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# Geomechanics in reservoir simulation

Historically, reservoir simulation has accounted for rock mechanics by simple use of a time-invariant rock compressibility cR , spatially constant or variable. In reality, rock mechanics is intimately coupled with fluid flow.

Rock mechanics is coupled with fluid flow in two aspects.

• Porosity changes are a direct result of the deformation of the skeleton, which is a complex function of both pressure and stress (effective stress state)
• Permeability of the media is also a function of effective stresses in the reservoir.

Therefore, rigorous reservoir simulation should include simultaneous solution of multiphase flow and stresses as well as the appropriate dependencies between these processes. While these couplings physically exist to some extent in all reservoirs, they can be often ignored or approximated when the reservoir behaves elastically. However, the changes in porosity and permeability are more pronounced when rock failure occurs, such as in compacting reservoirs or in high-pressure injection operations, and these processes require use of more complex, coupled geomechanical modeling.

## Types of coupling

There are essentially two main types of coupling between reservoir flow and stress:

• Pore volume coupling
• Flow properties coupling

The first led to the development of various simplified compaction modeling techniques, while the latter is reflected in the "pressure-dependent permeability" options available in many simulators. In a coupled geomechanical model, both can be treated more rigorously.

### Pore-volume coupling

The porosity in the reservoir model is traditionally treated as a function of pressure via rock compressibility:

....................(1)

and changes in block pore volume Vp are computed as Vp = Vb0Ф, where Vb0 is the block (bulk) volume. In reality, pore volume changes are a result of complex interaction of fluid (pore) pressure, stresses acting on the element of the rock and temperature. The deformation of the rock solid (also called the skeleton) caused by combination of stress and pressure changes results in changes in the bulk volume of an element Vb, which is computed at any conditions as

....................(2)

where εv is the volumetric strain at these conditions, and Vb0 is the bulk volume at reference conditions at which the volumetric strain = 0. Then the true porosity is given by

....................(3)

where Vp is the pore volume of the element. In Eq. 3, both pore volume and bulk volume are variable; therefore, true porosity and pore volumes are both functions of pressure, temperature and stress:

....................(4)

which shows the coupling between fluid flow and geomechanics (stress modeling).

In stress modeling, the changes in volumetric strain and porosity are calculated from complex constitutive relations of the material, which define both the stress-strain and volumetric behavior.[1][2] To compute pressure changes correctly in the reservoir simulator, it is necessary to force the changes in pore volume to be the same as computed by the stress model, which is the essence of the "volume coupling." This can be achieved in two ways. The rigorous solution is for the reservoir model to recalculate the block sizes based on the stress solution, and use the true porosity. However, reservoir simulators do not allow for modifying the bulk volume. In this case, one can define a pseudoporosity Ф* = Vp / Vb0 , which will give the correct pore volume.[3] In either case, the usual treatment of porosity by rock compressibility must be replaced by the coupling with stress-strain solution. The porosity changes become more complex when failure of the skeleton is reached either in compression (rock compaction) or in shear.

### Flow-properties coupling

The primary mechanism is the dependence of permeability on stress, usually of the form

....................(5)

This relationship is well established for many types of reservoirs[4] and generally becomes stronger as permeability decreases. It is dominant (and more complex) in fractured reservoirs where stress-dependent fracture aperture and reopening/creating fractures under injection can cause large, anisotropic changes. The tensor character of permeability may be important in these applications. In soft formations and unconsolidated sands, deformations can lead to porosity dilation, which will also lead to permeability enhancement. On reloading, there is a hysteretic effect. Two to three orders of magnitude enhancements in permeability can occur because of injection at fracture pressure (e.g., in microfractured rock[5] or coal seams).

The permeability changes are a function of some measure of effective stress, but are often simplified (and laboratory data reported) as a function of pressure. The creation of new fractures can cause a transition from initially single-porosity media to dual-porosity, and in such a case will also have an effect on relative permeabilities.

## Modeling of reservoir compaction and/or dilation

Modeling reservoir deformation is of considerable importance in soft and/or thick reservoirs where the results of compaction may provide an important production mechanism, cause well failures, and/or cause ground subsidence or heave with environmental consequences. Review of the compaction mechanics and its consequences for field development is provided by Settari[6]. Initial approach to modeling compaction was based on modifications of reservoir models.[7][8][9][10] The common feature of such reservoir compaction models is that the compaction is treated as a 1D problem (uniaxial strain) by assuming that (a) only vertical deformations take place, and (b) each vertical column of blocks deforms independently. In such models, the porosity changes are calculated by modifying the conventional compressibility cR as a function of pressure only, in the form of "compaction tables." The tables are based on results of uniaxial strain laboratory experiments, and the stress problem is not solved. The compaction of the reservoir is then obtained analytically assuming uniaxial deformation. The relation between reservoir compaction and surface subsidence can be then obtained by an independent solution of a stress problem using the computed compaction as a boundary condition. Dilation (increase of porosity) is an important geomechanical mechanism occurring during steam injection into unconsolidated sands. This process has been also modeled by the "compaction-dilation" tables.[11] In chalk reservoirs, increased water saturation in waterflooding causes weakening of the rock and therefore Sw is an additional variable in Eq. 4.[12]

The major drawback of the use of compaction tables is that the dependence on stress indicated in Eq. 4 must be either ignored, or the change of stress must be estimated in terms of change in pressure. A more accurate modeling approach is to couple in some fashion the reservoir simulator with stress-strain (geomechanics) solution. Such models typically combine the solution of the multiphase flow in the reservoir and elastoplastic solution of the deformations in a much larger domain including the reservoir, sideburden, underburden, and overburden. The majority of coupled models use the iterative coupling[13]; the different variations and their shortcomings are discussed next.

Coupled models have much larger computing requirements compared with a reservoir model of the same reservoir, primarily for two reasons: first, the larger number of unknowns per gridblock, and second, the stress-solution grid must be usually much larger laterally than the reservoir grid to eliminate the effect of boundary conditions as well as extend up to surface to provide subsidence solution. Combination of these factors leads to computing times typically of one order of magnitude larger compared with conventional simulation, and even more if elastoplastic solution is required for the stresses.

## Modeling of stress-dependent flow properties

The primary flow-dependent property is permeability, and the problems to model its dependency in Eq. 5 are similar to modeling the pore-volume coupling (Eq. 4). However, the problem is somewhat easier because stress-dependent permeability (or transmissibility) does not affect mass-balance formulation.

Again, the traditional approach is to use tables of k vs. pressure in an uncoupled model. However, the problem remains one of replacing the dependency on effective stress by one on pressure only. Even in a single-phase, single-porosity gas-flow case, different assumptions about the stress change during depletion can lead to large errors in well decline.[14] Different strategies for converting the stress-dependent data to pressure tables are based on local constrainment assumptions.[15]

In coupled models, the permeability dependency can be usually computed explicitly on a timestep basis, and "loose" coupling can be used. In fact, a "coupled" model that deals only with flow properties coupling and ignores the pore-volume coupling can be run successfully even if the stress solution is done in larger intervals of time compared to the reservoir. Such models have been used extensively to study permeability changes in waterfloods, particularly in fractured or jointed media.[16][17] Here, the advantage of coupled modeling is in its capability to predict the permeability changes from the geomechanics of reopening of fractures or failure (dilation) of joints.[18] The development of anisotropy is dictated by the orientation of fractures or faults, and requires a "full tensor" treatment of transmissibilities in the flow model. In the stress strain model, different methods of pseudoizing the fracture/joint networks into a continuum are used, which include predicting permeability as a function of effective stress and/or strain. While the need for the tensor transmissibilities in such models has been recognized,[19] in injection processes dual-porosity media can be created. Therefore, reservoir description may be changing in time because of geomechanics; this aspect has been ignored in coupled models to date.

The same principles can be also applied to model hydraulically induced fractures being represented by dynamically changing transmissibility multipliers in the potential fracture plane.[20] The effective stress dependency (as opposed to pressure in an uncoupled model) allows capturing the changes of the fracture propagation pressure with time, which can be large, in particular in steam injection. Another application is the prediction of production/injection-induced slippage on faults, which can induce communication between reservoir fault blocks and/or seismicity.

## Types of coupled models

Coupled models can be either of the following:

• Fully coupled (i.e., all unknowns solved simultaneously)
• Modular (reservoir simulator and stress code)

In the latter case, different coupling strategies can be used, with consequences for running speed and accuracy. The majority of coupled models use a conventional finite-difference reservoir simulator coupled with a finite-element (FEM) stress simulator. However, attempts have been made to develop fully coupled FEM codes,[21] and a fully coupled geomechanics was implemented in a commercial model using a finite-difference stress solution.[22] Considering the proliferation and sophistication of the geomechanics codes available outside the petroleum engineering, the modular approach offers the best solution.[4][12] Generally, the reservoir simulator is the "host" or "master." Commercial stress simulators are, in principle, easy to couple to it (in particular if only permeability coupling is considered).

Because of the extreme complexity and large computing requirements of coupled models, different simplifications have been developed. The main types of models (in the order of increasing rigorousness, but also computing time) are as follows:

### One-way coupling

Pressure and temperature changes are passed from the reservoir code to the geomechanics module, but no information is passed back on timestep basis. The geomechanics does not improve the flow solution, but the model can be useful for predictions of wellbore stability for infill drilling, fracturing pressures, and so on. Manual adjustment of "compaction tables" is possible manually through restarts. Such manually coupled solutions[23] of the stress problem (at intervals of time) or one-way coupling[24] were often used in early coupled modeling. The method can be satisfactory when the reservoir fluid system is highly compressible (i.e., in gas reservoirs), but can lead to errors when the porosity is strongly coupled to flow.

### Loose coupling

Reservoir and geomechanics modules are run sequentially on a timestep basis, passing converged solutions of flow and stress variables to each other. Pore volumes and permeabilities in the flow model are computed as a function of p, T, and σavg with stress variables lagged a timestep. However, the relationships are "distilled" into tables similar to the "compaction tables," but now a function of effective stress. The advantages are functional similarity to the uncoupled reservoirs with "compaction tables," no need for iteration during timestep, and the possibility of updating the stress solution less frequently than the reservoir solution. However, complex constitutive models of the solid (e.g., plasticity) may be difficult to represent.

### Iterative coupling

This method is shown schematically in Fig. 1. Iteration is carried out between the reservoir and stress solution at every timestep until the pore volumes and permeabilities calculated from the stress model and those used by the reservoir model agree. In each iteration, the previous guess of the Vpn+1 for the end of the timestep is used to converge the flow solution, and the changes of p and T over timestep are then used to solve for the new deformations and stresses, which in turn provide updated estimate of Vpn+1. The changes of permeability are also iterated on. Therefore, each "geomechanical" iteration costs the equivalent of a timestep solution of the previous methods. The original formulation of the coupling iteration[13] is always convergent, and its efficiency has been recently improved.[25]

The iterative coupling, when converged, is equivalent to a fully coupled code while it retains flexibility. In many problems, it is not necessary to fully converge the timestep, and if used with only one iteration per timestep, the method is similar to the loose coupling, except that the porosity is determined directly by the constitutive model of the solid rather than by a simplified relationship.

### Full coupling

This requires simultaneous formulation of the flow and stress variables and therefore results in larger matrices. The advantage is that consistent approach to discretization can be used, and the model is integrated from the point of view of code development. However, it is very costly to redevelop all the features of the physics and numerics now readily available in stress codes. Moreover, in published fully coupled models, the approach for solving the resulting matrix problem has been to partition the matrix in the same fashion as in the iterative coupling[26] and to apply the geomechanical iteration at the matrix-solution level. Therefore, the fully coupled formulation, which results in larger, strongly nonlinear matrix equations, does not reduce the difficulty of the problem, and it may need to use geomechanical iteration in the solution process as the best strategy. These aspects need further study.

## Future trends and needs

As a result of much larger computing requirements, coupled models lag behind conventional models in the size of the problems that can be currently handled. Therefore, they are a prime candidate for the use of massively parallel hardware and will require large future development effort in parallelization. Because of the increased understanding of the complexities of the geomechanics, the current trend is also toward more strongly coupled models with fewer simplifications. This further increases the computing requirements.

Given that not all problems require use of geomechanics, and the cost of the study may increase dramatically, it is important to be able to identify when coupled simulation is needed, and what approximations can be made without compromising the answers. There are no simple rules, but there is a growing need to conduct a "screening" process at an early stage of a reservoir study to determine if geomechanics is an issue. This process requires an integration of reservoir, production, and completion engineering data as well as field experience.

Finally, coupled geomechanical modeling is the future tool for truly integrated reservoir management. Conventional reservoir simulation studies ignore numerous constraints placed on the development scenarios from the point of view of drilling, completion, and operations. These constraints can be incorporated into coupled models, and additional modules can be integrated (e.g., long-term wellbore stability and sand production predictions, subsidence management, 4D seismic interpretation, and so on[27]).

## Nomenclature

 p = polynomial T = time, time units or transmissibility, md•ft/cp V = vapor phase or volume Vp = pore volume, bbl Ф = porosity, fraction

## Subscripts

 R = reservoir

## References

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