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

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Original Paper - Production Engineering
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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.

Keywords

Re-injection Fracturing Rock properties Permeability damage Acquifer 

List of symbols

ST

Skin factor

μ

Viscosity

Pinj

Injection Pressure

q

Flow rate (m3/s)

k

Permeability

kσ

Permeability damage factor

\(\eta\)

Total collision probability

\(\eta\) l

Collision probability due to interception

\(\eta\) D

Collision probability due to diffusion

\(\eta\) lm

Collision Probability due to impaction

\(\eta\) s

Collision probability due to sedimentation

\(\eta\) E

Collision probability due to surface forces

dp

Particle diameter

dg

Grain diameter

\(\phi\)

Effective porosity

\(\rho_{p}\)

Particle density

\(\rho_{f}\)

Fluid density

\(U,u\)

Darcy’s velocity

g

Gravity acceleration (m/s2)

T

Absolute temperature (K, °C)

\(C(r,t)\)

Volumetric concentrations of suspended particles (ppm)

\(\sigma (r,t)\)

Volumetric concentrations of the deposited particles (ppm)

ko

Absolute permeability

\(\lambda\)

Filtration coefficient

L

Depth of the porous media

\(\varepsilon_{r}\)

Scaled length in radial direction

\(\varepsilon_{z}\)

Scaled length in axial direction

t

Time (yrs)

\(\tau\)

Scaled time

\(\in\)

Scaled concentration of suspended solids

S

Scaled concentration of deposited particles

\(\lambda_{o}\)

Initial filtration coefficient

\(\alpha_{c}\)

Clean bed collision efficiency

I

Injectivity index

J

Inverse of Injectivity Index

Tr

Transition time

n

Number of particles attached to one grain

\(J_{d}\)

Impedance during one phase suspension flow

Kror

Relative permeability of residual oil

m

Slope of Impedance straight line during deep bed filtration for one Phase suspension flow

mc

Slope of Impedance straight line during external cake formation for one phase suspension flow

p

Pressure (M/LT2, Pa)

q

Total flow rate per unit reservoir thickness, L2/T

r

Reservoir radius (L, m)

rw

Well radius (L/m)

rd

Damage zone radius (L, m)

Rc

Contour radius (L, m)

Sor

Residual oil saturation

Swi

Initial water saturation

T

Time (T, s)

T

Dimensionless time

Ttr

Dimensionless transition time

U

Total flow velocity (L/T, m/s)

\(\alpha\)

Critical porosity fraction

\(\beta\)

Formation damage coefficient

\(\phi\)

Porosity

Definition of terms and acronyms

Produced water

Water associated with crude oil exploration and production

Produced water re-injection

Sending back produced water from the surface into the subsurface

Non-fresh water hydrocarbon aquifer

Crude oil bearing formation

Reservoir

A permeable subsurface rock that contains petroleum

Formation

Refers to the reservoir bearing fluids e.g. oil, gas and water

Produced water constituents

Heavy metals, suspended solids, dissolved solids, hydrocarbon traces etc.

Injection pipe

Produced water transfer medium from surface to subsurface

Well bore

Point of contact of injection pipe with formation/reservoir

Deep bed filtration

The flow and deposition of particles in the rock matrix

Injectivity decline

Index signifying the change in the injection rate of the injected fluid

Formation damage

Reduction in aquifer properties that are solely responsible for the transmissibility of reservoir fluids through the pore spaces (fracture in internal walls of the aquifer)

Adsorption kinetics

Attraction and retention of particle to the surface grain and the preference of this particle for a particular site within the reservoir

Hydrodynamic dispersion

Is a term used to include both diffusion and dispersion of particles within a medium

Geochemical reaction

This is the interaction of species constituents in the produced water and the formation of the host aquifer

Colloids

Colloidal particles are suspended particles carried in the fluid stream

Scales

Result of nucleation of colloids

Cakes

Deposition of scales in pore sites is referred to as cakes

Geomechanics

Involves the geologic study of the behavior of soil and rock

Corrosion

Loss in metal due to degradation, erosion or prevailing ambient conditions

Souring

Acidic smell/taste characteristic

Representative elementary volume

A pictured or drawn shape representative of the actual shape. Used in solving mathematical problems

Isotherms

Equations considered at constant temperature

Finite element method

Numerical method of solution whereby a problem is characterized by boundaries and solved within these boundaries

PW

Produced water

PWRI

Produced water re-injection

EOR

Enhanced oil recovery

E & P

Exploration and Production

REV

Representative elementary volume

TVD

Total vertical depth

BHP

Bottom hole pressure

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. 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. 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 mc 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

TVD @ Bottom, m

σHmin, Psi

Young’s Modulus, Psi

Poisson’s Ratio

Toughness psi-in1/2

Pressure, psi

Compressibility, psi−1

Permeability, md

Porosity

Formation Fluid Viscosity, cp

Coeff. Therm Exp (1/R)

Temp (F)

Biot’s Constant

2133.64

3750

9.2E+04

0.392

400

3134

1.05E−05

100

0.343

0.70

3.5E−06

95.6

1

2134.29

3751

8.6E+04

0.392

400

3134

1.07E−05

100

0.386

0.70

3.5E−06

95.6

1

2134.43

3752

1.8E+05

0.392

400

3134

1.03E−05

100

0.393

0.70

3.5E−06

95.6

1

2134.57

3752

3.5E+05

0.392

400

3135

1.05E−05

100

0.350

0.70

3.5E−06

95.7

1

2134.72

3729

7.7E+05

0.386

400

3135

9.53E−06

100

0.216

0.70

3.5E−06

95.7

1

2135.57

3657

2.3E+06

0.368

400

3135

3.65E−06

100

0.117

0.70

3.5E−06

95.7

1

2135.86

3790

1.1E+06

0.4

400

3136

3.27E−06

100

0.274

0.70

3.5E−06

95.7

1

2139.29

3795

4.6E+05

0.4

400

3138

5.82E−06

100

0.314

0.70

3.5E−06

95.9

1

2140.72

3726

1.0E+06

0.383

400

3141

5.48E−06

100

0.295

0.70

3.5E−06

96.1

1

2142.58

3906

2.6E+06

0.421

400

3143

3.34E−06

100

0.150

0.70

3.5E−06

96.2

1

2142.86

3890

1.1E+06

0.418

400

3145

4.76E−06

100

0.291

0.70

3.5E−06

96.3

1

2146.15

3691

3.7E+05

0.371

400

3147

5.70E−06

100

0.308

0.70

3.5E−06

96.4

1

2147.86

3785

7.2E+05

0.391

400

3150

4.31 E-06

100

0.289

0.70

3.5E−06

96.6

1

2148.29

3884

2.7E+05

0.413

400

3151

5.52E−06

100

0.351

0.70

3.5E−06

96.6

1

2166.76

3895

1.2E+05

0.411

400

3163

6.10E−06

100

0.371

0.70

3.5E−06

97.3

1

2167.34

3778

2.9E+05

0.379

400

3174

6.37E−06

100

0.265

0.70

3.5E−06

98.0

1

2175.37

3775

4.7E+05

0.376

400

3180

4.63E−06

100

0.310

0.70

3.5E−06

98.3

1

2185.71

3878

1.9E+05

0.394

400

3191

6.93E−06

100

0.331

0.70

3.5E−06

99.0

1

2194.96

3903

9.2E+04

0.394

400

3203

9.13E−06

1500

0.358

0.70

3.5E−06

99.7

1

2205.51

3937

1.0E+05

0.397

400

3215

8.62E−06

1500

0.347

0.70

3.5E−06

100.4

1

2208.97

3948

4.2E+05

0.395

400

3224

6.09E−06

100

0.320

0.70

3.5E−06

100.9

1

2209.84

4122

8.7E+05

0.429

400

3226

3.72E−06

1500

0.295

0.70

3.5E−06

101.0

1

2210.13

4195

3.9E+05

0.443

400

3227

5.81 E-06

1500

0.285

0.70

3.5E−06

101.1

1

2210.42

4100

1.5E+05

0.425

400

3227

9.06E−06

1500

0.307

0.70

3.5E−06

101.1

1

2221.17

4046

8.9E+04

0.411

400

3234

9.56E−06

1500

0.308

0.70

3.5E−06

101.5

1

2221.32

4003

1.9E+05

0.4

400

3241

9.24E−06

1500

0.290

0.70

3.5E−06

101.9

1

TVD @ Bottom, m

σHmin, Psi

Young’s Modulus, Psi

Poisson’s Ratio

Toughness psi-in1/2

Pressure, psi

Compressibility, psi−1

Permeability, md

Porosity

Formation Fluid Viscosity, cp

Coeff. Therm Exp (1/R)

Temp (F)

Biot’s Constant

2221.46

4020

3.5E+05

0.403

400

3241

8.56E−06

1500

0.278

0.70

3.5E−06

101.9

1

2221.75

4023

8.8E+05

0.404

400

3241

5.68E−06

1500

0.212

0.70

3.5E−06

101.9

1

2222.33

4025

1.5E+06

0.404

400

3242

4.00E−06

1500

0.186

0.70

3.5E−06

101.9

1

2222.48

4026

5.2E+05

0.404

400

3242

7.16E−06

1500

0.246

0.70

3.5E−06

102.0

1

2222.63

4039

2.7E+05

0.406

400

3242

6.99E−06

1500

0.254

0.70

3.5E−06

102.0

1

2233.68

4078

8.7E+04

0.411

400

3249

9.14E−06

1500

0.311

0.70

3.5E−06

102.4

1

2234.70

3933

2.6E+05

0.378

400

3257

4.33E−06

1500

0.293

0.70

3.5E−06

102.8

1

2236.30

4070

5.3E+05

0.405

400

3259

3.67E−06

1500

0.213

0.70

3.5E−06

102.9

1

2238.63

4338

8.2E+05

0.454

400

3262

5.11E−06

1500

0.248

0.70

3.5E−06

103.1

1

2239.51

4038

2.5E+05

0.397

400

3264

4.17E−06

1500

0.329

0.70

3.5E−06

103.2

1

2239.95

3942

6.1E+05

0.377

400

3265

3.43E−06

1500

0.320

0.70

3.5E−06

103.2

1

2241.71

4119

2.5E+06

0.411

400

3267

3.81E−06

1500

0.175

0.70

3.5E−06

103.3

1

2242.15

4320

1.1E+06

0.449

400

3268

3.23E−06

1500

0.283

0.70

3.5E−06

103.4

1

2245.38

4224

5.3E+05

0.431

400

3271

3.56E−06

1500

0.258

0.70

3.5E−06

103.5

1

2251.54

4281

2.4E+05

0.439

400

3278

3.94E−06

1500

0.318

0.70

3.5E−06

103.8

1

2251.98

4333

6.8E+05

0.446

400

3283

3.33E−06

1500

0.261

0.70

3.5E−06

104.1

1

2252.86

4366

2.3E+06

0.451

400

3284

3.33E−06

1500

0.143

0.70

3.5E−06

104.1

1

2253.01

4312

1.3E+06

0.442

400

3285

3.25E−06

1500

0.156

0.70

3.5E−06

104.2

1

2253.15

4313

7.5E+05

0.442

400

3285

3.46E−06

1500

0.200

0.70

3.5E−06

104.2

1

2255.94

4123

3.2E+05

0.406

400

3287

3.70E−06

1500

0.311

0.70

3.5E−06

104.3

1

2256.38

4235

9.9E+05

0.427

400

3290

5.31E−06

1500

0.210

0.70

3.5E−06

104.4

1

2257.99

4411

2.2E+06

0.457

400

3291

3.14E−06

1500

0.206

0.70

3.5E−06

104.5

1

2259.02

4134

1.1E+06

0.407

400

3293

3.28E−06

1500

0.206

0.70

3.5E−06

104.6

1

2259.76

4149

1.8E+06

0.409

400

3295

3.19E−06

1500

0.251

0.70

3.5E−06

104.6

1

2261.81

4104

7.6E+05

0.4

400

3297

3.73E−06

1500

0.266

0.70

3.5E−06

104.7

1

2264.89

4025

5.2E+05

0.382

400

3300

3.86E−06

1500

0.258

0.70

3.5E−06

104.9

1

2272.81

4197

3.4E+05

0.413

400

3309

3.78E−06

100

0.300

0.70

3.5E−06

105.3

1

2273.70

4159

2.1E+05

0.404

400

3315

5.63E−06

100

0.307

0.70

3.5E−06

105.6

1

2275.60

4207

3.9E+05

0.412

400

3317

3.48E−06

100

0.334

0.70

3.5E−06

105.7

1

2288.77

4205

2.7E+05

0.408

400

3329

3.73E−06

100

0.316

0.70

3.5E−06

106.3

1

2289.51

4313

4.4E+05

0.425

400

3339

3.41E−06

100

0.277

0.70

3.5E−06

106.3

1

2291.88

4266

1.8E+05

0.416

400

3342

3.88E−06

100

0.353

0.70

3.5E−06

106.9

1

2295.44

4281

4.5E+05

0.417

400

3346

3.84E−06

100

0.293

0.70

3.5E−06

107.1

1

2316.41

4182

3.8E+05

0.392

400

3364

3.56E−06

100

0.312

0.70

3.5E−06

108.0

1

Table 2

Invariant simulation data

Description

Value

Fluid loss model

Dynamic, calculate fracture skin, and include fluid loss history

Fracture geometry

3-Dimensional

Flow back

Off

Simulate to closure

On

Fracture fluid gradient

Include

Propagation parameters

Default Growth (+, −)

Fracture initiation interval

Minimum stress interval

Fracture friction model

On, with a = 24 and b = 1

Wall roughness

Off

Tip effects

Off

Flow path

Tubing

Surface line volume

0 bbls

Depth

2210 m MD

Maximum BHTP

6000 psi

Fixed depth

MD

Calculated (TVD or Angle)

Angle

Deviation survey

Based on 3-TXCO-3DA, MD:TVD, 22:22, 1332:1332, 2143.45:2133.6, 2334.98:2316.48, 2506:2473

Casing

9-5/8″ 47# set at 2506 m MD

Tubing

5-1/2″ 20# set at 2180 m MD

Downhole flow restrictions

None

Perforation size

0.5″

Additional near wellbore friction

None

Schedule type

Bottom hole

Stage type

Pad

Injection fluid type

KCL2

Injected fluid type

Water

Reservoir lithology type

Sandstone

In situ fluid

Water

Non-pay permeability

100 md

In situ fluid viscosity

0.7 cp

Irreducible water saturation

0

Deposited concentration ratio after transition

0.5

Permeability damage power

0.1

Cake porosity

0.25

Fractional deposition of TSS building cake

0.5

Cake build

1

Cake erosion

1

Table 3

Zone data

 

TVD Depths from Rig Floor, m

Perforations

Zone

Zone name

Top

Bottom

Top

Bottom

Top

2181.10

2187.02

2174.24

2193.24

Upper

2198.88

2204.83

2196.55

2209.63

Lower

2217.74

2223.72

2209.63

2231.36

Bottom

2237.71

2243.72

2231.36

2249.97

All

2181.10

2243.72

2175.19

2249.75

Table 4

Seawater versus PWRI case data

Variable

Seawater cases

PWRI cases

Injected fluid temperature, F

60

120

Injected fluid viscosity, cp

1.12

0.60

Internal filter cake permeability damage ratio

100

400

External filter cake permeability, mD

0.0100

0.0025

Table 5

Variable simulation data

Run

Case name

Ellipsoidal constant DP

Ellipsoidal pseudo-steady

Thermal stresses

Poroelastic stresses

Wellbore hydraulics model

Injection fluid temperature, F

Max time Step, Yrs

Number of fractures

Average injection rate, Mbbls/d

Injection time, yrs

Rock properties

Compressibility, 1/psi

Coefficient of thermal expansion, 1/R

Drainage area, acres

1

Base

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

2

TIMSTP005

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

3

TIMSTP002

 

 

60

0.2

1

25

20

RMA

RMA

3.5E−06

1200

4

TIMSTP001

 

 

60

0.1

1

25

20

RMA

RMA

3.5E−06

1200

5

TIMSTP0005

 

 

60

0.05

1

25

20

RMA

RMA

3.5E−06

1200

6

PERFTOP

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

7

PERFUPPER

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

8

PERFBOTTOM

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

9

PERFFOUR

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

10

PERFALL

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

11

PERFMOVE1

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

12

PERFMOVE2

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

13

PERFMOVE3

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

14

PERFMOVE4

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

15

PERFMOVE5

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

16

PERFMOVE6

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

17

PERFMOVE7

 

 

60

0.5

1

25

20

RMA

RMA

3.5E−06

1200

18

PWRI

 

 

120

0.5

1

25

20

RMA

RMA

3.5E−06

1200

19

PWRIALL

 

 

120

0.5

1

25

20

RMA

RMA

3.5E−06

1200

20

PWRIUNIFORM

 

 

120

0.5

1

25

20

See text

See text

3.5E−06

1200

21

UNIFORM

 

 

60

0.5

1

25

20

See text

See text

3.5E−06

1200

Run

Case name

Filtrate viscosity, cp

Total suspended solids, ppm

Filtration coefficient

Internal perm. damage factor

External cake permeability, md

Maximum cake thickness, in

Minimum cake thickness, in

Filter cake erosion ratio

Formation permeability, md

Perforated interval, m

1

Base

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

6

2

TIMSTP005

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

6

3

TIMSTP002

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

6

4

TIMSTP001

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

6

5

TIMSTP0005

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

6

6

PERFTOP

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

6

7

PERFUPPER

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

6

8

PERFBOTTOM

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

6

9

PERFFOUR

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

24

10

PERFALL

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

*

11

PERFMOVE1

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

3

12

PERFMOVE2

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

3

13

PERFMOVE3

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

3

14

PERFMOVE4

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

3

15

PERFMOVE5

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

3

16

PERFMOVE6

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

3

17

PERFMOVE7

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

3

18

PWRI

0.6

20

0.1

400

0.00,251

0.10

0.05

1E+01

1500

6

19

PWRIALL

0.6

20

0.1

400

0.00251

0.10

0.05

1E+01

1500

*

20

PWRIUNIFORM

0.6

20

0.1

400

0.00251

0.10

0.05

1E+01

1500

6

21

UNIFORM

1.12

20

0.1

100

0.01

0.10

0.05

1E+01

1500

6

Italics value show distinction from the class as filtrate viscosity is 0.6 while all others is 1.12, and the value of damage factor is 400, while others is 100

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

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

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

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.

Notes

Acknowledgements

The data were supplied by Department of Petroleum Resources (DPR) and Energy Technology Company in Houston, Texas, and this was well appreciated. Substantial data analysis was carried out by a simulation software supplied by Systems Engineering and Chemical and Petroleum Engineering faculty, and this was well appreciated as well. The authors express thanks to Division of Petroleum Regulator, Department of Petroleum Resources, and CNL/Energy Tech. Co. for access to data under their Local Technology Partnerships in PhD research work thesis and supported by the University of Lagos, Postgraduate School for granting publication of data for research and scholarly purpose.

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Authors and Affiliations

  • Kingsley E. Abhulimen
    • 1
  • S. Fashanu
    • 2
  • Peter Idialu
    • 2
  1. 1.Department of Chemical and Petroleum EngineeringUniversity of LagosLagosNigeria
  2. 2.Department of Systems EngineeringUniversity of LagosLagosNigeria

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