on salt gas storage caverns. So many researchers in scope of rock mechanics have studied time dependent behavior of salt storage caverns by using different constitutive models. Among constitutive models, Cvisc, Cpower and PWIPP models have been used much more than others. In this research the effect of utilization of these three models on dilation occurrence, a key parameter in stability analysis of storage caverns, has been studied. In this study, a cavern was modeled and the dilation safety factor changes for critical points of the cavern, were analyzed. Results have shown that dilation safety factors have a cyclic change when a cyclic cavern gas pressure is applied. In the cases of using Cpower and PWIPP constitutive models safety factors reach their minimum values at the maximum and minimum cavern gas pressures, however in the case of using Cvisc constitutive model, safety factors reach their minimum values only at the minimum cavern gas pressures. As It have been approved by researchers that in maximum cavern gas pressures, microcracks are created and developed, which may be a result of rock dilation, So it can be concluded that Cpower and PWIPP constitutive models are more capable than Cvisc constitutive model for predicting dilation occurrence around gas storage caverns because these models show reduction of dilation safety factors in maximum cavern gas pressures too. The used numeric method have been validated by means of analytic methods. 8. References [1] Truesdell, C. (004) The non linear theories of mechanics. Third edition, Springer. [] Ottosen, N. S. (1986) Viscoelastic-viscoplastic formulas for analysis of cavities in rock salt. journal of rock mechanics and mining science, vol 3, no 3, p 01-1. [3] Goodman, R. E. (1989) Introduction to rock mechanics" Second edition, Wiley. [4] "Creep material models. (014) Flac 3D manual, version 5, Itasca. [5] Mahnken, R., Kohlmeier, M. (001) Finite element simulation for rock salt with dilatancy boundry coupled to fluid permeation. computer methods in applied mechanics and engineering(190), p 459-478. [6] Wang,T., Yan,X., Yang,H. (013) A new shape design method of salt cavern used as underground gas storage. Applied Energy(104), p 50-61. [7] Van sambeek, L., Ratigan, J., Hansen, F.( 1993) Dilatancy of rock salt in laboratory tests. journal of rock mechanics and mining science and geomechanics, vol 30, no 7, p 735-738. [8] Slizowski, J., Pilecki, Z. (013) Site assessment for astroparticle detector location in evaporates of the polkovice sieroszowice copper oil mine, Poland. advances in high energy physics, article ID 461764. [9] Heusermann, S., Rolfs, O., Schmidt, U. (003) Non linear finite element analysis of solution mined storage caverns in rock salt using the lubby constitutive model. computers and structures(81),p 69-638. [10] Nomikos, P., Rahmannejad, R., Sofianos, A.(011) Supported axisymmetric Tunnels within linear viscoelastic Burgers rocks. Rock mechanical rock engineering (44), p 553-564. 1