Introduction

Reinjection of produced water into spent hydrocarbon aquifer also referred to as produced water reinjection (PWRI) is one of the earliest and most environment friendly methods to dispose produced water from production platforms. However, reinjection of produced water degrades the aquifer that results in injectivity decline, fracturing of the internal walls of the aquifer and later formation damage, as shown in Fig. 1. Thus, PWRI has reduced performance over a period, because the method cannot be sustained throughout the production life of the reservoir.

Fig. 1
figure 1

Collapsed features where fracture will be more prevalent

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Previous studies and models described processes and mechanisms that resulted in formation damage and cake formation which were well developed and documented in technical literatures. PWRI in aquifers is generally studied under two research domains: (1) internal filtration and (2) external cake build up (Bedrikovetsky et al. 2001; Bedrikovetsky et al. 2007; Wennberg and Sharma 1997; Farajzadeh 2002; Al-Abduwani 2005). Significant research works and models were advanced and documented in several technical literatures to predict injectivity and characterize formation damage system and well behavior (Pang and Sharma 1994, 1997; Ojukwu and van den Hoek 2004; Guedes et al. 2006; Yerramilli et al. 2013).

Precious studies show that formation damage and injectivity decline are two major drawbacks associated with PWRI performance in hydrocarbon aquifer. Some past and recent studies were focused on understanding formation damage mechanisms (Salehi and Settari 2008; Prasad et al. 1999; Davidson 1979; Marchesin et al. 2011; Abou-Sayed et al. 2005; Zhang et al. 1993; Todd 1979; Ochi et al. 2007; Nabzar et al. 1997; De Zwart 2007; Faruk 2010; Lawal et al. 2011; Lawal and Vesovic 2010; Wang and Le 2008; Li et al. 2012).

There are other studies and models available in technical literature targeted to predict injectivity decline from particulate mechanics and flow transport. Notable contributions in this regard include work of Barkman and Davidson (1972), Pang and Sharma (1994, 1997) as well as Wennberg and Sharma (1997). In theory, efficiency and sustainability of the PWRI were progressed by considering injectivity decline as an outcome of momentum and particulate transport phenomena in porous media (Mendez 1999). There are other model and studies reported in technical literatures by previous researchers that focused on the filtration coefficient as the sole determinant of injectivity decline and fracturing (Abou-Sayed et al. 2007; Ajay and Sharman 2007, Al-Abduwani et al. 2001; Altoef et al. 2004; Chang 1985; Clifford et al. 1991; Donaldson et al. 1977; Doresa et al. 2012; Folarin et al. 2013; Gong et al. 2013; Hustedt et al. 2006). None of these models hinted on possible geochemical reaction of produced water heavy metals and aquifer water constituents and effect of geochemical reaction, the focus of this research study.

Nevertheless, recent findings (Idialu 2014) suggest a significant role of adsorption, geochemical reaction and molecular transport kinetics in well behavior, cake formation and damage characterization in PWRI modeling, and field data analysis. Therefore, this paper considers the effects of geochemical reaction, adsorption kinetics, and hydrodynamic molecular transport in formation damage and injectivity decline modeling and developments. Performance of PWRI water injectivity decline as a function of injection water quality, rates, and pressures was found to be significant factors in well injection design and formulation of secondary and tertiary recovery strategies. The effect of geochemical reaction in scale formation to injectivity decline was considered in the PWRI model analysis while outlining also the role of adsorption kinetics and molecular transport. The justification of this work inspired by the significant and active research interest over the last decade in the use of produced water as a resource in reinjection as alternative secondary and tertiary recovery method could achieve the goals of the zero tolerance by regulatory authority to water disposal management to maintain marine life and environment sustainability.

Model development

The aquifer grid for produced water system and geometry of the PWRI schemes in well-reservoir formation, effects, and problems encountered were illustrated in Figs. 2 and 3, respectively. The implications arising from PWRI management are: (1) injectivity loss; (2) permeability loss; (3) loss of recovery; (4) loss in reservoir potential; (5) poor reservoir sweep (bypass oil and early water breakthrough); (6) excessive chemical treatment; and (7) discharge not meeting environmental regulations.

Fig. 2
figure 2

Generic aquifer grid system for produced water re-injection system. Cin Concentration of active constituents in produced water in Reservoir-Aquifer Control volume grid, Cout Concentration of active constituents in produced water out Reservoir-Aquifer Control volume grid. U x in, U y in, U z in is the velocity of produced water in Cartesian coordinates x, y, z in Reservoir-Aquifer Control volume grid. U xout , U yout , U zout is the velocity of produced water in Cartesian coordinates x, y, z out Reservoir-Aquifer Control volume grid

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Fig. 3
figure 3

Geometry of PWRI in well-reservoir formation

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The generalized improved PWRI model incorporated molecular transport, geochemical, and adsorption kinetics in Eq. 1 with boundary conditions presented in Eqs. 2, 3, and 4:

$$ \frac{\partial \phi (t)C}{\partial t} + \nabla \left( {C\upsilon } \right) - \nabla^{2} \left( {DC} \right) = - \left( {\sigma_{\text{n}} + R_{\text{d}} } \right) + R_{\text{n}} $$
(1)
$$ \begin{aligned} C = & C\left( {x, z, t} \right); \;0 \le x \le L; \;0 \le z \le Z; \;t \ge 0 \\ C(x,z,0) = & C_{0} \\ \end{aligned} $$
(2)
$$ C\left( {L, z, t} \right) = 0 $$
(3)
$$ \left[ {\frac{\partial C}{\partial r}} \right]_{(r = R,Z,t)} = 0\left[ {\frac{\partial C}{\partial z}} \right]_{(x,Z = Z,t)} = 0 $$
(4)

where C is the concentration of produced water active constituents, C 0 is the initial concentration of the active constituents, v is the produced water transport velocity in the geologic formation, ϕ(t) is the variable porosity, D is the molecular diffusivity, and t is time.

The significant control variables in improved PWRI model are as follows:

σ n deposition filtration term, R n geochemical reaction term, R d adsorption kinetics term, DC molecular transport term.

In this work, the geochemical reaction rate mechanism was described to follow second-order kinetics summarized as Eqs. 5 and 7 as follows.

At time t = 0

$$ \begin{aligned} C_{\text{Ao}} \quad C_{\text{Bo}} \quad \mathop{\longrightarrow}\limits{X}\quad C_{\text{co}} \hfill \\ C_{\text{Ao}} - X_{\text{r}} C_{\text{Ao}} \quad C_{\text{Bo}} - X_{r} C_{\text{Bo}} \quad C_{\text{co}} + X_{\text{r}} C_{\text{Co}} . \hfill \\ \end{aligned} $$
(5)

At time t = t

$$ C_{\text{A}} \quad C_{\text{B}} \quad C_{\text{C}} . $$
(6)

1 mol of Component A reacted with 1 mol of Component B to produce 1 mol of scale products:

$$ R_{\text{n}} = \frac{1}{{V_{\text{fm}} }}\frac{\partial C}{\partial t} = K_{\text{o}} \left( {\frac{{C_{\text{T}} }}{{1 + \sum {K_{i} C_{i} } }}} \right)\left( {C_{\text{A}} C_{\text{B}} - K_{1} C} \right) $$
(7)
$$ R_{\text{n}} = \frac{{\partial R_{\text{rn}} }}{\partial t} = K_{o} \left( {\frac{{C_{\text{T}} }}{{1 + \sum {K_{i} C_{i} } }}} \right)\left( {C_{\text{A}} C_{\text{B}} - K_{1} C} \right). $$
(8)

These important contributions in the improved model were used to standardize the general performance of produced water reinjection in hydrocarbon aquifers, with geochemical reaction, adsorption kinetics, and hydrodynamic dispersion transport that highlighted as the key performance indicators of the improved model, as illustrated in subsequent sections (see Fig. 4).

Fig. 4
figure 4

Micro pore particle retention kinetics

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Invasion Zone Front 1

To account for adsorption kinetics (Rd) in internal filtration modeling, three linear adsorption isotherms which are Linear; Langmuir, and Freundlich isotherms were considered. Single particle (suspended) linear adsorption is shown in

$$ C_{\text{s}} = K_{\text{a}} C . $$
(9)

To determine the active mass transfer coefficient (K a), the Arrhenius equation is introduced as follows:

$$ K_{\text{a}} = K_{\text{o}} \,e^{{ - \frac{\Delta H}{RT}}} . $$
(10)

The R d macroscale adsorption particle retention kinetics (adsorption) on the surface of grain were computed and presented as Eq. 9:

$$ \frac{{\partial C_{i} }}{\partial t} + \left( {v_{z} \frac{{\partial C_{i} }}{\partial z} + \frac{{v_{\theta } }}{r}\frac{{\partial C_{i} }}{\partial \theta } + \frac{{v_{r} }}{r}\frac{{\partial rC_{i} }}{\partial r}} \right) - \left( {D_{z} \frac{{\partial^{2} C_{i} }}{{\partial z^{2} }} + D_{r} \left( {\frac{{\partial^{2} C_{i} }}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial C{}_{i}}}{\partial r}} \right)} \right) = R_{\text{d}} . $$
(11)

With R d is given by Eq. 12:

$$ R_{d} = E_{\text{mp}} \left( {\frac{{1 - \varphi_{\text{mp}} }}{{\varphi_{\text{mp}} }}} \right)\left( {\frac{{3K_{g} }}{R}} \right)\left( {C_{i} - \frac{{C_{T} K_{i} C_{i} }}{{1 + \sum\nolimits_{1 = n}^{n} {K_{i} C_{i} } }}} \right) $$
(12)

where E mp is defined as trapping efficiency factor. For particulate transport systems at the PW invasion zone when t = 0 and 0 < t < t i , where t i is the residence time of the particle in invasion zone Li. Previous studies illustrated that the suspended particles adsorbed have different dynamics in each invasion zone in the aquifer, thus the transport equation becomes

$$ \begin{aligned} \left( {\frac{{\partial C_{i} }}{\partial t}} \right)_{{L_{i} }} + \left( {E_{\text{mp}} \left( {\frac{{1 - \varphi_{\text{mp}} }}{{\varphi_{\text{mp}} }}} \right)} \right)_{{L_{i} }} \left( {\frac{{\hat{k}k_{g} }}{R}\left( {C_{\text{T}} - \frac{{C_{\text{T}} k_{i} c_{i} }}{{1 + \sum {k_{i} c_{i} } }}} \right)} \right)_{{L_{i} }} = & \left( { - V_{\text{Z}} \frac{{\partial c_{i} }}{\partial z}} \right)_{{L_{i} }} + \left( { - \frac{{V{}_{\theta }}}{r}\frac{{\partial c_{i} }}{\partial \theta }} \right)_{{L_{i} }} + \left( { - \frac{{V_{r} }}{r}\left( {\frac{\partial }{\partial r}\left( {rc_{i} } \right)} \right)} \right)_{{L_{i} }} \\ & + \left( {D_{z} \frac{{\partial^{2} c_{i} }}{{\partial z^{2} }}} \right)_{{L_{i} }} + \left( {D_{r} \left( {\frac{{\partial^{2} c_{i} }}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial c_{i} }}{\partial r}} \right)} \right)_{{L_{i} }} \\ & 0 < L_{i} < L_{{T_{i} }} . \\ \end{aligned} $$
(13)

The improved PWRI model in generalized dimensionless form accounts for residual oil mobility S or and permeability k or, porous media particle retention adsorption factor in Eq. 12:

$$ \left( {\frac{\partial \zeta }{{\partial \tau^{ * } }} - \left( {\frac{\partial S}{{\partial \tau^{ * } }} + \left( {\frac{{q_{\text{o}} }}{{c_{\text{o}} }}} \right)\frac{{\partial \varPsi_{\text{D}} }}{{\partial \tau^{ * } }}} \right) + \left( {\frac{{R_{\text{o}} }}{{c_{\text{o}} }}} \right)\frac{{\partial \varPsi_{\text{r}} }}{{\partial \tau^{ * } }}} \right) + \alpha_{1} \left( \upsilon \right)\left( {\frac{\partial \zeta }{{\partial \varepsilon_{\text{r}} }}} \right) + \alpha_{2} \left( \upsilon \right)\frac{\zeta }{{\varepsilon_{\text{r}} }} + \alpha_{3} \left( \upsilon \right)\frac{\partial \zeta }{{\partial \varepsilon_{\text{z}} }} = \alpha_{4} \left( \upsilon \right)\frac{{\partial^{2} \zeta }}{{\partial r^{2} }} + \alpha_{5} \left( \upsilon \right)\left( {\frac{\partial \zeta }{{\partial \varepsilon_{\text{r}} }}} \right) + \alpha_{6} \left( \upsilon \right)\left( {\frac{{\partial^{2} \zeta }}{{\partial \varepsilon_{{{\bar{\text{z}}}}}^{2} }}} \right). $$
(14)

Iwaski (1937) proposed the filtration model in Eq. 15 which represented the rate of particle trapping:

$$ \frac{\partial \sigma }{\partial t} = \lambda \upsilon C $$
(15)

where υ is the superficial velocity, λ is defined as the filtration coefficient, a function of a large number of parameters, and C is the fraction of suspended particles per unit volume of suspension.

The rate of deposition was proportional to the concentration of suspended particles and fluid velocity, see Eq. 14:

$$ \frac{\partial \sigma }{\partial t} = \lambda \parallel U\parallel C_{s}. $$
(16)

The filtration coefficient was computed by the relation Eq. 17:

$$ \lambda = \frac{3}{2}\left( {\frac{1 - \phi }{{d_{s} }}} \right)\alpha_{c} n $$
(17)

where ∝ c represents the total collision probability of the bed efficiency and n is the collision probability of the filtration mechanisms.

The Filtration Coefficient Numerical Model was computed as Eq. 16:

$$ \left[ \begin{aligned} S_{i1} \hfill \\ S_{i2} \hfill \\ S_{i3} \hfill \\ \vdots \hfill \\ \vdots \hfill \\ S_{in} \hfill \\ \end{aligned} \right]_{K + 1} = \left[ \begin{aligned} S_{i1} \hfill \\ S_{i2} \hfill \\ S_{i3} \hfill \\ \vdots \hfill \\ \vdots \hfill \\ S_{in} \hfill \\ \end{aligned} \right]_{K} + \;\alpha_{0} \left[ \begin{aligned} \in_{i1} \hfill \\ \in_{i2} \hfill \\ \in_{i3} \hfill \\ \vdots \hfill \\ \vdots \hfill \\ \in_{in} \hfill \\ \end{aligned} \right]_{K} . $$
(18)

We assumed the following in the development of improved PWRI model:

  1. (a)

    The displacing fluid (water) and the deposited solids were considered incompressible.

  2. (b)

    The densities of the solid particles were considered equal in both dispersed and deposited states.

  3. (c)

    The linear velocity in ν r , ν z , and ν φ along the core is constant. In addition, we assumed a constant velocity with time. Therefore, the conservation of the total flux is dων = 0.

  4. (d)

    The kinetics of the particles was considered linear.

  5. (e)

    Dependency of the viscosity and concentration was considered negligible.

  6. (f)

    Thermal and shear stresses were considered negligible.

The injectivity index was computed as flow rate per unit of the pressure drop between the injector and the reservoir and computed, see formula as shown in

$$ \prod = \frac{q(t)}{\Delta p(t)}. $$
(17)

The impedance was computed as the inverse of the dimensionless injectivity index:

$$ J(T) = \frac{\prod \left( 0 \right)}{\prod (t)} = \frac{{q_{o} \Delta p(T)}}{\Delta p\left( 0 \right)q\left( T \right)}. $$
(19)

The impedance was computed as piecewise linear function of the dimensionless time for either deep bed filtration or external cake formation (Pang and Sharma 1994, 1997; Prasad et al. 1999):

$$ J_{\text{d}} \left( T \right) = 1 + mT\quad {\text{For}}\quad T < T_{\text{r}} $$
(20)
$$ J_{\text{d}} \left( T \right) = 1 + mT_{\text{r}} + m_{\text{c}} \left( {T - T_{\text{r}} } \right)\quad {\text{For}}\quad T > T_{\text{r}} . $$
(21)

The nucleation or transition time T r was represented as Eq. 22:

$$ T_{\text{r}} > \frac{{2\alpha \,r_{\text{w}} }}{{\lambda C_{\text{o}} R_{\text{C}}^{2} }}. $$
(22)

The impedance slope m during the deep filtration was computed by the formula of Eq. 22:

$$ m_{\text{c}} = \frac{{kk_{\text{rowr}} \varphi c_{o} }}{{k_{\text{c}} \left( {1 - \varphi_{\text{c}} } \right)X_{\text{w}} \left( { - {\text{In}}X_{\text{w}} } \right)}} $$
(23)
$$ m = \left( {\frac{{\beta \varphi c_{\text{o}} }}{{{\text{In}}X_{\text{w}} }}} \right)\left( {\lambda R_{\text{c}} } \right)\left( {\frac{1}{{\sqrt {X_{\text{w}} } }}} \right)\left( { - \exp ( - \lambda \left( {R_{\text{C}} - r_{\text{w}} } \right)} \right) - \lambda R_{C} \exp \left( {\lambda r_{\text{w}} } \right)\int\limits_{{\lambda r_{\text{w}} }}^{{\lambda R_{\text{c}} }} {\frac{{\exp \left( { - u} \right)}}{u}} {\text{d}}u $$
(24)

where

$$ u = \lambda R_{\text{c}} \sqrt X $$
(25)
$$ X = \ell^{2} = \left( {\frac{r}{{R_{\text{c}} }}} \right)^{2} $$
(26)
$$ X_{\text{w}} = \ell^{2} = \left( {\frac{{r_{\text{w}} }}{{R_{\text{c}} }}} \right)^{2} . $$
(27)

m c represent the slope of the external cake formation.

The damage section of the aquifer formation was computed as a ratio of differential in injection pressure over injection rate presented in Eq. 28:

$$ \frac{{\Delta P_{ \in } }}{q} = \frac{{\mu {\text{In}}\left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right)}}{{2\pi K_{\text{or}} K_{\sigma } }}. $$
(28)

The undamaged section was computed as Eq. 29:

$$ \frac{\Delta P}{q} = \frac{{\mu {\text{In}}\left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right)}}{{2\pi K_{\text{or}} }} $$
(29)
$$ {\text{Total}}\;{\text{Injectivity}}\;{\text{Pressure}}\;{\text{Gradient}} = \frac{\Delta P}{q} + \frac{{\Delta P_{ \in } }}{q} = (1 + K_{\sigma } )\frac{{\mu {\text{In}}\left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right)}}{{2\varPi hK_{\text{or}} }} $$
(30)

where

$$ \frac{{\Delta P_{\infty } }}{q} = \frac{{\mu {\text{In}}\left( {\frac{{r_{\text{c}} }}{{r_{\text{ec}} }}} \right)}}{{2\pi K_{\text{or}} }} $$
(31)
$$ {\text{Total Impedance}} = {\text{Damage Impedance }} + {\text{ Undamaged Impedance}} . $$
(32)

Figure 5 shows the damage section which represents area that has been affected by cake deposits, whereas undamaged section is unaffected by solids deposition.

Fig. 5
figure 5

Damage and undamaged section of a reservoir

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The total impedance was computed by

$$ \frac{\Delta P}{q} = \frac{\mu }{{2\pi K_{\text{or}} }}\left( {{\text{In}}\left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right) + K^{\prime}_{\sigma } } \right). $$
(33)

The dimensionless form of total impedance index was computed as Eq. 33 as follows:

$$ \frac{{\left( {\frac{\Delta P}{q}} \right)_{T} }}{{\left( {\frac{{\Delta P_{\text{O}} }}{{q_{\text{O}} }}} \right)}} = \frac{{\left( {\left( {{\text{In}}\left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right) + K^{\prime}_{\sigma } } \right)} \right)_{\text{T}} }}{{\left( {\left( {{\text{In}}\left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right)_{\sigma } } \right)} \right)_{\text{T}} }} $$
(34)

where

$$ j = 1 + K_{\sigma }^{'} \left( {\frac{1}{{{\text{In}}\left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right)}}} \right). $$
(35)

The injectivity index was computed as the flow rate per unit of the pressure drop between the injector and the reservoir (Eq. 35):

$$ \prod = \frac{q(t)}{\Delta p(t)}. $$
(36)

Based on preliminary field data obtained from a field operator and regulator in Nigeria, the model was solved using finite-element method and the injectivity and solid deposition simulated in COMSOL environment. Details of the finite-element method and COMSOL software algorithms applied to solve the mode were presented in subsequent sections.

Field data, numerical development, and computer simulation

The improved PWRI model was solved by finite-element method and injectivity and permeability damage simulated in the COMSOL Multiphysics software environment using the field data obtained from regulator for the Onshore Field in Nigeria. In the numerical model, a six-order six-point implicit differencing scheme was used and resulting numerical solution of the governing equations of the PWRI concentration field was solved by the Triadiagonal Matrix Algorithm (TDMA) method. The implicit finite scheme was then applied to the PWRI Model of Eq. 5 to give Eq. 36:

$$ \alpha^{\prime}_{1} \zeta_{ijk + 1} + \alpha^{\prime}_{2} \zeta_{i + 1jk + 1} + \alpha^{\prime}_{3} \zeta_{ij + 1k + 1} + \alpha^{\prime}_{4} \zeta_{i - 1jk + 1} + \alpha^{\prime}_{5} \zeta_{ij - 1,k + 1} = \alpha^{\prime}_{6} \zeta_{ijk} + \hat{a}_{r} \Delta q_{rijk} - \hat{a}_{d} \Delta q_{dijk} $$
(37)

where

the adsorption term in Eq. (37) was specified by Eq. (38):

$$ \Delta q_{dijk} = \left( {\frac{{a_{1} }}{{\hat{a}_{\text{d}} }}} \right)\zeta_{ijk + 1} - \left( {1 - \frac{{\hat{a}_{2} }}{{\hat{a}_{\text{d}} }}} \right)\zeta_{ijk} + \left( {1 - \frac{{\hat{a}_{2} }}{{\hat{a}_{\text{d}} }}} \right)\left( {C_{\text{T}} \frac{{k_{1} }}{{1 + k_{1} c_{\text{o}} \zeta_{ijk} }}} \right)\zeta_{ijk} . $$
(38)

The other terms in Eq. (36) was defined in line with the reservoir:

$$ \alpha^{\prime}_{1} = 1 - \Delta \tau^{ * } \left( {\frac{{\alpha_{1} }}{{\Delta \in_{{\bar{r}}} }} + \frac{{\alpha_{3} }}{{\Delta \in_{{\bar{z}}} }} - \frac{{2\alpha_{4} }}{{\left( {\Delta \in_{{\bar{r}}} } \right)^{2} }} - \frac{{2\alpha_{6} }}{{\left( {\Delta \in_{{\bar{z}}} } \right)^{2} }}} \right) $$
(39)
$$ \alpha_{2}^{\prime } = \left( {\frac{{\alpha_{1} \Delta \tau^{*} }}{{\Delta \in_{{\bar{r}}} }} - \frac{{\alpha_{4} \Delta \tau^{*} }}{{\left( {\Delta \in_{r} } \right)^{2} }} + \frac{{\alpha_{5} \Delta \tau^{*} }}{{\Delta \in_{r} }}} \right) $$
(40)
$$ \alpha^{\prime}_{3} = \left( {\frac{{\alpha_{3} \Delta \tau {}^{ * }}}{{\Delta \in_{z} }} - \frac{{\alpha_{6} \Delta \tau^{ * } }}{{\left( {\Delta \in_{z} } \right)^{2} }}} \right) $$
(41)
$$ \alpha^{\prime}_{4} = \left( {\frac{{\alpha_{4} \Delta \tau^{ * } }}{{\left( {\Delta \in_{r} } \right)^{2} }}} \right) $$
(42)
$$ \alpha^{\prime}_{5} = \left( {\frac{{\alpha_{6} \Delta \tau^{ * } }}{{\left( {\Delta \in_{z} } \right)^{2} }}} \right) $$
(43)
$$ \alpha^{\prime}_{6} = 1 - \Delta \tau^{ * } \left( {\alpha_{\text{o}} + \alpha_{2} } \right) $$
(44)

where

$$ \alpha_{1} \left( \upsilon \right) = \left( {\frac{{\upsilon_{\text{r}} }}{\upsilon }} \right)\left( {\frac{L}{{R_{\text{o}} }}} \right) $$
(45)
$$ \alpha_{2} \left( \upsilon \right) = \left( {\frac{{\upsilon_{r} }}{\upsilon }} \right)\left( {\frac{L}{{R_{o} }}} \right)\left( {\frac{1}{{\varepsilon_{r} }}} \right) $$
(46)
$$ \alpha_{3} \left( \upsilon \right) = \frac{{\upsilon_{\text{z}} }}{\upsilon } $$
(47)
$$ \alpha_{4} \left( \upsilon \right) = \left( {\frac{{D_{\text{er}} }}{\upsilon }} \right)\left( {\frac{L}{{R_{o}^{2} }}} \right) $$
(48)
$$ \alpha_{5} (\upsilon ) = \left( {\frac{{D_{\text{er}} }}{\upsilon }} \right)\left( {\frac{L}{{R_{0}^{2} }}} \right)\left( {\frac{1}{{\varepsilon_{\text{r}} }}} \right) $$
(49)
$$ \alpha_{6} \left( \upsilon \right) = \left( {\frac{{D_{ez} }}{\upsilon }} \right)\left( {\frac{1}{L}} \right). $$
(50)

Computation of Velocity in r and z direction:

$$ \upsilon_{r} = \frac{{q_{r} }}{2\pi rh} = \left( {\frac{{K_{\text{or}} K_{{\sigma_{r} }} }}{\mu }} \right)\int\limits_{{r_{w} }}^{{r_{e1} }} {\frac{1}{r}\frac{{{\text{d}}rP}}{{{\text{d}}r}}} + \int\limits_{{r_{e1} }}^{{r_{e} }} {K_{or} } \frac{1}{r}\frac{{{\text{d}}rP}}{{{\text{d}}r}} $$
(51)
$$ \upsilon_{z} = \frac{{q_{z} }}{{\pi r^{2} h}} = \left( {\frac{{K_{oz} K_{{\sigma_{z} }} }}{\mu }} \right)\int\limits_{{r_{w} }}^{{r_{e1} }} {\frac{{{\text{d}}P}}{{{\text{d}}z}}} + \int\limits_{{r_{e1} }}^{{r_{e} }} {K_{oz} } \frac{{{\text{d}}P}}{{{\text{d}}z}} $$
(52)
$$ \int\limits_{{r_{\text{w}} }}^{{r_{\text{e}} }} {\left( {\frac{q}{2\pi h}} \right)\frac{{{\text{d}}r}}{r}} = \left( {\frac{{K_{\text{or}} \left( {1 + K_{\sigma } } \right)}}{\mu }} \right)\Delta P $$
(53)
$$ \frac{\Delta P}{q} = \frac{{\mu {\text{In}}\left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right)}}{{2\pi K_{\text{or}} }} + \frac{{\mu {\text{In}}\left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right)}}{{2\pi K_{\text{or}} K_{\sigma } }} $$
(54)
$$ \frac{\Delta P}{q} = \frac{1}{{2\pi K_{or} }}\left( {1 + \frac{1}{{K_{\sigma } }}} \right). $$

The flow chart in Fig. 4 described the simulation algorithm using the COMSOL Multiphysics software.

Field water compatibility studies for fields in Gulf of Guinea, Nigeria field in Niger Delta

The data of PWRI case studies for Gulf of Guinea (Niger Delta, Nigeria) were provided in operator’s report planned for water flood for secondary enhanced recovery in the Niger Delta region of Nigeria. Five PWRI runs were assessed for this study. The limiting factors for injection rates were friction and pumping capabilities. Table 1 showed the data of field study conducted for PWRI programme in a field in Gulf of Mexico.

Table 1 Re-injection parameters
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From these studies, friction contributed significant part in PWRI performance which varied significantly depending on rate and tubing size. In addition, a number of other estimates were determined over a range of the variables of Young’s Modulus, Poisson’s Ratio, injection water temperature, and the difference in pressure between the reservoir pressure and flowing bottom hole pressure that were ran (Fig. 6).

Fig. 6
figure 6figure 6

Flow chart algorithm of simulation program for the PWRI model

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Impact of water quality on matrix injection

The WID (Water Injectivity Decline) simulator results presented in Figs. 7 and 8 outlined the significance of water quality on injection rates and pressure. The simulator was developed at the University of Texas. The output shows injectivity vs. time. The injectivity is dimensionless permeability, and the half-life is the amount of time, in days, at which the dimensionless permeability drops to half the original value. The output based on their results shown in Fig. 7 is for matrix injection with very good water quality (1 ppm of 1 micron sized particles). The results show futility of trying to inject below the fracture gradient since injection rate declines in the matter of a few days to a fraction of the original value. The permeability profile shows that the damage is shallow, even with good perm (200 md).

Fig. 7
figure 7

*source: Energy Tech Co, Houston, Texas, USA and petroleum regulator, Department of Petroleum Resources (DPR), as reported by (Idialu 2014)

1 ppm, 1 micron, Injectivity and perm profiles

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Fig. 8
figure 8

*source: Energy Tech Co, Houston, Texas, USA and petroleum regulator, Department of Petroleum Resources (DPR), as reported by (Idialu 2014)

60 md, 100 ft fracture, 5 ppm, 5 micron

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The field data runs reviewed showed that water quality has a significant impact on the half-life. With 5 ppm of 5 micron solids, the half-life is about a year for a 100′ fracture. This meant that the fracture will continue to grow at about this rate every year, assuming that it is confined to a single zone (Tables 2, 3).

Table 2 Sensitivity summary data
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Table 3 WID run summary
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Results and discussions

The discussions of results of findings are presented as follows:

Injectivity profile with time

In this section, the results of injectivity with time are presented for two different studies obtained for two different fields, with the actual field data run compared with simulated data obtained from this work. Figure 9 results show plot of injectivity with time for a typical PWRI data obtained for a field in the Gulf of Mexico (Texas Fields) and were supplied by the field operator, while Fig. 10 shows the simulated injectivity based on data obtained from operator Nigerian Oil Field. While the numerical values of the simulated and field study may differ, injectivity profile trends obtained in Fig. 9 compared favorably with injectivity obtained in Fig. 10 simulated on the COMSOL Multiphysics platform thereby validating the improved PWRI model. The plots establish that injectivity decline was spatially away from the produced water invasion zone in the host aquifer to settle at a threshold value. The transition time to cake formation for actual field run was 50 days while for our simulated run was observed to occur within 5 days. The blue line shown in Fig. 9 shows the injectivity decline for the field studies, while the grey line in Fig. 10 shows the injectivity decline profile simulated in COMSOL metaphysics environment where a correlation in trend was observed. The green line in Fig. 9 shows a steep change in injectivity near the well bore showing effect of geometry with respect to injection fracture with injection decline steepest at the well bore than further away.

Fig. 9
figure 9

Field studies, injectivity with time

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Fig. 10
figure 10

Injectivity profile with time (days)

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Effect of flow rate on injectivity with time

In this section, we show how higher injection rate and sweep volume could impact on injectivity decline and cake deposition. Figures 11 and 12 show the plots of the simulated injectivity decline against PWRI rates and observed to be inversely proportional to each other. From plots increased injection rate led to decreased injectivity decline leading to sustained impairment and transition to cake formation which was minimized to a constant residual value. The sweep volume erodes deposition and adsorption on walls of aquifer, because drag force was observed to have the effect of reducing solids deposition to a constant injection rate. Previous studies showed that friction contributed significant role in PWRI performance and varied significantly depending on rate and tubing size. Significant reduction in injectivity decline and fracturing could be attributed to lager drag force resulting from increased rates. These results were validated by Gulf of Mexico study presented in Table 1 where similar observations of plots of actual and simulated results trends were correlated. In addition, a number of other estimates over a range of the variables of Young’s Modulus, Poisson’s Ratio, injection water temperature, and the difference in pressure between the reservoir pressure and flowing bottom hole pressure were ran. The results in Fig. 11 showed futility of attempting to inject below the fracture gradient since injection rate declines in the matter of a few days to a fraction of the original value transient was followed by a steady state of constant injectivity beyond which decline remains constant after 15 days, transition time to cake formation irrespective of the injection rates. Injectivity decline increased as flow rate decreases and vice versa. Formation around the fracture is impaired by deep penetration of solids, (ii) an external filter cake is built on the fracture wall by oil and solids that remain in the fracture and (iii) filter cake growth eventually leads to plugging of the fracture. The injectivity decline was dependent on injection rate impact on produced water invasion zone flooding volume in aquifer formation with a lower sweep volume leading to higher injectivity decline and in increase of the produced water sweep volume rate leads to higher injectivity performance and, therefore, higher recovery.

Fig. 11
figure 11

Effect of flow rate on injectivity

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Fig. 12
figure 12

Injectivity against time and flow rate. \( \left( {\frac{{{\text{d}}p}}{{{\text{d}}g}}} \right) \) is the particle to grain ratio

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Effect of particle size and formation damage on injectivity

In this section, the effects of particle size on injectivity with time were studied and demonstrated. Figure 13 showed decrease in injectivity with time as the particle size decreases. The particle to grain size \( \left( {\frac{{{\text{d}}p}}{{{\text{d}}g}}} \right) \) of 0.6164 showed lower injectivity decline than a particle to grain size \( \left( {\frac{{{\text{d}}p}}{{{\text{d}}g}}} \right) \) of 0.2740. The smaller particles were able to penetrate the pores faster than larger grain particles in suspended solids thereby increasing chances for internal cake formation and external cake build up. The plots showed that injectivity index decreased from 1.135 to 1.1 in 30 days. The impact of particle to grain size is a function of adsorption capacity of particles on aquifer wall to form cake deposits significant in altering injectivity and formation damage alongside quality of constituents, injection rate of produced water which were established in the previous section.

Fig. 13
figure 13

Effect of particle size on injectivity with time. dp/dg is the particle to grain ratio

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Figure 14 outlined the variation of injectivity decline with velocity damage factor. The plots showed decreased injectivity as damage factor increased irrespective of time the produced water is transported in the reservoir. The damage factor is a numerical index of the reduction in permeability resulted from formation damage due to scaling. However, the extent of decline of injectivity with damage factor was observed to invariant with time. The injectivity was same irrespective of the time for any specific damage factor.

Fig. 14
figure 14

Variation of injectivity decline with damage factor

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Profile of permeability damage with distance

Figure 15 showed the profile of permeability on both fracturing and filtration phenomena. 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 were 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. However, from results, formation permeability was directly dependent on the formation grain size (dg). A comparison of the profile in Fig. 16 and the permeability of field data of Fig. 15 showed a good agreement for damage permeability, with a little allowance for lithological variation and other factors that may partially contribute to injectivity variation.

Fig. 15
figure 15

Field simulation data of profile of permeability with depth at different zones

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Fig. 16
figure 16

Profile of permeability damage with distance

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Effect of temperature variation on injectivity with time

Figure 17 outlines the significance of temperature variation as a key role in adsorption rate in the Arrhenius equation and subsequently injectivity decline. Higher temperature favors greater retention rates outlining the importance of adsorption rate in particle deposition. The fracture gradient was influenced as temperature changes which led to less injectivity as temperature increases. As cooler injection fluids reduce temperature, the rock becomes more brittle and this effect is strongly dependent on Young’s Modulus of elasticity. The profile is an exponential decrease in injectivity with time as temperature decreases.

Fig. 17
figure 17

Effect of temperature variation on injectivity with time

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Concentration variation with depth for first 5 days

Figure 18 showed the concentration of suspended and deposited solids decreases exponentially with depth. At an assumed depth 100 m, the concentration decrease reaches a minimum after which concentration remains constant. The effect of geochemical reaction scaling is apparent as concentration solids deposited was observed to be less than concentration in suspension.

Fig. 18
figure 18

Variation of concentration (suspended and deposited solids) with time for the first 5 days, of injection for calcite geochemical reaction index \( {\text{SI}} = 1.48 \)

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Figure 19 outlines simulation of injectivity performance with time for the reservoir temperature temp = 164 °F and Flow Line temp = 125 °F for calcite geochemical reaction which has a scaling index \( {\text{SI}} = 1.48 \). The significant results reveal Injectivity decline is exponential in time. For a water injection rate of 5000 bbls/day, injectivity decline is a maximum on the first day and remain constant for the remaining days as it progresses. The simulation results show potential calcite scaling of SI = 1.48 > 1 induces a faster time to injectivity decline.

Fig. 19
figure 19

Profile of injectivity with time for reservoir temp = 164 °F and flow line temp = 125 °F for calcite geochemical reaction index \( {\text{SI}} = 1.48 \)

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Figure 20 is the profile of concentration of suspended and deposited solids with radial distance on the 5th day for injection at 5000 bbl/day (TVD 44.2 m and time 1 day) for reservoir temp = 164 °F and flow line temp = 125 °F for Calcite Scaling Index \( {\text{SI}} = 1.48 \). Concentration decreases for deposited and suspended solids after 50–100 m may be the result of increased deposition precipitated due to increased geochemical scaling.

Fig. 20
figure 20

Profile of Concentration (suspended and deposited) with radial distance at TVD 44.2 m and at 5 days Reservoir Temp = 164 °F and Flow Line Temp = 125 °F for Calcite Geochemical Rxn Index \( {\text{SI}} = 1.48 \)

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Conclusion

An improved PWRI model incorporating the effect of geochemical reaction, adsorption kinetics and hydrodynamics molecular transport was presented to predict performance of produced water reinjection Schemes in hydrocarbon aquifer. The model was solved using a finite-element method with the injectivity and solids deposition simulated in COMSOL Multiphysics Software. 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. The injectivity decline was found to be a function of cake deposition resulting from geochemical reaction, adsorption kinetics coupled filtration scheme and molecular diffusion. In conclusion, we established that the transition time t r to cake nucleation and growth was a consequence aquifer capacity, filtration coefficients particle and grain size diameters and, more importantly, adsorption kinetics, geochemical reaction and produced water quality.