Original article

Study of parametric effect on saturation rate in imbibition phenomenon for various porous materials

  • Pratiksha A. More ,
  • Priti V. Tandel , *
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  • Veer Narmad South Gujarat University, Surat, Gujarat, India
*E-mail addresses: (P.A. More),
(P.V.Tandel).

Received date: 2022-04-08

  Revised date: 2022-05-24

  Accepted date: 2022-06-01

  Online published: 2022-06-09

Abstract

The mathematical model of imbibition phenomenon through homogeneous as well as heterogeneous porous media is presented in this study. Various types of porous materials including Fragmented Mixture, Touchet silt loam, and Glass Beads are investigated and discussed in terms of the relative permeability, capillarity, and heterogeneity of the material on saturation rate of a reservoir. In the present model, the comparison of saturation level for different time and distance level have been discussed between homogeneous and heterogeneous porous medium for various types of sands. The reduced differential transform method (RDTM) is used to obtain approximate analytical solution of the proposed model. Numerical and graphical results are presented for a wide range of time and distance.

Highlights

● Here Purcell relative permeability model of imbibition phenomenon in double phase fluid flow in porous media is discussed

● The imbibition phenomenon is one of the useful phenomena occurs in secondary oil recovery process in Petroleum Engineering

● This study proposes the effect of heterogeneity on saturation rate for various types of porous materials

● The effect of capillary pressure and relative permeability on saturation rate are analysed for various porous materials Reduced differential transform method is used to obtain saturation rates for different distance and time levels

Cite this article

Pratiksha A. More , Priti V. Tandel . Study of parametric effect on saturation rate in imbibition phenomenon for various porous materials[J]. Journal of Ocean Engineering and Science, 2024 , 9(3) : 278 -292 . DOI: 10.1016/j.joes.2022.06.002

1. Introduction

This work examines counter-current imbibition processe in two immiscible fluids via homogeneous and heterogeneous porous media using various porous materials such as Fragmented Mixture, Touchet Silt Loam, and Glass Beads. Flows in petroleum reservoirs have been studied extensively in the last few decades.Many scholars have worked on various sorts of porous medium models, employing a variety of methodologies to deal with the governing equations of these types of models [1], [2], [3], [4]. Oil recovery includes three phases: primary, secondary, and enhanced. The primary oil recovery technique drains oil that naturally flows to the well's bottom due to reservoir pressure and gravity. First-stage oil recovery averages 10%, thus secondary and tertiary methods are needed. In the secondary recovery, Fluids like water are injected into oil formations to drive the oil into the production wells, resulting in the imbibition of oil. Some substance (oil) is brought into contact with another fluid in a porous medium that is heterogeneous or homogeneous. Counter-current imbibition occurs due to difference in wetting ability of water and oil. When the flow of the originating fluid (oil) and the flow of the fluid (water) injected is in the opposite direction [5]. Numerous researchers examined this topic from a variety of perspectives. Patel et al. [6] investigated this phenomenon in heterogeneous porous media, taking into account capillary pressure. For this study, Patel et al. [7], [8], [9] took into consideration Corey's model and investigated the phenomenon in a heterogeneously porous medium under the influence of capillary pressure as well as an inclined heterogeneously porous medium under the consideration of different porous materials, among other things. Ghosh et al. [10] investigated non-classical flow simulations of spontaneous imbibition in spatially diverse reservoirs. Pooladi-Darvish et al. [11], [12] covered the experimental investigation of cocurrent and countercurrent flows in water-wet fractured porous media. Ashruf et al. [13] proposes a generalised model for spontaneous imbibition in a porous material with many layers. Khan et al. [5] used LeNN-WOA-NM Algorithm to analyse multi-phase flow through porous media for this phenomena. It is commonly believed, when discussing imbibition of saltwater into an oil-filled porous rock, that the flow rates of the seawater and oil are equal and moving in opposing directions. Recently, Meher et al. [14] discussed temporal-fractional porous medium model in fractured media. The proposed model is analysed for saturation profile with different parametric effect for various types of porous materials.
This paper proposes Purcell relative permeability model for imbibition phenomenon.The imbibition phenomenon plays an important role in petroleum technology while secondary oil recovery process. The oil recovery rate is proportional to saturation of injected fluid. The main purpose of the work is to study the parametric saturation behaviour for different types of sands. Effect of relative permeability, capillarity, and heterogeneity on a saturation rate for different types of porous materials like Fragmented Mixture, Touchet silt loam, and Glass Beads have been carried out in the present study. Reduced differential transform method is used to solve the governing second-order partial differential equation [15], [16], [17], [18], [19].
The remainder of the report is structured as follows: The model's mathematical construction is discussed in Section 2. Sections 3 and 4 give some basic definitions and properties of RDTM, as well as applications of the proposed approaches. Section 5 contains a discussion of the numerical results and their implications. Section 6 concludes with closing remarks.

2. Mathematical construction

The mathematical model assumes that a cylindrical piece of a porous matrix with an oil-formatted region has a length “L ” that is completely saturated with oil. Except for one end, a common interface (the lowermost section of the matrix), the supposed cylinder is enclosed by an impermeable surface. This end is near an injected water formation. The water saturates the right side of the imbibition face because of the difference in viscosity between oil and water. Due to the capillary pressure action generated by imbibition, it travels only a short distance “ l ” (without external effort) [20]. The problem under consideration is depicted schematically in the diagram below (Fig. 1):
Fig.1 Natural mechanism of imbibition phenomenon.
According to the Darcy's law [20],
Viw=kiwδiwKpiwVno=knoδnoKpno
where Viw , Vno are seepage velocities, kiw , kno are relative permeabilities, piw , pno are pressures, δiw , δno are viscosities of water and oil respectively. K is permeability of the medium and suffix iw stands for injected water, no stands for native oil. The equation of conservation of mass for double phase flow with respect to volume can be written as Patel and Meher [20]
t(ϕ(x)Siwρiw)+·(Viwρiw)=0t(ϕ(x)Snoρno)+·(Vnoρno)=0
where ϕ(x) is porosity of the medium and ρiw , ρiw are mass, Siw , Sno are saturation of wetting and non-wetting fluid.
If the compressibility of the injecting fluid is ignored, the values of ρ 's are constant, and Eq. (2) is simplified
t(ϕ(x)Siw)+·(Viw)=0t(ϕ(x)Sno)+·(Vno)=0
According to Patel et al. [6], the condition for imbibition can be described as follows:
Viw=Vno
Capillary pressure pc can written as follows:
pc=piwpno
Oroveanu [21] and Patel et al. [6] have shown that the variation laws in the porosity and permeability of a medium may be characterized as a function of x alone.
ϕ(x)=1mnxK(x)=Kcϕ(x)
where m , n and Kc are constants.
The pcSiw relationship (Purcell relative permeability model [22]) can be expressed as
pc(Siw)=pdSe1λ=pd(SiwSwr1Swr)1λ
kiw=(SiwSwr1Swr)2+λλ
where pd is entry pressure, Se is effective saturation, Swr is wetting phase residual saturation and λ is grain size distribution.
Combining Eqs. (1), (4) and (5), it obtains
Viw=K(x)kiwknokiwδno+knoδiw(pcx)
Hence, Continuity Eq. (3) with Eq. (9) can be expressed as
ϕ(x)Siwt+x[K(x)kiwknokiwδno+knoδiwpcxSiwx]=0
Using Eqs. (7) and (8) with Eq. (10), it gives
ϕ(x)Siwt+Kcpdδnox[ϕ(x)(SiwSwr1Swr)2+λλSiw(SiwSwr1Swr)1λSiwx]=0
where kiwknokiwδno+knoδiwknoδno [6] Simplifying Eq. (11), we get
Siwt+Kcpdδno[x[(SiwSwr1Swr)2+λλSiw(SiwSwr1Swr)1λSiwx]+(SiwSwr1Swr)2+λλSiw(SiwSwr1Swr)1λSiwx1ϕϕx]=0
Now, this phenomenon is studied for homogeneous and heterogeneous porous medium.
Case: 1 Heterogeneous porous media
If ϕ(x) is considered in variable form in Eq. (12), then the obtained equation is the governing equation for heterogeneous porous media. Here we have considered the variable form of ϕ(x) as
ϕ(x)=1mnx
Then using dimensionless variablesX=xL , T=KcpdδnoL2t and simplification of 1ϕϕX as
1ϕϕX=X[nmLXlogm]=nLm (Neglecting higher order term of X )
The dimensionless form of Eq. (12) for this case is
SiwT+X[(SiwSwr1Swr)2+λλSiw(SiwSwr1Swr)1λSiwX]+nLm(SiwSwr1Swr)2+λλSiw(SiwSwr1Swr)1λSiwX=0
Case:2 Homogeneous porous media
If ϕ(x) is considered as constant in Eq. (12), then obtained equation is form as governing equation for homogeneous porous media, which is given by
SiwT+X[(SiwSwr1Swr)2+λλSiw(SiwSwr1Swr)1λSiwX]=0
Initial saturation for the present study is considered as [20]
Siw(X,0)=Siw0=eX

3. Analysis of RDTM

In order to demonstrate the fundamentals of RDTM, we can look at the general nonlinear partial differential equation
Lμ(ξ,η)+Rμ(ξ,η)+Nμ(ξ,η)=g(ξ,η)
with initial conditionμ(ξ,0)=f(ξ) , where L=η , R is a linear operator which has partial derivatives, Nμ(ξ,η) is a nonlinear term and g(ξ,η) is an inhomogeneous term [15], [16], [17], [18], [19].
According to the RDTM and Table 1, we can construct the following iteration formula:
(σ+1)Ψσ+1(ξ)=Gσ(ξ)RΨσ(ξ)NΨσ(ξ)
where Ψσ(ξ) , RΨσ(ξ) , NΨσ(ξ) and Gσ(ξ) are the transformed functions of Lμ(ξ,η) , Rμ(ξ,η) , Nμ(ξ,η) and g(ξ,η) respectively.
Table 1 Reduced differential transformation [19].
Functional formTransformed form
μ(ξ,η) Ψσ(ξ)=1σ![σtσμ(ξ,η)]η=0
ω(ξ,η)=γμ(ξ,η) Ωσ(ξ)=γΨσ(ξ)
ω(ξ,η)=μ(ξ,η)±ν(ξ,η) Ωσ(ξ)=Ψσ(ξ)±χσ(ξ)
ω(ξ,η)=ξcηd Ωσ(ξ)=ξcϵ(σd)
ω(ξ,η)=ξcηdμ(ξ,η) Ωσ(ξ)=ξcΨσd(ξ)
ω(ξ,η)=μ(ξ,η)ν(ξ,η Ωσ(ξ)=q=0σχq(ξ)Ψσq(ξ)
ω(ξ,η)=qηqμ(ξ,η) Ωσ(ξ)=(σ+q)!σ!Ψσ+q(ξ)
ω(ξ,η)=ξμ(ξ,η) Ωσ(ξ)=ξΨσ(ξ)
From the initial condition, We write
Ψ0(ξ)=f(Ψ)
We can obtain Ψσ(ξ) by substituting Eq. (19) into Eq. (18) and performing easy iterative calculations on the results. Then the inverse transformation of the set of values {Ψσ(ξ)}σ=0j gives the approximation solution as,
μ˜j(ξ,η)=σ=0jΨσ(ξ)ησ
where j is order of approximation solution. Therefore, the exact solution of the problem is given by
μ(ξ,η)=limjμ˜j(ξ,η)
Fig.2 Water saturation in Fragmented mixture, Touchet silt loam and Glass beads for homogeneous porous media.
Fig.3 Water saturation in Fragmented mixture, Touchet silt loam and Glass beads for heterogeneous porous media.

4. Mathematical solution

The approximate analytical solution is obtained in the series form by applying the aforesaid procedure to Eqs. (14) and (15) and Table 1 for three different porous materials.
We use the same initial condition for both homogeneous and heterogeneous porous media in order to produce results that can be compared for this occurrence in both cases.
For the entire study, the values of parameters are considered as Table 2.
Table 2 Parametric value of parameters [23].
PropertyFragmented mixtureTouchet silt loamGlass beads
λ 2.891.827.3
Swr 0.2760.2700.085
pd 17.27529
K 11.30.606.30
ϕ 0.4430.4850.370
Case - 1: Fragmented mixture
Homogeneous porous media
From Eq. (15), the solution of this case is Siw(X,T)=ex+707927319054081e2x(386947554758585ex+479488912553371)t1267650600228229401496703205376+7121927425588105e3x(299456820267296259463482179584e2x+2226444746956558684688446652416ex+2069186555354528415061469298688)t291343852333181432387730302044767688728495783936+5970698885105605e4x(4020414302986372453574493618519795091225182208e2x+115874084358211362262350635500977319850278912e3x+13611333524874160931585745741662562232184078336ex+9701041888237317601517933139477977606173753344)t3411376139330301510538742295639337626245683966408394965837152256+...
Heterogeneous porous media
From Eq. (14), the solution of this case is Siw(X,T)=ex+2265367420973059e2x(431059576001063680ex+746564236845598592)t8112963841460668169578900514406400+4558033552376387e3x(185812358062216791304687846953058304e2x+2442779477239339723899661156370874368ex+2745982301439553412871328370467536896)t2116920130986472233456294786617302641572474603438080000+305699782917407e4x(3220880299373723714760843076157597984327923858571001856e2x+40048098141028500418335727483149241617866476807520256e3x+13826977945161411250283373710438254392167767446913024000ex+11267972202970802213858488286527084241133620739346268160)t342124916667422874679167211073468172927558038160219644501724391014400000+...
Case - 2: Touchet silt loam
Homogeneous porous media
Siw(X,T)=ex+735403554637737e2x(1360977908479383ex+4949010575469829)t5070602400912917605986812821504+8646177984353847e3x(411613526081981307314354782208e2x+8980658749393700598938232422400ex+24492705676112207752149306703872)t2182687704666362864775460604089535377456991567872+1673309727480099e4x(513344082783710237822654366476547635839426887680e2x+5041772242459909727963000091904977736008990720e3x+5100106795643674796258493305212716430881527431168ex+10666890028339531721005131587917143785668570251264)t36582018229284824168619876730229402019930943462534319453394436096+...
Heterogeneous porous media
Siw(X,T)=ex+2206210663913211e2x(93453816382250960ex+499850068122452736)t2028240960365167042394725128601600+8646177984353847e3x(8733615796407478794394665605922816e2x+368260892677638081445813598415749120ex+1244302926463528512423518208926941184)t214615016373309029182036848327162830196559325429760000+323090534843951e4x(469298896211069784160060568340683127255865785057280e2x+1741514353464988082519140397010087899986044911616e3x+6087333774368200317594762934650362879947812557029376ex+14789158165655108731069587658916460526622972288434176)t33213876088517980551083924184682325205044405987565585670602752000000+...
Case - 3: Glass beads
Homogeneous porous media
Siw(X,T)=ex+4995876357250537e2x(15336915991045080ex+4935451633107701)t649037107316853453566312041152512+6234707894193061e3x(52271331581416505721128998666240e2x+100926142766452068958194553061376ex+24358682822745473090869620899840)t246768052394588893382517914646921056628989841375232+8196986730653767e4x(8126452807773812314048823288254931543560892186624e2x+901891148854777333536065689404250330918081789952e3x+7144852754007363233998611827263912637321124511744ex+1322432110096588097255385301303676507452690399232)t326959946667150639794667015087019630673637144422540572481103610249216+...
Heterogeneous porous media
Siw(X,T)=ex+4995876357250537e2x(1932451414871680000ex+804478616196555264)t129807421463370690713262408230502400+8312943858924081e3x(1244789490279852698047587425452032000e2x+3717917247230561249232379693383024640ex+1044232373928275724136525417339158528)t23741444191567111470601433171753684530319187310018560000+245909601919613e4x(2573973049308220422927301705137411639176280281664454656e2x+150343450731793677747862299151321169819428890104299520e3x+2736615104161290427069775477105887707037396337313185792ex+564326744069957433601108956768521306624968308211843072)t3539198933343012795893340301740392613472742888450811449622072204984320000+...

5. Results and discussions

From the saturation rate of injected fluid at various distances and time levels, we can study the recovery rate of oil. If saturation level is increased, more oil can be recovered. In this section, we have discussed the saturation level at different time and distance level for various sands. Also, we have studied the behaviour of saturation level for fixed time and distance. In this section, we have also studied the variations in saturation level with the different parametric effect like capillary pressure, relative permeability, and heterogeneity.

5.1. Effect of types of porous media on saturation profile for different porous materials

Tables 3 and 4 show the saturation rate for Fragmented Mixture for different distances and time levels for homogeneous and heterogeneous porous media. The saturation rates at Touchet silt loam for homogeneous and heterogeneous porous media are shown in Tables 5 and 6 for varying distances and time levels, respectively. The saturation rates for Glass Beads are shown in Tables 7 and 8 for homogeneous and heterogeneous porous media, depending on the distance and time level being considered. Fig. 2 compares the saturation rate of all sands for homogeneous porous media for different spaces and times. Fig. 3 compares the saturation rate of all sands for heterogeneous porous media at different distances and times. At a fixed distance, X=0.4 and X=0.6 , Figs. 4, 6, and 8 show the variations in saturation rate in homogeneous and heterogeneous porous media for Fragmented Mixture, Touchet silt loam, and Glass Beads, respectively. We can see that the saturation level increases as time increases for all sands for a fixed distance, which is consistent with the underlying physical phenomenon. For simplicity, we have considered X=0.4 and X=0.6 only. Figs. 5, 7, and 9 indicate the saturation rate at a fixed T=0.004 and T=0.006 for all the porous materials in homogeneous and heterogeneous porous media. The saturation rate decreases with increasing distance for the specified time. For simplicity, we have considered T=0.004 and T=0.006 . Also, for all types of sands, the saturation rate is higher in the homogeneous porous media than in the heterogeneous porous medium.
Table 3 Water saturation in Fragmented mixture in homogeneous porous media.
X T0.0010.0020.0030.0040.0050.0060.0070.0080.0090.01
0.10.9052520.9056680.9060840.9065010.9069180.9073360.9077550.9081740.9085930.909014
0.20.8190870.8194440.8198020.8201600.8205180.8208770.8212370.8215970.8219570.822318
0.30.7411250.7414330.7417410.7420490.7423580.7426670.7429760.7432860.7435960.743907
0.40.6705850.6708510.6711170.6713830.6716490.6719160.6721830.6724510.6727180.672986
0.50.6067600.6069900.6072200.6074510.6076810.6079120.6081430.6083750.6086060.608838
0.60.5490110.5492100.5494100.5496100.5498100.5500100.5502110.5504120.5506120.550813
0.70.4967590.4969320.4971060.4972800.4974540.4976280.4978020.4979770.4981510.498326
0.80.4494800.4496320.4497830.4499350.4500860.4502380.4503900.4505420.4506940.450847
0.90.4067020.4068340.4069660.4070990.4072310.4073640.4074970.4076300.4077630.407896
10.3679950.3681110.3682270.3683430.3684590.3685750.3686920.3688080.3689240.369041
Table 4 Water saturation in Fragmented mixture in heterogeneous porous media.
X T0.0010.0020.0030.0040.0050.0060.0070.0080.0090.01
0.10.9051170.9053970.9056780.9059580.9062390.9065210.9068030.9070850.9073670.907650
0.20.8189690.8192080.8194470.8196860.8199250.8201650.8204050.8206450.8208860.821127
0.30.7410220.7412260.7414300.7416340.7418390.7420430.7422480.7424530.7426590.742864
0.40.6704940.6706690.6708440.6710190.6711940.6713690.6715440.6717200.6718960.672072
0.50.6066800.6068300.6069800.6071300.6072810.6074310.6075820.6077320.6078830.608034
0.60.5489410.5490700.5491990.5493280.5494570.5495870.5497160.5498460.5499750.550105
0.70.4966970.4968080.4969190.4970310.4971420.4972540.4973660.4974770.4975890.497701
0.80.4494250.4495210.4496180.4497140.4498110.4499070.4500040.4501000.4501970.450294
0.90.4066530.4067370.4068200.4069040.4069870.4070710.4071550.4072380.4073220.407406
10.3679520.3680250.3680970.3681700.3682420.3683150.3683880.3684610.3685340.368606
Table 5 Water saturation rate in Touchet silt loam in homogeneous porous media.
X T0.0010.0020.0030.0040.0050.0060.0070.0080.0090.01
0.10.9056050.9063750.9071470.9079220.9087000.9094800.9102620.9110470.9118350.912625
0.20.8193740.8200200.8206670.8213170.8219680.8226210.8232770.8239340.8245930.825254
0.30.7413590.7419010.7424450.7429900.7435370.7440850.7446350.7451860.7457390.746293
0.40.6707750.6712320.6716900.6721480.6726080.6730690.6735310.6739950.6744590.674925
0.50.6069150.6073000.6076860.6080730.6084600.6088490.6092380.6096290.6100200.610412
0.60.5491360.5494620.5497880.5501150.5504430.5507710.5511000.5514290.5517590.552090
0.70.4968610.4971360.4974130.4976900.4979670.4982450.4985230.4988020.4990820.499361
0.80.4495630.4497970.4500320.4502670.4505020.4507380.4509740.4512110.4514480.451685
0.90.4067690.4069680.4071680.4073680.4075680.4077690.4079700.4081710.4083730.408574
10.3680490.3682190.3683900.3685600.3687310.3689020.3690740.3692450.3694170.369589
Table 6 Water saturation in Touchet silt loam in heterogeneous porous media.
X T0.0010.0020.0030.0040.0050.0060.0070.0080.0090.01
0.10.9053750.9059150.9064550.9069980.9075410.9080870.9086340.9091820.9097320.910283
0.20.8191790.8196280.8200790.8205300.8209830.8214370.8218920.8223480.8228060.823264
0.30.7411920.7415670.7419430.7423200.7426970.7430760.7434550.7438350.7442160.744598
0.40.6706330.6709460.6712600.6715750.6718900.6722060.6725230.6728410.6731590.673477
0.50.6067930.6070550.6073180.6075810.6078450.6081100.6083750.6086400.6089060.609173
0.60.5490310.5492520.5494720.5496930.5499140.5501360.5503580.5505800.5508030.551026
0.70.4967700.4969550.4971400.4973260.4975120.4976980.4978840.4980710.4982580.498446
0.80.4494850.4496400.4497960.4499530.4501090.4502660.4504230.4505800.4507370.450895
0.90.4067010.4068320.4069640.4070960.4072280.4073600.4074920.4076250.4077580.407891
10.3679900.3681020.3682130.3683240.3684360.3685480.3686600.3687720.3688840.368996
Fig.4 Water saturation vs. time in Fragmented mixture for fixed distance (a) X=0.4 (b) X=0.6 .
Fig.5 Water saturation vs. distance in Fragmented mixture for fixed time (a) T=0.004 (b) T=0.006 .
Fig.6 Water saturation vs. time in Touchet silt loam for fixed distance (a) X=0.4 (b) X=0.6 .
Table 7 Water saturation in Glass beads in homogeneous porous media.
X T0.0010.0020.0030.0040.0050.0060.0070.0080.0090.01
0.10.9049750.9051130.9052510.9053890.9055280.9056660.9058040.9059420.9060800.906219
0.20.8188530.8189750.8190970.8192190.8193420.8194640.8195860.8197090.8198310.819954
0.30.7409270.7410350.7411430.7412520.7413600.7414690.7415770.7416860.7417940.741903
0.40.6704160.6705130.6706090.6707050.6708010.6708980.6709940.6710900.6711870.671283
0.50.6066160.6067020.6067870.6068730.6069590.6070440.6071300.6072160.6073020.607387
0.60.5488880.5489640.5490400.5491170.5491930.5492690.5493460.5494220.5494980.549575
0.70.4966530.4967210.4967890.4968570.4969250.4969940.4970620.4971300.4971980.497266
0.80.4493900.4494500.4495110.4495720.4496330.4496930.4497540.4498150.4498760.449937
0.90.4066240.4066780.4067330.4067870.4068410.4068960.4069500.4070040.4070590.407113
10.3679280.3679770.3680250.3680740.3681220.3681710.3682200.3682680.3683170.368366
Table 8 Water saturation in Glass beads in heterogeneous porous media .
X T0.0010.0020.0030.0040.0050.0060.0070.0080.0090.01
0.10.9049300.9050230.9051150.9052080.9053010.9053940.9054860.9055790.9056720.905765
0.20.8188120.8188940.8189760.8190570.8191390.8192210.8193030.8193850.8194660.819548
0.30.7408900.7409620.7410350.7411070.7411790.7412510.7413230.7413950.7414680.741540
0.40.6703840.6704480.6705110.6705750.6706390.6707030.6707670.6708310.6708940.670958
0.50.6065870.6066440.6067000.6067570.6068130.6068700.6069260.6069830.6070400.607096
0.60.5488620.5489120.5489620.5490120.5490620.5491130.5491630.5492130.5492630.549314
0.70.4966300.4966740.4967190.4967640.4968080.4968530.4968970.4969420.4969870.497031
0.80.4493690.4494080.4494480.4494880.4495270.4495670.4496070.4496470.4496860.449726
0.90.4066050.4066400.4066760.4067110.4067470.4067820.4068170.4068530.4068880.406923
10.3679110.3679430.3679740.3680060.3680370.3680690.3681000.3681320.3681640.368195
Fig.7 Water saturation vs. distance in Touchet silt loam for fixed time (a) T=0.004 (b) T=0.006 .
Fig.8 Water saturation vs. time in Glass beads for fixed distance (a) X=0.4 (b) X=0.6 .
Fig.9 Water saturation vs. distance in Glass beads for fixed time (a) T=0.004 (b) T=0.006 .

5.2. Effect of capillary pressure and heterogeneity on saturation rate

Figs. 10 -12 show the variations in capillary pressure at a fixed distance X=0.4 and X=0.6 in homogeneous and heterogeneous porous media for Fragmented Mixture, Touchet silt loam, and Glass Beads, respectively. We can see that capillary pressure increases as saturation rates decrease for all sands for a fixed distance. It demonstrates that capillary pressure has a more significant influence on heterogeneous than homogeneous porous media. Fig. 13 compares the capillary pressure of all sands for homogeneous porous media at different distances and times. Fig. 14 show the capillary pressure of all sands for heterogeneous porous media for different spaces and times. Saturation rates concerning capillary pressure have been explored numerically in detail in Table 9 for Fragmented Mixture, Touchet silt loam, and Glass Beads for homogeneous and heterogeneous porous media at fixed time T=0.004 . Table 10 shows the saturation rates concerning capillary pressure for Fragmented Mixture, Touchet silt loam, and Glass Beads for homogeneous and heterogeneous porous media at a fixed time of T=0.006 .
Fig.10 Comparison between capillary pressure vs. water saturation in Fragmented mixture for fixed (a) X=0.4 (b) X=0.6 , for homogeneous porous media and for fixed (c) X=0.4 , (d) X=0.6 , for heterogeneous porous media.
Table 9 Capillary pressure vs. water saturation in Fragmented mixture, Touchet silt loam, and Glass beads.
T=0.004
Fragmented mixtureTouchet silt loamGlass beads
Homogeneous porous mediaHeterogeneous porous mediaHomogeneous porous mediaHeterogeneous porous mediaHomogeneous porous mediaHeterogeneous porous media
Siw pc Siw pc Siw pc Siw pc Siw pc Siw pc
0.90650118.042970.90595818.048340.90792280.767080.90699880.831470.90538929.436850.90520829.43774
0.82016018.986250.81968618.991970.82131787.508680.82053087.577340.81921929.887750.81905729.88865
0.74204920.032010.74163420.038190.74299095.195520.74232095.269750.74125230.350930.74110730.35185
0.67138321.204850.67101921.211610.672148104.07210.671575104.15370.67070530.827480.67057530.82842
0.60745122.539280.60713022.546820.608073114.48530.607581114.57680.60687331.318640.60675731.3196
0.54961024.085690.54932824.094290.550115126.94780.549693127.05310.54911731.82590.54901231.82688
0.49728025.921420.49703125.931530.497690142.25650.497326142.38150.49685732.350990.49676432.352
0.44993528.173270.44971428.185640.450267161.73410.449953161.88910.44957232.895990.44948832.89703
0.40709931.068770.40690431.08480.407368187.78180.407096187.98640.40678733.463370.40671133.46444
0.36834335.074120.36817035.096920.368560225.35880.368324225.65590.36807434.056140.36800634.05727
Fig.11 Comparison between capillary pressure vs. water saturation in Touchet silt loam for fixed (a) X=0.4 (b) X=0.6 , for homogeneous porous media and for fixed (c) X=0.4 , (d) X=0.6 , for heterogeneous porous media.
Fig.12 Comparison between capillary pressure vs. water saturation in Glass beads for fixed (a) X=0.4 (b) X=0.6 , for homogeneous porous media and for fixed (c) X=0.4 , (d) X=0.6 , for heterogeneous porous media.
Fig.13 Comparison of capillary pressure in Fragmented mixture, Touchet silt loam and Glass beads for homogeneous porous media.
Fig.14 Comparison of capillary pressure in Fragmented mixture, Touchet silt loam and Glass beads for heterogeneous porous media.
Table 10 Capillary pressure vs. water saturation in Fragmented mixture, Touchet silt loam, and Glass beads.
T=0.006
Fragmented mixtureTouchet silt loamGlass beads
Homogeneous porous mediaHeterogeneous porous mediaHomogeneous porous mediaHeterogeneous porous mediaHomogeneous porous mediaHeterogeneous porous media
Siw pc Siw pc Siw pc Siw pc Siw pc Siw pc
0.90733618.03470.90652118.042770.90948080.658940.90808780.755640.90566629.435490.90539429.43683
0.82087718.97760.82016518.986190.82262187.395120.82143787.49820.81946429.886390.81922129.88774
0.74266720.022830.74204320.03210.74408595.074660.74307695.186080.74146930.349560.74125130.35094
0.67191621.194960.67136921.205110.673069103.94140.672206104.06380.67089830.826090.67070330.8275
0.60791222.528430.60743122.539740.608849114.34110.608110114.47840.60704431.317240.60687031.31867
0.55001024.073510.54958724.086410.550771126.78480.550136126.94260.54926931.824470.54911331.82594
0.49762825.907320.49725425.922480.498245142.06630.497698142.25360.49699432.349530.49685332.35104
0.45023828.156280.44990728.174810.450738161.50230.450266161.73460.44969332.894480.44956732.89605
0.40736431.047050.40707131.071070.407769187.48140.407360187.78760.40689633.461820.40678233.46344
0.36857535.043640.36831535.077780.368902224.93040.368548225.37470.36817134.054540.36806934.05622

5.3. Effect of relative permeability and heterogeneity on Saturation rate

Figs. 15 -17 show the variations in relative permeability at a fixed distance X=0.4 and X=0.6 in homogeneous and heterogeneous porous media for Fragmented Mixture, Touchet silt loam, and Glass Beads, respectively. We can see that for a fixed distance, relative permeability decreases as saturation rates decrease for all sands. It demonstrates that relative permeability has a greater influence on homogeneous than heterogeneous porous media. Fig. 18 compare the relative permeability of all sands for homogeneous porous media at different distances and time. Fig. 19 shows the relative permeability of all sands for heterogeneous porous media for different distances and times. Saturation rates for relative permeability have been explored numerically in detail in Table 11 for Fragmented Mixture, Touchet silt loam, and Glass Beads for homogeneous and heterogeneous porous media at fixed time T=0.004 . Table 12 shows the saturation rates for relative permeability for Fragmented Mixture, Touchet silt loam, and Glass Beads for homogeneous and heterogeneous porous media at a fixed time of T=0.006 .
Fig.15 Comparison between relative permeability vs. water saturation in Fragmented mixture for homogeneous porous media for fixed (a) X=0.4 (b) X=0.6 , for homogeneous porous media and for fixed (c) X=0.4 , (d) X=0.6 , for heterogeneous porous media.
Fig.16 Comparison between relative permeability vs. water saturation in Touchet silt loam for fixed (a) X=0.4 (b) X=0.6 , for homogeneous porous media and for fixed (c) X=0.4 , (d) X=0.6 , for heterogeneous porous media.
Fig.17 Comparison between relative permeability vs. water saturation in Glass beads for fixed (a) X=0.4 (b) X=0.6 , for homogeneous porous media and for fixed (c) X=0.4 , (d) X=0.6 , for heterogeneous porous media.
Fig.18 Comparison of relative permeability in Fragmented mixture, Touchet silt loam and Glass beads for homogeneous porous media.
Fig.19 Comparison of relative permeability in Fragmented mixture, Touchet silt loam and Glass beads for heterogeneous porous media.
Table 11 Relative permeability vs. water saturation in Fragmented Mixture, Touchet silt loam, and Glass beads.
T=0.004
Fragmented mixtureTouchet silt loamGlass beads
Homogeneous porous mediaHeterogeneous porous mediaHomogeneous porous mediaHeterogeneous porous mediaHomogeneous porous mediaHeterogeneous porous media
Siw kiw Siw kiw Siw kiw Siw kiw Siw kiw Siw kiw
0.9065010.7913860.9059580.7902340.9079220.7535270.9069980.7512360.9053890.8701870.9052080.869942
0.8201600.6168320.8196860.6159230.8213170.5547520.8205300.5530920.8192190.7554650.8190570.755253
0.7420490.4745710.7416340.4738560.7429900.4021790.7423200.4009830.7412520.6547890.7411070.654604
0.6713830.3593070.6710190.3587470.6721480.28610.6715750.2852450.6707050.5664710.6705750.566311
0.6074510.2665980.6071300.2661620.6080730.1987510.6075810.1981460.6068730.4890280.6067570.488889
0.5496100.1927220.5493280.1923860.5501150.1339320.5496930.1335090.5491170.4211540.5490120.421033
0.4972800.1345680.4970310.1343120.4976900.0866960.4973260.0864060.4968570.3616980.4967640.361593
0.4499350.0895430.4497140.0893510.4502670.0531020.4499530.0529080.4495720.3096510.4494880.30956
0.4070990.0554970.4069040.0553570.4073680.0300180.4070960.0298930.4067870.2641220.4067110.264042
0.3683430.0306720.3681700.0305750.3685600.0149540.3683240.0148790.3680740.2243280.3680060.224259
Table 12 Relative permeability vs. water saturation in Fragmented mixture, Touchet silt loam, and Glass beads.
T=0.006
Fragmented mixtureTouchet silt loamGlass beads
Homogeneous porous mediaHeterogeneous porous mediaHomogeneous porous mediaHeterogeneous porous mediaHomogeneous porous mediaHeterogeneous porous media
Siw kiw Siw kiw Siw kiw Siw kiw Siw kiw Siw kiw
0.9073360.793160.9065210.7914280.9094800.7573930.9080870.7539350.9056660.870560.9053940.870192
0.8208770.6182080.8201650.6168410.8226210.557510.8214370.5550060.8194640.7557860.8192210.755467
0.7426670.4756360.7420430.4745610.7440850.4041350.7430760.4023310.7414690.6550640.7412510.654788
0.6719160.3601270.6713690.3592860.6730690.2874770.6722060.2861870.6708980.5667090.6707030.566469
0.6079120.2672270.6074310.2665710.6088490.199710.6081100.1987970.6070440.4892330.6068700.489024
0.5500100.19320.5495870.1926940.5507710.1345910.5501360.1339530.5492690.421330.5491130.421149
0.4976280.1349270.4972540.1345410.4982450.087140.4976980.0867030.4969940.3618510.4968530.361693
0.4502380.0898070.4499070.0895190.4507380.0533940.4502660.0531020.4496930.3097820.4495670.309646
0.4073640.0556870.4070710.0554770.4077690.0302020.4073600.0300140.4068960.2642350.4067820.264116
0.3685750.0308030.3683150.0306570.3689020.0150630.3685480.014950.3681710.2244260.3680690.224323

5.4. Rate of convergence

Theorem: If r=0Hntr is given series,
[1] If 0<κ<1 Hr+1Hrκ then given series is convergent.
[2] If κ>1 Hr+1Hrκ then given series is divergent [24].
Proof. Let (C[l],,) be the Banach Space of all continuous functions on l with the norm . .Also, assume that H0(x)M ,where M is a positive integer. Let {Φn}r=0 be a partial sum
Φr=H0+H1+H2+...+Hr
If we prove, {Φr}r=0 is Cauchy sequence in Banach Space then sequence of partial sum is convergent in Banach Space. Therefore, Now We will prove that sequence of partial sum is Cauchy sequence. We take
Φr+1Φr=Hr+1κHr...κr+1H0κr+1M
Therefore, r,sN,rs, We have
ΦrΦs=(ΦrΦr1)+(Φr1Φr2)+...+(Φs+1Φs)ΦrΦr1+Φr1Φr2+...+Φs+1Φs1κrs1κκs+1H0
Now here 0<κ<1 , We obtain
limr,sΦrΦs=0
Hence {Φr}r=0 is Cauchy sequence in Banach Space. Therefore, given series is convergent.
From the Eq. (18), the solution of the problem is of the series form
Siw(X,T)=Siw0(X,T)+Siw1(X,T)T+Siw2(X,T)T2+...
So, the convergence of series solution for Fragmented Mixture in homogeneous is discussed as below:
Homogeneous porous media
Siw(X,T)=Siw0(X,T)+Siw1(X,T)T+Siw2(X,T)T2+Siw3(X,T)T3+...=M0+M1+M2+M3+...
M1M0=0.0046<1,M2M1=0.0068<1,M3M2=0.0103<1
Similarly, the convergence of series solution for Fragmented Mixture in heterogeneous is discussed as below: M1M0=0.0031<1,M2M1=0.0059<1,M3M2=0.0091<1
Similarly, the convergence of series solution for Touchet silt loam and Glass beads in homogeneous and heterogeneous porous media are discussed as below:
M1M0=0.0085<1,M2M1=0.0160<1,M3M2=0.0236<1
M1M0=0.0059<1,M2M1=0.0136<1,M3M2=0.0204<1
M1M0=0.0015<1,M2M1=0.0014<1,M3M2=0.0021<1
M1M0=0.0010<1,M2M1=0.0012<1,M3M2=0.0019<1
Hence, the obtain series solution is convergent.

6. Conclusion

From the present study, we can say that saturation rates are higher in homogeneous porous media than in heterogeneous porous media for all types of porous material. Also, the saturation rate for Touchet silt loam is higher than other sands. It is observed that saturation rate increases as capillary pressure decreases, and saturation rate decreases as relative permeability decreases. If we maintain the capillary pressure level as low as possible, we can recover more oil through the reservoir, and high relative permeability gives a high rate of saturation and more oil recovery. The study of the effect of the relative permeability, capillarity, and heterogeneity on saturation rate with different types of materials in homogeneous as well as heterogeneous porous media concludes that saturation strongly depends on the type of external force, type of materials, relative permeability, capillarity, and the types of the porous medium. Also, the obtained results show that RDTM is a very powerful method to handle the non-linear partial differential equations. The findings are intriguing and useful for hydrologists and geologists working on reservoir problems and secondary oil recovery techniques.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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