www.elsevier.com/locate/enggeo

Engineering Geology 8

Hydro-geomechanical modelling of seal behaviour in overpressured

basins using discontinuous deformation analysis

M. Rouainia a,*, H. Lewis c, C. Pearce b, N. Bicanic b, G.D. Couples c, M.A. Reynolds c

a School of Civil Engineering and Geosciences, University of Newcastle, Newcastle NE1 7RU, UKb Department of Civil Engineering, University of Glasgow, Glasgow G12 8T, UK

c Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh EH14 4AS, UK

Received 1 December 2004; received in revised form 2 November 2005; accepted 4 November 2005

Abstract

A coupled hydro-geomechanical modelling environment, developed to evaluate the coupled responses of fluid flow in

deforming discontinuous media, is described. A staggered computational framework is presented, where the two simulations

tools, HYDRO and DDA, communicate via the mapping of an equivalent porosity (and related permeabilities) from the rock

system to the fluid phase and an inverse mapping of the pressure field. Several algorithmic and modelling issues are discussed, in

particular the computational procedure to map the current geometry of the discontinuous rock blocks assembly into an equivalent

porosity (and permeability) field. A generic, geometrically simple, overpressured reservoir/seal system is analysed for illustration.

Further examples investigate discontinuous, fractured configurations in flexure causing a degree of spatial variability in the induced

stresses. Model predictions show that the combination of hydraulic and mechanical loads causes a dilational opening of some pre-

existing fractures and closure of others, with strong localisation of the modified flow pattern along wider fracture openings.

D 2005 Elsevier B.V. All rights reserved.

Keywords: Discontinuous deformation analysis; Porous media; Hydro-geomechanical modelling; Overpressured basins; Seal failure

1. Introduction

The existence of overpressure in hydrocarbon reser-

voirs is a major reservoir management concern from both

a safety and an economic perspective. Hydrocarbons are

generated in the deep organic-rich mudrocks and tend to

migrate to porous and permeable reservoir rocks. In

locations where low permeability sealing rocks overlie

the reservoir rocks a hydrocarbon trap is created (see Fig.

1). In most cases the sealing layer actually acts as a leaky

0013-7952/$ - see front matter D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.enggeo.2005.11.004

* Corresponding author. Tel.: +44 191 222 3608; fax: +44 191 222

5322.

E-mail address: m.rouainia@ncl.ac.uk (M. Rouainia).

seal with some flux of hydrocarbons out of the reservoir

(Couples, 1999). This sealing layer is often characterised

by some matrix permeability and small-aperture frac-

tures and, when overpressures are created in the reser-

voir, has a higher fluid pressure below the seal than

above it: this is the scenario investigated here. Such a

leaky seal system operates at considerable depths and

pressures, with small, but complex, block interactions,

and is quite impractical for reproduction in a testing

laboratory. This paper describes a numerical method of

investigating the behaviour of such systems, illustrating

the use of this approach in some very simple reservoir-

seal scenarios.

On the computational front, continuum approaches

have been used successfully for many subsurface geo-

2 (2006) 222–233

Fig. 1. Schematic vertical diagram through the top of a reservoir, highlighting the overpressured reservoir and fractured mudrocks (the seal).

M. Rouainia et al. / Engineering Geology 82 (2006) 222–233 223

logical and hydrogeological problems, but the use of

continuum approaches to model saturated materials

with discrete discontinuities has met with only limited

success. A number of discontinuum-based, or discon-

tinuum-incorporating simulators have been developed

in the disciplines of geomechanics, petroleum, and

environmental engineering (e.g., Lee et al., 2000;

Gurpinar and Cossack, 2000). Typically, most of

these tools focus on only a few components of the

complete multi-physics problem, or greatly simplify

the interactions between these components. For exam-

ple a finite element stress analysis crudely coupled

with a finite difference flow program can be used for

predicting surface subsidence. Other studies (e.g.,

Aifantis, 1980; Barenblatt et al., 1960; Elsworth and

Bai, 1992; Smart et al., 2001; Lewis and Ghafouri,

1997; Koutsebeloulis et al., 1994) use the double

porosity approach — where one porosity represents

the fracture network and the other a continuum porous

medium. This approach treats matrix and fracture

systems as essentially independent of one another,

communicating through a leakage, or transfer, term,

but still demonstrates a strong coupling between fluid

flow and solid deformability.

In this paper a framework for modelling deform-

able discontinuous media is coupled with a continuum

formulation for flow through fractured media. In such

a system there is considerable flexibility in defining

initial block and fracture geometries. Furthermore,

solid deformability and fracture/matrix fluid flow are

represented in such a way that fracture apertures re-

flect both the deformation of solids and changes in the

fluid pressure. Fractures are represented deterministi-

cally. The discontinuous medium is represented by the

Discontinuous Deformation Analysis DDA method

(Shi, 1988) and the fluid flow system by a continuum

finite element steady-state flow model HYDRO. The

fluid system is assumed to obey Darcy’s law which

employs a fixed finite element mesh and responds to

pressure (potential energy) boundary conditions (e.g.

Garven and Freeze, 1985). The DDA method repre-

sents the blocky, dry, discontinuous deformable solid

phase responding to force or displacement boundary

conditions and to the initial state of stress. Block

contact constraints are imposed through an implicit

augmented Lagrangian format. These two linked fra-

meworks, denoted here as HYDRO–DDA, communi-

cate via mapping of an equivalent porosity field from

the solid to the fluid phase and an inverse mapping of

the calculated pressure field (Rouainia et al., 2001).

This linkage allows us to investigate the interaction

between the pore-fluid and mechanical loads.

HYDRO–DDA is currently realised as a two dimen-

sional simulator.

2. The HYDRO–DDA components

HYDRO–DDA is the product of a staggered cou-

pling of the two modelling environments HYDRO and

DDA. The principle of the interface is illustrated

schematically in Fig. 2, where HYDRO calculates

the fluid pressures and flow rates in a 2-D model

containing a number of different materials with spe-

cific permeabilities. Fluid pressures are then passed to

the DDA environment. The pressure field is interpo-

lated to the current positions of the DDA block ver-

tices, which furnishes the equivalent forces used in the

calculation of the further deformation of the loaded

assembly of blocks. These equivalent forces – which

represent effective stress behaviour – produce, in gen-

eral, a change in the configuration of the solid block

assembly — i.e. they induce block rotations, transla-

tions and/or straining. These changes in block geom-

etry are returned to HYDRO as changes in porosity

and permeability. The new fluid flow solution is found

for the current level of permeabilities, which leads to

Fig. 2. Staggered calculation environment HYDRO–DDA and the flow chart showing the functional relationships between HYDRO and DDA.

M. Rouainia et al. / Engineering Geology 82 (2006) 222–233224

new fluid pressures. These in turn lead to new equiv-

alent forces and this computational loop continues

until a steady-state solution is reached within a certain

convergence norm.

In the following, DDA and HYDRO are addressed

as distinct tools and then the development and imple-

mentation of the HYDRO–DDA interface is discussed.

The performance of HYDRO–DDA is illustrated with a

simple model of four regular fracture-bounded blocks.

Its capabilities are demonstrated first for a rock-mass

model consisting of six fracture-bounded blocks and

then for symmetric and asymmetric flexural configura-

tions approximating a fractured seal overlying a reser-

voir with a strong fluid-potential gradient. All

configurations are expected to develop some degree

of spatial variability in the induced stresses, which

can lead to a localisation of any enhanced fluid flow

due to hydro-geomechanical coupling.

2.1. Discontinuous Deformation Analysis

method — DDA

In the DDA method, an assembly of bodies is mod-

elled by a number of deformable blocks of an arbitrary

M. Rouainia et al. / Engineering Geology 82 (2006) 222–233 225

shape that are able to move independently of one

another. In the low-order scheme adopted here, each

block has a uniform state of stress and strain. In two

dimensions, the first order approximation of the dis-

placement (u, v) at any point (x, y) of a block i is

interpolated as:

u

v

� �¼

1 0 � y� yoð Þ x� xoð Þ 0 y� yoð Þ0 1 � x� xoð Þ 0 y� yoð Þ x� xoð Þ

� �df g

¼ Tð Þ df g ð1Þ

where d is the vector of variables associated with an

individual block comprising the rigid body transla-

tions and rotation at the centroid of the block, uo, voand co and the normal and shear strains, ex, ey and

exy, respectively:

dT ¼ uo; vo; co; ex; ey; exy� �T

: ð2Þ

The equilibrium formulation in the DDA method is

provided by the principle of potential energy minimi-

zation. For a system of n blocks, the total potential

energy P is given by

P ¼ dTpkpqdq � dTp f p p; q ¼ 1; . . . 6n ð3Þ

where P comprises contributions from block strain

energy and energy terms emanating from point and

gravity loads, block to block contacts, initial stresses

and boundary constraints. For more information re-

garding the derivation of these contributions, the read-

er is referred to Shi (1988) and Shi and Goodman

(1990). Minimisation of the above energy potential

yields

BPBdq

¼ kpqdq � f q ¼ 0: ð4Þ

Consequently, Eq. (4) takes the partitioned form:

k11 k12 k13 . . . k1n

k21 k22 k23 . . . k2n

k11 k12 k13 . . . k3n

. . . . . . . . . . . . . . .

kn1 kn2 kn3 . . . knn

0BBBB@

1CCCCA

d1d2d3:dn

0BBBB@

1CCCCA

f 1f 2f 3. . .

f n

0BBBB@

1CCCCA ð5Þ

where fi is a 6�1 vector of forces acting on block i and

di contains the variables associated with block i. The

off-diagonal submatrices kij, (i p j) contain the stiffness

components associated with the contact between block i

and j and kii refers also to the components of the

material stiffness of block i.

The inter-block contact conditions are such that

no tension, with or without sliding, can develop

between blocks and no penetration of one block

into another is possible, resulting in different con-

tributions to the DDA equilibrium equations. The

penalty method, the Lagrangian method and the

augmented Lagrangian method are all considered to

impose these constraints.

The advantage of the penalty method is its simplic-

ity. In this method a penalty spring of stiffness, p, is

added between the vertex of one block and the side of

the other. The penalty stiffness is arbitrarily large in

order to adequately impose the contact constraint and

as a consequence, the system stiffness matrix is often

ill-conditioned.

In the Lagrangian method, the contact constraints

are imposed by means of the Lagrangian multipliers,

k, which represent the contact forces between blocks.

This method ensures the contact constraints are satis-

fied exactly, but leads to an increase in the number of

system equations. The solution adopted here was to

use the augmented Lagrangian method, advocated by

Lin (1995), which can be interpreted as a mixed

method that includes the Lagrangian multiplier and

the penalty method as limiting cases. The size of the

system matrix does not increase and k is obtained

through an iterative procedure. The iterative process

is completed if a convergence criterion for the zero

penetration condition is satisfied, within some toler-

ance norm.

2.2. Steady state fluid flow model — HYDRO

Assuming nonnegativeness of the dissipation and

assuming isothermal conditions, fluid flow through

porous media is governed by Darcy’s law. Accordingly,

a hydraulic gradient gives rise to a proportional flow:

q ¼ � khdjh ð6Þ

where q is the hydraulic induced Darcy flux [m/s],

defined as the volumetric flow rate per unit cross-

sectional area of medium and h is the hydraulic head

[m]. The hydraulic conductivity may be written as:

kh ¼kcl

ð7Þ

where k is the intrinsic permeability [m2] of the

medium, and c [N/m3] and l [Pa s] are the specific

weight and the dynamic viscosity of the fluid, with

approximate values, for water of 104 and 10�3,

respectively.

In the case of flow through an isotropic and homo-

geneous medium, the hydraulic conductivity is charac-

ig. 3. An element in the HYDRO FE mesh which covers both matrix

nd fracture materials is assigned an equivalent porosity using the

olygon clipping algorithm.

M. Rouainia et al. / Engineering Geology 82 (2006) 222–233226

terised by a scalar quantity, i.e. kh =kI where k is a

constant. In two dimension it holds that

kh ¼k 0

0 k

� �:

The finite element approximation (e.g. Smith and

Griffiths, 1998) is used to solve Eq. (6) for the hydrau-

lic head, with suitable boundary conditions and fluid

properties. We have chosen a triangular mesh formula-

tion for HYDRO. By adopting a suitable reference

frame, hydraulic heads are interchangeable with fluid

pressures. Here, the smallest hydraulic head value (as a

boundary condition) defines the reference point. Using

fresh-water densities, a 1-m hydraulic head difference

equates to 10 kPa pressure difference.

3. HYDRO–DDA interface

The HYDRO–DDA interface is the product of a

staggered coupling of the two computational environ-

ments outlined above. Of primary interest are: (a) cre-

ating the initial fracture network; (b) calculation of

equivalent porosity values for the HYDRO finite ele-

ment mesh to represent the original and the updated

DDA block and gap positions; and (c) appropriate

transfer of the pressure field (from HYDRO) to equiv-

alent forces (or face pressures) acting on the edges and

vertices of every DDA block.

3.1. Initial fracture network

Both DDA and HYDRO need an initial geometry

that takes the fracture configuration into account. DDA

requires block geometry and a specified initial fracture

aperture while HYDRO needs a finite element mesh by

which appropriate matrix or fracture porosities and

permeabilities can be assigned to individual elements.

The first step is to create an initial block and fracture

geometry such that the fractures have zero apertures

and are simply the interfaces between the DDA blocks.

However, fractures typically have some finite width and

the HYDRO–DDA interface is used to shrink the

blocks, producing a predetermined initial fracture aper-

ture. This is a required geometry input for DDA as

implemented here.

HYDRO requires a finite element mesh that both

allows initial assignment of porosities for fracture and

matrix regions to be simple and permits efficient mod-

ification of the values assigned to each element when

the fracture boundaries have moved. The block geom-

etry with fracture aperture is combined with a polygon

clipping algorithm (see Murta, 1998) to define the

initial porosity map for HYDRO. The finite element

mesh generation also uses this initial block and fracture

geometry such that mesh refinement along block

boundaries, producing a mesh with significantly smal-

ler elements around each fracture.

3.2. Equivalent porosity

In the HYDRO–DDA framework described here, the

area and the position of the solid material blocks can

change from one iteration to the next, while the finite

element mesh that represents the fluid continuum

remains fixed. The fluid system is affected by all

changes in block geometry because of the way that

fracture-aperture changes are related to the porosity

and permeability field. Therefore, a given finite element

belonging to the fixed mesh for the fluid phase may be

lying completely within a fracture (/ =100%), com-

pletely within the matrix (porosity / =/matrix) or par-

tially inside the fracture and partially inside the matrix

(/matrixb/ b100%). The HYDRO–DDA interface uses

a polygon clipping method to determine the equivalent

porosity for every finite element in the fixed fluid mesh.

The algorithm for polygon clipping (see Fig. 3) is

extremely fast and efficient and allows the interface

to determine, for each element, the area lying within a

fracture, Afracture, compared to the total area of the

element, Aelement, so that the equivalent porosity is

determined as:

/element ¼Afracture

Aelement

1� /matrixð Þ þ /matrix ð8Þ

Algorithmically, the block geometry is defined as the

clipping polygon set and each finite element is the

subject polygon; the algorithm returns a polygon defin-

ing the difference between the clipping polygon set and

the subject polygon. The result is a porosity value per

element that can vary between the background matrix

porosity and 100%.

F

a

p

M. Rouainia et al. / Engineering Geology 82 (2006) 222–233 227

The fluid flow calculations also need hydraulic conduc-

tivity values. It is common to assume that porosity and

hydraulic conductivity are related. Often the relation-

ships used for the prediction are both complex and

tailored to specific conditions and rock types. In the

current version of HYDRO–DDA we chose a very

simple relationship that is patterned after the Kozeny–

Carman formula (see Boudreau, 1997):

kelement ¼/3element

180 1� /2element

� d2g ð9Þ

where dg is the mean diameter of the sediment particles.

Since we only have one material type in our models, the

term dg becomes a scaling parameter. To avoid singu-

larities in the open fractures (where porosity is 100%),

we place an upper bound on the calculation so that the

maximum permeability is limited to 105 mD.

3.3. Transfer of pressure distribution to blocks

The fluid flow analysis calculates fluid head, which

is related to pressure at the nodes of the fluid finite

element mesh. Pressures at the centroid of each element

are then calculated via a weighted average of the nodal

pressures. The DDA part of the calculation is then

driven by forces acting on the vertices of the DDA

blocks. The distributed pressures acting on the block

edges are then converted to forces at the vertices.

Where a block vertex lies within a fluid element, the

pressure at the centroid of the fluid element is applied

Fig. 4. HYDRO–DDA interface: conversion of f

as force components to the vertex of the block (see Fig.

4). The normal forces acting on the vertices associated

with edge 1–2 of a block may be written as:

F1 ¼L

2p1m þ 1

3p2m � p1mð Þ

� �; and

F2 ¼L

2p1m þ 2

3p2m � p1mð Þ

� �

where L ¼ x2 � x1ð Þ2 þ y2 � y1ð Þ2h i1

2

is the edge

length and p1m and p2m are the pressures at the centroid

of the corresponding finite elements in the fluid mesh

where the two block vertices lie, respectively. The

horizontal and vertical components, Fx and Fy, of the

force components F1 and F2 (Fig. 4) are determined,

added to the external mechanical load and given as

boundary conditions to the discontinuous deformation

analysis. This produces a changed element geometry

according to block system kinematics, and therefore

leads to changes in the porosity and permeability of

the system, and consequently causes a change in the

fluid pressure distribution.

4. Exploration of the HYDRO–DDA environment

This section explores some of the multi-physics

characteristics of coupled mechanical–hydraulic sys-

tems that can now be addressed by the HYDRO–

DDA simulation environment. We use as our focus

the behaviour of a fractured seal layer overlying an

luid pressure into equivalent vertex forces.

M. Rouainia et al. / Engineering Geology 82 (2006) 222–233228

overpressured hydrocarbon reservoir (Fig. 1) because

such a system is both a simple representation of com-

mon subsurface geological situations and because it is

highly sensitive to the rock system/fluid system inter-

action addressed by HYDRO–DDA. Subsurface reser-

voir rocks and their seals are very typically layered,

with layers potentially having quite different mechani-

cal properties. Layers, or beams, very often support

different loads and contain different patterns of open

or closed fractures. Interfaces between the layers are

often partially welded and/or frictional.

4.1. Asymmetric block cases

4.1.1. A range of loading scenarios

This simplified model problem is performed to il-

lustrate the combined hydro-geomechanical modelling

interface, HYDRO–DDA. The model has a rectangular

shape with a width of 5 m and a height of 5 m as shown

in Fig. 5. The mechanical behaviour of blocks is con-

sidered to be elastic with the following material prop-

erties: Young’s modulus E =104 MPa, Poisson’s ratio

v =0.35, unit weight of 1. All joints have a friction

angle of 308 and a cohesion of 300 kPa. The finite

element fluid mesh consists of 9164 three noded ele-

ments and the mesh is much finer in the areas which

correspond to zones of potentially open fractures. The

permeability of the intact rock is chosen to be 105 times

smaller than the initial permeabilities of the fractures.

Block movements reduce most of the fracture apertures

so that the contrast is generally much smaller than 105.

The interactions between the mechanical and hy-

draulic systems of the asymmetric block configuration

are investigated by creating four individual models with

Fig. 5. Schematic diagram illustrating the HYDRO–DDA boundary cond

hydraulic head boundary conditions where p2Np1.

different fluid and mechanical boundary loads. The

mechanical loading is set to be either a left–right short-

ening, or an up–down shortening. Likewise the fluid

loads are applied to either the left and right sides or to

the top and base using an imposed head gradient to

generate flow from right-to-left or bottom-to-top, re-

spectively. These four combinations create cases where

the mechanical and hydraulic energy gradients are ei-

ther crossing or parallel.

Fig. 6(a–d) shows results from the HYDRO–DDA

computation. It can be observed that the flow is faster in

the fractures than in the porous rocks, but that the

difference between fracture and matrix flow is small

in parts of the models. In some locations, the openness

of the fracture system leads to a homogenisation of the

pressure (head) distribution, but in others the fractures

and matrix are quite similar. The intentionally asym-

metrical configuration of the six-block system reveals

that the coupled performance of such systems is depen-

dent on the details of the flow network fracture topol-

ogy, which is itself dependent on the coupling between

the fluids and the block mechanics. It is not always true

that the permeability is enhanced along the direction of

the mechanical load, and it is not true that fractures

always have a major impact on the flow system.

4.2. Flexure models

A further two models investigate flexural systems.

Flexing of rock layers is quite common in subsurface

reservoirs causing spatial variability in the induced

stresses. This could, in turn, lead to localisation of

any enhanced fluid flow due to mechanical and hydrau-

lic coupling. Equally, the existence of a fluid pressure

itions (a) mechanical loading and model pinning location, and (b)

Fig. 6. Comparison of HYDRO–DDA models based on the six-block configuration. Each image (a through d) shows: the deformed block

configuration (aperture changes are almost impossible to discern at this scale, but some of the larger block displacements are obvious); contours of

the fluid hydraulic head (m); and vectors representing the flux. Flux vectors smaller than 1% of the maximum flux are not shown (to reduce visual

clutter). Load refers to the axis along which the model has been shortened (external boundary conditions).

M. Rouainia et al. / Engineering Geology 82 (2006) 222–233 229

gradient, especially if deformation enables the pressure

to be transferred to a new region of the deforming

material, could lead to deformation enhancement, and

to progressive failure.

4.2.1. Symmetric flexure model

This flexure problem depicts a fractured layer of

rock that is assumed to be pinned – i.e. without rota-

tional constraints – at its two bottom corners, while

uniform fluid pressure acts along the base and the top of

the layer, of 11 and 6 MPa, respectively (see Fig. 7).

The fluid pressure and boundary conditions cause the

layer to bend upwards with a nearly constant-curvature,

resulting in a symmetric flexural shape. The layer has a

Fig. 7. Simple, symmetric flexure case with central fractured region.

Fluid pressure along the base greater than that applied at the top.

simple, pre-existing symmetric open fracture pattern

distributed through its central region, but neither end

portion is fractured.

Several examples have been run with different bot-

tom-to-top fluid pressure values and gradients. Only

two examples are shown here. Each model problem

using this configuration shows that this combination

of hydraulic and mechanical loads causes a dilational

opening of some or all of the existing fractures with

flow also occurring through the matrix blocks. It also

causes strong localisation of the flow along some of

fractures of the array (Fig. 8), though these fractures are

not necessarily also the most dilated ones. The net

effect of the fracture is to enhance the permeability

and hence the flow, due to the increased apertures

through parts of the fractured region (Fig. 9).

4.2.2. Asymmetric flexure model

A different flexural model has been developed to

consider a nonsymmetric mechanical loading case. In

this configuration, the layer itself is composed as above,

but the mechanical loading is designed to cause asym-

metric flexure of the layer (Fig. 10). The right boundary

Fig. 8. Example results for simple-flexure model configuration. The top image for each case shows the block configuration along with vectors

representing the flux. Note that the blocks on the left and right sides of each model are welded together (no fractures). The two cases illustrated (20

and 5 m hydraulic head differentials) indicate that the fracture flux and the background (matrix) flux are more different in the higher-pressure case.

The lower portions of the figure plot the hydraulic head contours. Note that there is no indication of pressure anomalies (compare Fig. 6).

M. Rouainia et al. / Engineering Geology 82 (2006) 222–233230

of the layers in this model set is fixed, while the left

boundary of the layers is displaced downwards and

subjected to end moments to induce a concave–convex

shape-change onto the model boundary.

The applied fluid pressure along the base is uniform

and always larger than the uniform pressure along the

top of the model. The models show a considerable

degree of interdependence between the mechanical

and hydraulic system. The imposed boundary fluid

pressures lead to changes in the fracture pattern that

experience an enhanced fluid flux (Fig. 11). Note that

the pattern of higher fluid flux does not necessarily

identify all areas of increased fracture aperture; i.e. an

open fracture without a pressure gradient does not

exhibit a rapid flow. The effective, or upscaled, hydrau-

lic conductivity of the central, fractured region is not a

simple function of the fluid pressure gradient, but in-

stead is suggestive of the actions of multiple processes.

Fig. 9. Plot of effective hydraulic conductivity (or permeability) of the fractu

the hydraulic head differential. The plots are normalised to the unfractured ca

differential becomes larger (causing more flexure).

Further work is required to develop a more robust

understanding of these interactions. Additional models

(not shown) have explored some variations in the me-

chanical loading. Although the fluid flow patterns of

those models indicate some sensitivity to the details of

the loading, the overall response is very similar to that

illustrated here, suggesting that the local deformation

state within the central region is relatively insensitive to

the far-field conditions. That conclusion is similar to

other published work on flexural systems (see Lewis et

al., 2001).

The DDA scheme that we are using represents the

mechanical state in each block as a uniform strain. This

means that spatial heterogeneity of stress cannot be

reproduced if the blocks are too large. The fracture-

bounded blocks of the flexure models presented here

are overly large, relative to the spatial variability of

stress indicated by continuum models of this deforma-

red (central) portions of the models depicted in Fig. 9, as a function of

se. Note the increase in effective flow properties as the hydraulic head

Fig. 12. Model showing the impact of increased fracture density

(compare with Fig. 11). The DDA blocks to either side of the

fractured region are welded together. Note the hydraulic head is

more uniform compared to the low fracture density model (Fig. 11)

suggesting uniform fluid-flow through the fracture network.

Fig. 10. Loading case for asymmetric flexure models.

M. Rouainia et al. / Engineering Geology 82 (2006) 222–233 231

tion process. We show here one result from a model

configuration that has a much greater density of frac-

tures, but is otherwise similar to the models described

above. The results from the higher-density model (see

Fig. 12) are generally similar to those seen before, but

the details of the fluxes and pressure field show more

spatial variability.

5. Discussion and conclusion

Several algorithmic and modelling aspects of the

HYDRO–DDA coupled hydro-geomechanical model-

ling environment, using a staggered calculation of the

fluid pressure and the deformation of fractured porous

media, are discussed. The presented numerical applica-

tion indicates the potential of using HYDRO–DDA in

the analysis of overpressure retention and especially in

Fig. 11. Example results for asymmetric flexure model configuration. The top image for each case shows the block configuration along with vector

representing the flux. Note that the blocks on the left and right sides of each model are welded together (no fractures). The lower portions of the

figure plot the hydraulic head contours. The four cases illustrated (20, 15, 10, and 5 MPa mechanical loads) show an interaction between the

hydraulic head differential and the mechanical loading. Note that these models develop significant pressure anomalies (compare Fig. 8).

,

the maintenance of overpressure by fractured mudrock

seals.

The simulations produced using HYDRO–DDA

show that, even in very simple systems (both geomet-

rically and in terms of loading) complex, non-linear

behaviour results. This is both intuitively satisfying

and reproduces the generalities of the natural systems

they represent.

The non-linearity produced by the six-block models

reinforces concerns about overly simple predictions of

s

M. Rouainia et al. / Engineering Geology 82 (2006) 222–233232

fluid system/rock system interactions when the fluid

and mechanical loads are known or thought to be

simple. There are two elements to this problem. First

these results strongly suggest it could be unsafe to

presume presence or absence of open fractures just

because the mechanical loading has a significant exten-

sional or contractional element. Second subsurface sys-

tems have only one true boundary: the land surface and

most models of interest are deep enough that this

boundary is scarcely relevant. Normal practise is to

assume a boundary some distance from the area of

interest and to assign simple loadings to this boundary

such that they match the perceived average mechanical

setting for that area. In many situations this is the only

available approach, but in some circumstances it can be

appropriate. But the non-linearity exhibited by the

HYDRO–DDA simulations adds to concerns over

whether such simplifying assumptions can be applied

appropriately in all cases.

The flexure examples, particularly the asymmetric

model, also show marked non-linearity. For the sym-

metric flexure model, the fracture aperture and flow

distribution are nearly symmetric about the models

centreline but they vary through the region that might

otherwise be expected to flex uniformly. The assigned

fracture systems are simple but reasonable, but there are

other realistic patterns that could also be considered.

Furthermore, any new fracturing that may occur as

deformation proceeds may change the response signif-

icantly. It is also clear that a larger number of blocks or

a better description of their deformability could better

capture spatial variability of the mechanical state.

In all cases the system behaviour was easy to predict

in general. This situation is encouraging in light of the

need to upscale the mechanical and fluid behaviour of

these systems to allow the simulation of large portions

of hydrocarbon reservoirs (see Christie, 1996; Pickup et

al., 1995).

The HYDRO–DDA calculation environment is

shown to be useful in developing new understanding

of interactions and system responses when there is a

coupling between fluid flow and the deformation of

blocky rock masses. HYDRO–DDA differs from

other approaches (e.g. Zhang and Sanderson, 2002) in

that it accounts for the fluid flow that occurs through

the matrix. When the matrix permeability is extremely

low, the UDEC scheme (as used by Zhang and Sander-

son, 2002) may be an appropriate simplification. How-

ever, the ability of the matrix materials to impact the

pressure distribution, and hence the rock mechanical

response, should not be ignored. Even if fluid flux is

low in the matrix, there is still the potential for the

pressures to be affected by the matrix continuum. The

ability to run models with or without matrix flow will

allow that question to be addressed.

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