Modelling of imbibition phenomena in two-phase fluid flow through fractured porous media

Hardik S. Patel; Ramakanta Meher

doi:10.1515/nleng-2016-0057

Article Publicly Available

Modelling of imbibition phenomena in two-phase fluid flow through fractured porous media

Hardik S. Patel and Ramakanta Meher

Published/Copyright: November 30, 2016

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From the journal Nonlinear Engineering Volume 6 Issue 1

Abstract

In this paper, the counter-current imbibition phenomenon in two phase fluid through fracture porous media is discussed and Adomian decomposition method is applied to find the saturation of wetting phase and the recovery rate of the reservoir. A simulation result is developed for the saturation of wetting phase in fracture matrix and in porous matrix for some interesting choices of parametric value to study the recovery rate of the oil reservoir with dimensionless time. This problem has a great importance in the oil recovery process.

Keywords: Counter–current imbibition; Fracture porous media; Capillary pressure; Adomian decomposition method

MSC 2010: 76S05; 74R10

1 Introduction

Modelling of fluid flow in fracture porous media with deformable matrix is a subject of great interest as well as one of the most challenging problems in the application of oil reservoir problems. Over the past four decades, the study on the fluid flow in fractures and porous media in under ground petroleum reservoir is much focussed. When a well is drilled, the pressure inside the formation pushes the oil deposits from the fissures and pores where it collects and into the well where it can be recovered. Oil recovery by imbibition process is accomplished by contacting water with the porous solid. If the rock is water-wet, the displacement of oil by water without any external forces gives rise to a pressure gradient is known as imbibition. The rate of water imbibition varies with the square root of absolute permeability and interfacial tension between two liquids. It is a function of viscosity of both oil and water and a complex function of relative permeability and capillary pressure. The effectiveness of this process depends on several parameters; including matrix block size, rock porosity and permeability, fluid viscosities, interfacial tensions, and rock wettability (Graham and Richardson [5]).

Several authors discussed the imbibition recovery is a function of time and Aronofsky et al. [2] first proposed that oil recovery by spontaneous imbibition as a function of time could be modeled by a simple exponential function:

R=R∞1−e−yT

Where R is the recovery, R_∞ is the ultimate recovery and y is a constant that best matches the data with a value of approximately 0.5. It was proposed for strongly water-wet media and ignores the effects of wettability and here we have defined the dimensionless time as T=KcβL2μwt to study the recovery rate with dimensionless time.

Many researchers studied this phenomenon with different approaches. Graham and Richardson et.al [5], Scheidegger et.al [20], Bokserman et. al. [3] described the physics of oil-water motion in porous medium. Verma et.al [23] employed a perturbation procedure and similarity methods to obtain an explicit analytical solution of the imbibition phenomena and studied the stabilization of fingers in a slightly heterogeneous cracked porous medium. Mehta and Verma [7] used a singular perturbation approach and discussed analytically the phenomenon of imbibition in homogenous porous media. Standnes [21] discussed the co-current and counter-current spontaneous imbibition experimentally and studied the impact of boundary conditions on oil recovery. Tavassoli, Zimmerman and Blunt [22] discussed the gravitational effect and Fischer, Wo and Morrow [4] studied the effect of viscosity ratio analytically on counter-current imbibition in a weakly water-wet system. Mirzaei-Paiaman, Masihi and Standnes [9] studied the non-equilibrium effects during spontaneous imbibition and found its negligible effect on oil recovery. Saboorian-Jooybari, Ashoori and Mowazi [19] developed an analytical time-dependent matrix/fracture shape factor for counter-current imbibition and studied the simulation analysis of fractured reservoirs. Patel and Meher [16] discussed simulation of fingering phenomena in fluid flow through fracture porous media with inclination and gravitational effect. Mirzaei-Paiaman and Masihi [11, 12] studied the scaling of oil/gas recovery rate from fractured porous media by counter-current and by co-current spontaneous imbibition. Hamidpour, Mirzaei-Paiaman, Masihi and Harimi [10] studied experimentally the effect of some important factors on non-wetting phase recovery during spontaneous imbibition and also studied the analysis of spontaneous imbibition with resistive gravity forces: displacement characteristics and scaling. Patel et.al [15] discussed the counter-current imbibition phenomena in a heterogeneous porous media with out any external effect. Patel and Meher [17, 18] studied approximate analytical study of counter-current imbibition phenomenon in a heterogeneous porous media and study on recovery rate for counter-current imbibition phenomenon with Corey’s model arising during oil recovery process.

Fig. 1

Block 1 and Block 3 are Matrix blocks and Block 2 is Fracture block

The main objective in this paper is to provide a rigorous treatment of the theory of fluid flow and deformation in fractured porous media saturated with two immiscible fluids using dual porosity model with the consideration of the fracture deformation with the porous block and the fracture network combine. The governing differential equation is formulated by using a systematic macroscopic approach based on the theory of conservation equation of mass and Darcy’s velocity. Gravity forces included with the consideration of Corey’s model in the governing equations and it is furthermore argued that they are also able to properly compensate the impact of variations on saturation rate of wetting phase as well. Analytical solution for the flow equations is presented for the counter-current imbibition phenomena in a fracture porous media by using Adomian decomposition method to study the saturation of wetting phase and a simulation result is developed to study the recovery rate as a function of time, T of the reservoir. Finally the convergence analysis of the method is done by using fixed point theory in a suitable Hilbert space.

2 Mathematical model

For the sake of mathematical model: We consider here that a finite cylindrical piece of porous matrix of an oil formatted region having length ‘L’ containing viscous oil that is completely surrounded by an impermeable surface except for one end (common interface) of the cylinder which is labeled as the Imbibition face and this end is exposed to an adjacent formation of ‘injected’ water. When the reservoir oil (non wetting phase) is come into contact with water (wetting phase) then there is a spontaneous flow of the wetting phase (Water) into the medium and a counter flow of the resident fluid i.e. nonwetting phase (oil) from the medium initiated by imbibitions. Due to the difference in viscosities of water and oil, the water saturation on the right side of imbibitions face will travel only a small distance ‘l’ due to capillary pressure effect (without external force).

Fig. 2

schematic diagram of the problem under consideration

The conservation equation of mass of oil and water can be expressed as,

∂∂tϕ(x)Siρi+∇.viρi−ρiqi=0(1)

Where i=o,w,x∈ℜ3,t≥0,ϕ(x) denotes the porosity of the porous medium, s_i denotes its saturation for each phase i, ρ_i is its specific mass, and v_i is its volumetric rate of flow (or, Darcy velocity) which is given by the two phase extension of Darcy’s law

vi=−K(x)kiμi∇pi−ρig(2)

Where i=o,w,x∈ℜ3,t≥0,K(x)[m2] denotes the absolute permeability tensor of the porous medium, p_i is its pressure, μ_i is its viscosity and k_i is its relative permeability depends on the saturation vector.

If the compressibility of fluid is neglected, then ρ_i′s are constant and the conservation equation becomes:

∂∂tϕ(x)Si+∇.vi−qi=0,i=0,w(3)

The imbibition condition for counter-current imbibitions and capillary pressure can be expressed due to Scheidegger [20] as

vw=−vo(4)

Pc(Sw)=po−pw(5)

The analytical linear relationship between capillary pressure and phase saturation Meher [8] can be written as

Pc=βC0+Sw−2(6)

Where β and C₀ are constant of proportionality.

Since the problem is dealing with the heterogeneous fracture porous media so, the porosity and permeability of heterogeneous porous media, according to Oroveanu [14] as

ϕ(x)=1a−bxK(x)=Kcϕ(x)(7)

where a − bx ≥ 0.

Using Corey’s Model [13], the standard relationship between the relative permeability and phase saturation is given by

kw=Sw4(8)

Now eq. (2) with eq. (4) and eq. (5) gives,

∂pw∂x=−koμ0∂pc∂x−kwρwμw+koρoμogkwμw+koμo(9)

Upon substituting eq. (9) in eq. (2) for phase w, it obtains

vw=K(x)kokwkwμo+koμw∂pc∂x−gρo−ρw(10)

Now the conservation equation with eq. (10) can be written as

ϕ(x)∂Sw∂t=βKcμw∂∂xϕ(x)∂Sw2∂x+Kcgρo−ρwμw∂(ϕ(x)Sw4)∂x+q(11)

where kokwkwμo+koμw≈kwμw=Sw4μw (Scheidegger [20] and P_c = β(C₀ + S_w⁻² Meher [08]).By using the dimensionless variables

X=xL,T=KcβμwL2t

By assuming the value of the source term as q=q(x,t)=1v+wx,v,w≥1.

The dimensionless forms of eq. (12) can be written as

∂Sw∂T=∂2Sw2∂X2+1ϕ∂ϕ∂X∂Sw2∂X+C∂Sw4∂X+1ϕ∂ϕ∂XSw4+Dϕ1v+wXL(12)

where C=gρo−ρwLβ.

Simplification of 1ϕ∂ϕ∂X, as

1ϕ∂ϕ∂X=∂∂Xlogϕ=∂∂XbaLX−loga(NeglectinghigherordertermofX)=bLa.(13)

It leads eq. (13) in to the form as

∂Sw∂T=∂2Sw2∂X2+bLa∂Sw2∂X+C∂Sw4∂X+bLaSw4+Da−bXLv+wXL(14)

With suitable initial condition S_w(X,0)=e^−X.

3 Analysis of the method

For the purposes of illustration of the ADM, in this study we shall consider eq. (13) in an operator form as

LTSw(X,T)=LXXN1SW+bLaLXN1SW+CLXN2SW+bLaN2SW+Da−bXLv+wXL(15)

where N₁S_w(X, T) = Sw2, N₂S_w(X, T) = Sw4 and S_w0(X) can be solved subject to the corresponding initial condition S_w(X,0) = f(X) = e^−X.

Following Adomian [1] defined the linear operators LT=∂∂T,LXX=∂2∂X2andLX=∂∂X the definite integration inverse operator LT−1 and the nonlinear term as NS_w.

Operating the inverse operator and following the analysis of Adomian decomposition, we set the recursive relation of eq. (14) as

∑n=0∞Swn(X,T)=e−X+LT−1LXX∑n=0∞An+bLaLT−1LX∑n=0∞An+CLT−1LX∑n=0∞Bn+bLa∑n=0∞Bn+DLT−1a−bXLv+wXL(16)

which gives the recurrence relation as

Sw0=Sw(X,0)=e−X+DLT−1a−bXLv+wXLSw,k+1=LT−1AkXX+bLaLT−1AkX+CLT−1BkX+bLaBkk≥0

gives the approximate analytical solution of problem eq. (13), and it can be written in the series form up to three terms as:

Sw(X,T)=Sw0+Sw1+Sw2+⋯

Sw(X,T)=e−X+Da−bXLv+wXL+2−e−X−DLv+wXL−Da−bXLLwv+wXL22+2e−X+Da−bXLv+wXL+e−X+2DL2wv+wXL2+2Da−bXLL2w2v+wXL3+2bae−X+Da−bXLv+wXL−e−X−DLv+wXL−Da−bXLLwv+wXL2+4Ce−X+Da−bXLv+wXL3−e−X−DLv+wXL−Da−bXLLwv+wXL2+2bCae−X+Da−bXLv+wXL4T+…(17)

This represents the saturation of wetting phase during counter-current imbibition phenomena in a fractured porous media.

4 Convergence analysis of the Adomian decomposition method

We recall the following theorem from[6] which guarantees the convergence of Adomian’s method for the general operator equation given by LS_w + RS_w + NS_w=g.

Consider the Hilbert space H = L²((α, β) × [0,T]) defined by the set of applications:

Sw:α,β×0,T→R

with

∫α,β×0,TSW2(η,ξ)dηdξ<+∞(18)

Let us denote

LSw=∂Sw∂T,N1Sw=Sw2,N2Sw=Sw4TSw=RSw+NSw=∂2Sw2∂X2+bLa∂Sw2∂X+C∂Sw4∂X+bLaSw4+Da−bXLv+wXL

Theorem 1

Let TS_w = − RS_w − NS_w be a hemi continuous operator in a Hilbert space H and satisfy the following hypothesis: (H1):TSw−TSw∗,Sw−Sw∗≥kSw−Sw∗2,k>0,∀Sw,Sw∗∈H

(H₂): Whatever may be M < 0, there exist constant D(M) < 0 such that for S_w, Sw∗ ∊ H with ∥ S_w ∥ ≤ M, ∥ Sw∗ ∥ ≤ M, we have (TS_w − TSw∗,w) ≤ D(M) ∥S_w − Sw∗) ∥ w ∥ for every w ∊ H.

Then, for every g ∊ H, the nonlinear functional equation LS_w + RS_w + NS_w = g admits a unique solution S_w ∊ H. Furthermore, if the solution S_w can be represented in a series form given by Sw=∑n=0∞Swnλn then the Adomian decomposition scheme corresponding to the functional equation under consideration converges strongly to _TS_w ∊ H, which is the unique solution to the functional equation.

Proof

Verification of hypothesis (H₁)

TSw−TSw∗=−∂2(Sw2−Sw2∗)∂X2+bLa∂(Sw2−Sw2∗)∂X+C∂(Sw4−Sw4∗)∂X+bLa(Sw4−Sw4∗)(TSw−TSw∗,Sw−Sw∗)=−∂2(Sw2−Sw2∗)∂X2+bLa∂(Sw2−Sw2∗)∂X+C∂(Sw4−Sw4∗)∂X+bLa(Sw4−Sw4∗),Sw−Sw∗(19)

Since ∂2/∂X2and∂/∂X are differential operator in H, then there exist constants “δ₁” and “δ₂” respectively, such that, according to Schwartz inequality, we get

∂2(Sw2−Sw2∗)∂X2+bLa∂(Sw2−Sw2∗)∂X+C∂(Sw4−Sw4∗)∂X+bLa(Sw4−Sw4∗),Sw−Sw∗≤δ1Sw2−Sw2∗+bLaδ2Sw2−Sw2∗+Cδ2Sw4−Sw4∗+CbLaSw4−Sw4∗.Sw−Sw∗(20)

Now, we use mean value theorem, then we have

∂2(Sw2−Sw2∗)∂X2+bLa∂(Sw2−Sw2∗)∂X+C∂(Sw4−Sw4∗)∂X+bLa(Sw4−Sw4∗),Sw−Sw∗≤δ1Sw2−Sw2∗+bLaδ2Sw2−Sw2∗+Cδ2Sw4−Sw4∗+CbLaSw4−Sw4∗.Sw−Sw∗≤δ1M+bLaδ2M+Cδ2M3+CM3bLaSw−Sw∗2(21)

For ∥S_w∥ ≤ M and ∥Sw∗∥ ≤ M.

Therefore

−∂2(Sw2−Sw2∗)∂X2+bLa∂(Sw2−Sw2∗)∂X]+C∂(Sw4−Sw4∗)∂X+bLa(Sw4−Sw4∗),Sw−Sw∗≥−δ1M+bLaδ2M+Cδ2M3+CM3bLaSw−Sw∗2(22)

Substituting eq. (22) in eq. (19),

(TSw−TSw∗,Sw−Sw∗)≥kSw−Sw∗2(23)

Where k=−δ1M+bLaδ2M+Cδ2M3+CM3bLa.

Hence we find the hypothesis (H₁).

For hypothesis (H₂),

−∂2(Sw2−Sw2∗)∂X2+bLa∂(Sw2−Sw2∗)∂X+C∂(Sw4−Sw4∗)∂X+bLa(Sw4−Sw4∗),V=M+bLaM+CM3+bLaM3Sw−Sw∗V=E(M)Sw−Sw∗V(24)

Where E(M)=M+bLaM+CM3+bLaM3 and therefore (H₂)holds.

The proof is complete.

Table 1

Parameters Values

Parameters	Values (Units)
Constant β	6895 N/m²
Permeability K	10⁻¹¹m²
Porosity ϕ	0.3
Viscosity of water μ_w	0.894 Ns/m²
Density of water ρ_w	1000 Kg/m³
Density of oil ρ_o	980 Kg/m³
Acceleration due to gravity g	0.101972 m/s²

5 Results and Discussion

5.1 Phase Saturation in Fracture Porous Media and Matrix Porous Media

Fig. 3 and 4 discusses the saturation rate of wetting phase in fracture porous media and in matrix porous media keeping dimensional distance X fixed. Here It is observed that the saturation rate be more in fracture porous media as compared to the porous matrix near the imbibition face X = 0 implies the saturation rate be increases as the injected water reaches the fracture porous media from the porous matrix.

Fig. 3

Comparison of Fracture Porous Media and Matrix Porous Media keeping distance X = 0.5 fixed.

Fig. 4

Comparison of Fracture Porous Media and Matrix Porous Media keeping distance X = 0.9 fixed.

5.2 Effect of initial condition on phase saturation in Fracture Porous Media and Matrix Porous Media

Fig. 5 and 6 discusses the effect of initial condition on the saturation rate of wetting phase in Fracture Porous Media and in porous Matrix. Here It is observed that the saturation rate be more at the interface and closer to the imbibition face X=0 and decreases as it away from the imbibition face for both fracture porous media and for porous matrix.

Fig. 5

Effect on initial condition in Fracture porous media

Fig. 6

Effect on initial condition in Matrix Porous media

5.3 Comparison of capillary pressure vs. saturation of water in Fracture Porous Media and Matrix Porous Media

Fig. 7 and 8 discusses the effect of capillary pressure on saturation of wetting phase in fracture porous media and in porous matrix. Here it is observed that the capillary pressure of wetting phase decreases with saturation rate implies saturation rate will be maximum with less capillary pressure for both fracture porous media and porous matrix. The numerical value for saturation and capillary pressure has been discussed in Table 2 for both fracture and porous matrix.

Fig. 7

Comparision of capillary pressure vs. Water saturation in Fracture porous media

Fig. 8

Comparision of capillary pressure vs. Water saturation in Matrix porous media

Table 2

Comparison of the numerical values for saturation vs. capillary pressure for Fracture porous media and Matrix porous media.

Saturation Vs. Capillary Pressure T = 0.005
Fracture Porous Media			Matrix Porous Media
	S_w	P_c		S_w	P_c
X = 0.1	0.9198380183	8149	X = 0.1	0.9175522034	8190
X = 0.2	0.8313931571	9975	X = 0.2	0.8291057692	10030
X = 0.3	0.7515518858	12207	X = 0.3	0.7492868275	12281
X = 0.4	0.6794608729	14935	X = 0.4	0.6772346016	15033
X = 0.5	0.6143542708	18268	X = 0.5	0.6121778552	18398
X = 0.6	0.5555442556	22341	X = 0.6	0.5534248762	22512
X = 0.7	0.5024126721	27316	X = 0.7	0.5003547156	27541
X = 0.8	0.4544036415	33393	X = 0.8	0.4524094986	33688
X = 0.9	0.4110170098	10815	X = 0.9	0.4090876562	41200
X = 1.0	0.3718025331	49878	X = 1.0	0.3699379523	50382

5.4 Comparison of permeability of water vs. saturation of water in Fracture Porous Media and Matrix Porous Media

Fig. 9 and 10 discusses the effect of permeability on saturation of wetting phase in fracture porous media and in porous matrix. Here it is observed that the saturation rate will be more with higher permeability and the permeability of the medium in fracture porous media be more as compared to the porous matrix. The numerical value for saturation and permeability has been discussed in Table 3 for both fracture and porous matrix.

Fig. 9

Comparision of capillary pressure vs. Water saturation in Fracture porous media

Fig. 10

Comparision of capillary pressure vs. Water saturation in Matrix porous media

Table 3

Comparison of the numerical values for saturation vs. permeability for Fracture porous media and Matrix porous media.

Saturation Vs. Permeability T = 0.005
Fracture Porous Media			Matrix Porous Media
	S_w	K_w		S_w	K_w
X = 0.1	0.9198380183	0.7158885604	X = 0.1	0.9175522034	0.7087990550
X = 0.2	0.8313931571	0.4777775979	X = 0.2	0.8291057692	0.4725412747
X = 0.3	0.7515518858	0.3190331967	X = 0.3	0.7492868275	0.3152044869
X = 0.4	0.6794608729	0.2131364907	X = 0.4	0.6772346016	0.2103568006
X = 0.5	0.6143542708	0.1424542881	X = 0.5	0.6121778552	0.1404463516
X = 0.6	0.5555442556	0.0952521189	X = 0.6	0.5534248762	0.0938068832
X = 0.7	0.5024126721	0.0637150957	X = 0.7	0.5003547156	0.0626775466
X = 0.8	0.4544036415	0.0426350928	X = 0.8	0.4524094986	0.0418915914
X = 0.9	0.4110170098	0.0285390283	X = 0.9	0.4090876562	0.0280069297
X = 1.0	0.3718025331	0.0191095024	X = 1.0	0.3699379523	0.0187290416

Table 4 and 5 discusses the numerical values for saturation rate of wetting phase in both fracture matrix and porous matrix.

Table 4

The numerical value for saturation rate of wetting phase in Fracture porous media.

Fracture Porous Media
S_w(X, T)	T=0.001	T=0.002	T=0.003	T=0.004	T=0.005	T=0.006	T=0.007	T=0.008	T=0.009	T=0.010
X=0.1	0.90999	0.91245	0.91491	0.91738	0.91984	0.92230	0.92476	0.92722	0.92969	0.93215
X=0.2	0.82333	0.82534	0.82736	0.82938	0.83139	0.83341	0.83543	0.83744	0.83946	0.84147
X=0.3	0.74495	0.74660	0.74825	0.74990	0.75155	0.75320	0.75485	0.75650	0.75816	0.75981
X=0.4	0.67405	0.67541	0.67676	0.67811	0.67946	0.68081	0.68216	0.68352	0.68487	0.68622
X=0.5	0.60993	0.61103	0.61214	0.61325	0.61435	0.61546	0.61657	0.61768	0.61878	0.61989
X=0.6	0.55192	0.55282	0.55373	0.55464	0.55554	0.55645	0.55736	0.55826	0.55917	0.56008
X=0.7	0.49944	0.50019	0.50093	0.50167	0.50241	0.50316	0.50390	0.50464	0.50538	0.50612
X=0.8	0.45197	0.45258	0.45319	0.45380	0.45440	0.45501	0.45562	0.45623	0.45684	0.45744
X=0.9	0.40903	0.40952	0.41002	0.41052	0.41102	0.41151	0.41201	0.41251	0.41301	0.41351
X=1.0	0.37017	0.37058	0.37099	0.37139	0.37180	0.37221	0.37262	0.37303	0.37343	0.37384

Table 5

The numerical value for saturation rate of wetting phase in Matrix porous media.

Matrix Porous Media
S_w(X, T)	T=0.001	T=0.002	T=0.003	T=0.004	T=0.005	T=0.006	T=0.007	T=0.008	T=0.009	T=0.010
X=0.1	0.90731	0.90982	0.91236	0.91494	0.91755	0.92021	0.92290	0.92563	0.92841	0.93123
X=0.2	0.82075	0.82280	0.82488	0.82698	0.82911	0.83126	0.83345	0.83566	0.83791	0.84018
X=0.3	0.74247	0.74415	0.74584	0.74755	0.74929	0.75104	0.75281	0.75461	0.75643	0.75827
X=0.4	0.67167	0.67304	0.67443	0.67582	0.67723	0.67866	0.68010	0.68156	0.68304	0.68453
X=0.5	0.60764	0.60876	0.60989	0.61103	0.61218	0.61334	0.61451	0.61570	0.61689	0.61810
X=0.6	0.54972	0.55063	0.55156	0.55249	0.55342	0.55437	0.55533	0.55629	0.55726	0.55824
X=0.7	0.49733	0.49808	0.49883	0.49959	0.50035	0.50113	0.50190	0.50269	0.50348	0.50427
X=0.8	0.44994	0.45055	0.45116	0.45178	0.45241	0.45304	0.45367	0.45431	0.45495	0.45560
X=0.9	0.40707	0.40757	0.40807	0.40858	0.40909	0.40960	0.41012	0.41064	0.41116	0.41169
X=1.0	0.36829	0.36870	0.36911	0.36952	0.36994	0.37036	0.37078	0.37120	0.37163	0.37206

6 Recovery rate

Fig. 11 and Table 6 discusses the recovery rate of the reservoir in fracture porous media and it shows that the recovery rate be more and increases with time in fracture porous media.

$Fig. 11 Recovery rate vs. dimension time for heterogeneous fracture porous media$

Fig. 11

Recovery rate vs. dimension time for heterogeneous fracture porous media

Table 6

Oil recovery rate for heterogeneous fracture porous media.

Dimensionless Time (T)	Dimension Time (t) (s)	Recovery rate for fracture porous media
0.001	1.30 × 10⁹	6.911
0.002	2.59 × 10⁹	13.30
0.003	3.88 × 10⁹	19.24
0.004	5.19 × 10⁹	24.87
0.005	6.48 × 10⁹	30.02
0.006	7.78 × 10⁹	34.86
0.007	9.07 × 10⁹	39.33
0.008	1.04 × 10¹⁰	43.61
0.009	1.17 × 10¹⁰	47.51
0.010	1.30 × 10¹⁰	51.14

7 Conclusion

Here we discussed the saturation rate as well as the recovery rate for a counter-current imbibition phenomenon in a heterogeneous porous matrix and in fractures. The effect of fractures on the saturation rate rendered the problem highly nonlinear. The significant part of this study is the determination of saturation of wetting phase and the recovery rate of the oil reservoir. It is found that the saturation rate be maximum in fractures as compared to a porous matrix. The stability of the used method has been proved by using fixed point theory in a suitable Hilbert space. The simulation results for the saturation of wetting phase is shown in table 4 and 5 and the recovery rate of the reservoir in table.6 with the choice of suitable parametric values. Table-4 and 5 reveals that the saturation rate be maximum in fractures as compared to heterogeneous porous matrix implies the recovery rate of the oil reservoir be maximum in presence of a fractures in porous media which is physically consistent with the real world phenomena.

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Received: 2016-9-6

Accepted: 2016-11-4

Published Online: 2016-11-30

Published in Print: 2017-3-1

Articles in the same Issue

https://doi.org/10.1515/nleng-2016-0057

Keywords for this article

Counter–current imbibition; Fracture porous media; Capillary pressure; Adomian decomposition method