Injection-Induced Poroelastic Response of Porous Media Containing Fine Particles, Incorporating Particle Mobilization, Transport, and Straining

Abstract

Fine-migration-induced permeability damage in geological reservoirs is a critical problem in several environmental and energy operations. This study presents a novel numerical model aimed at predicting the hydro-mechanical response of a saturated porous formation containing fines to injection-induced fluid flow. The proposed model contains two novel components: incorporation of spatiotemporal fluid flow in the porous formation, and integration of the coupled interactions between the poroelastic response of the saturated porous formation and fine mobilization, migration, and straining. The modified particle detachment model has been adopted, using the maximum retention function for a monolayer of size-distributed fines. The results from the proposed model reveal that incorporation of the fines migration, mobilization, and straining has a substantial impact on in situ pore pressures (~ 65% higher pore pressures were predicted at wellbore vicinity after 1 h injection when incorporating particle straining). The findings also depict a lower fine-migration-induced permeability damage when incorporating the poroelastic response of the porous layer to injection flow.

Article Highlights

• Coupled interactions between geomechanical response of porous media and migration of in situ fines under injection.

• Higher injection-induced pore pressures when incorporating particle mobilization and straining.

• Reduction in fine-migration-induced permeability damage due to poroelastic response of injection layer.

Appendix

The shape functions $N_{i}$ ($i = 1,2 \ldots 8$) for 8-node rectangular quadrilateral elements are

$$\left\{ {\begin{array}{*{20}l} {N_{1}^{8} = \frac{1}{4}\left( {1 - \xi } \right)\left( {1 - \eta } \right)\left( { - \xi - \eta - 1} \right)} \hfill \\ {N_{2}^{8} = \frac{1}{2}\left( {1 - \xi } \right)\left( {1 - \eta^{2} } \right)} \hfill \\ {N_{3}^{8} = \frac{1}{4}\left( {1 - \xi } \right)\left( {1 + \eta } \right)\left( { - \xi + \eta - 1} \right)} \hfill \\ {N_{4}^{8} = \frac{1}{2}\left( {1 - \xi^{2} } \right)\left( {1 + \eta } \right)} \hfill \\ {N_{5}^{8} = \frac{1}{4}\left( {1 + \xi } \right)\left( {1 + \eta } \right)\left( {\xi + \eta - 1} \right)} \hfill \\ {N_{6}^{8} = \frac{1}{2}\left( {1 + \xi } \right)\left( {1 - \eta^{2} } \right)} \hfill \\ {N_{7}^{8} = \frac{1}{4}\left( {1 + \xi } \right)\left( {1 - \eta } \right)\left( {\xi - \eta - 1} \right)} \hfill \\ {N_{8}^{8} = \frac{1}{2}\left( {1 - \xi^{2} } \right)\left( {1 - \eta } \right)} \hfill \\ \end{array} } \right.$$

(34)

where $\xi$ and $\eta$ are dimensions in the local coordinates.

The shape functions $N_{i}$ ($i = 1,2,3,4$) for 4-node rectangular quadrilateral elements are

$$\begin{array}{*{20}c} {N_{1}^{4} = \frac{1}{4}\left( {1 - \xi } \right)\left( {1 - \eta } \right),} & {N_{2}^{4} = \frac{1}{4}\left( {1 - \xi } \right)\left( {1 + \eta } \right),} & {N_{3}^{4} = \frac{1}{4}\left( {1 + \xi } \right)\left( {1 + \eta } \right),} & {N_{4}^{4} = \frac{1}{4}\left( {1 + \xi } \right)\left( {1 - \eta } \right)} \\ \end{array}$$

(35)

The shape functions of solid displacement, pore pressure and fine particle are given by

$$\left\{ {\begin{array}{*{20}l} {{\mathbf{N}}_{u} = \left[ {\begin{array}{*{20}c} {N_{1}^{8} } & {N_{2}^{8} } & {N_{3}^{8} } & {N_{4}^{8} } & {N_{5}^{8} } & {N_{6}^{8} } & {N_{7}^{8} } & {N_{8}^{8} } \\ \end{array} } \right]} \hfill \\ {{\mathbf{N}}_{p} = {\mathbf{N}}_{F} = \left[ {\begin{array}{*{20}c} {N_{1}^{4} } & {N_{2}^{4} } & {N_{3}^{4} } & {N_{4}^{4} } \\ \end{array} } \right]} \hfill \\ \end{array} } \right.$$

(36)

The strain–displacement transformation matrix ${\mathbf{B}}$ is

$$B = \left[ {\begin{array}{*{20}c} {\frac{{\partial N_{1}^{8} }}{\partial x}} & 0 & {\frac{{\partial N_{2}^{8} }}{\partial x}} & 0 & \cdots & {\frac{{\partial N_{8}^{8} }}{\partial x}} & 0 \\ 0 & {\frac{{\partial N_{1}^{8} }}{\partial y}} & 0 & {\frac{{\partial N_{2}^{8} }}{\partial y}} & \cdots & 0 & {\frac{{\partial N_{8}^{8} }}{\partial y}} \\ {\frac{{\partial N_{1}^{8} }}{\partial y}} & {\frac{{\partial N_{1}^{8} }}{\partial x}} & {\frac{{\partial N_{2}^{8} }}{\partial y}} & {\frac{{\partial N_{2}^{8} }}{\partial x}} & \cdots & {\frac{{\partial N_{8}^{8} }}{\partial y}} & {\frac{{\partial N_{8}^{8} }}{\partial x}} \\ \end{array} } \right]$$

(37)

Similarly, the matrices $\nabla {\mathbf{N}}_{p}$ and $\nabla {\mathbf{N}}_{F}$ are defined as

$$\nabla {\mathbf{N}}_{p} = \nabla {\mathbf{N}}_{F} = \left[ {\begin{array}{*{20}c} {\frac{{\partial N_{1}^{4} }}{\partial x}} & {\frac{{\partial N_{2}^{4} }}{\partial x}} & {\frac{{\partial N_{3}^{4} }}{\partial x}} & {\frac{{\partial N_{4}^{4} }}{\partial x}} \\ {\frac{{\partial N_{1}^{4} }}{\partial y}} & {\frac{{\partial N_{2}^{4} }}{\partial y}} & {\frac{{\partial N_{3}^{4} }}{\partial y}} & {\frac{{\partial N_{4}^{4} }}{\partial y}} \\ \end{array} } \right]$$

(38)

In Eqs. (37) and (38), the shape function derivatives with respect to the global coordinates is defined as

$$\begin{array}{*{20}c} {\left[ {\begin{array}{*{20}c} {\frac{{\partial N_{i}^{8} }}{\partial x}} \\ {\frac{{\partial N_{i}^{8} }}{\partial y}} \\ \end{array} } \right] = {\mathbf{J}}^{ - 1} \left[ {\begin{array}{*{20}c} {\frac{{\partial N_{i}^{8} }}{\partial \xi }} \\ {\frac{{\partial N_{i}^{8} }}{\partial \eta }} \\ \end{array} } \right]} & \& & {\left[ {\begin{array}{*{20}c} {\frac{{\partial N_{j}^{4} }}{\partial x}} \\ {\frac{{\partial N_{j}^{4} }}{\partial y}} \\ \end{array} } \right] = {\mathbf{J}}^{ - 1} \left[ {\begin{array}{*{20}c} {\frac{{\partial N_{j}^{4} }}{\partial \xi }} \\ {\frac{{\partial N_{j}^{4} }}{\partial \eta }} \\ \end{array} } \right]} \\ \end{array}$$

(39)

where $i = \begin{array}{*{20}c} {1,} & {2,} & \cdots & 8 \\ \end{array}$ and $j = \begin{array}{*{20}c} {1,} & {2,} & {3,} & 4 \\ \end{array}$. ${\mathbf{J}}$ is the Jacobin matrix and defined as

$${\mathbf{J}} = \left[ {\begin{array}{*{20}c} {\frac{\partial x}{{\partial \xi }}} & {\frac{\partial y}{{\partial \xi }}} \\ {\frac{\partial x}{{\partial \eta }}} & {\frac{\partial y}{{\partial \eta }}} \\ \end{array} } \right]$$

(40)

where

$$\begin{array}{*{20}c} {\frac{\partial x}{{\partial \xi }} = \sum\limits_{i = 1}^{8} {\frac{{\partial N_{i}^{8} }}{\partial \xi } \times x_{i} } ,} & {\frac{\partial x}{{\partial \eta }} = \sum\limits_{i = 1}^{8} {\frac{{\partial N_{i}^{8} }}{\partial \eta } \times x_{i} } ,} & {\frac{\partial y}{{\partial \xi }} = \sum\limits_{i = 1}^{8} {\frac{{\partial N_{i}^{8} }}{\partial \xi } \times y_{i} } ,} & {\frac{\partial y}{{\partial \eta }} = \sum\limits_{i = 1}^{8} {\frac{{\partial N_{i}^{8} }}{\partial \eta } \times y_{i} } } \\ \end{array}$$

(41)