1 Introduction

This paper discusses mathematically the phenomenon of imbibition in an inclined heterogeneous porous media with the consideration of capillary pressure and different porous materials. When the reservoir oil (non-wetting phase) comes 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. non wetting phase (oil) from the medium initiated by imbibition. Due to the difference in viscosities of water and oil, the water saturation on the right side of imbibition face will travel only a small distance ‘l’ due to capillary pressure effect (without external force) initiated by imbibition. Oil recovery by imbibition process is accomplished by contacting water with the porous solid. The water is then imbibed into the pore matrix, where upon reaching the regions of oil saturation, the water will encroach along the solid surface causing a portion of trapped oil to be displaced. Water imbibition is an alternative enhanced oil production strategy capable of recovering a portion of this trapped oil through a replacement mechanism that exchanges water for oil. The effectiveness of this process depends on several parameters; including matrix block size, rock porosity and permeability, fluid viscosities, interfacial tensions, and rock wettability.

The following empirical function first proposed by Aronofsky et al. (1958) to study the recovery rate of the reservoir:

Where R is the recovery, R_∞, is the ultimate recovery and γ 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=KpdϕμwL2t for studying the recovery rate with dimensionless time.

Zeybek et al. [21] discussed the capillary imbibition in porous structures and studied the simulation results numerically with the study of the effect of heterogeneity of permeability and wettability on counter - current and co - current imbibition. Zhou et al [22] investigated the effect of wettability and aging time on oil recovery and concluded that wettability has a significant impact on the imbibition rate. Hughes and Blunt [5] found that the pattern of displacement and the rate of imbibition is depends on the relationship between capillary number, contact angle and initial wetting phase saturation. Pooladi-Darvish and Firoozabadi [11] modelled the co - and counter - current imbibition in water-wet rocks. According to their findings, when porous media are partially covered by water, flow is dominated by co-current imbibition and concluded that the oil recovery from co - current imbibition is higher than the one from the counter - current. Shah and Verma [19] studied the multiphase flow through slightly dipping porous media with magnetic fluid. Karpyn et al. [7] identified three distinct flow intervals during spontaneous imbibition in layered sandstone with a single longitudinal fracture, where counter-current flow was dominant at early and intermediate times; while both co - current and counter - current flow mechanisms coexisted at late times. Hatiboglu and Babadagli [6] conducted imbibition experiments on aged Berea sandstone and limestone and investigated the effect of different pore structures. Darvishi et al. [3] investigated the effect of permeability on spontaneous imbibition using carbonate cores. They reported that water imbibed more easily in cores with higher permeability, even in oil-wet cores at connate water saturation. Meher et al. [8] discussed the series solution for porous medium equation arising in fingero-imbibition phenomenon during oil recovery process and concluded that at a fix distance saturation of wetting phase is increases with time. Patel et al. [13] discussed the Imbibitions phenomena in a multiphase flow through porous media and also discussed it under special conditions as homogeneous medium with capillary pressure, involving magnetic fluid, homogeneous medium inclined at small angle involving magnetic fluid. Desai et al. [4] discussed this phenomena in a homogeneous porous media by using Homotopy perturbation method and concluded that if permeability and capillary pressure is more, than the saturation of water will advances faster in the porous medium which will result oil recovery in lesser time. Similarly if the viscosity of the fluid and the porosity of the medium are high, the saturation of water will advances slowly into the medium which will result to oil recovery after a longer time. Mishra [9] discussed this phenomena in homogeneous porous media by using Homotopy perturbation transform method and observed that the saturation of water advances with the time and reaches to a constant value after a very long time. Meher et al. [10] discussed this phenomena with capillary pressure mathematically and concluded that the saturation of water be increases exponentially with distance X for any time T > 0. Patel et al. [14] studied this phenomena in heterogeneous porous media and concluded that the saturation rate as well as the recovery rate be more in homogeneous porous matrix as compared to heterogeneous porous media. Patel and Meher [15] studied the fingering phenomena in a fluid flow through fracture porous media with inclination and gravitational effect and developed a simulation result of saturation of wetting phase with the consideration of inclination effect for some interesting choices of parametric data and studied the recovery rate of the oil reservoir with dimensionless time and concluded that the saturation of wetting phase be more with zero inclination and in homogeneous porous media but as the inclination of the porous matrix be increases, the saturation rate be less as compared to zero inclination. Patel and Meher [16,17] studied approximate analytical study of counter - current imbibition phenomenon in a heterogeneous porous media and a study on recovery rate for counter - current imbibition phenomenon with Corey’s model arising during oil recovery process. Patel and Meher [18] studied simulation of counter - current imbibition phenomenon with Corey’s Model in double phase flow through heterogeneous porous medium with capillary pressure.

To analyse the behavior of saturation of wetting phase in imbibition phenomena, an advance analytical approach i.e. Adomian Decomposition Method is applied here to study the behavior of solution along with its stability analysis in counter current case. The most advantage of Adomian Decomposition Method is that it assumes that the unknown function can be expressed as an infinite series and the non-linear operator can be decomposed in to a special series of polynomials referred as adomian polynomials and it converges faster than other numerical method. We defined a dimensionless time with almost all the parameters considered. These include porosity, permeability, size, shape, boundary conditions, wetting and non-wetting phase, relative permeability, wettability, gravity and different porous materials. The definition of the dimensionless time based on theoretical analysis of the fluid flow mechanisms that governed counter - current imbibition. A general analytical solution to the relation between recovery and saturation rate for counter - current imbibition is derived. The analytical solution predicts a correlation between the imbibition rate and recovery by counter - current imbibition in most fluid/fluid/rock systems.

Here we studied the effect of different porous materials like volcanic and fine sand on initial water saturation and the sensitiveness of imbibition phenomena to initial water saturation in an inclined heterogeneous porous media with the consideration of gravitational effect. The purpose of this study is to extend the comprehensive analysis of counter-current imbibition phenomena in a heterogeneous porous media done by Patel et al. [14] to an inclined heterogeneous porous media with the consideration of different types of porous materials like volcanic sand and fine sand and compared the obtained results with homogeneous porous media. Analytical solution for the flow equations is presented for the counter - current imbibition phenomena in an inclined heterogeneous porous matrix by using Adomian decomposition method to study the saturation rate of wetting phase and the simulation result is developed to study the recovery rate as a function of dimensionless time, T of the reservoir. The results obtained here are in perfect agreement with the physical situation. This can be realized by conducting an experiment with the help of a capillary porous matrix having different porous material filled with oil. The saturation rate for different porous materials and its effect on capillary pressure and relative permeability can be verified from the expression obtained for saturation.it is of great significance in oil recovery, where it can be responsible to increase oil production up to 40%-50% in some cases.

2 Mathematical model

In secondary oil recovery process when water is injected into the oil reservoir through one well, it displaces the oil so that it can be extracted from a neighbouring well since the water is less viscous than the oil and the permeability of the rock is often highly heterogeneous. For the sake of mathematical model: We consider here that a finite cylindrical piece of porous matrix having an oil formatted region having length L and heterogeneous in nature that is completely surrounded by an impermeable surface except for one end (common interface) of the cylinder which is labelled as the imbibition face and this end is exposed to an adjacent formation of ‘injected’ water and inclined at an certain angle with the base surface. When the reservoir oil (non wetting phase) is come into contact with water (wetting phase) there is a disturbances happens at the interface causing imbibition due to the viscosities differences of both phases and the water saturates on the right side of imbibition face through a small distance ‘l’ due to the capillary pressure effect(without external force) causes the oil to imbibe on the left side through a small distance initiated by imbibition. The schematic diagram of the phenomena is shown in fig-1

Fig. 1

schematic diagram of Imbibition phenomena

The conservation of mass for multi-phase flow with respect to volume can be formulated as

Wherei = o, w, x ∈ ℜ³, t ≥ 0, φ(x) denotes the porosity of the porous medium, S_i is the saturation for each phasei, 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

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, k_i is its relative permeability and µ_i is its viscosity of the porous media, α is a inclination angle.

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

The imbibition condition for counter–current imbibition and capillary pressure can be expressed (Patel et al. [12]) as

Since the problem is dealing with the heterogeneous porous media so, the porosity and permeability of heterogeneous porous media can be expressed as

The most famous p_c - S_w relationships determined experimentally are those of (Brooks and Corey [2]):

Where p_d is the entry pressure, it represents the minimum pressure needed by a non-wetting fluid to displace the wetting fluid which initially fills in the porous medium, S_e is called effective saturation. The parameter λ describes grain size distribution in the medium.

The relative permeability, k_w of the wetting and non-wetting phases in the domain are governed by the following (Brooks and Corey [2]):

Combining eq. (2), (4) and (5), we get

Hence the conservation eq. (3) with eq. (8) can be written as

By using eq. (6), (7) and (b) with eq. (9), it yields

Where kokwkwμo+koμw≈kwμw(Patel et al. [12])

Simplifying eq. (10) becomes

Using the dimensionless variables X=xLandT=KcpdμwL2t.

Simplification of 1ϕ∂ϕ∂X as 1ϕ∂ϕ∂X=∂∂Xlogϕ=∂∂XbaLX−loga(Neglecting higher order term of X) = bLa The dimensionless forms of eq. (11) can be written as

Where C=gρo−ρwLλpd.

Equation (12) describes the counter-current imbibition phenomena in a heterogeneous porous media with inclination effect for different porous materials.

3 Analysis of the Method

For the purpose of illustration of the Adomian decomposition method, in this study we shall consider eq. (13) in an operator form as

Where N1Sw=Sw−Swr1−Swr2+3λλ∂∂SwSw−Swr1−Swr−1λ∂Sw∂X,N2Sw=∂∂SwSw−Swr1−Swr2+3λλ+bLaSw−Swr1−Swr2+3λλand S_w0 can be solved subject to the corresponding initial condition S(X, 0)=f(X)=e^-X.

Following Adomian [1] denned the linear operators LT=∂∂TandLX=∂∂Xthe definite integration inverse operator LT−1 and the nonlinear term as N₁S_w and N₂S_w.

Operating the inverse operator and following the analysis of Adomian decomposition, we set the recursive relation of the Eq. (13) as

Which gives the recurrence relation as

and it can be written in the series form up to four terms as

Equation (16) represents the saturation of wetting phase during counter-current imbibition phenomena in a heterogeneous porous media with inclination effect for different porous materials.

4 Numerical results and Discussion

4.1 Effect of inclination on initial water saturation for different porous materials

Fig. 2 and 3 discusses the variation of initial water saturation in heterogeneous porous media for volcanic and fine sand at fixed distance. It shows that the initial water saturation rate be more for α = 0° as well as in volcanic sand and less for α = 0° and in fine sand as compared to volcanic sand.

Fig. 2

Comparison of saturation vs. dimensionless times for different inclination α = 0°, 15°, 30° in Volcanic Sand

Fig. 3

Comparison of saturation vs. dimensionless times for different inclination α = 0°, 15°, 30° in Fine Sand

4.2 Effect of heterogeneity on initial water saturation in volcanic sand

Figure 4 discusses the variation of initial water saturation in homogeneous as well as in heterogeneous porous media in volcanic sand for different inclined plane α = 0°, α = 15° and α = 30°. It shows that the initial water saturation rate be more in homogeneous porous media and for α = 0° as compared to heterogeneous porous media in volcanic sand.

Fig. 4

comparison of saturation vs. dimensionless times for homogeneous and heterogeneous porous media in Volcanic Sand for different inclined plane α = 0°, 15°, 30°

4.3 Effect of heterogeneity on initial water saturation in fine sand

Figure 5 discusses the variation of initial water saturation in homogeneous as well as in heterogeneous porous media for fine sand for different inclined plane α = 0°, α = 15° and α = 30°. It shows that the initial water saturation rate be more in homogeneous porous media and for α = 0° as compared to heterogeneous porous media in fine sand.

Fig. 5

comparison of saturation vs. dimensionless times for homogeneous and heterogeneous porous media in Fine Sand for different inclined plane α = 0°, 15°, 30°

4.4 Effect of Capillary pressure on initial water saturation in heterogeneous inclined porous matrix in volcanic sand at different inclination

Figure 6 discusses the variation of capillary pressure with initial water saturation in heterogeneous porous media in volcanic sand at different inclined plane. It shows that the capillary pressure be more in inclined plane having α = 30° as compared to zero inclined plane in volcanic sand.

Fig. 6

Capillary pressure vs. Saturation in Heterogeneous Porous Media for Volcanic Sand at different inclined plane α = 0°, 15°, 30°.

4.5 Effect of Capillary pressure on initial water saturation in heterogeneous inclined porous matrix in Fine sand at different inclination

Figure 7 discusses the variation of capillary pressure with initial water saturation in heterogeneous porous media in fine sand at different inclined plane. It shows that the capillary pressure be more in fine sand and in inclined plane having α = 30° as compared to zero inclined plane and in fine sand.

Fig. 7

Capillary pressure vs. Saturation in Heterogeneous Porous Media for Fine Sand at different inclined plane α = 0°, 15°, 30°.

4.6 Effect of Relative permeability on initial water saturation in heterogeneous inclined porous matrix in Volcanic sand at different inclination

Figure 8 discusses the variation of Relative permeability with initial water saturation in heterogeneous porous media in volcanic sand at different inclined plane. It shows that the value of relative permeability be more at zero inclined plane implies saturation rate be more as compared to the inclined plane having α = 30° in volcanic sand.

Fig. 8

Relative permeability vs. Saturation in Heterogeneous Porous Media for Volcanic Sand at different inclined plane α = 0°, 15°, 30°.

4.7 Effect of Relative permeability on initial water saturation in heterogeneous inclined porous matrix in Fine sand at different inclination

Figure 9 discusses the variation of Relative permeability with initial water saturation in heterogeneous porous media in fine sand at different inclined plane. It shows that the value of relative permeability be more in fine sand at zero inclined plane implies the saturation rate be more as compared to the inclined plane having α = 30° but less saturation rate as compared to volcanic sand.

Fig. 9

Relative permeability vs. Saturation in Heterogeneous Porous Media for Fine Sand at different inclined plane α = 0°, 15°, 30°.

5 Recovery rate

Figure 10 discusses the variation of saturation rate in homogeneous and heterogeneous porous media with different porous materials which shows that the recovery rate be more in volcanic sand as compared to heterogeneous porous media and in find sand implies the recovery rate be optimum in the presence of volcanic sand in heterogeneous inclined porous media as compared to fine sand.

Fig. 10

Comparison of recovery rate for Volcanic sand and Fine sand in Homogeneous and Heterogeneous porous media

It is found here that the dependence of the type of porous matrix, effect of inclination and gravitational effect on saturation rate rendered the problem highly nonlinear. The significant part of this study is to study the advantage of the proposed mathematical expression in the determination of saturation of wetting phase and the recovery rate of this phenomenon with the inclusion of inclined plane and different porous materials with suitable choices of parametric values. It is found that there is an impact of inclination, heterogeneity and types of porous materials on saturation of wetting phase in counter - current imbibition phenomena and it shows that the saturation rate be more in case of homogeneous and for zero inclination as well as in volcanic sand as compared to heterogeneous inclined plane porous media in fine sand. The saturation rate be increases with time provided the recovery rate be more in case volcanic sand as compared to fine sand as shown in Fig. 10.

6 Conclusion

Here we studied the saturation rate as well as the recovery rate for a counter - current imbibition phenomenon in a heterogeneous inclined porous media for two types of porous materials like volcanic sand and fine sand. The simulation results for the saturation of wetting phase with time is shown in Table 2,3 and 4 and recovery rate is shown in table 9 with the choices of suitable parametric values which shows in table 1 that the saturation rate be maximum in homogeneous porous matrix with zero inclination as compared to heterogeneous inclined porous matrix implies recovery rate of oil reservoir be maximum and around 45% in homogeneous porous matrix as well as in volcanic sand as compared to heterogeneous inclined porous matrix and fine sand which is physically consistent with the real world phenomena.