Modeling fracturing pressure parameters in predicting injector performance and permeability damage in subsea well completion multi-reservoir system

Abstract

The significance of fracturing parameters which are aquifer integrity, rock properties, thermal stress, fracturing pressure and produced water quality to alter permeability damage, cake formation and injectivity performance was highlighted in a robust improved internal filtration—hydraulic model and permeability reduction model incorporating a $R_{\text{AT}} (c)$ function. The studied system is an injection well multi-reservoir formations. Field data obtained from the log and field reports and improved model were used to simulate injector, fracturing and permeability damage performance. Thus, data requirements in the $R_{\text{AT}} (c)$ function which are rock properties, water quality, aquifer integrity, fractures rates and pressures parameters were assessed for its impact on injector performance and permeability damage simulated in MATHLAB and COMSOL multi physics environment. The profile of injector performance and damage reservoir permeability to changes in rock properties and aquifer integrity were demonstrated to have a profound influence on both fracturing phenomena. Thus, sustainable re-injection scheme was shown as a direct consequence of rock mechanics parameters, well hydraulics aquifer integrity that largely depends on the initial concentration of active constituents of the produced water as well as physic-chemical properties of the host aquifer.

Introduction

Produced water re-injection in multi-reservoir and hydrocarbon aquifer systems above fracturing pressure is a necessary water flood strategy commonly employed for disposal of produced water in subsea well peripheral water flood project. There are several leading publications in the field produced water injection modeling, fractured modeling, injectivity decline and their outcome of particulate mechanics and flow studies (Pang and Sharma 1997; Barkman and Davidson 1972; Wennberg and Sharma 1997).

Regardless of the source, produced water handling and injection is still the single biggest operating costs for producers in mature fields (Ajay and Sharman 2007; Salehi and Settari 2008). Studies related to investigation of rate of fracture height and length growth due to injection above fracture pressure are required to evaluate injection strategies where necessary (Prasad et al. 1999). Water injection is the outcome of stricter offshore regulatory requirements accounting for 500 million bbl. of water/day injected into the subsurface formation, annual operating costs in the range of $100 billion US Dollars.

Current models for predicting internal filtration and injectivity decline in water injection studies for secondary recovery were only limited to mass balance of suspended solids, settling particles equation, particle capture kinetics and Darcy’s law accounting for permeability damage to particle retention. Other analytical models are limited to both particle retention and water–oil mobility alteration (Belfort et al. 1994; McDowell-Boyer et al. 1986).

Improved models and field data to describe the role of geochemical reaction, adsorption-scale kinetics were recently published to impact cake formation, permeability damage and injection performance (Obe et al. 2017). Nonetheless, the well-established field data for fractured modeling in most cases show more than these parameters including filtration coefficient $\lambda$ characterizing the intensity of the particle capture by the porous rock, while formation damage coefficient $\beta$ shows permeability decrease due to particle capture (Pang and Sharman 1994; Al-Abduwani et al. 2001; Guo 2000; Meyer et al. 2003a, b).

Formation damage has been studied under two subject domains; internal filtration and external cake build up. Several articles have provided models and understanding in the field of injectivity decline for characterization of the formation damage system and consequent well behavior prediction. The combined effect of particle suspension injection and total oil–water mobility variation on well injectivity was studied (Altoef et al. 2004). Explicit formulas for injectivity decline due to both effects were derived and applied their model for a deep water offshore reservoir.

The filtration and formation damage coefficients and filter cake permeability from the well injectivity history were determined from the linear dependence of impedance index (the inverse of injectivity index) on injected water volume for deep bed filtration and external cake formation. Researchers considered the effect of particle/pore size distribution, injected solid concentration, wellbore narrowing, particle invasion (Pang and Sharma 1997; Barkaman and Davidson 1972; Donaldson et al. 1977), but fell short to highlight its impact on injector performance and permeability damage and fail to relate rock in situ stresses, aquifer integrity and produced water quality as important in these assessments, which is the objective of our study.

Several other models exist to describe fracturing, injectivity decline, formation damage, particulate mechanics and this have been published elsewhere (Abou-Sayed et al. 2005, 2007; Al-Abduwani 2005; Bedrikovetsky et al. 2007; Chang 1985; Clifford et al. 1991; Davidson 1979; De Zwart 2007; Dong et al. 2010; Donaldson et al. 1977; Doresa et al. 2012; Farajzadeh 2002; Faruk 2010; Folarin et al. 2013; Furtado et al. 2005; Gong et al. 2013; Greenhill 2002; Guedes et al. 2006; Hustedt et al. 2006; Iwasaki 1937; Khatib 2007; Khodaverdian et al. 2009; Lawal and Vesovic 2010; Lawal et al. 2011; Li et al. 2011, 2012; Ojukwu and van den Hoek 2004; Sahni and Kovacevich 2007; Shuler and Subcaskey 1997; Souza et al. 2005; Todd 1979; Van den Hoek et al. 1996; Wang and Le 2008; Wang et al. 2011; Sharma et al. 2000; Yerramilli et al. 2013; Zeinijahromi et al. 2011; Zhang et al. 1993).

In this study, the significance of rock stresses mechanics, aquifer integrity and produced water quality in altering permeability damage, fracturing, cake formation and injectivity decline were highlighted in a robust improved internal filtration—hydraulic model. Thus, our solution accounted for suspended particle propagation C (X, T), retained particle accumulation S (X, T), aquifer integrity related to grain/particle size ration and “In Situ Rock Stress and Wellbore Stability highlighted in the Frade Field, Brazil,” Frade CPDEP Phase 2 report DR-AP-RP-021,209 by GeoMechanics International, Inc. (Guo 2000, Meyer et al. 2003a, b).

Reduced model for PWRI and fracturing performance

As an improvement over the filtration model for cake formation, fracturing well hydraulics and aquifer integrity residual oil mobility and correction for good completion geometry, rock mechanics formation damage coefficient including geochemical reaction, leak off parameters and retention kinetics were introduced as $R_{AT}$.

$$\frac{\partial \phi C}{\partial t} + \left( {U_{r} \frac{\partial C}{\partial r} + \frac{{U_{r} C}}{r} + U_{z} \frac{\partial C}{\partial z}} \right) - \left( {\left( {D_{r} \frac{{\partial^{2} C}}{{\partial r^{2} }}} \right) + \frac{1}{r}D_{r} \frac{\partial C}{\partial r} + D_{z} \frac{{\partial^{2} C}}{{\partial z^{2} }}} \right) = \frac{\partial \sigma }{\partial t} + R_{AT}$$

(1)

$R_{AT} (c,t,\phi ) = \left( {\sum\nolimits_{i = 0}^{N} {w_{i} } R_{{Feff_{i} }} } \right) = (1 - \sum\limits_{i}^{N} {k_{ori} } )\frac{\partial \phi C}{\partial t}$, a variable that is a function of concentration, transition time to cake formation, and effective porosity that highlights the contribution of rock properties, aquifer integrity, fracturing pressure and water quality related to impacts in geochemical reaction and adsorption kinetics.

Subject to the Robin type boundary condition

$$C(r = 0,t) = C_{o}$$

(2)

$$C(r = r_{n} ,z = z_{n} ,t_{n} = 0) = 0^{\prime}$$

(3)

$$\left[ {\frac{\partial C}{\partial r}} \right]_{r = Rc,t} = 0\left[ {\frac{\partial C}{\partial z}} \right]_{r = Zc,t} = 0$$

(4)

Dimensionless form of the boundary condition

$$\zeta (\varepsilon_{r} = 0,\tau \ge 0) = 1$$

(5)

$$\zeta (\varepsilon_{r} = 1,\varepsilon_{z} = 1,t_{n} = 0) = 0$$

(6)

$$\left[ {\left( {\frac{{c_{o} }}{{R_{o} }}} \right)\frac{\partial \zeta }{{\partial \varepsilon_{{\bar{r}}} }}} \right]_{{\varepsilon_{r} = 1,\tau }} = 0\left[ {\left( {\frac{{c_{o} }}{L}} \right)\left( {\frac{\partial \zeta }{{\partial \varepsilon_{{\bar{z}}} }}} \right)} \right]_{z = 1,\tau } = 0$$

(7)

The generalized equations of the internal filtration model are converted to dimensionless form parameters define as follows:

$$\varepsilon_{{\bar{r}}} = \frac{r}{{R_{c} }}$$

(8)

$$\varepsilon_{{\bar{z}}} = \frac{z}{L}$$

(9)

$$\tau = \left( {\frac{\upsilon }{\phi l}} \right)t$$

(10)

$$S = \frac{\sigma }{{\phi c_{o} }}$$

(11)

$$\zeta = \frac{c}{{c_{o} }}$$

(12)

$$\varLambda \left( S \right) = \lambda \left( \sigma \right)L$$

(13)

$$r_{D} = \frac{{q_{di} }}{{\phi c_{di} }}$$

(14)

$$Now:\frac{\partial c}{\partial t} = \frac{\partial c}{\partial \tau }\frac{\partial \tau }{\partial t} = \left( {\frac{\upsilon }{\phi L}} \right)c_{o} \frac{\partial \zeta }{\partial \tau }$$

(15)

$$\left( {\frac{\upsilon }{\phi L}} \right)c_{o} \frac{\partial \zeta }{\partial \tau } = \frac{\partial \zeta }{{\partial \tau^{ * } }}$$

(16)

$$\text{where:}\,\frac{\partial \tau }{\partial t} = \left( {\frac{\upsilon }{\phi L}} \right)$$

(17)

Equation 1 is re-expressed in dimensionless form as:

$$\left( {\frac{\partial \zeta }{{\partial \tau^{ * } }} - \left( {\frac{\partial S}{{\partial \tau^{ * } }} + \frac{{\partial \varPsi_{D} }}{{\partial \tau^{ * } }}} \right) + \frac{{\partial \varPsi_{r} }}{{\partial \tau^{ * } }} + \frac{{\partial \varPsi_{kff} }}{\partial \tau }} \right) + \alpha_{1} \left( \upsilon \right)\left( {\frac{\partial \zeta }{{\partial \varepsilon_{r} }}} \right) + \alpha_{2} \left( \upsilon \right)\frac{\zeta }{{\varepsilon_{r} }} + \alpha_{3} \left( \upsilon \right)\frac{\partial \zeta }{{\partial \varepsilon_{z} }} = \alpha_{4} \left( \upsilon \right)\frac{{\partial^{2} \zeta }}{{\partial r^{2} }} + \alpha_{5} \left( \upsilon \right)\left( {\frac{\partial \zeta }{{\partial \varepsilon_{r} }}} \right) + \alpha_{6} \left( \upsilon \right)\left( {\frac{{\partial^{2} \zeta }}{{\partial \varepsilon_{{\bar{z}}}^{2} }}} \right)$$

(18)

The partial differential equations are solved by the Tridiagonal Matrix Algorithm (TDMA) method. In the model, a second-order six-point implicit finite scheme has been used to obtain a numerical of the governing equations involving the concentration field:

$$\alpha^{\prime}_{1} \zeta_{ijk + 1} + \alpha^{\prime}_{2} \zeta_{i + 1jk + 1} + \alpha^{\prime}_{3} \zeta_{ij + 1k + 1} + \alpha^{\prime}_{4} \zeta_{i - 1jk + 1} + \alpha^{\prime}_{5} \zeta_{ij - 1,k + 1} = \alpha^{\prime}_{6} \zeta_{ijk} + \hat{a}_{r} \left( {\varPsi_{rijk + 1} - \varPsi_{rijk} } \right) - \hat{a}_{d} \left( {\varPsi_{dijk + 1} - \varPsi_{dijk} } \right)$$

(19)

where:

$$\alpha^{\prime}_{1} = 1 - \Delta \tau^{ * } \left( {\frac{{\alpha_{1} }}{{\Delta \in_{{\bar{r}}} }} + \frac{{\alpha_{3} }}{{\Delta \in_{{\bar{z}}} }} - \frac{{2\alpha_{4} }}{{\left( {\Delta \in_{{\bar{r}}} } \right)^{2} }} - \frac{{2\alpha_{6} }}{{\left( {\Delta \in_{{\bar{z}}} } \right)^{2} }}} \right)$$

(20)

$$\alpha^{\prime}_{2} = \left( {\frac{{\alpha_{1} \Delta \tau^{ * } }}{{\Delta \in_{{\bar{r}}} }} - \frac{{\alpha_{4} \Delta \tau^{ * } }}{{\left( {\Delta \in_{r} } \right)^{2} }} + \frac{{\alpha_{5} \Delta \tau^{ * } }}{{\Delta \in_{r} }}} \right)$$

(21)

$$\alpha^{\prime}_{3} = \left( {\frac{{\alpha_{3} \Delta \tau {}^{ * }}}{{\Delta \in_{z} }} - \frac{{\alpha_{6} \Delta \tau^{ * } }}{{\left( {\Delta \in_{z} } \right)^{2} }}} \right)$$

(22)

$$\alpha^{\prime}_{4} = \left( {\frac{{\alpha_{4} \Delta \tau^{ * } }}{{\left( {\Delta \in_{r} } \right)^{2} }}} \right)$$

(23)

$$\alpha^{\prime}_{5} = \left( {\frac{{\alpha_{6} \Delta \tau^{ * } }}{{\left( {\Delta \in_{z} } \right)^{2} }}} \right)$$

(24)

$$\alpha^{\prime}_{6} = 1 - \Delta \tau^{ * } \left( {\alpha_{o} + \alpha_{2} } \right)$$

(25)

where:

$$\alpha_{1} \left( \upsilon \right) = \left( {\frac{{\upsilon_{r} }}{\upsilon }} \right)\left( {\frac{L}{{R_{o} }}} \right)$$

(26)

$$\alpha_{2} \left( \upsilon \right) = \left( {\frac{{\upsilon_{r} }}{\upsilon }} \right)\left( {\frac{L}{{R_{o} }}} \right)\left( {\frac{1}{{\varepsilon_{r} }}} \right)$$

(27)

$$\alpha_{3} \left( \upsilon \right) = \frac{{\upsilon_{z} }}{\upsilon }$$

(28)

$$\alpha_{4} \left( \upsilon \right) = \left( {\frac{{D_{er} }}{\upsilon }} \right)\left( {\frac{L}{{R_{o}^{2} }}} \right)$$

(29)

$$\alpha_{5} \left( \upsilon \right) = \left( {\frac{{D_{er} }}{\upsilon }} \right)\left( {\frac{L}{{R_{o}^{2} }}} \right)\left( {\frac{1}{{\varepsilon_{r} }}} \right)$$

(30)

$$\alpha_{6} \left( \upsilon \right) = \left( {\frac{{D_{ez} }}{\upsilon }} \right)\left( {\frac{1}{L}} \right)$$

(31)

$$\tau^{ * } = \left( {\frac{1}{{\left( {1 - k_{or} R_{AT} } \right)}}} \right)$$

(32)

For the implicit finite difference scheme, multiply by ∆τ ^* and rearranging yields;

$$\alpha^{\prime}_{1} \zeta_{ijk + 1} + \alpha^{\prime}_{2} \zeta_{i + 1jk + 1} + \alpha^{\prime}_{3} \zeta_{ij + 1k + 1} + \alpha^{\prime}_{4} \zeta_{i - 1jk + 1} + \alpha^{\prime}_{5} \zeta_{ij - 1,k + 1} - \alpha^{\prime}_{6} \zeta_{ijk} = \hat{a}_{r} \left( {\varPsi_{rijk + 1} - \varPsi_{rijk} } \right) - \hat{a}_{d} \left( {\varPsi_{dijk + 1} - \varPsi_{dijk} } \right)$$

(33)

Rearranging, for $i = 1,n,k = 1,n,for\,j = 1,n$, then the defining matrix equation

$$A\varOmega_{ik + 1} + B\varOmega_{i - 1k + 1} + C\varOmega_{i + 1k + 1} = D\varOmega_{ik} + \Delta \bar{\varXi }_{ik} + \bar{c}_{o} + \bar{d}_{0}$$

(34)

Prediction of geomechanical rock failure derived from rock stress factors evolves from the Mohr–Coulomb failure criterion. Mechanical decementation responses are governed by a phenomenon called rock fracture arching which is the resistance to withhold forces applied due to mechanical and hydrodynamic stresses. Radial stress gradient is derived from one of the equations of equilibrium in spherical coordinates as presented in Eq. 35

$$\frac{{\partial \sigma_{r} }}{\partial r} + \frac{1}{r}\left( {2\sigma_{r} - \sigma_{\theta } - \sigma_{\phi } } \right) = 0$$

(35)

A simplified spherical symmetry of the stressed field was assumed such that two tangential stresses are equal that is:$\sigma_{\theta } = \sigma_{\phi }$ resulting Eq. 36

$$\frac{{\partial \sigma_{r} }}{\partial r} + \frac{2}{r}\left( {\sigma_{r} - \sigma_{\theta } } \right) = 0$$

(36)

By Mohr–Coulomb criterion, radial and tangential stresses are related by:

$$\sigma_{\theta } - P_{f} = C_{o} + \left( {\sigma_{r} - P_{f} } \right)\tan^{2} \beta$$

(37)

At the cavity wall, P _f  = P _w  = σ _r , therefore:

$$\sigma_{\theta } - \sigma_{r} = C_{O} = 2S_{o} \tan \beta$$

(38)

An expression for normal stress gradient is given by Eq. 3

$$\left[ {\frac{\partial \sigma }{\partial r}} \right]_{{r = R_{c} }} = \left[ {\frac{{2C_{o} }}{r}} \right]_{{r = R_{c} }} = \left[ {\frac{{4S_{o} \tan \beta }}{r}} \right]_{{r = R_{c} }}$$

(39)

where C _o is uniaxial compressive strength. S _o is cohesive strength. R _c is cavity radius (Fig. 1).

Fig. 1

Stability diagram for production cavities as reported by (Morita et al. 1987a, b: SPE)

The basic equations that for constituting rock fracture models are: (1) Mechanical equilibrium eq. (2) Constitutive equation for the porous medium. (3) Continuity equation for fluid. (4) Darcy’s law. Extending the paradigm for rock fracture prediction models is progressed by a rock fracture production factor $k_{L}$ derived from the

Mohr–Coulomb Failure Criterion is segmented into three stages (1) Formation failure (2) Rock fracture erosion due to flow (3) Rock fracture transport (Fig. 2).

Fig. 2

Mohr–coulomb criterion in τ − σ′ space, and Mohr’s circle critical stress state

Rock failure occurs when the shear stress on a given plane within the rock reaches a critical value;

$$\tau_{\hbox{max} } = S_{o} + \sigma^{\prime}\tan \phi$$

(40)

Figure 2 shows the angle 2β, which gives the position of the point where the Mohr’s circle touches the failure line. Shear stress at this point of contact is given by Eq. 41:

$$\left| \tau \right| = \frac{1}{2}\left( {\sigma^{\prime}_{1} - \sigma^{\prime}_{3} } \right)$$

(41)

Normal stress is given by:

$$\sigma^{\prime} = \frac{1}{2}\left( {\sigma^{\prime}_{1} + \sigma^{\prime}_{3} } \right) + \frac{1}{2}\left( {\sigma^{\prime}_{1} - \sigma^{\prime}_{3} } \right)\cos 2\beta$$

(42)

Also, β and φ are related thus:

$$\beta = \frac{\pi }{4} + \frac{\phi }{2}$$

(43)

β is the angle of failure criterion. The maximum normal stress is related to the minimum normal stress

$$\sigma^{\prime}_{1} = 2S_{o} \left( {\frac{\cos \phi }{1 - \sin \phi }} \right) + \sigma_{3} \left( {\frac{1 + \sin \phi }{1 - \sin \phi }} \right)$$

(44)

The maximum stress is further given by:

$$\sigma^{\prime}_{1} = C_{o} + \sigma^{\prime}_{3} \tan^{2} \beta$$

(45)

Rock failure in petroleum production from mature fields represents significant equipment maintenance and work over costs challenges. Rock failure models documented in technical literature is solved using the mass balance equation of fluidized solids in conjunction with the erosion criterion and mass balance of the flowing fluids. However, equilibrium equation and, therefore, the mechanical responses of the reservoir, are not well captured. Rock stress failure is a two-stage process. The first stage is fractured rock stone decementation. Before rock fracture stone is decemented, rock fracturing cannot occur. Simulation of aquifer decementation requires the solution of equilibrium equation along with a suitable constitutive equation. Models based on coupled erosion-geomechanical model concepts are limiting. Therefore, there must be two conditions to produce rock fractures: (1) rock failure is mainly determined by the rock shear stress, and (2) aquifer production flow rate is mainly controlled by the fluid shear stress. Equation 46 is the Mohr–coulomb criterion correlation use in determining the range of the failure plane for which rock fracture production can be predicted. Mohr–Coulomb model is extended using rock fracture factor, K _Ls in a defining equation, where rock fracturing factor of 0 represents (minimum threshold of failure or rock fracturing) and rock fracturing factor of 1 is maximum safe zone when K _L < 0 to limit extensive rock fracture data requirement in the development of predictive models:

Necessary condition for rock fracture is given by:

$$k_{Rs} = \left[ {1 - \left( {\frac{{\tau_{\hbox{max} } \,}}{{{\text{Fluid}}\,\,{\text{Shear}}\,\,{\text{Stress}}\,\tau_{P} + {\text{Rock}}\,\,{\text{Shear}}\,\,{\text{Stress}}\left| \tau \right|}}} \right)} \right] .$$

(46)

The rock shear stress $\left| \tau \right|$ and maximum shear stress $\tau_{\hbox{max} }$ are represented by the Mohr–Coulomb Failure criterion

Sufficient condition for rock fracture is given when necessary condition is attained:

The fluid pressure shear stress $\tau_{p}$ derived from the Darcy equation greater that than rock stresses-maximum stresses lead to rock fracture occurring (Figs. 3, 4). Rock fracture is only produced when the fluid shear stress is greater than the residual stress from the maximum rock stress—rock shear stresses $0 \le k_{\text{fLs}} \le 1$.

Fig. 3

Diagram for damage and undamage section of reservoir

Fig. 4

Flow chart numerical simulation model

$$k_{\text{fs}} = \left[ {1 - \left( {\frac{{\tau_{\hbox{max} } \, - {\text{Rock}}\,{\text{Shear}}\,{\text{Stress}}\left| \tau \right|}}{{{\text{Fluid}}\,\,{\text{Shear}}\,\,{\text{Stress}}\,\tau_{P} }}} \right)} \right]$$

(47)

$$\tau_{p} = \hat{k}\left( {k_{r} \nabla^{2} p + \nabla p\nabla k_{r} } \right)$$

(48)

The region of rock fracturing is represented as $\tau_{\hbox{max} } - {\text{Rock}}\,{\text{Shear}}\,{\text{Stress}}\left| \tau \right| < {\text{Fluid}}\,\,{\text{Shear}}\,\,{\text{Stress}}\,\tau_{P}$, $0 \le k_{Ls} \le 1$

$- 1 \le k_{Ls} < 0$ is region of. $\tau_{\hbox{max} } > \left( {{\text{Rock}}\,{\text{Shear}}\,{\text{Stress}}\left| \tau \right|} \right)$, $0 \le k_{Ls} < - m$ represents the region of no rock fracturing or safe region.

$$\tau_{\hbox{max} } = \left[ {{\text{Rock}}\,{\text{Shear}}\,{\text{Stress}}\left| \tau \right|} \right] + \left[ {{\text{Fluid}}\,\,{\text{Shear}}\,\,{\text{Stress}}\,\tau_{P} } \right]\left( {1 - k_{Ls} } \right)$$

(49)

$$S_{o} = \tau_{\hbox{max} } - \sigma^{\prime}\tan \phi$$

(50)

where the fluid shear stress is computed from Eq. 14 becomes the sufficient condition

$$\tau_{z} = \hat{k}\left( {k_{r} \nabla^{2} p + \nabla p\nabla k_{r} } \right)$$

(51)

$$\left| \tau \right| = \frac{1}{2}\left( {\sigma^{\prime}_{1} - \sigma^{\prime}_{3} } \right)\quad \sigma^{\prime} = \frac{1}{2}\left( {\sigma^{\prime}_{1} + \sigma^{\prime}_{3} } \right) + \frac{1}{2}\left( {\sigma^{\prime}_{1} - \sigma^{\prime}_{3} } \right)\cos 2\beta$$

(52)

In this paper, concept of rock failure factor or rock failure producing factor (k _LS) to predict and quantify rock fracture produced in a reservoir field leads to the conclusion that the rock fails when rock shear stress is greater than or equal to the maximum rock shear stress. This is a necessary condition for rock fracture production must be failure of the rock; i.e., the rock shear stress must be greater than or equal to the maximum shear stress. If this condition is not met, rock fracture cannot be produced, regardless of the value of fluid shear stress. Fluid shear stress mainly controls the rock fracture production rate and not the rock failure, and this becomes the sufficient condition that rock fracture is produced. Fluid shear stress can be considered at the sufficient condition for rock fracture flow; therefore:

• 1. The lowest fluid shear stress yields the most rock fracture propagation (k _LS = 0, fluid shear stress = 0) which leads to not much fluid flow.

• 2. The highest fluid shear stress yields the least rock fracture propagation (k _LS = 1, fluid shear stress ≫ rock shear stress) which leads to more fluid flow

The most interesting result in the paper is that the value of fluid shear stress controls the rock fracture propagation rate. The combined effect of rock failure and fluid shear stress leads to rock failure propagation leading to fractured rocks.

Permeability damage reduction model

As particles are trapped in the pore throats permeability declines, which in return leads to a reduction in injectivity. Several relationships have been suggested to relate the decline in permeability to the concentration of deposited particles (17, 18). Wennberg and Sharma (1997) proposed a permeability reduction model starting with the Carman Kozeny equation:

$$\kappa = \frac{{\mathop \phi \nolimits^{3} }}{{5(1 - \phi )^{2} }}\frac{1}{{s^{2} }}\frac{1}{\tau }$$

(53)

Here, S is the specific surface area based on the solids volume and τ is the tortuosity of the porous medium. They further postulate that the permeability reduction due to particle deposition can be split into 3 parts: reduced porosity, increased surface area and increased tortuosity. The reduced permeability model can thus be expressed as Eq. 54:

$$\frac{k}{{k_{0} }} = k_{\text{dp}} k_{\text{ds}} k_{\text{dt}}$$

(54)

where

$$k_{\text{dp}} = \frac{{\phi^{3} }}{{\phi_{0}^{3} }}\frac{{(1 - \phi_{0}^{2} )}}{{(1 - \phi^{2} )}}$$

(55)

$$k_{ds} = (\frac{{1 + \sigma /(1 - \varphi_{0} )}}{{1 + \sigma /(1 - \varphi_{0} )(d_{g} /d_{p} )}})^{2}$$

(56)

$$k_{dt} = \frac{1}{(1 + \beta \sigma )}$$

(57)

The damage factor β accounts for trapped particles deposit in the pores. Β is normally greater than 0.

The permeability distribution is determined by the extent and distribution of particles trapped in the pore spaces. Payatakes et al. indicate that the pressure drop increase is a linear function of the extent of the particle deposition in the case of dilute suspension injection. This suggests that the following equation holds for small particle sizes

$$k(C) = \frac{{k\left( {x,t)} \right)}}{{k_{m} }} = \frac{1}{{1 + \beta \sigma \left( {x,t} \right)}}$$

(58)

where $\beta$ is a constant and represents the damage factor.

The average dimensionless permeability between the injected face and the injection front of the core can be obtained by expanded model including the $R_{\text{AT}} (c)$ function and permeability damage factor.

$$k^{\prime}\left( C \right) = \frac{{k\left( {r,z,t} \right)}}{{k_{m} }} = \frac{{K_{O} .e^{{ - R_{\text{AT}} }} }}{1 + \beta \sigma }$$

(59)

$$K_{O} = k_{\text{dp}} .k_{\text{ds}}$$

where σ can be determined by Eq. 42 below:

$$\frac{\partial \sigma }{\partial t} = \lambda \nu C$$

(60)

Injectivity performance related to fracturing pressure

The sustaining or fracturing pressure equation derived from mass balance injector-production performance is given as Eq. 43 below

$$\rho c_{T} \frac{\partial \phi P}{\partial t} + \rho \nabla \lambda P = i - q_{i}$$

(61)

$i =$ injection rate; $q_{i}$ = production rate

For cylindrical coordinates:

$$\left( {c_{T} \phi \frac{\partial P}{\partial t} + \lambda \nabla P} \right) + \left( {Pc_{T} \left( {\frac{\partial \phi }{\partial t}} \right) + P\nabla \lambda } \right) = \left( {\frac{{i - q_{i} }}{\rho }} \right)$$

(62)

$$\left( {c_{T} \phi \frac{\partial P}{\partial t} + \lambda_{r} \frac{\partial P}{\partial r} + \lambda_{z} \frac{\partial P}{\partial z} + \frac{{\lambda_{r} P_{r} }}{r}} \right) + \left( {Pc_{T} \left( {\frac{\partial \phi }{\partial t}} \right) + P_{r} \frac{\partial \lambda }{\partial r} + P_{z} \frac{\partial \lambda }{\partial z} + \frac{{\lambda_{r} P_{r} }}{r}} \right) = \left( {\frac{{i - q_{i} }}{\rho }} \right)$$

(63)

Measure of interconnectivity

$$\frac{\partial \phi }{\partial t} = K_{I} \frac{\partial \lambda }{\partial t}$$

(64)

$$\left( {c_{T} \phi \frac{\partial P}{\partial t} + \lambda_{r} \frac{\partial P}{\partial r} + \lambda_{z} \frac{\partial P}{\partial z} + \frac{{2\lambda_{r} P_{r} }}{r}} \right) + \left( {Pc_{T} K_{I} \left( {\frac{\partial \lambda }{\partial t}} \right) + P_{r} \frac{\partial \lambda }{\partial r} + P_{z} \frac{\partial \lambda }{\partial z}} \right) = \left( {\left( {\frac{{i - q_{i} }}{\rho }} \right)} \right)$$

(65)

where

$$K_{I} = \frac{\partial \phi }{\partial \lambda }$$

(66)

Measure the rate of flow ingress and egress

$$\left( {c_{T} \phi \frac{\partial P}{\partial t} + \lambda_{r} \frac{\partial P}{\partial r} + \lambda_{z} \frac{\partial P}{\partial z} + \frac{{2\lambda_{r} P_{r} }}{r}} \right) + \left( {Pc_{T} K_{I} \left( {\frac{\partial \lambda }{\partial t}} \right) + P_{r} \frac{\partial \lambda }{\partial r} + P_{z} \frac{\partial \lambda }{\partial z}} \right) = b_{i}$$

(67)

$b_{i}$ the permeability damage factor

$$b_{i} = \left( {\frac{{i - q_{i} }}{\rho }} \right)$$

(68)

The injectivity index model is defined as the flow rate per unity of the pressure drop between the injector and the reservoir. Injectivity decline is computed as in Eq. 69

$$\prod = \frac{q(t)}{\Delta p(t)}$$

(69)

The impedance is equal to the inverse of the dimensionless injectivity index

$$J(T) = \frac{\prod \left( 0 \right)}{\prod (t)} = \frac{{q_{o} \Delta p(T)}}{\Delta p\left( 0 \right)q\left( T \right)}$$

(70)

The impedance is a piecewise linear function of the dimensionless time for either deep bed filtration or external cake formation (Ajay and Sharman 2007) and now extended by a variable $R_{AT} (c)$ at transition point T _r.

$$J_{d} \left( T \right) = 1 + mT + R_{AT} T_{r} \quad {\text{for}}\; T \le T_{r}$$

(71)

$$J_{d} \left( T \right) = 1 + mT_{r} + m_{c} \left( {T - T_{r} } \right)\quad {\text{for}}\,T > T_{r}$$

(72)

$$T_{r} > \frac{{2\alpha r_{w} }}{{\lambda C_{o} R_{c}^{2} }}$$

(73)

$$m_{c} = \frac{{kk_{\text{rowr}} \phi C_{o} }}{{k_{c} \left( {1 - \phi_{c} } \right)X_{w} \left( { - In\left( {X_{w} } \right)} \right)}}$$

(74)

The impedance slope m during the deep filtration is given by the formula below

$$m = \left( {\frac{{\beta \phi c_{o} }}{{InX_{w} }}} \right)\left( {\lambda R_{c} } \right)\left( {\frac{1}{{\sqrt {X_{w} } }}} \right)\left( { - \exp ( - \lambda \left( {R_{C} - r_{w} } \right)} \right) - \lambda R_{C} \exp \left( {\lambda r_{w} } \right)\int\limits_{{\lambda r_{w} }}^{{\lambda R_{c} }} {\frac{{\exp \left( { - u} \right)}}{u}} {\text{d}}u$$

(75)

where

$$u = \lambda R_{c} \sqrt X$$

(76)

$$X = \ell^{2} = \left( {\frac{r}{{R_{c} }}} \right)^{2}$$

(77)

$$X_{w} = \ell^{2} = \left( {\frac{{r_{w} }}{{R_{c} }}} \right)^{2}$$

(78)

The slope m_c during the external cake formation is: The computation of the velocity is given

$$\upsilon_{r} = \frac{{q_{r} }}{2\pi rh} = \left( {\frac{{K_{or} K_{{\sigma_{r} }} }}{\mu }} \right)\int\limits_{{r_{w} }}^{{r_{e1} }} {\frac{1}{r}\frac{drP}{dr}} + \int\limits_{{r_{e1} }}^{{r_{e} }} {K_{or} } \frac{1}{r}\frac{drP}{dr}$$

(79)

$$\upsilon_{z} = \frac{{q_{z} }}{{\pi r^{2} h}} = \left( {\frac{{K_{oz} K_{{\sigma_{z} }} }}{\mu }} \right)\int\limits_{{r_{w} }}^{{r_{e1} }} {\frac{dP}{dz}} + \int\limits_{{r_{e1} }}^{{r_{e} }} {K_{oz} } \frac{dP}{dz}$$

(80)

$$\int\limits_{{r_{w} }}^{{r_{e} }} {\left( {\frac{q}{2\pi h}} \right)\frac{dr}{r}} = \left( {\frac{{K_{or} \left( {1 + K_{\sigma } } \right)}}{\mu }} \right)\Delta P$$

(81)

$$\frac{\Delta P}{q} = \frac{{\mu In\left( {\frac{{r_{e} }}{{r_{w} }}} \right)}}{{2\pi K_{or} }} + \frac{{\mu In\left( {\frac{{r_{e} }}{{r_{w} }}} \right)}}{{2\pi K_{or} K_{\sigma } }}$$

(82)

$$\frac{\Delta P}{q} = \frac{1}{{2\pi K_{or} }}\left( {1 + \frac{1}{{K_{\sigma } }}} \right)$$

(83)

$$\frac{\Delta P}{q} = \frac{1}{{2\pi K_{or} }}\left( {1 + \frac{1}{{K_{\sigma } }}} \right)$$

(84)

$${\text{Total}}\,{\text{Impedance}}\, = \,{\text{Damage}}\,{\text{Impedance}}\, + \,{\text{Undamaged}}\,{\text{Impedance}}$$

(85)

$$\frac{\Delta P}{q} = \frac{{\mu In\left( {\frac{{r_{e} }}{{r_{w} }}} \right)}}{{2\pi K_{or} }} + \frac{\mu }{{2\pi K_{or} }}K^{\prime}_{\sigma }$$

(86)

$$\frac{\Delta P}{q} = \frac{\mu }{{2\pi K_{or} }}\left( {In\left( {\frac{{r_{e} }}{{r_{w} }}} \right) + K^{\prime}_{\sigma } } \right)$$

(87)

Dimensionless form

$$\frac{{\left( {\frac{\Delta P}{q}} \right)_{T} }}{{\left( {\frac{{\Delta P_{O} }}{{q_{O} }}} \right)}} = \frac{{\left( {\left( {In\left( {\frac{{r_{e} }}{{r_{w} }}} \right) + K^{\prime}_{\sigma } } \right)} \right)_{T} }}{{\left( {\left( {In\left( {\frac{{r_{e} }}{{r_{w} }}} \right)_{\sigma } } \right)} \right)_{T} }}$$

(88)

The final form of injectivity model is presented in Eq. 89

$$j = 1 + K^{\prime}_{\sigma } \left( {\frac{1}{{In\left( {\frac{re}{{r_{w} }}} \right)}}} \right)$$

(89)

Field data model analysis and computer simulation

The studied field is a multi-reservoir, faulted anticline, heavy oil accumulation at a depth ranging from approximately 2200–2600 m subsea, in Campos Basin block BC-4. Water depth within the areal extent of the field ranges from 1050 to 1300 m. Studied field was developed as an all subsea well peripheral water flood project, with all injection below the various oil water contacts. The project uses vertical or deviated water injection wells and long, horizontal open-hole gravel pack production wells. At the time of this evaluation, a final decision has not been made regarding injection completion selection and also regarding whether produced water will be processed for overboard discharge or re-injected into the reservoir; therefore, this study will examine multiple completion geometries and the effects of alternative produced water strategies. The field data as reported in (Idialu 2014) were sourced in field report Wehunt (2002), Guo (2000), Meyer et al. 2003a, b.

Modeling methods

The simulation profiles for the water injection project are presented below and obtained from a Field Injection Study report Wehunt (2002). The values for all invariant simulation data are listed in 2 (Tables 1, 2, 3, 4). Additional information regarding what the various parameters are and how they function within the program is available from the program documentation. Details of the PWRI, well prognosis and simulation results for the effects of completion geometry, rock mechanics, filtration parameters, well hydraulics, leak off properties, operations, produced water re-injection parameters, reservoir properties are provided in “Appendix A”. Details of the field report and data could be found in Wehunt (2002), Guo (2000), Meyer et al. (2003a, b). The reports highlight significance of (1) Completion geometry, (2) Rock mechanics (3) Filtration Properties (4) Total suspended solids (5) Filtration coefficient (6) Internal cake permeability damage factor (7) External filter cake permeability (8) Filter cake erosion ratio (9) Other leak off properties (10) Formation permeability (11) Injection fluid viscosity (12) compressibility (13) Aquifer oil saturation (14) Other assumptions (15) Boundary conditions, “ellipsoidal coupling, constant pressure B. C.” was used for all runs except one. Ellipsoidal coupling, pseudo-steady state” was used for the other run. The fracture geometry was very insensitive to this parameter, and no plots are provided for this case. (16) Drainage Area; The BASE Case value was 1200 acres. Sensitivity cases were calculated for 750 acres and 2000 acres. The fracture geometry was very insensitive to this parameter, and no plots are provided for this case (17) Number of Fractures (18) Operations (19) Startup Procedure (20) Slurry Rate (21) Downtime (22) Wellbore Hydraulics in altering fracturing, permeability damage and injectivity. Results for this section are listed under the “Other Assumptions” category in Table 5 of their report.

Table 1 Layered properties model

Table 2 Invariant simulation data

Table 3 Zone data

Table 4 Seawater versus PWRI case data

Table 5 Variable simulation data

Results and discussions

The results of model simulation based on the field data provided in “Field data model analysis and computer simulation” section were based on the field report and data obtained from Wehunt (2002), Guo (2000), Meyer et al. 2003a, b.

Injector Performance and permeability damage as a function of aquifer integrity

Figures 5 and 6 show injector performance with time related to fracturing hydraulics pressure and aquifer system. Figures 5 and 6 show field data simulation of a known field using Meyer fracturing simulator. Figures 5, 6, 7 and 8 show performance based on our software simulator in MATHLAB and COMSOL Multiphysics

Fig. 5

Height of fracturing with time

Fig. 6

BASE case fracture height versus time

Fig. 7

Injector performance with time

Fig. 8

Concentration profiles with time

Figures 5, 6, 7, 8 and 9 show the profile of permeability on both fracturing and filtration phenomena on the outlay in injector performance and concentration of cake build up. The profile decreased with time and increased uniformly with radial distance from produced water invasion zone. From the analysis of the results in the absence of particle deposition, low permeability formation was observed to be more likely fractured as the net fracturing pressure was observed to be inversely proportional to permeability, for a given injection rate. In addition, particle filtration and formation damage were governed by the interactions of particles in the injected water within the reservoir. In general, formation plugging is severe as the formation permeability decreased (Fig. 10).

Fig. 9

Plot of fracturing pressure on impedance with time

Fig. 10

source: Energy Tech Co, Houston, Texas and Department of Petroleum Resources [Nigeria] as reported by Idialu (2014)

Profile of injectivity decline of produced water from the Bakasap formation.

Figure 11 shows profile of permeability and injectivity for 49 days for a particular field in Bakasap formation. The results were reported from the field and log data obtained and showed permeability damage with depth showing similar profile with Fig. 12, our simulated profile using COMSOL Multiphysics

Fig. 11

Permeability damage with depth and time

Fig. 12

Field Studied case fracture height versus time

Case 1: WID Simulation Data and Results

Figures 12, 13, 14 and 15 show fracture height with time and increase based on log data of PWRI case thermal and fractured profiles of decreased injector performance at different rates based on report Meyer et al. 2003a, b.

Fig. 13

Field Studied case thermal and fracture profiles

Fig. 14

Thermal/water and fractured fronts

Fig. 15

Fracture height versus time

Figure 16 shows injectivity decline for different injection rates and shows a decrease with time and showing effect of fracturing pressure injector performance.

Fig. 16

Injectivity decline with time for different injection rates

Thermal and Pore Pressure Effects on Injectivity Performance

Profiles in Figs. 17 and 18 show effect of thermal gradient in reservoir further to injectivity decline. Higher temperature favors the reduction of particle deposition in reservoir. This validates established technique in the industry called stimulation whereby heat injected into the reservoir clean pore spaces of deposition. The simulation profiles showed that fracture gradient was more likely to be influenced by pore pressure and temperature changes. As cooler injection fluids reduce temperature, the rock becomes more brittle, strongly dependent on Young’s Modulus of elasticity. Injection flow rate is an important parameter in permeability impairment. The higher the linear velocity, the greater the depth of particle penetration. Smaller velocities and larger particle concentration results in larger permeability declines and thus greater decline is experienced. From the graph above, it is seen that the increase in the fluid flow rate results in the internal cake forming faster.

Fig. 17

Injectivity with time at different rates

Fig. 18

Injectivity with time at different temperature

The results of model simulation based on the field data provided in “Field data model analysis and computer simulation” section were based on the field report and data obtained from Wehunt (2002), Guo (2000), Meyer et al. 2003a, b.

Figure 19 shows Bekasap Formation of the Kotabatak field as well as produced water from the Bekasap formation from other fields in the areas such as Kasikan, Lindai, Langgak, Petapahan; that the higher pressures seen are a good indication of the maximum pressure expected before fracture extension occurs, in this case about 2900 psi. Results of Fig. 19 show that lower pressures seen that either have low injection rates or have recently had a fracture extension. In either case, they are an indication of what the reservoir pressure would be (about 2000 psi). Similarly, this is the pressure ultimately seen by a hydraulic fracture conducted on a producing well.

Fig. 19

PERFTOP case thermal and fracture profiles

Figure 20 shows the output of the WID (water injectivity decline) simulator, using data input to roughly simulate a Kotabatak injector, for a case with a 20-foot fracture. Note that injection proceeds steadily for about a year and then suddenly drops. This corresponds to the behavior seen in Fig. 6 for Well 190, where a pressure spike occurs about once a year. Injection rate climbs slightly for a majority of that year, followed by a swift decline in injection and increase in pressure. This cycle has been repeated several times, which can be interpreted as fracture growth/extension occurring about once a year, at least for this well.

Fig. 20

PERFTOP case fracture height versus time

Injectivity performance and permeability damage on rock properties

Profiles in Fig.  21 show that well injectivity varies during water injection basically due to two competitive factors: formation damage by the suspended particles which results into deposition and thus injectivity decline. As shown in Figure 30, for different injection rates, injectivity decline exponentially decreases with time and increases as the produced water injection rates increase. Injectivity index decline decreases with damaged factor exponentially are plotted in Fig. 22. Injectivity decreases from 1.126 to zero when damaged factor is 1.0 indicating the effects of cake deposits in pore blocking and permeability damage. The injectivity decline experienced in the reservoir has been linked to the volume of oil produced. From the graph, it is observed that the injectivity decline experienced increases as the production rate reduces. This is better explained by suggesting that mobility ratio, voidage factor and reservoir permeability has a profound influence on both fracturing and filtration phenomena. Even in the absence of particle deposition, low permeability formation is more likely to be fractured as the net fracturing pressure is inversely proportional to permeability, for a given injection rate. In addition, particle filtration and formation damage are governed by the interactions of particles in the injected water with the reservoir rock. In general, formation plugging will be more severe as the formation permeability decreases. It should be noted here that the formation permeability is directly dependent upon the formation grain size (dg). Particle deposition around the wellbore and the fracture face, modeled using filtration theory. This influence is via an increase in injection pressure due to additional skin resistance across the face of the fracture or near wellbore. This additional flow resistance is due to combination of internal and external cakes. The pressure increase due to skin resistance is inversely proportional to the area of fracture face with differing particle size, we find out that (1) overall damage is related to the mean pore throat size (2) the pore damage with 0–3 microns exhibit damage throughout the entire reservoir length (3) as particle size increase, the damage is gradually shifted toward the injection end of the pore and to an external cake. Particles of sizes ranging from 0.05–7 cause formation damage. The larger particles cause a rapid decline in permeability with the damage region being shallow. Smaller particles enter the core and cause a gradual permeability decline.

Fig. 21

Injectivity decline with damage factor

Fig. 22

Injectivity decline with time (day)

Figures 23 and 24 injector performance profiles showed the effect of ratio of particle size to reservoir pore size on injectivity decline as the ratio increases, injectivity decline decreases as well, and all injectivity decline decreases with time. When suspended particles in a carrier fluid are flowed through a porous medium, the operative plugging mechanism depends on the characteristics of the particle, the characteristics of the formation, and the nature of the interaction between the particle and the various reservoir materials. With differing particle size, we find out that (1) overall damage is related to the mean pore throat size (2) the pore damage with 0–3 microns exhibit damage throughout the entire reservoir length (3) as particle size increases, the damage is gradually shifted toward the injection end of the pore and to an external cake. Particles of sizes ranging from 0.05–7 cause formation damage. The larger particles cause a rapid decline in permeability with the damage region being shallow. Smaller particles enter the core and cause a gradual permeability decline. The particle/pore size ratio is the most important parameter in the filtration process. It can be seen that the larger the particle/pore size ratios tend to cause rapid, but shallow damage. As shown from the graph, varying the damage factor used for the simulation would have little or no effect on the outcome of the simulation. The injectivity decline experienced even with these varying factors and days showed that the decline has very little dependence on these factors.

Fig. 23

Injectivity decline with damage factor in days

Fig. 24

Injectivity decline with time for different damage factors

Conclusion

An improved internal filtration model incorporating the effect of adsorption kinetics, geochemical reaction and hydrodynamics, well hydraulics and aquifer integrity residual oil mobility and correction for well completion geometry and rock mechanics formation damage coefficient introduced as $R_{\text{AT}}$ variables that highlights the contribution of the combination of well geometry, leak off, geochemical reaction, filtration parameters, well hydraulics and rock mechanics and other hydraulic parameters effects factors. The model injectivity and fracturing was solved using the finite element method simulated in COMSOL Multiphysics Software. To simulate the model, well-known implicit finite difference discretization scheme was employed to the improvements in advection–dispersion–geochemical reaction process incorporating the variable $R_{\text{AT}}$ in a dimensionless time constants. The attendant banded linear systems of equations were solved in MATHLAB environment using decomposition approach. Using preliminary field data obtained from re-injection sites in the Injection Field Project, our simulation showed that permeability decline is exponential function in time of $R_{\text{AT}}$ factors signifying of aquifer integrity, rock mechanics properties, thermal stress, particle to grain ratio, retention kinetics, filtration parameters, well hydraulics, and produced water quality in R _AT function alters permeability damage, fracturing, cake formation and injectivity decline in an improved robust improved internal filtration—hydraulic model. However, at a specific length in the aquifer, the concentration profile of the active specie follows an exponential distribution in time. Meanwhile, injectivity decline decreases exponentially with radial distance in the aquifer. Clearly, injectivity decline is a function of fracturing mechanics for injector performance and cake deposition resulting in permeability damage g from an adsorption coupled filtration scheme. In this regard, it is established that the transition time t _r to cake nucleation and growth is a consequence aquifer capacity, filtration coefficients particle and grain size diameters and more importantly adsorption kinetics and produced water quality.

Appendix A: Well schematics for target wells

See Figs. 25, 26, 27 and 28.

Fig. 25

Well 217

Fig. 26

Well 210

Fig. 27

Well 38

Fig. 28

Well 238