Evaluation of water coning phenomenon in naturally fractured oil reservoirs
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Abstract
Water coning is a complex phenomenon observed in conventional and unconventional reservoirs. This phenomenon takes place due to the imbalance between viscous and gravitational forces during simultaneous production of oil and water. In a fractured reservoir, controlling of water coning is challenging due to the complexity originates from large number of uncertain variables associated with such reservoirs system. This paper presents a fully coupled poroelastic multiphase fluidflow numerical model to provide a new insight and understanding of water coning phenomenon in naturally fractured reservoir under effect of various rock and fluid properties. These properties include anisotropy ratio, fracture permeability, mobility ratio, and production rate. The simulation workflow of the developed numerical model is based on upstream flux weighted finite element discretization method and a new hybrid methodology, which combines singlecontinuum and discrete fracture approach. Moreover, the capillary pressure effect is included during the discretization of the partial differential equations of multiphase fluid flow. The numerical system is decoupled using implicit pressure and explicit saturation (IMPES) approach. Discretization of water saturation equation using standard finite element method produces solution with spatial oscillations due to its hyperbolic nature. To overcome this, Galerkin’s least square technique (GLS) is employed to stabilize the equation solutions. The developed numerical scheme is validated successfully against Eclipse100 and then applied to a case study of fractured reservoir taken from Southern Vietnam. The results showed that the break through time is very sensitive to the distributions of fracture network, anisotropy ratio between fracture horizontal, vertical permeability, and mobility ratio. Furthermore, it has been concluded that aquifer strength has a little effect on coning behavior during oil production process.
Keywords
Water coning phenomenon Naturally fractured reservoir Finite element modellingIntroduction
As a result of invading the wellbore by water cone, excessive water will be produced and kills the producing well or limiting its economic life (Beattie and Roberts 1996). The shape and the nature of the cone in conventional reservoirs depend on numerous factors such as completion interval, production rate, anisotropy ratio, and mobility ratio (Inikori 2002). While, in naturally fractured reservoirs, the problem becomes more complicated due to the existence of two interacting porous media (fracture and matrix) which results in the formation of two cones. The first cone is the fastest and developed in fractures, while the slowest one is developed in the matrix porous media. The relative position of the two cones is rate sensitive and is a function of rock properties (Namani et al. 2007). If the well flows above the critical rate, the viscous force dominates and the cone will break into the wellbore. Therefore, the production flow rate plays a key role in limiting coning effects.
Hoyland et al. (1989) presented an analytical solution for critical flow rate calculations. The presented correlation is validated against numerical solutions. The authors concluded that the critical flow rate is independent on shape of water/oil relative permeability curve between endpoints, water viscosity, and wellbore radius, but it has a strong dependency on completion interval and permeability ratio k _{ v } /k _{ h }.
AlAfaleg and Ershaghi (1993) used an empirical correlation for homogenous reservoirs to calculate the critical production rate and breakthrough time for naturally fractured reservoirs. Al Afaleg stated that no correlation or even simulation study has the capability to predict the estimation of critical production rate and breakthrough time if the fracture network is not accurately distributed.
Saad et al. (1995) performed an experimental work to assess the problem of water coning in naturally fractured reservoirs. The outcome of their experimental work was that the main factor influencing the breakthrough time is the difference in viscosities between oil and water phase. Furthermore, they stated that capillary forces may be neglected if the distance between oil water contact and fluid entry is sufficiently large compared to capillary rise.
Bahrami et al. (2004) assessed the coning phenomenon using actual water and gas coning data from Iranian natural fractured reservoirs. By analyzing the results, it was concluded that the ratio k _{ v } /k _{ h } is the main factor controlling coning occurrence. Bahrami et al. (2004) presented a developed method suitable for calculation of breakthrough time and water cut at each specific oil production rate. They mentioned that the water invades the wellbore through fracture network and the breakthrough time is strongly dependant on fracture porosity. In addition, their study proved that the breakthrough time is very sensitive to horizontal and vertical fracture permeability. PerezMartinez et al. (2012) evaluated the occurrence of coning in naturally fractured reservoirs using fine coning radial grid with onemeter thick layers concentric around the well and 2inch thick layers in the annulus, with and without cement. They found that water coning takes place in fractured porous medium with permeability up to 10 Darcy’s in both good and poorly cemented wells. Moreover, they developed a new correlation to determine maximum height of water coning, the breakthrough time, and well shut in time to reverse water cone.
Most of the previous studies simplify simulation of coning phenomenon by assuming that the dominant forces is only viscous force and capillary forces are therefore neglected. In the current study, the capillary forces are included during the derivation of the multiphase flow equations. The fluid production or injection alters the pressure state in fractured reservoirs that causes rock deformation and leads to generation of seismic activities. Moreover, the rock deformation leads to the change of porosity and permeability, which affects fluid flow, oil recovery, and coning phenomenon. To simulate these physical processes accurately, coupled effects need to be considered during simulation of mechanical and fluidflow responses. For the above reasons, this study presents a new poroelastic numerical model to evaluate coning phenomenon in naturally fractured reservoirs. The model is for twophase fluid flow through matrix and fractured medium. Furthermore, the model used a new hybrid methodology for fluidflow simulation. In hybrid methodology, a threshold value for fracture length is defined. Fractures, which are smaller than the threshold value, are used to generate the gridbased permeability tensor. The reservoir domain is divided into a number of grid blocks, and the fluidflow simulation is carried out using the singlecontinuum approach in the nominated blocks. Fractures, which are longer than the threshold value, are explicitly discretized in the domain using tetrahedral elements and the fluidflow is modeled using the discrete fracture approach.
The model is validated against Eclipse100 using horizontal fractures. In addition, the current paper presented a real case study of fractured reservoir taken from Southern Vietnam to evaluate parameters that affect water coning phenomenon.
Eclipse100 is used as a dual porosity/permeability approach to simulate fluid flow in naturally fractured reservoirs. The dual porosity/permeability approach has a lot of limitations which include (1) the fluid distribution within the matrix blocks remains constant during the simulation period, (2) the model cannot be applied to disconnected and discrete fractured (oriented fractures) media and a small number of large scale fractures can be considered for flow simulation. Therefore, the developed numerical model that is presented in this paper overcomes all the abovementioned limitations by account flow through matrix porous media and discrete fractures by taking into account all fracture properties which include fracture orientation, radius, and location.
Methodology
Finite element method formulation
The weighted residual method is used to derive the weak formulation of the governing equation of fluid flow through a fractured system. Standard Galerkin method is applied to discretize the weak forms with appropriate boundary conditions (Zimmerman and Bodvarsson 1996; Woodbury and Zhang 2001).
Galerkin finite element method
Boundary conditions
Governing equations for multiphase poroelastic numerical model
In general, behavior of twophase fluidflow system through fractures network and matrix porous medium is controlled by generalized Darcy’s law and continuity equation of fluid flow for each fluid phase.
Saturation equations
The discretization of these equations has been implemented using standard finite element method; also the partial differential equation used for calculating the saturation changes in fractures and matrix was discretized using Galerkin’s least square technique (GLS) to stabilize the equation solutions. To obtain the numerical solution of this highly nonlinear equations system, suitable initialization and boundary conditions should be designated at first, then some of auxiliary functions are employed which are known as the constitutive relationships.
Auxiliary equations
Saturation equation
Relative permeability function of twophase fluid flow
Reference  Permeability function 

Corey (1954)  \(k_{rw} = s_{e}^{ 4}\) \(k_{{rnw}} = ({\text{1}}  s_{e} )^{{\text{2}}} ({\text{1}}  s_{e} ^{{\text{2}}} )\) 
Brooks and Corey (1964)  \(k_{{rw}} = s_{e} ^{{\left( {{\text{2}}/\lambda } \right) + {\text{3}}}}\) \(k_{rnw} = \left( { 1  s_{e} } \right)^{ 2} \left( { 1  s_{e}^{{\left( { 2/\lambda } \right) + 1}} } \right)\) 
Van Genuchten (1980)  \(k_{rw} = \sqrt {\mathop s\nolimits_{e}^{{}} } \left( {1  (1  \mathop s\nolimits_{e}^{1/\gamma } )^{\gamma } } \right)^{2}\) \(k_{rnw} = \sqrt {\mathop {1  s}\nolimits_{e}^{{}} } \left( {1  \mathop s\nolimits_{e}^{1/\gamma } } \right)^{2\gamma }\) 
Finite element discretization
Validation of the developed numerical model
Reservoir fluid properties used in the simulation of water coning
Water density (lbm/ft^{3})  63.02 
Water viscosity (cp)  0.96 
Oil density (lbm/ft^{3})  45.00 
Oil viscosity (cp)  1.233 
Formation volume factor (RB/STB)  1.0915 
Reservoir model properties used in the simulation of water coning
Reservoir thickness (ft)  100 
Drainage radius (ft)  2098 
Perforation interval (ft)  20 
Well radius  0.20 
Fracture porosity  0.001 
Fracture permeability (md)  300 
Matrix porosity  0.18 
Matrix permeability (md)  1.25 
Initial water saturation  0.22 
Production rate (BPD)  3000 
Simulation work
As can be seen form Figs. 9 and 10, the oil production rate and water cut at the producer resulted from Eclipse100 and the developed numerical simulation model is in agreement. Oil rate decreases with time, due to the increase of water rate production as a result of moving oil–water contact toward the producing interval. The model was simulated without the effect of stresses because the main aim was to validate the numerical model to be used in realfield applications. All fractures that used in this model are horizontal fractures where Eclipse 100 can handle.
Water flooding test in arbitrarily oriented fracture system
Case study
Generation of discrete fracture map of a typical basement reservoir
In this paper, objectbased simulation technique is used to generate subsurface discrete fracture maps (Gholizadeh and Rahman 2012). In this model, fractures are treated as objects and placed in the domain stochastically. The number of generated fractures is controlled by fracture intensity and fractal dimension parameters. The fractures are treated as objects with varying radii, dips, and azimuth angles.
Study of water coning phenomenon
The coning phenomenon is studied using the generated fractured map (see Fig. 13a, b) under effect of different rock and fluid properties. Therefore, the model is initialized by applying vertical and horizontal stresses on x, y, and z directions as shown in Fig. 13b. Because of the lack of information about far field stresses, the horizontal stresses are estimated based on overburden stress (Zoback 2010).
Reservoir input data for Daihung fractured basement reservoir (Farag et al. 2009)
Parameter  Value 

Reservoir dimensions  500 m × 500 m × 250 m 
Matrix permeability  10^{−4} mD 
Matrix porosity  2 % 
Fracture aperture  0.04 mm 
Initial fracture intensity  0.15 m^{−1} 
Fractal dimension (D)  1.25 
Initial reservoir pressure  5063 psia 
Fluid viscosity  1.38 cp 
Fluid compressibility  10^{−5} psi^{−1} 
Horizontal stresses  4000 psi 
Vertical stress  6800 psi 
The permeability tensors for short fractures with length smaller than 100 m are calculated using the method that has been described in the first part of the current paper. Fracture with length longer than 100 m is discretized explicitly in the domain using the triangular elements. The reservoir parameters that used for the sensitivity analysis are (1) anisotropy ratio k _{ v }/k _{ h }, (2) fracture permeability, and (3) oil production rate.
Anisotropy ratio k _{ v }/k _{ h }
Anisotropy ratio values used in different simulation runs for evaluating coning phenomenon
Run  Anisotropy ratio k _{ v }/k _{ h } 

1  2 
2  1 
3  0.6 
Oil production rate
Mobility ratio
Aquifer strength
Conclusion
A 3D multiphase poroelastic numerical model is developed to assess the water coning phenomenon in naturally fractured reservoirs. The developed numerical model has the ability to simulate large numbers of discrete fracture in the reservoir domain. Several parameters have been used to assess their effects and contribution to the water coning phenomenon in naturally fractured reservoirs. These parameters are anisotropy ratio, mobility ratio, oil production, and the aquifer strength.

Increase in the vertical permeability [i.e., increase in anisotropy ratio (k _{ v }/k _{ h })] leads to increase of water cut and water saturation at the producing interval without any significant effect on the oil production rate from the fractured reservoir.

Decrease of oil production rate leads to decrease of produced water cut. The physics explanation is to the fact that at low rate, water cut is controlled by fast moving cone inside the fractures.

Aquifer strength has a little effect on produced water cut.

Investigation of the effective parameters is necessary to understand the mechanism of water coning in naturally fractured reservoirs. Simulation of this phenomenon helps to optimize the conditions in which the breakthrough time of water cone is delayed.
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