Advances in numerical simulation techniques play an important role in helping mining engineers understand those parts of the rock mass that cannot be readily observed. The Material Point Method (MPM) is an example of such a tool that is gaining popularity for studying geotechnical problems. In recent years, the original formulation of MPM has been extended to not only account for simulating the mechanical behaviour of rock under different loading conditions, but also to describe the coupled interaction of pore water and solid phases in materials. These methods assume that the permeability of mediums is homogeneous, and we show that these MPM techniques fail to accurately capture the correct behaviour of the fluid phase if the permeability of the material is heterogeneous. In this work, we propose a novel implementation of the coupled MPM to address this problem. We employ an approach commonly used in coupled Finite Volume Methods, known as the Two Point Flux Approximation (TPFA). Our new method is benchmarked against two well-known analytical expressions (a one-dimensional geostatic consolidation and the so-called Mandel-Cryer effect). Its performance is compared to existing coupled MPM approaches for homogeneous materials. In order to gauge the possible effectiveness of our technique in the field, we apply our method to a case study relating to a mine known to experience severe problems with pore water.

Numerical modelling is an essential tool for understanding the induced stress changes in underground hard rock mines. Mines rely on these tools for numerous purposes, of which arguably the most important is the planning of mining operations that mitigate seismic risk. Since engineers cannot physically observe the solid rock mass, numerical techniques must be capable of providing reliable predictions of the complex behaviour of the material. Many of the available commercial software packages simulate models that only allow for a single solid phase and frequently neglect groundwater’s influence on the stress regime in the rock mass [

A standard modelling technique that is often used in the mining industry is based on boundary elements. The Boundary Element Method (BEM) [

The original formulation MPM was proposed by Sulsky et al. [

The MPM has been successfully used to study many problems, and its application to geotechnical questions [

Many coupled MPM formulations use the Biot theory approach [

Both Al-Kafaji [

The remainder of the paper is structured as follows. In

In a single-point formulation of the two-phase MPM, the medium is represented by a solitary particle layer which means that each particle must carry information about both the solid and fluid phases. Most existing literature that uses this approach assumes that the continuum is defined as a homogeneous porous solid material with constant permeability and porosity [

To delve a little deeper into the source of the problem, we focus on the process by which the fluid phase properties are mapped onto the nearby nodes, see

where _{s}

The summation is taken over all _{p}

where

so that the total nodal fluid phase mass is expressed as

On the other hand, when the material is heterogeneous, the situation changes as

When using domain discretisation approaches to solve partial differential equations, a relatively large domain is divided into smaller elements, otherwise known as finite volumes (FVs). Finite Volume methods are usually the industry-preferred method for solving problems involving porous media as they are based on the conservation of continuity of fluxes [

The total fluid flux through the surface of a finite volume can be written as

where

where the sum is taken over the six faces of the cube. The integral term on the right-hand side of

To demonstrate how the flux at the interface of two cells are calculated in practice, consider two cells _{1}, cell _{2} and that the shared face

where _{q}_{A}

where _{B}_{B|q} should be the same, this enables us to deduce that

where _{j}_{j}

which shows that the flux through

To improve matters, here we propose a modification of

We base many of our design decisions guided by the strategy employed in the usual TPFA. First, we remark that in examining node Q, we will only consider contributions from particles located in cells 1, 3 and 5; that is, cell C together with those cells that have Q has a node and which share an interface with C (i.e., the cells to the north and the west of C). For each of these, the cell pore pressure _{c}

where _{p}_{c}_{c}

where _{c}

The standard shape functions of an isoparametric hexahedron can be calculated using the schematic idea that is sketched in

Next, the conductivity term is evaluated independently for each cell that shares an interface with

With these key changes in mind, we can now derive the appropriately modified version of

where the terms _{i}_{i}

As has been kindly pointed out to us by an anonymous referee, if

With these changes taken into account we can modify

where

We have developed result

where subscript

Given this modified expression for the Darcy velocity, we need to embed it within the cycle of calculations that are required to complete a time step within the modified MPM.

In this section, we will explain how we deploy the dynamic MPM in our modelling experiments. The Lagrangian particle layer present in our solid skeleton and the momentum balance equations for the fluid-solid mixture are solved on a regular Eulerian background mesh composed of trilinear shape functions

Our medium is heterogeneous, which means we can not assume constant conductivity as commonly adopted in the majority of MPM applications [

To initiate our model, suppose that at time

With this particle state specified, we can derive:

the volume of the mixture

the fluid phase mass

the particle mass _{s}_{f}

The method then proceeds as follows. For each particle p, we can identify the particular grid cell that contains p and then links are created between p and the eight nodes of that cell. Reciprocal links from the nodes back to the particle are formed, and the combined information is stored in a bi-directional map

For each grid node

respectively. The force exerted due to the solid-fluid mixture is

where ^{T}, _{s}

where the frequency independent damping term is defined to be

and is added to enhance the numerical stability of the process and to encourage the convergence to a quasi-static solution [

and boundary conditions for

At this point in the process, our novel MPM formulation deviates from many of the established two-phase MPM strategies. In these standard versions, the nodal Darcy velocities are calculated by appealing to

Having updated the state of each node, we can revert to the particles. For each particle, we use _{n}

while the strain increments for the solid phase are

The fluid phase strain increment for each particle requires the calculation of the average pore pressure and conductivity of the cell which are set to be

Each cell is then revisited to deduce the associated nodal Darcy velocities

and the pore pressure is advanced using

where _{f}

The explicit nature of the proposed iteration scheme means that the timestep _{p}

Fortunately, for a simulation that is kept near quasi-static equilibrium, this stringent constraint on the timestep can often be relaxed somewhat. Basson et al. [

where ^{t} is a mass density scaling factor. Here we set ^{t} = 1 if model is not in equilibrium but set

Before using our technique in practical contexts, it is essential to validate its performance against well-known and standard examples. For this purpose, we selected two problems [

in the two subsections that follow.

The first example taken from Bandara [

where _{1} will influence the results as the system settles towards its final steady-state, this test is beneficial for monitoring _{2} as the effect of gravity on the fluid phase will dominate _{c}

Our experiment was conducted on a grid of dimensions

Young’s Modulus | |
---|---|

Poisson’s Ratio | 0.3 |

Solid phase density | 2143 kg/m^{3} |

Solid phase porosity | 0.3 |

Solid phase conductivity | |

Fluid density | 1000 kg/m^{3} |

Fluid Bulk modulus |

We remark that in the course of the calculation, we noted the presence of oscillations in both the pore pressure and effective stress profiles during the system’s evolution and before settling to equilibrium. This phenomenon is illustrated in

In this second example, we attempt to validate the performance of our proposed technique by applying it to the standard Cryer sphere problem [_{1} in _{1} as the drag term

An MPM model was constructed to investigate this effect. A two-dimensional half-circle can visualise the inherent symmetry of the whole three-dimensional sphere (see

Young’s modulus | |
---|---|

Poisson’s ratio | 0.1 and 0.3 |

Solid phase density | 2143 kg/m^{3} |

Solid phase porosity | 0.3 |

Solid phase conductivity | |

Fluid density | 1000 kg/m^{3} |

Fluid Bulk modulus |

Ernest Henry is a copper and gold producing mine situated approximately 40 km North-East of Cloncurry in Queensland, Australia. Initially, the mine design was open pit; however, it introduced underground operations; presently, the deepest part is at about 1.2 km. The mine is seismically active, and its largest recorded event had a moment magnitude of 2.1. Groundwater is a significant problem for engineers working at the mine as they constantly have to drain the deeper levels to prevent flooding. Even though the host rock contains some pore water, it is thought that most of the water is contained in faults that act as the fluid’s principal transport mechanisms. Thus the porosity and conductivity are significantly more than that of the host rock.

On 28 September 2019, the mine drilled an exploration hole in the deepest part of the mine. The hole encountered a geological fault, at which point fluid was ejected from the hole. Several hours later, a large seismic event occurred on the same fault, but several hundreds of metres higher and near development areas. A schematic of the hole and the event location is shown in

Was the event triggered by the reduction in pore pressure in the fault?

If pore pressure is released in the fault, how does it affect the nearby geology?

The fluid that is trapped within the faults is at high pressure, but this does not imply that it necessarily moves at high speed. The high pressure arises owing to the overburden weight of the fluid, and the porous nature of the rock means that Darcy’s law is applicable. The Institute of Mine Seismology (IMS) was tasked to investigate these questions. An in-house developed version of the standard coupled MPM was used to explore the issues raised. IMS has successfully used their version of the MPM in various three-dimensional numerical studies owing to its ability to simulate large deformation, dynamic problems twinned with the relative ease of building complex mining geometries. It is the prospect of this occurrence at Ernest Henry mine that led to the investigation of a way to simulate the coupled solid-fluid interaction of an heterogeneous medium as it is assumed that the permeability/conductivity of the faults is noticeably different than that of the host rock. We assume that the host rock permeability is very low compared to the rock mass that describes the fault zones and so, to this extent, can be legitimately regarded as a heterogeneous permeability rock experiment. This is a circumstance in which current MPM implementations will likely fail, and we try to overcome this problem using our new method.

A high-resolution single-point coupled MPM model was constructed for the mine. The scheme incorporated an adaptive grid of the type used in the work of [

Several assumptions were made in the model:

All materials were assumed to be poroelastic, however, it was assumed that only the two faults were filled with water and that the host rock was dry. This means that fluid flow was confined to be only within the faults.

Initial pore pressure in the faults was set to approximately 20 MPa. This is a typical value seen in practice, but precise pressure could not be used as it was not measured by the mine.

Draining in the model started approximately 150 metres below the lowest mining level. To reduce the computational complexity, no restriction was placed on the fluid’s outflow rate at the location where the hole intersected the Fault 6.

To reduce pressure oscillations caused by the drag term (see ^{d} was non-zero in

Young’s modulus [GPa] | 50 | 30 | 30 |

Poisson’s ratio | 0.22 | 0.20 | 0.20 |

Density [kg/m^{3}] |
2,850 | 2,700 | 2,700 |

Porosity | 0.01 | 0.1 | 0.1 |

Conductivity [m/s] | 0 | 0.01 | 0.01 |

Density [kg/m^{3}] |
1000 | ||

Bulk Modulus [GPa] | 2.2 | ||

0.0553 |
5 | 212 | |

0.0372 |
18 | 303 | |

0.0234 |
72 | 107 |

Initially, the model was solved for quasi-static equilibrium by assuming no mining has taken place and using the material properties and stress conditions stated. It was assumed that all materials were dry, which set the rock mass’s undisturbed state. The model was then resolved for the quasi-static equilibrium by removing all particles from the system within the mining excavations. This step represents the current mining state, and it was again assumed that all materials were dry. These two initial steps are common in stress-strain modelling for mining simulations for single-phase problems. Several common stress parameters were selected to quantify the faults as:

The major principal stress magnitude

The normal stress to the fault plane

The in plane shear stress

The excess shear stress, otherwise known as the Coulomb stress [

The excess shear stress (ESS) is often used in elastostatic modelling to determine whether a fault may be at risk of slipping within the context of the elastic model. Areas on the fault that display negative (or zero) ESS values are considered stable, whereas areas of positive ESS are liable to slip, or may have already slipped at a prior point.

The draining phase was initiated by modifying the particles that characterise the faults. It was assumed that these were fully saturated, and a 20 MPa pore pressure was assigned to each particle. To preserve quasi-static equilibrium, the diagonal terms of the particle stress matrix was reduced by 20 MPa, thus ensuring that the total particle stress remains unaltered and

The results as outlined in

Video clips of the simulations of each modelling parameters used in this experiment can be obtained from:

http://downloads.imseismology.org/numericalmodelling/mpm/ ess.m4v Link to video clip showing Excess Shear Stress changes as the F6 drains

http://downloads.imseismology.org/numericalmodelling/mpm/shear-stress. m4v Link to video clip showing

http://downloads.imseismology.org/numericalmodelling/mpm/sigma1. m4v Link to video clip showing

http://downloads.imseismology.org/numericalmodelling/mpm/normal-stress. m4v Link to video clip showing

In each of the clips we show how the draining of the F6 fault changes the properties of both itself and the nearby Angry man fault. These clips are free to download and redistribute.

A fully coupled two-phase three-dimensional MPM has been developed to model stress changes caused by fluid-filled rock in mining. This has been accomplished by modifying existing coupled MPM approaches to incorporate the description of mediums with heterogeneous permeability. Two verification examples were used to test the accuracy of the numerical model, and good agreement was observed in both cases. These benchmarks provide confidence that our methodology has value when modelling real mining cases in which a seismic event is likely triggered due to pore pressures changes caused by draining of a nearby fault. In this experiment, we have shown that even though the effective stress changes on the fault were relatively small, it was nonetheless still sufficient to play a role as a provocative mechanism for the event.

The case study at Ernest Henry mine highlights the importance of accounting for the effect of pore water in mining simulations, an effect that is often neglected. However, as mining methods become increasingly complicated, so the need for more advanced multi-phase numerical simulations grows. This is especially true for simulations of mine tailings dams for which the presence of fluid can play a central role in determining the outcome of experiments.

Our method can benefit from further improvement in two areas; a more advanced soil model might be implemented for dealing with problems involving partial saturation, and the numerical fluctuations in pore pressure could be reduced when dealing with dynamic problems. Further research in both these aspects must be pursued.

An intriguing avenue for future work would be an investigation of the relationship between our formulation and the innovative technique known as peridynamics. This is a novel non-local method that has been suggested as a vehicle for simulating hydraulic fractures [

In summary then, our new model appears to show promise for modelling multi-phase coupling mechanisms in realistic mining situations. What we have described here is a first step in this direction, and extensions to further real case studies is the obvious way forward. As the catalogue of simulations increases, this would enable us to predict those circumstances in which mine sites might benefit from our ideas. One particular attraction of our suggestions is the fact that they can be incorporated into conventional MPM methods with only marginal extra computational cost.

We would like to thank Ernest Henry mine for allowing the use of their data in our case study. We are grateful to the reviewers for numerous suggestions and additional references that have significantly improved the quality of the manuscript.