## Abstract

An areal model of a fractured/faulted reservoir with 49 wells is developed that incorporates fully-coupled geo-mechanics and fluid flow. It is a generic example of a pattern waterflood although it is inspired by a parallel study of the Gullfaks reservoir in the North Sea, in which stress-related, fault-related and long-range correlations in rate fluctuations are observed. Based on this model, three scenarios are examined in terms of different initial stress states prior to production, each of which involves 36 months of production and injection in the presence of fracture sets and faults. The results support the concept that the long-range, stress-related and fault-related characteristics of correlations in rate fluctuations, observed not only in the Gullfaks data, but also in several other fields worldwide, are symptomatic of a system near a geomechanical critical point. These characteristics are not observed in models that are sub-critical. Short-range rate correlations are likely to exist where there are highly permeable zones between producers and injectors. Long-range rate correlations occur only within critically-stressed regions where there is active shearing or fault reactivation. The modelling results are consistent with field evidence suggesting that incipient shearing is an important mechanism coupled with reservoir flow behaviour.

A range of phenomena associated with reservoirs from original hydrocarbon migration, to production-related surface subsidence and induced seismicity are related to deformation, faulting/fracturing, and their interaction with fluid flow (Maillot *et al.* 1999; Bruno 2002; Barkved *et al.* 2003). To understand the geomechanical influences that affect the temporal and spatial correlations of well-rate fluctuations, fully coupled geomechanical–flow modelling approaches are used to investigate the characteristics of correlations in well-rate fluctuations based on the finite element method.

The models simulate fluid flow and the geomechanical reactions of reservoirs during production, such as reactivation of faults and pre-existing discrete fracture networks, creation of new fractures and rock matrix deformation. Any of these dynamic events can result in changes with time in the conductivities of faults, fractures and rock matrix, and hence in the total permeability field of the reservoir, which will be reflected in changes in the production and injection rates at individual wells. The models are therefore able to incorporate the geomechanical influences on flow rates at wells, and so allow examination of these influences on spatial and temporal correlations in flow rates at pairs of wells as the rates fluctuate due to an imposed noise. Fracture sets are modelled as planes of altered stiffnesses and potential failure without specific location in individual finite elements; they could be taken to represent a range of sizes, from micro-fractures to fractures at the scale of the element. Mechanical properties are assumed for the fracture sets, faults and intact rock. Initial stresses applied across the model determine whether or not the simulated reservoir is at a critically-stressed state prior to production and injection.

The overall deformation of the fractured/faulted reservoir is controlled by potential Coulomb failure of the individual fractures and faults and also of the surrounding intact rock. With this complexity and the time varying nature of the conditions, in some cases approaching a critical point, it is very difficult or impossible to employ analytical techniques. Here, a coupled geomechanical–flow numerical modelling approach based on the finite element model is used to investigate the hydro-mechanical responses of a faulted and fractured reservoir.

## Stress state within and surrounding fractured/faulted reservoirs

Geological evidence suggests that the hydraulic properties of fractures and faults evolve continuously in space and time (Evans *et al.* 1997; Ngwenya *et al.* 2000; Main *et al.* 2001). We will show from our investigations that this is particularly true within and around a ‘fractured/faulted’ reservoir during a production period. By ‘fractured/faulted’ we imply a reservoir containing either fractures or faults or one containing both. We are not specific in this labelling because, although our model contains only one particular configuration of faults and fractures in this study (Fig. 1), we do not yet have cause to suggest that the exact configuration of the structural features, among the many natural possibilities, is a key component in the behaviour that we will describe.

Due to reservoir depletion and/or injection, local pressure variation around a well changes the effective stress state and may either promote or suppress deformation in surrounding areas, which leads to the further changes in stress and pressure within the adjacent regions. This is now widely recognized in seismic hazard studies (e.g. Scholz 1990; Harris 1998).

Most shallow seismicity is associated with slip along pre-existing faults (Rummel *et al.* 1986; Spicak *et al.* 1986). For a single fault the shear strength may be expressed by the Coulomb failure criterion
1
where *τ* is the shear stress along the fault, *σ*_{n} is the normal total stress acting across the fault surface, *P* is the fluid pressure, *ϕ* is the friction angle and *C* is the cohesion.

Variations of local stress state within the rock mass make analysis of the stability of a system of faults according to the Coulomb criterion a complex calculation. More importantly, failure locally within the system changes the stress and may either promote or suppress deformation in surrounding areas. This principle also applies to the stress state within a faulted reservoir due to production/injection operations.

The stress state surrounding a system of faulted rocks in equilibrium can be expressed with the principal effective stresses (*σ*′_{1} and *σ*′_{3}) and orientation (*θ*), where the Coulomb failure stress (CFS) (Harris 1998) is
2

A change in the Coulomb failure stress, ΔCFS, can be due to any change in the principal stresses: 3

If ΔCFS>0, the region becomes closer to instability (fault reactivation and or new fault initialization), whereas if ΔCFS<0, it moves farther from instability.

During the production and/or injection of a fractured/faulted reservoir, the variation of the effective stress path is complex. In very general terms, production results in an increase in the effective principal stresses due to reservoir depletion, and fluid injection reduces the effective principal stresses. However, the change of the effective stresses at a specific location is strongly related to other conditions, such as the variation of mechanical properties, the presence of faults/fractures and the changes in fluid pressure gradients with evolution of permeability. There are at least three possible effective stress paths to bring a stable stress state to an unstable stress state: (a) *σ*′_{1} increases and *σ*′_{3} decreases; (b) both the *σ*′_{1} and *σ*′_{3} decrease, and (c) the *σ*′_{1} increases and *σ*′_{3} is kept constant.

In fact, the effective stress changes within a faulted reservoir are much more complex than the three stress paths above. The complex stress paths may be understood by reference to a state boundary surface, as discussed by Zhang & Sanderson (2001). In principle, the state boundary surface represents a surface, below which a fault/fracture system is stable and the fault/fracture system is unstable if the stress state reaches it. Any stress state which would touch this surface on application of a small positive stress change is at a critical stress state.

The stress state in a fractured/faulted reservoir during production/injection is variable from place to place and is changing progressively. The stress paths are further complicated by the interaction among stress, strain, pressure and permeability, which results in both changes of magnitude and direction. Studies have revealed a strong correlation between the directionality of reservoir fluid flow and the local earth stress in reservoirs (Heffer & Lean 1993; Heffer & Koutsabeloulis 1993).

Changes in pore pressure in a reservoir affect not only the effective stress state, but also in many cases the total stress state (Addis 1997; Hillis 2001). The stress path is defined as change in total horizontal stress, *S*_{h}, induced by change in the pore pressure, *P*_{p}. Usually, a stress path parameter, *A*_{p}, is defined as *A*_{p}=Δ*S*_{h}/Δ*P*_{p} for an extended region or whole reservoir: its value generally depends upon the deformational responses of the surrounding rock mass including nearby faults. For simplicity in this study, in which average pressure changes are relatively small, it is assumed that the parameter, *A*, is unity, and the uncertainty in effective stress has been addressed in different cases by using different stress ratios.

## Fully coupled modelling of fluid flow and stress/strain behaviour

There are several assumptions about the variables involved in coupling between flow and stress/strain (Koutsabeloulis & Hope 1998; Maillot *et al.* 1999; Settari & Walters 1999). In this study, the coupling between flow and stress/strain involved the modification of the permeability of fractures and/or faults. In this way, the development of an extensional normal strain of fractures and/or faults increases the permeability of the fractures/faults, and the modification of permeability has an impact on the pressure distribution, which leads to the change of the effective stresses: those effective stresses must be compatible with the fracture strains in a consistent scheme.

In coupled geomechanical–flow numerical modelling, the pressure predictions due to fluid flow changes are used by a stress simulator to provide predictions of deformation and to update pore/aperture volumes, which are in turn used by a flow simulator.

The effective stress calculations are performed at pre-selected times known as ‘stress steps’. The simulations are termed iterative if at a given stress step pore/aperture volumes are repeatedly updated in response to pressure predictions from the flow simulator. The simulations are termed explicit if only one pore/aperture volume update is permitted at a given stress step.

In this current study, a fully coupled simulation is performed, in which the deformation, pressure and stress are simultaneously calculated at each stress step. The fully coupled approach offers internal consistency for solution of the hydro-mechanical equations, i.e. the fluid-flow equations and force-balance equation of continuum mechanics, and bypasses the need for explicit and iterative coupling strategies.

Reservoir rock behaviour is often time-dependent, with reservoir pressure and the hydraulic boundary conditions playing a part. To account for these effects, it is necessary to combine the equations governing the flow of fluid through the reservoir rock with the equilibrium and constitutive equations of the reservoir rock in the finite element model. To achieve that, this model makes use of the principle of effective stress.

The principle of effective stress is written as:
4
where *σ* is the total stress, *σ*′ is the effective stress, *p* is pressure, {*m*} is equal to unity for the diagonal components of stress and zero for the off-diagonal components, *K*_{T} is the bulk modulus of the reservoir material (solid phase plus voids), *K*_{S} is the bulk modulus of the solid phase, and Δ indicates an increment.

The Darcy velocity of fluid flow through the reservoir rock is defined by Darcy's law.
5
where [*K*] is the permeability tensor.

The equation of continuity and compressibility for the fluid is given as:
6
where *Q* represents any fluid flow of the production and injection, ε is strain, *v* is flow velocity, *n* is the porosity and *K*_{W} is the bulk modulus of the fluid.

The hydraulic behaviour of fractured/faulted rocks is considerably different from that of rocks with just matrix porosity, particularly if the fractures are connected. In this case, the fracture networks provide fast flow channels and the pores mainly provide fluid storage. Understanding fluid flow through fractures/faults is essential to the analysis of the hydro-mechanical properties and behaviour of fractured/faulted reservoirs. This is because the effective (bulk) permeability of fractured/faulted reservoir rock may be dominated by the permeability of fracture/fault networks, which is usually highly anisotropic. Generally, the effective permeability can be variable during reservoir depletion and injection, a function of fluid pressure and stress state that is non-linear and spatially variable.

A considerable amount of experimental and theoretical research suggests that there is an approximate cubic law for flow through fractures (Witherspoon *et al.* 1980; Raven & Gale 1985), which can be expressed as:
7
where *Q* is the flow per unit length normal to the direction of the flow, *d* is the hydraulic aperture of fractures, and d*p*/d*x* is the pressure gradient in the direction of the flow.

The permeability of a fracture system can also be highly dependent upon the stress state due to changes in the connectivity of fractures close to a percolation threshold (Zhang & Sanderson 1998). Significant permeability changes in fractures/faults can occur due to inelastic deformation despite injection pressures being much lower than the confining stress (the minimum total principal stress). In this case, the increase in permeability in the reservoir rocks can be due to dilation normal to the surface of the faults/fractures (even in a compressive stress regime), which is caused by shearing along the faults/fractures, rather than hydro-fracturing. Such shearing need not actually produce localized dynamic slip to significantly affect the permeability.

Another important feature of fracture/fault flow is that flow tends to an irreducible, aperture-dependent limit at the highest normal stress and smallest aperture. This means that a fracture is still to some extent permeable even under a very high normal compressive stress.

In this study, an assumed relationship between permeability change and fracture normal strain change based on the above considerations is used. It is assumed that the fractures/faults possess a base-level permeability prior to production and injection, which corresponds to the residual permeability of the fractures/faults. This implies that the apertures of fractures/faults have closed to the irreducible limit under the reservoir depth, but a minimum hydraulic aperture still exists. Such deformation-related hydraulic behaviour under high normal stress was also observed by Goodman (1976), Bandis *et al.* (1983) and Cook (1992).

In the areal model of this study, eight sets of fractures with an interval of 22.5° were simulated within the reservoir rock. A nominal permeability enhancement function was used to simulate the permeability increase of fractures due to fracture dilation.
8
9
where *K*_{F} is the original permeability along fractures, Δ*K*_{F} is the enhancement of permeability along the fracture, *ε*_{n} is the strain change normal to the fracture, *ε*_{nmax} is a given value for the upper limit of the strain change, A is a constant for the maximum enhancement of permeability, and f( ) is an assumed function (see Fig. 2). It is assumed that the original permeability, *K*_{F}, is the residual value, which implies that the apertures of fractures within the reservoir rock have the maximum closure at the reservoir depth prior to production and injection operation. In addition, a maximum permeability is assumed where the increase of fracture normal strain is larger than 2%.

## Model geometry, mechanical properties and boundary conditions

To understand the temporal and spatial correlations between producers and injectors, in terms of their pressure, stress, deformation, permeability and flow-rates, a 2D plane strain areal model has been developed to investigate the well rate correlations during oil field production. The model has a size of about 21 km by 21 km with 49 wells (25 producers and 24 injectors) in the central region of 8 km by 8 km. The distance between the wells is 1 km. There are 8 fracture sets within the intact rock with directions of 0, 22.5, 45, 67.5, 90, 112.5, 135 and 157.5°, respectively. In addition, there are 3 major faults in the north–south direction and 7 smaller faults in the east–west directions, as shown in Figure 1. The simulation of the smaller east–west faults is to understand the influence of the intersections of these faults with the major north–south faults on the initialization of shear zones. The assumed hydraulically-reactive width of these faults, 100 m, is governed by the element size of the model, but such an extent of damage zone is not atypical for faults that are a few kilometres in length.

The gradient with depth of the maximum horizontal total stress (*S*_{H}) is 24 kPa m^{−1} and the gradient of the minimum horizontal total stress (*S*_{h}) is 13.5 or 17.5 kPa m^{−1} in different cases. The initial reservoir pressure gradient is 11.5 kPa m^{−1}. The more anisotropic horizontal stress state is chosen in order to place some of the fracture sets in a condition of critical stress in a strike-slip sense. This is consistent with the concept that much of the lithosphere is at or near a critically-stressed state, both locally (Barton *et al.* 1995; Zoback *et al.* 1996; Sanderson & Zhang 1999) and globally (Main & Al-Kindy 2002). For an assumed depth at 1000 m, the total maximum and minimum horizontal stresses and reservoir pressure are 24 MPa, 13.5 or 17.5 MPa and 11.5 MPa, respectively. The maximum horizontal principal stress axis, *S*_{H}, is at N0/180°E or N10/190°E in different scenarios.

A poro-elastoplastic (Mohr-Coulomb) constitutive model is applied to the reservoir intact rock and a Coulomb failure criterion is applied to the fracture sets and faults. The mechanical properties used for the intact rock, fracture sets and faults are shown in Table 1. The effective (bulk) permeability of the reservoir rock is assumed to be uniform prior to production with an initial value of 100 mD. However, a much lower initial permeability of 1 mD for fault-grids is assumed so that the faults serve as permeability baffles. The effective (bulk) permeabilities of the intact rock and fault-grids can increase if the fractures and faults develop an extensional normal strain due to plastic shearing, as detailed in the previous section, and enhancement of permeability can be different depending upon the properties of the fractures/faults. A boundary condition of zero displacement is assumed, which, due to the large extent of the model, has negligible influence on the results.

Rate fluctuations in an oilfield have two causes: (i) operator actions (wells on-/off-stream, changes to choke settings, workovers, platform downtime, etc.) that have direct effects on individual well pressures and rates; combined with (ii) inter-well responses that depend upon the physics of the reservoir behaviour. The rationale of the modelling is to simulate the operator-induced perturbations with random noise input to the well pressures that is spatially and temporally uncorrelated; and to analyse the well flow rates for correlations caused by responses through the reservoir. Therefore, the production/injection of the 49 wells subjected to monthly variation of well pressure is simulated for 36 months. During this period, the pressure at each of the 24 injectors varies randomly, with a Gaussian distribution, about an average well pressure that is 2.067 MPa (300 psi) higher than reservoir pressure. During the same time, the pressure at the 25 producers also varies randomly about an average well pressure that is 2.067 MPa lower than reservoir pressure. Due to the variation of well pressure and induced permeability change, the flow-rate at each of these wells changes monthly according to its pressure and permeability variation. Then, the correlations over time of flow-rates between each pair of the 49 simulated wells are analysed with the non-parametric Spearman-rank method. In this the history of rates from each well is ranked and the correlation coefficient formed from the consequent time series of ranks from each pair of wells.

To investigate the effects of stress state on well rate-correlations, three scenarios have been investigated:

In Case 1, the direction of the maximum horizontal principal stress

*S*_{H}is at the N10/180°E (Fig. 3) and the ratio of total stresses, S_{h}/*S*_{H}, is 0.56, providing a critical stress state prior to production under the given mechanical conditions (note that the non-zero values of cohesion chosen for the faults and fractures contribute slightly to their stability).In Case 2, the maximum horizontal principal stress

*S*_{H}is in the north–south direction and the ratio of total stresses, S_{h}/*S*_{H}, is 0.56, also providing a critical stress state prior to production under the given mechanical conditions, but with active faulting and permeability enhancement different in detail from those in Case 1.In Case 3, the maximum horizontal principal stress

*S*_{H}is also in the north–south direction. The total minimum horizontal stress is higher and the total stress ratio of*S*_{h}/*S*_{H}is 0.73, at which the stress state prior to production is sub-critical under the given mechanical conditions.

## Well-rate correlations under different stress conditions

Each of these three production/injection scenarios is run for a modelled period of 36 months. In all cases, the same mechanical properties, boundary conditions, production and injection pressure histories, and well configurations are applied. The differences in the three cases are the direction of the maximum horizontal principal stress and the initial stress state, governed by the chosen *S*_{h}/*S*_{H} ratio. In Cases 1 and 2, the initial stress is close to the critical state, at which a small change of the effective stress due to fluid pressure changes in the reservoir is likely to trigger global hydro-mechanical reaction of the reservoir, irrespective of whether the change is at a local scale or a global scale (Zhang *et al.* 2007). In Case 3, the initial stress state prior to production is in a subcritical state. The initial stress state in the three cases is illustrated in Figure 4.

Shear failure in the model is governed, not only by the assumed initial stress state, but also by the cohesions and orientations of the individual faults and fractures and the complexity of the stress state that evolves during production and injection.

### Case 1: *S*_{H} at N10/190°E with *S*_{h}/*S*_{H} ratio of 0.56

The pressure changes at the producers and injectors cause changes in effective stresses that, because of near-criticality, result in the development of plastic shear strains around wells. Figure 5 shows the developed plastic shear strains around two producers P19 and P20 and two injectors I18 and I19 (see Fig. 3) after the production of 1 month and 4 months. Although a relatively small change in pressure occurs, significant plastic shear strains develops around the two injectors due to the critical stress state prior to production. It is apparent that the developed plastic shear strains are larger around injector I19 than around injector I18. This is due to a major fault near injector I19 and the proximity to the intersection of the major fault and a smaller fault, exemplifying fault-related reservoir rock failure. Note that the direction of the developed plastic shear bands is in the NE direction, which makes an angle of about 30 degrees with the *S*_{H} direction, exemplifying reservoir rock failure related to stress direction.

The changes in plastic shear strains with production/injection between 1 and 4 months that are detectable by close examination of the details of Figure 5 may seem slight, but, due to the high sensitivity of fracture permeability to strain, cause significant permeability changes (see Fig. 6) and therefore significant flow rate changes at the wells. Particular permeability enhancement is observed along the plastic shear bands developed around injector I19, which therefore support high and time-dependent flow velocities, as shown in Figure 7. Generally, significant changes in well rates are governed by fault-related, stress-related and time-dependent geomechanical phenomena.

Figure 8 shows the Spearman rank correlation coefficients of rates at 49 wells during 36 months of production/injection. For most of the wells, the rate correlation coefficients are less than 0.5. However, significant positive correlation coefficients exist between some well pairs. For each of 49 wells, there are 48 correlation coefficients between 48 pairs of wells. From the approximate t-statistics calculated from the correlation coefficients with 34 degrees of freedom, the correlation coefficients corresponding to the 1% and 5% significance levels are 0.424 and 0.329 respectively. A correlation coefficient of 0.2 is significant only at the 24% level.

A map of permeability after 36 months of production/injection in Case 1 is shown in Figure 9 with the well rate correlation coefficients (the highest value at a well) indicated by the colour code of the circular well symbols. Figure 10 shows the flow velocity pattern of the reservoir at the end of month 36 with the same indicators of rate correlation coefficients. The three wells showing highest positive rate correlations are related to a fast flow channel over a short range.

An interesting feature of short-range rate correlations is their dependence on the type of wells involved. Note that there is a highly permeable zone between injectors I16 and I19 (see Figs 3 & 9), but the rate correlation coefficient between the two is only 0.1. This suggests that short-range rate correlations may not exist between injector pairs or between producer pairs. One possible reason is because the sensitivity of flow variation between a pair of wells is related to not only the permeability, but also the pressure gradient between them. The permeability between two injectors and between two producers might be high, but the pressure gradient between two producers and between two injectors is relatively low. Alternatively, for short inter-well distances between well pairs, the hydraulic interference is negative (increasing rate at one well tends to reduce rate at the offset well), whilst the geomechanical interference is positive (strain dilatation at one well tends to produce dilatation at the offset well). It is possible that those opposing trends tend to cancel each other. For either reason, correlated flow variation is not significant between two close injectors or between two close producers: short-range rate correlations (hydraulic links) only exist between an injector–producer pair.

Figure 11 shows the plastic shear strain pattern of the reservoir at the end of month 36 with selected rate correlation coefficients indicated. Here, high positive rate correlations are marked only between non-neighbouring wells. It is apparent that such long-range high positive rate correlations occur near the plastic zones (the major faults and the created plastic shear bands). This suggests that long-range rate correlations are predominantly related to geomechanical links. The long-range rate correlations occur between injectors and producers, between producers and between injectors, and appear to involve a different mechanism from that of short-range rate correlations. There may be no direct flow between a pair of wells that have a long-range correlation insofar as some of the well pairs with high correlations at long-range are separated by low permeability faults.

As demonstrated by Zhang *et al.* (2007), a small change in effective stress may trigger global geomechanical and hydraulic reaction, no matter whether the change is at a local scale or a large scale. If a pair of wells separated by a long distance are within a system at a critical stress state, the hydraulic change around one well (for example pressure change and flow change) may cause the geo-mechanics to change locally, then the local change in geo-mechanics may trigger a geomechanical change around the other well, despite the large separation. The geomechanical change around the remote well may, in turn, lead to the change in its hydraulic responses, such as permeability, pressure and flow. Thus, the mechanisms of long-range rate correlations are local hydraulic responses plus long-range mechanical responses plus local hydraulic responses (local-hydraulic+long-mechanical+ local-hydraulic). In this way, correlated flow rates may occur between a pair of wells over a long distance, even though no direct hydraulic links exist between them. Such long-range rate correlations require the presence of a critical stress state encompassing both wells, when the rocks are strongly susceptible to small perturbations (metastability) and may respond at a long distance from perturbing loads.

### Case 2: S_{H} in the north–south direction with S_{h}/S_{H} ratio of 0.56

In Case 2, the direction of the maximum horizontal principal stress is north–south, parallel to the major faults (Fig. 12). The ratio of the total minimum horizontal principal stress (*S*_{h}) to the total maximum horizontal principal stress (*S*_{H}) is still 0.56. Under these conditions, the stress state prior to production around the reservoir is again largely close to the critical state. The pressure changes around the wells cause plastic failure around the injectors, and the permeability in these regions changes. However, the induced plastic shear zones and highly permeable channels are locally different from those in Case 1.

Figure 13 shows the Spearman rank correlation coefficients of rates at 49 wells during 36 months of production/injection. Similar to Case 1, most of the well pairs have a rate correlation coefficient less than 0.5. Again, significantly higher positive correlation coefficients exist between some well pairs. Figures 14 and 15 show the correlation coefficients projected on the permeability pattern and the flow velocity pattern, respectively. The plastic shear strain pattern of the reservoir at the end of month 36 with the indicated rate correlation coefficients is shown in Figure 16.

The high correlations that are related to fast flow channels are marked in Figure 15, and the high positive rate correlations between non-neighbouring wells are marked in Figure 16. Again, short-range correlations are related to hydraulic links and occur between adjacent producer–injector pairs only. Long-range correlations are not related to hydraulic links and are likely to occur near plastic zones (the major faults and the newly-created plastic shear bands). The long-range rate correlations may occur between injector–producer, producer–producer, or injector–injector pairs.

### Case 3: S_{H} in the north–south direction with S_{h}/S_{H} ratio of 0.73

In the previous scenarios, the stress prior to production around the reservoir is close to the critical state. In this case, the same horizontal stress direction in Case 2 is applied, but a higher *S*_{h}/*S*_{H} of 0.73 is used. Under these conditions, the stress prior to production is well below the critical state, which we term sub-critical. The same production and injection schedule is performed. Spearman rank correlation coefficients of rates at 49 wells during 36 months of production/injection (Fig. 17) show that unlike Cases 1 and Case 2, no significantly high positive correlation coefficients exist between well pairs. Figure 18 shows the mean effective stress around the wells at the end of month 36. Essentially, due to the subcritical stress state prior to production, the effective stress changes due to pressure change around the wells during production and injection violate neither the Mohr-Coulomb failure criterion for the intact rock, nor the Coulomb failure criterion for the fractures and faults. As a result, no plastic shear failure occurs during the period of production and injection, and no plastic-failure-induced permeability enhancement develops. Thus, the permeability of the reservoir rock around the wells is still uniform, and the faults still serve as permeability barriers. Under such hydraulic conditions, the flows around the wells are determined by the pressure change only, as shown in Figure 19.

Neither strong geomechanical links nor strong hydraulic links exist between the wells. For such a reservoir, at a stress state which is not critical prior to production, no strong correlations are expected, as shown in Figure 17. Almost no well pair has a rate correlation coefficient larger than 0.5, except for producers P4 and P12 which have a coefficient of 0.53. The results in this case demonstrate that reservoir stress state prior to production plays an important role in the hydraulic and geomechanical responses of a reservoir, and thus strongly influences the correlations between the wells.

## Discussion

In this study, the injection pressure is much lower than the confining stress (the minimum total principal stress), so no hydraulic fracturing occurs. The increase in permeability in the reservoir rocks is due to the dilatation normal to the surface of the faults/fractures, which is caused by the shearing along the faults/fractures. This suggests that strong rate correlations might occur over broad areas characterized by compressive stress regimes in which normal components of fault displacements are likely to be large due to significant dilation. The spatial decay of the correlations in rate fluctuations is very different between the near-critical and subcritical cases. Proportional cumulative correlation coefficients are calculated for the three cases. These are defined by ranking the well pairs in order of their separation distances, accumulating the Spearman rank correlation coefficients with increasing distance, and then dividing by the cumulative number of well pairs that are spaced within each distance. Figure 20 shows these calculated values against distance for the three cases. For the near-critical cases, 1 and 2, the spatial decay of correlation approximates a power-law, albeit only over 1 order of magnitude in distances. The exponent of the power-law is ∼−0.7±0.1. For the sub-critical case 3, the correlations are an order of magnitude lower at short distances, and decay rapidly even further at larger separation distances. The distance at which the rapid decay begins is calculated to correspond to a dimensionless diffusion time *t*_{D}=*Kt*(*S _{σ}μr*

^{2}) of about 0.25 (

*K*is permeability;

*t*is time, taken as the one month time-step;

*S*is the storage at constant stress; μ is the fluid viscosity;

_{σ}*r*is the distance); this is the approximate dimensionless time which limits the influence of Darcy diffusion, beyond which correlations due to Darcy flow would indeed be expected to rapidly decrease. In contrast, the near-critical cases show a continued trend of correlation to greater distances. In terms of the cumulative number of correlated well pairs normalized by the cumulative total number of well pairs with distance, the field data also show apparent power-law decay (Fig. 21), although the exponent is ∼−0.15 to −0.25. The field data show no sign of dropping away at any distance in the manner of the numerical Case 3.

The interaction between deformation and permeability is likely to be the most important mechanism for rate correlation in fractured/faulted reservoir rocks. In the current study, only the permeability changes of fractures/faults are accounted for in the assessment of rate correlations. However, where the permeability of reservoir rocks is predominantly controlled by pore permeability, a more comprehensive model including updating permeabilities of intact rocks should be developed.

In spite of the limitations above, this geomechanical model can be used as an improved predictive tool for planning and managing waterfloods, particularly for fractured/faulted reservoirs. In conjunction with an assessment of the correlations in rate fluctuations, the geomechanical approach can assist in reservoir waterflood design, provide predictive power for production decisions, and examine the potential for rapid responses to geomechanical events. This is more important in fields with a small number of wells, where statistical techniques can only sparsely sample the full hydro-mechanical patterns within the field; and also in immature developments, where the hydro-mechanical patterns are yet to be established.

Long-range, fault-related and stress-related correlations in rate have been observed in several fields to date (Heffer *et al.* 1997). The commonality of their characteristics suggests that geomechanical influences are an intrinsic component of reservoir fluid flow. However, it may be that the stress states of the fields hitherto analysed are particularly close to critical, and that other fields are more stable; in other fields permeability may not be so sensitive to shearing, or deformation may not occur by brittle shear. In some fields the stress state and fault properties may be such as to cause the faults to be the dominant sites of shearing, in contrast with the new shear bands created during production in this model; such a change might well alter the detail of the characteristics of the rate correlations, although not the broad pattern. Further field analyses combining statistical and geomechanical assessments are necessary in order to define the limits of the hydro-mechanical behaviour described in this paper.

The implication that geomechanics is generally intrinsic to reservoir flow behaviour at many scales carries with it the corollary that coupled geomechanical–flow modelling should at least be considered seriously for valid prediction of reservoir flows. Coupled modelling can be applied in different contexts:

As an investigation tool for conceptual studies. At this level, detailed information of a reservoir may not be required and the emphasis is put on the recovery mechanisms and the effects of individual parameters, such as stress condition, geomechanical properties, well configurations, lithology, porosity-permeability characteristics, degree of fracturing, sedimentology, trap type, reservoir geometry, etc. This type of modelling would enhance the understanding of recovery mechanisms.

As an aid in reservoir waterflood design to examine the directionality of reservoir flows. Due to the complexity of the stress field in some real reservoirs, the directionality of flow between a pair of wells is controlled not only by the far-field maximum horizontal stresses, but also by the presence of faults/fractures near the wells. It may also be significantly altered by contrasts in the areal pattern of offtakes and injections leading to strong pressure, and therefore stress, gradients; thermoelastic stress changes can also be significant. Coupled modelling can be used to predict the flow directions around wells within a local region, which is particularly important in regions where there are faults between wells. Under a given

*in-situ*stress condition and production and injection schedule, the flow directions between each pair of wells can be predicted.As a predictive tool for reservoir-specific production-injection simulations. At this level, detailed information for a reservoir is necessary, including the field geological setting, geomechanical properties, hydraulic parameters, the

*in-situ*stress condition, faults and the production/injection schedules. The results from this type of modelling may explain the different production efficiency of some producers and predict changes during the lifetime of a reservoir. In addition, these results can provide information to identify hydraulic compartments where no correlations exist. In this way, this geomechanical approach has the potential to predict the short- to medium-term oil production by examining the short- and long-range of rate correlations together with the geomechanical responses

## Conclusions

A fully coupled stress–flow model which allows the dynamic responses of geomechanical–flow behaviour at individual wells to be investigated, including pressure, stress, strain, permeability and fluid flow, applied to three scenarios near or below a critical stress state, has shown the following.

Support for the concept that the characteristics of correlations in rate fluctuations seen in a variety of field data (viz: long-range, stress-related and fault-related) are symptomatic of a system near a geomechanical critical point. These characteristics are not observed in models that are subcritical.

Short-range rate correlations are likely to exist where there are highly permeable zones between producers and injectors. This suggests hydraulic links are the dominant mechanism for short-range rate correlations, by which the direct communication of fluid flow between producers and injectors occurs.

Long-range rate correlations occur only within critically-stressed regions where there is active faulting or fault reactivation. Long-range effects are likely to be caused by non-linear geomechanical responses, particularly shearing, rather than by direct hydraulic links. In this case, direct hydraulic links between producers and injectors are not required. Therefore, direct communication of fluid flow between the producers and injectors may not exist.

The implication that geomechanical influences are an intrinsic component of reservoir flow in at least a large proportion of fields examined to date, carries the corollary that geomechanical modelling, potentially coupled with fluid flow, should at least be seriously considered for unbiased reservoir prediction in various aspects of field management and investment decisions.

## Acknowledgments

The work was carried out as part of the COFFERS project with financial support, secured through the Industry Technology Facilitator (ITF), from the following nine organizations: Amerada Hess, BG Group, BP, ConocoPhillips, DTI, Kerr-McGee, Statoil, Shell and Total. The authors would like to thank J. Walsh, R. Hillis and M. Ameen for their valuable comments.

- © The Geological Society of London 2007