# Oil saturation distribution in polluted aquifer result from quadrupole wells flow field configuration

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## Abstract

A new physico-mathematical approach is developed for modeling oil **s**aturation distribution within a plain polluted aquifer subjected to a flow field governed by a quadrupole wells system. This system consists of two extraction and two injection (2E/2I) wells, located in a contaminated aquifer, which are used for groundwater remediation applications. The development of the model is done by considering a two-dimensional, nonlinear diffusion-convection transport equation, applying a geometrical self-similar scheme. A closed-form analytical solution is obtained for the steady oil saturation distribution in the aquifer for the above-mentioned flow field. The mathematical solution developed here is designed to simulate hydraulic interactions of 2E/2I wells with the oil polluted aquifer.

## Keywords

Aquifer Oil saturation Quadrupole wells configuration Two-dimensional saturation distribution Geometrical self-similar solution## 1 Introduction

Remediation of contaminated groundwater may involve the injection of water into the aquifer to stimulate degradation of oil contaminants, coupled with the removal of water from extraction wells to increase circulation. Ideally, the water should be mixed throughout the oil contaminated area to optimize the reaction rate. However, due to the laminar flows characteristic in porous media, mixing in groundwater is an inherently slow process governed by molecular diffusion. In contrast, mixing occurs more readily in rotational flows, such as the turbulent flow in streams [9] using groundwater circulation well (GCW) systems technology.

Subsurface double-well pumping serves to impede the groundwater flow and is often used in conjunction with groundwater extraction wells, as components of pump and treat systems. Such wells are also used to contain contaminated groundwater and in groundwater treatment systems. They are frequently used with surface caps to produce an essentially complete containment structure. Anderson and Mesa [2] presented an analytic solution describing the discharge vector for a two-dimensional, steady groundwater flow past an impermeable barrier, embedded in a regional flow field. In their study, the hydraulic containment of contaminant plumes was examined by combining a vertical barrier and extraction wells. A closed-form self-similar solution is also obtained by Pistiner and Shapiro [13] and Pistiner [11] which describes the saturation distribution within a two-dimensional isotropic and anisotropic aquifer of a pollutant discharging from an underground source. Later, Pistiner [12] proposed a self-similar solution for a two-dimensional steady-state oil plume distribution in an aquifer near an impermeable barrier. Satkin and Bedient [16] evaluated the effectiveness of seven different well patterns for restoring contaminated groundwater under eight generic hydrogeologic conditions. It was found that the doublet configuration is to minimize cleanup time, the volume of water circulated, and the volume of water treated. Antonacci et al. [3] specified several criteria for case studies of groundwater contamination events, analyzed by large-scale numerical models, results from sink and/or source conditions, which have a major impact on the concentration distribution in the aquifer. Zeidan [17] described in here study groundwater pollution and remediation schemes in the Nile Delta, together with numerical study which involve the use of injections and extractions wells. Bear and Cheng [5] presented methodology and procedure for constructing complete conceptual and mathematical models of groundwater contaminant transport by using pumping and injection wells. These models are used as essential tools for the planning and management for sustainable use of groundwater resources. The use of the mathematical models is based on physical assumptions, subjected to acceptable approximations and limitations.

In this study, we present a closed-form analytic solution describing the effects of 2E/2I wells, lying in a quadrupole scheme, on the distribution of an oil contamination plume in the polluted aquifer. The steady-state distribution of oil contaminants in a saturated aquifer is analyzed and simulated by applying a geometrical self-similar model. The model employs the basic equations of the two-phase flow. In addition we assume some power law relationship which is supposed to exist between oil saturation, capillary pressure and relative permeability. The self-similar model pertains to cases wherein both space coordinates in the oil transport equations can be replaced by a single variable. This solution is relevant in a preliminary design of remedial systems and in particular, we examine the plume dimension and the oil flux near the four wells in order to control the oil plume distribution in the aquifer. This calculation yields valuable similarity solutions. On this basis, the model is calibrated (e.g., without using data such as fluid densities and viscosities, soil permeability, etc.), by matching hypothetical saturation field data, taken from several observation wells and is further used to calculate the actual saturation in every location in the aquifer.

It should be noted that analytical solutions are usually solved under certain assumptions (e.g. aquifer homogeneity, neglect of gravity, saturation profiles of a certain shape) which make them appear less applicable to real world problems. However, analytical solutions have a value of their own, simply by allowing a quick assessment of what roughly to expect and, more importantly, can be used to validate implementations of numerical models.

## 2 Physico-mathematical model

Consider a two-dimensional homogeneous and isotropic porous layer (e.g., an aquifer), on a regional scale, that is, over a planar horizontal extent much larger than its thickness. Consider a case where a large portion of the aquifer is saturated with oil contaminant, i.e. the irreducible water saturation is 0.5 while the oil saturation in the contaminated aquifer reached the maximal possible value, i.e. *S* ≈ 0.5 [8]. This zone will be defined in this study as the “free layer” zone, which is characterized by connected oil blobs [10]. The remediation system of this zone contains two pairs of extraction and injection wells. As such, water is extracted from deep strata in the contaminated aquifer via two wells. The extracted water is re-injected into the aquifer through a pair of injection wells, located very close to the extraction wells. As a result, oil is displaced toward the extraction wells by the water and the oil saturation near the injection wells is reduced and decreases down to the value *S* = 0.5. This zone, where the oil saturation is smaller than 0.5 (*S* < 0.5) will be defined in this study as the transition zone, which is characterized by large amount of disconnected oil blobs. Although the capillary forces that hold residual oil in pores are relatively strong, they can be overcome to some degree by a sweep scenario involved injecting water on one side of the aquifer and extracting of from the other side. As a result, the water will sweep (removing) oil contamination from those zones which are closed to the wells.

_{w}and q

_{o}of the water and the oil in the transition zone in the absence of the gravity force, are modeled by Darcy’s law. We can write this for each fluid in a Cartesian system, using the double index summation convention

*x*

_{1}=

*x*(and

*x*

_{2}=

*x*) in the form

*S*is the macroscopic (averaged over the layer height) oil saturation in the porous medium (0 <

*S*< 1) (i.e., the water saturation equal to 1 −

*S*);

*K*

_{rw}(

*S*),

*K*

_{ro}(

*S*) are the water and oil relative permeabilities, respectively. Other quantities appearing in (1a) and (1b) are: \(\kappa_{ij}\)—the Cartesian components of the permeability tensor,

*µ*

_{o},

*µ*

_{w}—the oil and water viscosities, respectively,

*P*

_{o}and

*P*

_{w}—the oil and water pressures respectively, which are related via the capillary pressure

*P*

_{c}

*ϕ*yields

*t*is the time variable. The pair Eqs. (3a) and (3b) automatically gives the mass balance equation

*F*(

*S*) and the capillary dispersion term

*ψ*(

*S*) are given by

*P*

_{c}(

*S*)/d

*S*→ ∞ as the oil saturation approaches zero, we may assume [15] that

*m*in (8) were found to depend on the porous medium properties [14].

*S*is small, especially near the wells where the sweep up of residual oil supposed to be very effective. Accordingly, the fractional flow curve

*F*(

*S*) and the function

*ψ*(

*S*) can be represented by power-law dependence [14], usually valid for low oil saturation

Here *γ*_{ow} is the oil–water surface tension, *F*_{0} is a dimensionless factor of order one, and *ψ*_{0} is of the order of \(\tilde{P}_{c} /\gamma_{ow}\) where \(\tilde{P}_{c}\) is the characteristic capillary pressure in the aquifer. Parameters *m* and *n* were found to depend on the pore-size index of the porous medium and the oil viscosities [14]. From the experimental data on fractional flow rates and permeabilities it was found that *m* and *n* typically vary in the following ranges: 1 < *n *< 3, 1.5 < *m *< 4. It should be noted that (9b) assumes negligible changes in *K*_{rw}(*S*) (appears in (7b)) for low oil saturation.

We assume that a steady-state oil saturation distribution is established a sufficiently long time (i.e.,*t *→ ∞) after the beginning of the Extractions/Injections process [13]. In such a case, the term on the left-hand side of (6) vanishes (i.e., \(\partial S/\partial t \to 0\)) and Eq. (6) describe a steady-state distribution on the oil saturation *S*(*x*, *y*) within the porous medium. Both terms (c.f., Eq. (6)) of capillary dispersion and oil convection, in the *x* and the *y* directions, manifests themselves in such a way that the distribution of the oil saturation in the aquifer may exists under a steady-state condition. This condition exists as long as the 2E/2I process is going on and the aquifer area, far from both wells, remains in a “free layer” state and behaves as an infinite source of oil to the transition zone. Despite the fact that the model is more appropriate to a case where the oil saturation is not high, we will assume that the iso-saturation line *S *= 0.5 may serve as the boundary between the “free layer” zone and the transition zone (e.g., 0 < *S *< 0.5). This iso-saturation line will be obtained as a part of the solution.

*x*,

*y*) (shown in Fig. 1) and the specific discharge components, resulting from a quadrupole flow field are given by [6]

*Q*is the discharge intensity of the water (c.f. in volume per time) into and from the wells. The fluxes represented in (11a) and (11b), are created by stream lines that connect two extraction wells lying on the

*x*axis (at

*x*=+0 and

*x*= − 0) and two injection wells lying on the

*y*axis (at

*y*=+0 and

*y*= − 0). As such, the regions near the

*y*axis will be defined in this study as the upstream zone (positive and negative directions with respect to

*y*= 0) and the regions near the

*x*axis will be defined in this study as the downstream zone (positive and negative directions with respect to

*x*= 0).

*L*is a characteristic length of the aquifer which can be obtained by the introduction of (12), (13a) and (13b) into (6), (9a), (9b), (11a) and (11b) as follows:

*F*

_{o}(

*n*−

*m*+ 1) appearing in (14c) is a positive dimensionless factor [11].

In the next section we will develop a similarity solution, which is valid in the transition zone at the range of \(0 < \tilde{S}(\hat{x},\hat{y}) \le 0.5\).

## 3 Self-similar model

*f*(

*ξ*) is a similarity function and is valid for the particular case

*g*(

*ξ*) and

*h*(

*ξ*) are the flux components and \(f' \equiv \frac{{{\text{d}}f}}{{{\text{d}}\xi }}\). Using (16b), (19a) and (19b) we obtain the following expressions

*λ*is a constant of integration, which is related to the flux parameter to be determined hereafter. Introduction of (26a), (17a) and (17b) into (25a) we obtain

*u*(

*ξ*) is yet an unknown function of

*ξ*. Substituting (28) in (27) we obtain the following linear ODE

*z*(

*ξ*) is defined now as follows

*C*and

*C*

_{1}are constants to be determined below and \(a = \sqrt {1 - \lambda }\). Using the properties of the hypergeometric series, we obtain from (32), (32a) and (32b) the expression for \(\frac{{{\text{d}}u(z)}}{{{\text{d}}z}}\)

*F*

_{3}(

*z*) and

*F*

_{4}(

*z*) are given by

*C*we use the following property of Eq. (27)

*ξ*= 0, as follows

*f*(

*ξ*) in (37) must be equal to the denominator of the solution for

*f*(−

*ξ*) as follows

*C*→ ∞. As a result, (37) reduces to

*ξ*

_{b}≤

*ξ*≤ +

*ξ*

_{b}, as will be explained later. Using the results of (38b), together with (16a) we obtain the following symmetrical property (with respect to the axis \(\hat{x} = 0\) and \(\hat{y} = 0\)) of the saturation profile

## 4 Some more properties of the similarity function *f*(*ξ*)

*λ*must be greater than zero (i.e.,

*λ*> 0) to assure that the oil saturation \(S(\hat{x},\hat{y})\), given in (16a), is positive. In addition to this we obtain from (27) (and from (38a) as well)

*f*(

*ξ*) at infinity can be obtain from (40) as follow

*f*(

*ξ*) shows that the similarity solution

*f*(

*ξ*) is not uniformly valid everywhere since there is no physical meaning to a negative value for

*f*(

*ξ*). However, the result (43) shows that there is a boundary parameter

*ξ*

_{b}which divided between the region that possesses “physical meaning” exists in the range −

*ξ*

_{b}<

*ξ*<+

*ξ*

_{b}and the “no physical meaning” region lying at |

*ξ*

_{b}| <

*ξ*< |∞| which is part of the “free layer” zone. Hence, the boundary parameter

*ξ*

_{b}, is the point at which the function

*f*(

*ξ*) passes from a positive value to a negative value and possesses the following property

*λ*on

*ξ*

_{b}. It can be observed that as

*λ*becomes large, the boundary parameter

*ξ*

_{b}becomes large as well with the following dependence

## 5 Calculation of the oil fluxes on \(\hat{x} = 0\)

*ξ*= 0 (i.e., \(\hat{x} = 0\)) are obtained by the introduction of (42a) and (42b) into (47a) and (47b)

The result (49a) indicates that the \(\hat{y}\) axis behaves as an impermeable barrier for the oil saturation flux in the \(\hat{x}\) direction. In addition to this, the result (49b) indicates that for *λ *> 1, the oil saturation flux possesses two branches on the \(\hat{y}\) axis. The first one is directed from \(\hat{y} = 0\) to the \(+ \hat{y}\) direction and the other branch is directed from \(\hat{y} = 0\) to the \(- \hat{y}\) direction, namely the upstream directions, as was previously defined.

## 6 Results and discussion

*λ*[i.e.,

*m*=

*n*= 2 for all figures and hence \(\tilde{S} = S\), see (12)]. It can be observed that as the oil flux parameter

*λ*becomes larger, the dimension of the saturation zone is reduced. On the contrary, when the value of the oil flux is low, smaller amounts of oil are transported by relatively large amounts of water from the injection wells toward the extraction wells, resulting in a larger transition zone. That is to say, the convection mechanism takes place over the dispersion process as the flux parameter becomes higher. As such, iso-saturation paths tend to possess relatively steep gradients which assure the domination of the convection over the dispersion mechanism (e.g., the arc over the area of oil–water menisci increases and the oil convective transport acts as a significant oil transport mechanism, comparable to the capillary dispersion.)

*observation*wells are lying at two known distances from the origin (

*x*= 0,

*y*= 0). Such saturation data can be obtained by using elevation measurements of the water and the oil from the observation well [8]. Based on such data, we will illustrate how the actual position of the iso-saturation lines in the

*x*−

*y*domain can be determined. Toward this end, we use the flowing hypothetical data as follows

- (a)
*S*_{1}= 0.1 (at*x*_{1}= 0 m,*y*_{1}= 5 m), - (b)
*S*_{2}= 0.3 (at*x*_{2}= 10.0 m,*y*_{2}= 10.0 m),

*a*) into Eq. (16a) and (42a) we obtain

- (c)
\(\frac{{S_{1} }}{{y_{1}^{2} }} = \frac{\lambda }{{L^{2} }}\)

- (d)
\(\frac{{S_{2} }}{{S_{1} }} \times \frac{{y_{1}^{2} }}{{y_{2}^{2} }} = \frac{{(1 + \xi_{2}^{2} )^{2} }}{{1 - \xi_{2}^{2} + 2\sqrt {\lambda - 1} \xi_{2} \tanh \left( {2\sqrt {\lambda - 1} \cot^{ - 1} \xi_{2} } \right)}},\)

- (e)
\(\xi_{2} = \frac{{x_{2} }}{{y_{2} }}\).

- (f)
*λ*≈ 8.115.

- (g)
\(L = y_{1} \sqrt {\frac{\lambda }{{S_{1} }}} \approx 45\) m.

**actual**position of the iso-saturation lines as can be observed in Fig. 4.

The existence of the self-similar functional form (16a) and (16b) thus reflects the tendency of the solution \(\, \tilde{S}(\hat{x},\hat{y})\) to “overlook” the actual saturation specified in large distance \(\hat{y}\) while it eventually tends towards the similarity solution for small distances from the origin, in the transition zone. The free layer zone can be related to the zone where the similarity solution possesses no physical meaning (at |*ξ*_{b}| < *ξ *< |∞|).

## 7 Summary and conclusions

The steady-state distribution of oil contaminants in a saturated aquifer is analyzed and simulated by applying a geometrical self-similar model. The model employs the basic equations of the two-phase flow. In addition we assume some power law relationship which is supposed to exist between oil saturation, capillary pressure and relative permeability. The self-similar model pertains to cases wherein both space coordinates in the oil transport equations can be replaced by a single variable.

The influences of the extraction and the injection wells, located in quadrupole configuration and without regional flow, are lumped via the single oil flux parameter *λ*. It is found that decreasing the oil flux parameter led to the domination of the capillary dispersion processes compared with the convection process. This leads to an increasing of the transition zone dimensions.

It was shown that the obtained solution for \(\tilde{S}(\hat{x},\hat{y})\) in the form (16a) is complete and valid in the transition zone, which is enclosed by the iso saturation line *S *= 0.5, defined in (16a) and (16b). The obtained solution does not generally reproduce the actual saturation beyond the transition zone and is blows up as \(\hat{y}\) become large. However, it eventually tends towards the similarity solution (16a) and (16b) in the transition zone where *S *≤ 0.5. In addition to this, it was proved that the size and the steady state oil saturation distribution in the transition zone is depends on the oil flux only and it is independent on the size of the free layer zone (which is unknown).

It was also shown how to use the model jointly with data collected from several observation wells to evaluate the extension of the oil in the transition zone and to check more comprehensive numerical treatments of the oil–water transport in aquifers.

## Notes

### Compliance with ethical standards

### Conflict of interest

The author declare that there is no conflict of interest.

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