application of numerical modelling and genetic...

Research ArticleApplication of Numerical Modelling and GeneticProgramming in Hydrocarbon Seepage Prediction andControl for Crude Oil Storage Unlined Rock Caverns

Ebrahim Ghotbi Ravandi,1 Reza Rahmannejad,2 Saeed Karimi-Nasab,2 and Amir Sarrafi3

1Mineral Industries Research Center, Shahid Bahonar University of Kerman, Kerman, Iran2Mining Engineering Department, Shahid Bahonar University of Kerman, Kerman, Iran3Chemical Engineering Department, Shahid Bahonar University of Kerman, Kerman, Iran

Correspondence should be addressed to Ebrahim Ghotbi Ravandi; [email protected]

Received 23 November 2016; Revised 2 March 2017; Accepted 15 March 2017; Published 14 June 2017

Academic Editor: Micol Todesco

Copyright © 2017 Ebrahim Ghotbi Ravandi et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Seepage control is a prerequisite for hydrocarbon storage in unlined rock caverns (URCs) where the seepage of stored productsto the surrounding host rock and groundwater can cause serious environmental and financial problems. Practically seepagecontrol is performed by permeability and hydrodynamic control methods. This paper employs numerical modelling and geneticprogramming (GP) for the purpose of seepage prediction and control method determination for the crude oil storage URCs basedon the effective parameters including hydrogeologic characteristic of the rock and physicochemical properties of the hydrocarbons.Several levels for each parameter were considered and all the possible scenarios were modelled numerically for the two-phasemixture model formulation. The corresponding seepage values were evaluated to be used as genetic programming data base togenerate representative equations for the hydrocarbon seepage value. The coefficients of determination (𝑅2) and relative percenterrors of the proposed equations show their ability in the seepage prediction and permeability or hydrodynamic control methoddetermination and design. The results can be used for crude oil storage URCs worldwide.

1. Introduction

Underground storage of hydrocarbons in unlined rock cav-erns (URCs) ismore secure, safe, and economical than above-ground storage and has several environmental and opera-tional advantages. The main concern associated with URCsis the seepage of stored products, vapor, and VOCs of themto the surrounding host rock and groundwater which cancause serious environmental problems such as groundwatercontamination and accumulation of flammable gas near thesurface as well as economic and financial losses. Therefore,seepage control is a fundamental prerequisite for URCswhere minimum product seepage is required. Practicallyseepage control fromanURC is performed by permeability orhydrodynamic control methods. Permeability control meansapplying techniques such as grouting or freezing to controland decrease hydrocarbon seepage bymaintaining a specified

low permeability and sealing of the rockmass. However thesetechniques are very time-consuming and expensive [1]. Byhydrodynamic control, it is meant that there is groundwaterin the rock mass with the static head that exceeds the internalstorage pressure resulting in positive groundwater gradienttowards the cavern to prevent the escape of the storedproducts [2]. Aberg [3] postulated that no seepage will occurif the water pressure gradient towards the cavern is positiveand greater than unity. There is no standard for acceptableseepage value and how much sealing work is required. Itdepends on the environmental and economic (operational)aspects. As a general rule, 24m3/24 hr period in a cavernof 100,000m3 is considered to be an acceptable limit [1].Liquid hydrocarbons (e.g., crude oil and gasoline) are notstored under pressure and their pressure inside caverns ishydrostatic. Hydrocarbons vapors and gases pressure insidecaverns varies as a function of the temperature, oil level, and

HindawiGeofluidsVolume 2017, Article ID 6803294, 11 pageshttps://doi.org/10.1155/2017/6803294

https://doi.org/10.1155/2017/6803294

2 Geofluids

components, usually from 0.5 to 3 bar (105 Pa) [4]. Typically,75% and 25% of a cavern space are considered for liquid oilsand gases, respectively [2].

Hydrocarbons seepage (especially crude oil) from under-ground unlined rock caverns to the surrounding saturatedporous media is barely investigated in the literature [5,6]. In this paper prediction equations for the hydrocar-bons (crude oil and gas) seepage value from the IranianURCs in terms of m3/24 hr are represented using geneticprogramming based on the data gathered by numericalmodelling of the hydrocarbon seepage for a variety of con-ditions using COMSOL and the two-phase mixture modelformulation (see Supplementary Material available online athttps://doi.org/10.1155/2017/6803294). By applying and solv-ing the proposed equations for the seepage values less thanthe allowable one, prejudgment can be done and the seepagecontrol technique (e.g., permeability or hydrodynamic con-trol) can be selected.

2. Two-Phase Mixture Model

Two-phase mixture model which was first mentioned byWang and Beckermann [7] uses mixture variables to reducethe number of partial differential equations (PDEs) of clas-sical two-phase fluid flow formulations in porous media.Therefore, it is more convenient to use the appropriatenumerical schemes for the two-phase mixture model [7].

In the mixture model, the two phases are regarded asconstituents of a binary mixture and the mixture variablessuch as mixture density 𝜌 and mixture velocity vector 𝑢 aredenoted without index. With introduction of the mixturequantities (see [7]) the conservation ofmass with the porosity(𝜙) is defined as

𝜕𝜙𝜌𝜕𝑡 + ∇ ⋅ (𝜌𝑢) = 0. (1)

Conservation of momentum using Darcy velocity is asfollows in which the dynamic viscosity 𝜇 and the pressure 𝑝are also mixture quantities (see [7]):

𝑢 = −𝐾𝜇 (∇𝑝 − 𝜌𝑘𝑔∇𝑧)1𝜇 = 𝑠𝑤 𝑘𝑟𝑤𝜇𝑤 + 𝑠𝑛𝑤 𝑘𝑟𝑛𝑤𝜇𝑛𝑤𝜌𝑘 = 𝜌𝑤 𝑘𝑟𝑤𝜇𝑤 + 𝜌𝑛𝑤 𝑘𝑟𝑛𝑤𝜇𝑛𝑤∇𝑝 = 𝑘𝑟𝑤𝜇𝑤 ∇𝑝𝑤 +

𝑘𝑟𝑤𝜇𝑤 ∇𝑝𝑛𝑤,

(2)

where 𝜌𝑘 is the kinetic mixture density, 𝐾 is the intrinsicpermeability, 𝑠 is the saturation and the subscripts of 𝑤 and𝑛𝑤 are related to the wetting and nonwetting phase, respec-tively, 𝑘𝑟 is the relative permeability, 𝑔 is the gravitationalacceleration vector, and 𝑧 is the depth.

The diffusive mass flux 𝑗 connects the mixture meanvelocity with the velocity of the individual phases:

𝜌𝑤𝑢𝑤 = 𝑘𝑟𝑤𝜇𝑤 𝜌𝑢 + 𝑗𝜌𝑛𝑤𝑢𝑛𝑤 = 𝑘𝑟𝑛𝑤𝜇𝑛𝑤 𝜌𝑢 − 𝑗.

(3)

Wang and Beckerman [7] introduced the diffusive flux (𝑗) asfollows:

𝑗 = 𝐷𝐶∇𝑠𝑤 + 𝑓 (𝑠𝑤) 𝐾Δ𝜌]𝑛𝑤𝑔, (4)

where ] is the kinematic viscosity,𝑓 is the hindrance functionfor phase migration and separation, and 𝐷𝐶 is the capillarydiffusion coefficient as a function of the wetting phasesaturation:

𝐷𝐶 = 𝑘𝑟𝑤𝜇𝑤 𝐾(𝑠𝑤 − 1) 𝜕𝑝𝑐𝜕𝑠𝑤 , (5)

where 𝑝𝑐 is capillary pressure. The transport equation is asfollows, where 𝑐𝑤 is the fluid content of the wetting phase:

𝜕𝜙𝑐𝑤𝜕𝑡 + ∇ ⋅ (𝑐𝑤𝑢) = ∇ ⋅ (𝐷𝐶∇𝑐𝑤)𝑐𝑤 = 𝜌𝑤𝑠𝑤.

(6)

Several scientists have tried to derive a functional correlationfor the relative permeability and the capillary pressure (𝑝𝑐)as a function of the wetting phase saturation based on theexperimental data. Brooks and Corey (1964) developed anempirical correlation utilizing the entry capillary pressure(𝑝𝑒𝑐) and the wetting phase saturation that empirically hasbeen found to be appropriate for the drainage process asfollows [8, 9]:

𝑝𝑐 = 𝑝𝑒𝑐𝑠𝑤−1/𝜆𝑘𝑟𝑤 = 𝑠𝑤(3+2/𝜆)𝑘𝑟𝑛𝑤 = (1 − 𝑠𝑤)2 [1 − 𝑠𝑤((2+𝜆)/𝜆)]

(7)

where 𝜆 is the pore size distribution index. Its value is usuallyconsidered to be 2 for the carbonated rocks [10].

Exact solutions for two-phase fluid flow problems inporous media which involve gravity, capillarity, and fluidflow in two or three dimensions (multidimensional flow)are impossible due to inherent nonlinearity and the need tosolve for multiple dependent variables along with a varietyof unknowns. Solving practical problems requires a suitablenumerical method [7]. A lot of authors have used numericalmethods and software tools to model single- or two-phasefluid flow in porous and fractured media [11–15]. In thementioned literature the effect of gravity, capillary pressures,and multidimensional flow is usually neglected and notconsidered simultaneously.

https://doi.org/10.1155/2017/6803294

Geofluids 3

3. Validation of Numerical Modelling ofTwo-Phase Mixture Model by COMSOL

Neglecting the capillary pressure and gravity effects, the five-spot problem is the standard porous media problem wherea square computational domain is initially saturated withthe nonwetting phase (oil) and the wetting phase (water) isinjected through a well at a lower corner of the domain at aconstant rate (or pressure) and displaces the oil. The nonwet-ting phase is produced at the same rate through a well in theopposite upper corner. In order to evaluate the computationalefficiency and accuracy of the mixture model formulationnumerical modelling with COMSOL, a verification exampleof the computational domain with the dimensions 300m× 300m for the five-spot problem is given to comparethe numerical results with the fully coupled (classical) for-mulation. Boundary and initial conditions are depicted inFigure 1. Dirichlet pressure boundary conditions are 5m ×5m injection and production wells.The other boundaries areimpermeable and Neumann no-flow boundary conditions.The nonwetting and wetting phase density and viscosity areconsidered 1000Kg/m3 and 0.001 Pa⋅m, respectively. Intrinsicpermeability, porosity, and pore size distribution index are10−7m2, 0.2, and 2, respectively. Figure 1 shows the compar-ative study of the numerical modelling results for the fullycoupled [16] andmixturemodel formulations referring to thetime 𝑇 = 2000 days and time step of 1 day. As it can beseen fromFigure 1 the results (wetting phase saturation frontsand contour lines) of the fully coupled and mixture modelformulations are in good agreement with each other.

4. Genetic Programming

Genetic programming (GP) as an extension of the geneticalgorithms (GA) was introduced by Koza [17]. The maindifference between genetic programming and genetic algo-rithms is the representation of the solution. Genetic algo-rithms create a string of numbers that represent the solutionbut genetic programming creates computer programs (CPs)in the lisp or scheme computer languages as the solution [18].GP can be used to find a relationship between input and out-put data in the form of mathematical expression representedby the functions generated in the training (learning) process.If the error rate reaches a certain threshold, the training canbe stopped and the testing (validation) can be applied toverify the effectiveness of the best function.

Genetic programming uses four steps to solve problems[18]:

(1) Generate an initial population of random composi-tions of the functions and terminals of the problem(computer programs).

(2) Execute each program in the population and assignit a fitness value according to how well it solves theproblem.

(3) Create a new population of computer programs byapplying the following genetic operations:

(i) Copy the best existing programs (reproduc-tion).

(ii) Create new computer programs by mutation.(iii) Create new computer programs by crossover.

(4) The best computer program that appeared in anygeneration (the best-so-far solution) is designated asthe result of genetic programming.

Figure 2 represents the genetic programming flowchart.

5. Methodology

The purpose of this study is to employ numerical modellingand genetic programming to predict the hydrocarbon seep-age from the URCs (Iranian URCs in the limestone rocks)based on the allowable seepage value (m3/24 hr) to be able todecide on the seepage control technique selection. To reachthe stated goal, two parts of numerical modelling were donefor the oil and gas seepage from the URCs. The influencingparameters on the hydrocarbons seepage including hydro-geological properties of the rock mass and physicochemicalproperties of the hydrocarbons were considered in severallevels and using full factorial design all the hypothetical caseswere modelled using the finite element based commercialsoftware COMSOL version 5.1 and the two-phase mixturemodel formulation for the time 𝑇 = 24 hr and time step of0.1 hr. The corresponding seepage values were evaluated asdata base of genetic programming.Themodelling parametersand their corresponding values are shown in Table 1. Dueto the symmetry of the problem and to save time, half ofa cavern was modelled. Cavern dimension and initial andboundary conditions for numerical modelling of the gasand oil seepage are shown in Figure 3. Hydrocarbon flowis driven by the difference of Dirichlet pressure boundaryconditions which is hydrostatic (𝑃nw = 𝑃gas + 𝜌oil𝑔ℎ) for theoil seepage modelling and constant (𝑃gas) for the gas seepagemodelling and hydrostatic pressure of groundwater. Dirichletboundary condition, 𝑃w = 0 and 𝑆w = 1, is considered forthe groundwater table. Γnoflow boundaries are impermeableand given as Neumann no-flow boundary conditions. Rockmass is initially fully saturated with the water and the waterpressure in the domain is hydrostatic. The free quad finiteelements mesh was used for the modelling. Grids in thearea of seepage passage were refined to smaller elements tohave more accuracy. Mesh dependency tests were carriedout for each case and the meshes eventually used werejustified by the quality of the results. Three and four levelsof the groundwater level were considered for each of thegas and oil seepage modelling, respectively, where minimumgroundwater level was considered to be 2m above the caverncrown and maximum level is 1m below the level that noseepage will occur. In order to overestimate the hydrocarbonseepage to have higher factor of safety and for the sake ofsimplicity, several assumptionswere taken into account in theoil and gas seepage modelling as follows:

(i) The equivalent-continuum approach modelling wasused and no distinction was made between fracturesand thematrix blocks and fluids were assumed to flowthrough the whole system.

4 Geofluids

0.7

0.95

0.8

0.65

0.5

0.35

0.2

0.05

0.75

0.3 0.05

0.5

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.9

0.85

0.8

0.70.050.5

0.9

0.880.85

0.8

Fully coupled formulation Mixture model formulation

300

m

300 m

Γno�ow

Γno�ow

Γ no

�ow

Γ no

�ow

Γin

Γout

Y≅

160

m

Y≅

160

m

X ≅ 160 m

X ≅ 160 m−200

20406080

100120140160180200220240260280300320

Y(m

)

0 300200100X (m)

0

100

200

300

Y(m

)

3000 200100X (m)

0 200 300100X (m)

0

100

200

300

Y(m

)

0

100

200

300

Y(m

)

0 300200100X (m)

Pw = 199998.75 Pa

Snw = 1.0

Snw = 1.0

Pw = 2 × 105 PaSnw = 0.0

SwSw

Figure 1: Initial and boundary conditions for the five-spot problem as well as water saturation (𝑆w) fronts and contour lines for the time𝑇 = 2000 days and the time step (Δ𝑡) of 1 day for the fully coupled and mixture model formulation with COMSOL.

Geofluids 5

Table 1: Values and domain of the selected parameters.

Intrinsicpermeability (m2) Porosity Irreducible water

saturationNonwet phasedensity (Kg/m3)

Nonwet phaseviscosity (Pa⋅s) Gas pressure

(bar) 𝐾𝑧/𝐾𝑥

Oil

1𝑒 − 13 0.15–0.2 0.05–0.3(3)∗ 800–950(3) 0.1–5.42 — —5𝑒 − 14 0.12–0.17 0.05–0.3(3) 800–950(3) 0.1–5.42 — —1𝑒 − 14 0.1–0.15 0.05–0.3(3) 800–950(3) 0.1–5.42 — —5𝑒 − 15 0.08–012 0.05–0.3(3) 800–950(3) 0.1–5.42 — —1𝑒 − 15 0.05–0.1 0.1–0.3(3) 800–950(3) 0.1–5.42 —

Gas1𝑒 − 14 0.1–0.15 0.05–0.3 3.125 2𝑒 − 5 1-2(3) 0.5–1(3)5𝑒 − 15 0.08–0.12 0.05–0.3 3.125 2𝑒 − 5 1-2(3) 0.5–1(3)1𝑒 − 15 0.05–0.1 0.1–0.3 3.125 2𝑒 − 5 1-2(3) 0.5–1(3)

∗The number in the parentheses represents the number of levels for the parameter.

Start

Generate an initial population comprisingcomputer programs (CPs)

Evaluate each program and assign its �tness according to how well it solves the problem

From temporary population select the programs according to their �tness

Create new population by performing genetic operators (reproduction, crossover, and mutation)

Evaluate the performance of new population

Report the best CP so far as the result

Is the termination criterion satis�ed?

No

Yes

Figure 2: Genetic programming flowchart.

(ii) The porous medium, representing the rock, wasconsidered homogeneous and isotropic for the oilseepage and homogenous and anisotropic (𝐾𝑥 =𝐾𝑦 ≥ 𝐾𝑧) for the gas seepage modelling.

(iii) The rock and the fluids were considered incompress-ible.

(iv) The relative permeabilities and the capillary pressurefunction of the wetting and nonwetting phases were

considered based on Brooks and Corey’s colorationand 𝜆 = 2.

(v) The capillary pressure was consumed equal to theentry capillary pressure in the gas seepage modellingand Klinkenberg effect was neglected.

(vi) The gases were considered ideal and solubility of thegas in the water and pressure drop of the gas wereneglected.

(vii) Water density and viscosity were considered1000Kg/m3 and 0.001 Pa⋅m, respectively.

The allowable seepage value per unit length (1m) of the halfof the cavern with the stated dimension (Figure 3) wouldbe 0.042m3/24 h. Therefore, the permeability values werechosen in a domain in which seepage values are close tothe allowable seepage value. Corresponding porosity for eachpermeability valuewas considered based onArchie’s formulasas follows [19]:

𝐾 = 2.55 (10𝜙)5.65𝐾 = 9.35 (10𝜙)5.65 , (8)

where 𝐾 is in millidarcy, mD.Since hydrogeologic characteristics of the limestone rocks

of Iran are poorly referenced, the entry capillary pressurewas measured by 𝐽 function of capillary pressure datain the Edwards formation, Jourdanton field for limestonewhich is close to the limestone in Iran petrophysically andmineralogically [20]. Each capillary entry pressure value wasobtained by the 𝐽 function using a specific permeability andits corresponding value of porosity (𝜙) as follows [21]:

𝐽 (𝑆𝑤) = 0.21645𝑃𝑐𝜎 √𝐾𝜙 . (9)

The values of interfacial tension (𝜎) for the oil-water systemand the gas-water system were considered 48 and 50 dyn/cm,respectively [22]. The corresponding values of irreduciblewater saturation for each porosity value were obtained byHolmes [23] equation:

𝜙𝑄 × 𝑆𝑤𝑐 = constant, (10)

6 Geofluids

Crude oil

Water

Crude oil

Water

Gas Gas

20m

15 m

50m

Γno�ow

Γ no�

ow

Γ no�

owΓ n

o�ow

Γ no�ow

Γno�ow

Γ no�

ow

Γ no�

ow

Γ no�

ow

Γ no�

owP=Pga

s

−202468

101214161820222426283032

−202468

101214161820222426283032

100 20 30 40(b)

0 10 20 30 40(a)

Sw = 1 Pw = 0Sw = 1 Pw = 0

S w=0

S w=

0P

=Pga

s+�휌 n

wgℎ

Sw = 1 Pint = �휌wgℎSw = 1 Pint = �휌wgℎ

Figure 3: Scheme of the caverns dimensions and initial and boundary conditions for the numerical modelling of oil (a) and gas (b) seepage.

Table 2: Corresponding parameters of the oil and gas seepagemodelling example.

Parameter ValueGas pressure (𝑃gas) 2 barWater level above the oil level 6mPorosity 0.15Residual water saturation 0.05Oil density 950Kg/m3

Oil viscosity 5.42 Pa⋅mGas density 3.125 Kg/m3

Gas viscosity 2𝑒 − 5Pa⋅m

where the maximum value was considered as 0.3. 𝑄 andconstant were considered 1 and 0.005–0.06, respectively [23].Oil and vapor gas density and viscosity and 𝐾𝑧/𝐾𝑥 ratio forthe limestone were considered based on [24–26]. Figures 4and 5 show the examples of the oil and gas seepage modellingfor the parameters presented in Table 2 to give a view of thenumerical modelling.

GPTIPS, an open-source MATLAB toolbox for geneticprogramming (GP) technique, was used to generate predic-tion equations based on the hydrocarbons seepage value after24 hr corresponding to each numerical modelling. To reachthe stated goal, the whole data set was separated into two,training and testing set (80% and 20% of the whole dataset, resp.). The training set was used to evaluate the finalor optimum computer programs (CPs) while the testing setwas employed to validate the reliability of the GP model. Theoptimum combination of the values for the set of parameters

such as population size, number of generations, function set,mutation rate, crossover rate is achieved by the performanceof several trials. The mean absolute percent error (MPE)between the seepage values evaluated by numericalmodelling(COMSOL) and the values returned by the GP generatedequation was used in the evaluation stage as the fitnessfunction. The mathematical phrase of the best and simplestgenerated computer program (CP) by GP for the seepagevalue of the gas and oil was considered as final equation.

6. Results and Discussion

To have more accurate formulations, two equations werepresented for the gas seepage where the ground water levelis low and the gas seeps from all parts of the gas filled spaceof the cavern and where the gas seepage is not from all partsof the gas filled section due to high water level above thecavern. Two equations were presented for the oil seepage fortwo permeability value intervals as well.

Table 3 represents the simplest and the best mathemat-ical phrases generated by the GP where 𝐾 is the intrinsicpermeability, in mD, 𝑃𝑑 is the pressure difference of oil atits free level and groundwater hydrostatic pressure, in metersof water, 𝑤𝑙 is the water level above the cavern crown, in m,𝑃gas is the gas pressure, in bar, 𝜌 is oil density, in g/cm3, 𝜇 isdynamic viscosity of oil, in Pa.m, and𝐾𝑧/𝐾𝑥 is the vertical tohorizontal permeability ratio.

Predicted seepage values by the GP equations versusactual values of the seepage for the training and test sets(80 and 20% of data set) as well as relative percent errors ofequations (∗)–(∗ ∗ ∗∗) in Table 3 for the whole data set areshown in Figures 6 and 7, respectively.

Geofluids 7

Table3:GPequatio

nsof

theo

ilandgasseepage

value(m3/24h

r).

Equatio

nnu

mber

Descriptio

nMathematicalph

rase

Meanpercent

error

Max

percent

error

Coefficiento

fdeterm

ination

(𝑅2 )∗

Adjuste

d𝑅2

(adj_𝑅2

)∗

(∗)Oilseepage

𝐾=100–

10md

2.608𝐾3 108

+(−3.

847𝑃 𝑑2 108

−7.176 106

)𝐾2

+(0.00

01643𝑃𝑑−7.

176𝑆 𝑤𝑐 106

+0.000

1391log

(𝐾𝜙)𝜌 oil

+0.000

4071𝑆 𝑤𝑐𝑃 𝑑s

in𝜌 oil

+2.608 108

)𝐾+0.

000512

2𝑃 𝑑−0.

001781

3.32

1799.4

99.2

(∗∗)

Oilseepage

𝐾=10–1

mD

0.00035

02𝑆𝑤𝑐(𝐾𝑃𝑑+1.

961)+

6.479𝐾(

𝐾𝑃 𝑑+1.

961)

106−3.

49 (𝐾+

8.366 )/1

05sin

sin(𝐾 )

−4.428

exp𝜌 oil(

𝐾−𝐾𝑃 𝑑

)105

−2.206

(𝐾+16.5

5 )/106

sin𝜇 oil

+0.000

6279

2.2

13.2

99.8

99.7

(∗∗∗)

Gas

seepage,

from

allp

arts

(0.098

53 𝑤 𝑙−0.

0878

𝑃 gas𝐾−

0.00377

8arcsin(𝐾 𝑧

/𝐾 𝑥)log

(𝑤 𝑙)+

0.00051

39𝑃 gas𝑆𝑤𝑐log(𝑃

gas))𝐾

+0.002

057−

3.782𝜙

√exp𝑃 gas

−0.066

9𝑃 gas𝑤𝑙2

5.9

1898

96

(∗∗∗∗)

Gas

seepage,

notfrom

all

parts

0.00017

87𝐾cosh

√𝑤𝑙+0.

001714

𝐾(𝑃gas−𝑤𝑙cos(𝜙)

)−0.137

1𝜙tanh(tan

h𝐾𝑧/

𝐾𝑥 𝑃 gas)

+0.000

2673𝐾𝑤𝑙

𝐾𝑧/𝐾𝑥

+0.007

6𝐾𝑃gas−5.

055 10512

2992

90

∗𝑅2andadj_𝑅2ared

escribed

inAp

pend

ix.

8 Geofluids

0.90.80.70.60.50.40.30.20.1

0.90.80.70.60.50.40.30.20.1

0.90.80.70.60.50.40.30.20.1

3

2.5

2

1.5

1

0.5

Pressure contour (Pa)

5x

5x

×105

−10 −5−15 3025 350 1510 4020 455

02468

10121416182022242628

−202468

101214161820222426

−10 40 4520 3525 3010 150 5−5

−202468

101214161820222426

−10 20 25 3015 3510 450 5 40−5

−202468

101214161820222426

−10 35−5 5 4020 25 301510 450

−202468

101214161820222426

4520 35302510 15 4050−10 −5

−4−2

Time = 24 h, oil saturation, K = 10−12 m2

Time = 0, oil saturation Time = 24 h, oil saturation, K = 10−13 m2

Figure 4: Finite element mesh as well as pressure and oil saturation contours of the example of oil seepage modelling.

By considering some parameters (e.g., oil density, viscos-ity, and gas pressure) as given values and solving the proposedequations for the seepage value less than the allowable one,prejudgment can be done and required permeability value orgroundwater level above the cavern can be determined.

The results showed that, in order to control the oil seepageby permeability control technique such as grouting, thepermeability of the rock must be less than 100mD (10−13m2)with proper water level above the cavern; otherwise theseepage of the stored products would be much more thanallowable seepage value. Since grouting cost to have the safevalue of permeability is much expensive and complicated,it is better to use hydrodynamic control technique and tolocate the cavern deep enough below the groundwater levelwith a good margin of safety so that no seepage will occur.

Since groundwater level decreases due to water leaking tothe cavern, it has to be maintained at its original level.Therefore, the cavern must be equipped with the artificialsystem for supplying water which can be done by injectingwater through water curtain systems above the cavern orvertical wells.

7. Conclusion

In this paper prediction equations for the hydrocarbons (oiland gas) seepage from the Iranian unlined rock caverns(URCs) are presented using genetic programming on thebasis of the data gathered by numerical modelling of hydro-carbon seepage for the time 𝑇 = 24 hr and different physico-chemical properties of the hydrocarbons and hydrogeological

Geofluids 9

Pressure contour (Pa)

3x

0.90.80.70.60.50.40.30.20.1

0.90.80.70.60.50.40.30.20.1

3

2.5

2

1.5

1

0.5

×105

5 352015 30 40250 10−5−10−2

02468

101214161820222426

5 352015 30 40250 10−5−10−2

02468

101214161820222426

5 352015 30 40250 10−5−10−2

02468

101214161820222426

Time = 24 h, gas saturation, K = 10−14 m2

Time = 0, gas saturation

Figure 5: Pressure and gas saturation contours of the example of gas seepage modelling.

properties of the host rock.The seepage control technique canbe selected andprejudgment can be done by the application ofthe presented equations. Since the dimensions of the cavernfor numerical modelling (Iranian caverns) are common, theresults can be used for crude oil storage unlined rock caverns(URCs) worldwide.The coefficient of determination (𝑅2) andrelative percent errors of the proposed equations show theirability in the hydrocarbon seepage predication fromURCs inthe carbonated rocks with the intrinsic permeability between10−13 and 10−15m2 (1–100mD).

Appendix

Coefficient of Multiple Determination (𝑅2)The coefficient of multiple determination (𝑅2) measureshow successful the fit is in explaining the variation of data.Thus the closer 𝑅2 is to unity indicates the better modelperformance and fit.

𝑅2 = 1 − ∑𝑁𝑖=1 (𝑌𝑖 − �̂�𝑖)2∑𝑁𝑖=1 (𝑌𝑖 − 𝑌𝑖)2 , (A.1)

where 𝑁 is the number of data, 𝑌𝑖 and �̂�𝑖 are the actualand predicted values, respectively, and 𝑌𝑖 is the average of(𝑌1, . . . , 𝑌𝑁).

The adjusted 𝑅2, the degree of freedom, is a modifiedversion of 𝑅2 that has been adjusted for the number ofpredictors in the model and is generally the best indicator ofthe fit qualitywhen you add additional coefficients to amodel.It is always lower than or equal to 𝑅2.

𝑅2−adj = 1 − (1 − 𝑅2) (𝑁 − 1)𝑁 − 𝑝 − 1 , (A.2)

where 𝑝 is the number of predictors.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

10 Geofluids

Actual y (test values)Predicted y (test values)

Actual y (training values)Predicted y (training values)

2500 150 20050 350100 300Data point

00.10.20.30.4

y=

oil

seep

age

00.10.20.30.4

y=

oil

seep

age

906030 40 5020100 70 80Data point

(a)



2500 150 20050 350100 300Data point

00.10.20.30.4

y=

oil

seep

age

00.10.20.30.4

y=

oil

seep

age

906030 40 5020100 70 80Data point

(b)



20 40 60 800 100 140120Data point

00.05

0.10.15

0.2

y=

gas s

eepa

ge

00.05

0.10.15

0.2

y=

gas s

eepa

ge

0 5 15 20 25 3010 35Data point

(c)



100 20015050 2500Data point

200 30 4010 6050 70Data point

00.020.040.06

y=

gas s

eepa

ge0

0.020.040.06

y=

gas s

eepa

ge

(d)

Figure 6: Predicted values of the oil and gas seepage (m3/24 h) by the GP equations of (∗)–(∗ ∗ ∗∗) in Table 3 ((a)–(d), resp.) versus actualvalues for the training and test sets.

02468

1012141618

erro

r (%

)

50 100 150 200 2500 300 400350 450Data number

Rela

tive p

erce

nt

(a)

02468

101214

erro

r (%

)

150 200 25050 100 300 4504000 350Data number

Rela

tive p

erce

nt

(b)

02468

1012141618

erro

r (%

)

20 40 60 140800 120 160100 180Data number

Rela

tive p

erce

nt

(c)

05

101520253035

erro

r (%

)Re

lativ

e per

cent

200150 250 30050 100 3500Data number(d)

Figure 7: Relative percent errors of the equations (∗)–(∗ ∗ ∗∗) in Table 3 ((a)–(d), resp.) for the whole data set.

Geofluids 11

Acknowledgments

The authors would like to acknowledge the financial supportfrom Iranian Oil Terminals Company and thank Profes-sor Seyed Majid Hassanizaeh from Utrecht University forthoughtful comments and helpful discussions.

References

[1] C.-O. Morfeldt, “Storage of petroleum products in man-madecaverns in Sweden,” Bulletin of the International Association ofEngineering Geology, vol. 28, no. 1, pp. 17–30, 1983.

[2] S. M. Haug, “Storage of oil and gas in rock caverns belowthe groundwater table- general design development,” in Under-ground Constructions for the Norwegian Oil and Gas Industry,pp. 19–25, Norwegian Tunnelling Assossiation, 2007.

[3] B. Aberg, “Prevention of gas leakage from unlined reservoirs inrock,” in Rockstore, vol. 77, pp. 399–413, Pergamon Press, 1977.

[4] N. O. Midtlien, “Cavern storage excavtion-sture,” in Under-ground Constructions for the Norwegian Oil and Gas Industry,pp. 47–56, Norwegian Tunnelling Assossiation, 2007.

[5] H. Liu, J. C. Lee, and B. R. Li, “High precision pressurecontrol of a pneumatic chamber using a hybrid fuzzy PIDcontroller,” International Journal of Precision Engineering andManufacturing, vol. 7, pp. 7–13, 2007.

[6] C. Yu, S. C. Deng, H. B. Li, J. C. Li, and X. Xia, “Theanisotropic seepage analysis of water-sealed underground oilstorage caverns,”Tunnelling andUnderground Space Technology,vol. 38, pp. 26–37, 2013.

[7] W. Chao-Yang and C. Beckermann, “A two-phase mixturemodel of liquid-gas flow and heat transfer in capillary porousmedia-I. Formulation,” International Journal of Heat and MassTransfer, vol. 36, no. 11, pp. 2747–2758, 1993.

[8] M. K. Zahoor, M. N. Derahman, andM. H. Yunan, “Implemen-tation of Brooks and Corey correlation in water wet case withimmobile wetting phase,” NAFTA, vol. 60, pp. 435–437, 2009.

[9] K. Li, Theoretical Development of the Brooks-Corey CapillaryPressure Model from Fractal Modeling of Porous Media, SPE,2004.

[10] O. B. Baker, H. W. Yarranton, and J. Jensen, Practical ReservoirEngineering andCharacterization, Gulf Professional Publishing,1st edition, 2015.

[11] J. Rutqvist, Y.-S. Wu, C.-F. Tsang, and G. Bodvarsson, “Amodeling approach for analysis of coupled multiphase fluidflow, heat transfer, and deformation in fractured porous rock,”International Journal of Rock Mechanics and Mining Sciences,vol. 39, no. 4, pp. 429–442, 2002.

[12] V. Reichenberger, H. Jakobs, P. Bastian, and R. Helmig, “Amixed-dimensional finite volume method for two-phase flowin fractured porous media,” Advances in Water Resources, vol.29, no. 7, pp. 1020–1036, 2006.

[13] M.Diaz-Viera, “COMSOL implementation of amultiphase fluidflow model in porous media,” in Proceeding of the COMSOLConference, 2008.

[14] M. F. El-Amin and S. Shuyu, “Effects of gravity and inlet/outletlocation on a two-phase cocurrent imbibition in porousmedia,”Journal of Applied Mathematics, vol. 2011, Article ID 673523,2011.

[15] D. Droste, F. Lindner, C. Mundt, and M. Pfitzner, “Numericalcomputation of two-phase flow in porous media,” in Proceedingof the COMSOL Conference, pp. 1–6, 2013.

[16] Y. Cao, E. Birgitte, and R. Helmig, “Fractional flow formulationfor two phase flow in porous media,” in International ResearchTraining Group GRK 1398/1, Non-Linearities and Upscaling inPorous Media, 2007.

[17] J. R. Koza, Genetic Programming: on the Programming of Com-puters by Means of Natural Selection, MIT Press, Cambridge,Mass, USA, 1976.

[18] E. Ghotbi Ravandi, R. Rahmannejad, A. E. Feili Monfared,and E. Ghotbi Ravandi, “Application of numerical modelingand genetic programming to estimate rock mass modulusof deformation,” International Journal of Mining Science andTechnology, vol. 23, no. 5, pp. 733–737, 2013.

[19] S. Gholinezhad and M. Masihi, “A physical-based model ofpermeability/porosity relationship for the rock data of Iransouthern carbonate reservoirs,” Iranian Journal of Oil & GasScience and Technology, vol. 1, pp. 25–36, 2012.

[20] J. W. Amyx, D. M. Bass, and R. L. Whiting, Petroleum ReservoirEngineering Physical Properties, McGraw-Hill, New York, NY,USA, 1960.

[21] T. Ahmed, Reservoir Engineering Handbook, Elsevier, 4th edi-tion, 2010.

[22] J. H. Schon, “Physical Properties of Rocks,” in Handbook ofPetroleum Exploration and Production, Elsevier B. V., Amster-dam, The Netherlands, 2011.

[23] M. Holmes, A. Holmes, and D. Holmes, “Relationship betweenporosity and water saturation: methodology to distinguishmobile from capillary bound water two different rock types,”American Association of Petroleum Geologists (AAPG), pp. 1–11,2009.

[24] M. Sattarin, H. Modarresi, and M. Teymori, “New viscositycorrelations for dead crude oils,” Petroleum & Coal, vol. 49, pp.33–39, 2007.

[25] N. P. Cheremisinoff and A. Davletshin,AGuide to Safe Materialand Chemical Handling, Jon Wiley & Sons, 2010.

[26] P. A. Domenico and F. W. Schwartz, Physical and ChemicalHydrogeology, John Wiley & Sons, 1990.

Submit your manuscripts athttps://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

ClimatologyJournal of

EcologyInternational Journal of


EarthquakesJournal of


Mining


Journal of

Hindawi Publishing Corporation http://www.hindawi.com Volume 201

International Journal of

OceanographyInternational Journal of


Journal of Computational Environmental SciencesHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal ofPetroleum Engineering


GeochemistryHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of

Atmospheric SciencesInternational Journal of


OceanographyHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Advances in


MineralogyInternational Journal of


MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

Paleontology JournalHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

ScientificaHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Geological ResearchJournal of


Geology Advances in

application of numerical modelling and genetic...

Documents