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Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004
Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004

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Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004

Paper Number: ARMA-04-502

... transportation viscosity solid particle solid concentration complex reservoir concentration case high concentration slip velocity final geometry operation fracture numerical study reinjection operation geometry

**equation**ABSTRACT: In order to evaluate the growth of fractures for cuttings re...
Abstract

ABSTRACT: In order to evaluate the growth of fractures for cuttings re-injection, a solid transport model is included in a solidfluid coupled hydraulic fracturing simulator. The fracture geometry is a critical factor affecting the safety of the re-injection operation, and solid particle flow in the fractures is known to have a dominant effect on the fracture propagation. To improve the accuracy of the simulation, the finite element method (FEM) is introduced for modeling the particle motion in the fracture fluid. In the model, opening of the fracture, interaction between multiple particles, and change in viscosity by the solid concentration are taken into account. Numerical examples shown here reveal that the fracture geometry is highly dependent on the concentration of the solid due to the change of gravity and slurry viscosity. The injected solid concentration is one of the few controllable parameters, thus the results suggest the feasibility of geometry control. INTRODUCTION Hydraulic fracturing technique is widely used in the petroleum industry for stimulating wells. Another application of the technology is drill cuttings reinjection, in which huge fractures are created in formations around wellbores to contain the slurrified solid waste produced by the drillings [1]. The major concern of cuttings re-injection is a breakthrough of fracture into adjacent formations and surfaces. If a fracture propagates into usable aquifers, petroleum reservoirs, or the surface or seabed, it can cause the grave environmental pollution and operational risk. Although this operation requires careful design of the fracture growth, there are few controllable factors, and those that are controllable are also restricted by operation margins. A numerical study using a solid transport model in the fracture shows that the solid concentration of the injected cuttings slurry influences the fracture growth significantly through the leak-off character of the formation [2]. The authors have developed another numerical simulator of hydraulic fracturing, in which the true three-dimensional geometry and interaction of multiple fractures are considered [3]. The solid transport model is added to the fully coupled model of fluid flow in the fracture and opening of the fracture in an elastic medium. For the cuttings slurry problem, we need an accurate solution for the case of a high concentration of solid particles. In this paper, we demonstrate a solid transport model that considers the effect of the fracture wall and in the interaction of multiple particles. Furthermore, some numerical results for different concentrations of injected solids are shown to exhibit the effect of this parameter on the final geometry of the fracture. The slurry viscosity and vertical pressure gradient can be manipulated by varying the solid concentration in the slurry, so the fracture growth is controllable by this parameter. NUMERICAL MODELING A fully coupled model of a hydraulic fracturing simulator is developed for the design of well stimulation in complicated stress state and well and fracture geometries [3]. The coupled solution of the fluid pressure and fracture opening is computed using the displacement discontinuity (DD) methodfor solid, and the finite element method (FEM) for Newtonian or non-Newtonian fluid.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004

Paper Number: ARMA-04-494

... interaction

**equation**ABSTRACT: This paper presents a fully coupled reservoir-geomechanics model with erosion mechanics to address wellbore instability phenomena associated with sand production within the framework of mixture theory. A Representative Elementary Volume (REV) is chosen to comprise of...
Abstract

ABSTRACT: This paper presents a fully coupled reservoir-geomechanics model with erosion mechanics to address wellbore instability phenomena associated with sand production within the framework of mixture theory. A Representative Elementary Volume (REV) is chosen to comprise of five phases, namely solid grains ( s ), fluidized solids ( fs ), oil fluid ( f ), water ( w ) and gas ( g ). The particle transport and balance equations are written to reflect the interactions among phases in terms of mechanical stresses and hydrodynamics. Constitutive laws (mass generation law, Darcy's law, and stress-strain relationships) are written to describe the fundamental behaviour of sand erosion, fluid flow, and deformation of the solid skeleton respectively. Subsequently, the resulting governing equations are solved numerically using Galerkin's method with a generic nonlinear Newton-Raphson iteration scheme. Numerical examples in a typical light oil reservoir are presented to illustrate the capabilities of the proposed model in the absence of the gas phase. It is found that there is an intimate interaction between sand erosion activity and deformation of the solid matrix. As erosion activity progresses, porosity increases and in turn degrades the material strength. Strength degradation leads to an increased propensity for plastic shear failure that further magnifies the erosion activity. An escalation of plastic shear deformations will inevitably lead to instability with the complete erosion of the sand matrix. The self-adjusted mechanism enables the model to predict both the volumetric sand production and the propagation of wormholes, and hence instability phenomena in the wellbore. 1. INTRODUCTION The production of formation sand has plagued the oil and gas industry for decades because of its adverse effects on wellbore stability and equipment, while it has also been proven to be a most effective way to increase well productivity. When hydrocarbon production occurs from shallow and geologically young (or so-called unconsolidated / weakly consolidated) formations that have little or no cementation to hold the sand particles together, the interaction of fluid pressure and stresses within the porous granular material can lead to the mechanical failure of the formation and unwanted mobilization of sand. It has been reported that 10%- 40% sand cuts normally stabilize in time to levels less than 5% in heavy oil reservoirs [1], while an average of 40% productivity increase was achieved through sand management in light oil reservoirs [2]. When sand is produced from reservoir formations, it can cause a number of problems. These include the instability of wellbores, the erosion of pipes, the plugging of production liners, the subsidence of surface ground, and the need for disposal of sand in an environmentally acceptable manner. Each year, these issues cost the oil industry hundreds of millions of dollars. Furthermore, sand production and control becomes extremely crucial in offshore operations where a very low tolerance to sand production is allowed. Hence, it is imperative to find an efficient computational model that has the predictive capability to assist field operators to understand this unique process. The ultimate goal is to design an economical well-production strategy in which sand production and operating costs may be reduced to some extent with maximum hydrocarbon productivity.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), June 5–9, 2004

Paper Number: ARMA-04-471

... thermo-mechanical coupling during injection operations (time scale of months/years). In fact, the phenomena of the variation of injectivity with injection water temperature and reservoir seismicity can be attributed to thermal stresses. In this paper a three-dimensional integral

**equation**formulation is...
Abstract

ABSTRACT: Poro-mechanical, thermal, and chemical processes can play a significant role when developing enhanced geothermal systems. These processes occur on various time scales and the significance of their interaction varies with the problem of interest. Of particular importance is the thermo-mechanical coupling during injection operations (time scale of months/years). In fact, the phenomena of the variation of injectivity with injection water temperature and reservoir seismicity can be attributed to thermal stresses. In this paper a three-dimensional integral equation formulation is presented for calculating thermally induced stresses associated with cooling of a fracture in a geothermal reservoir. The procedure is then implemented in a computer program and is used to treat the problem of injection into an infinite fracture. The thermally induced stresses are calculated using actual field data for an injection experiment. The resulting calculations are found to be consistent with those based on a semi-analytical solution as well as field observations. 1. INTRODUCTION Thermally-induced stresses significantly contribute to seismicity in petroleum and geothermal fields [1, 2]. The variation of injectivity with injection water temperature and reservoir seismicity in geothermal fields have been attributed to thermally-induced stresses. It has been has found that half the earthquakes in The Geysers field seem to be associated with cold water injection [2]. The mechanism by which seismicity occurs is well understood namely, shear slip on natural fractures resulting from a reduction in effective stress acting across the fracture. The magnitude of the thermal stresses associated with advective cooling has been estimated analytically [3] using an axisymmetric model of injection into a planar reservoir and a 1D heat flow in the rock mass. It has been shown that one- and two-dimensional heat flow models underestimate heat transfer to the fluid from the crack [4]. Thus, rock cooling and the associated thermal stresses should be studied using three-dimensional heat transfer and stress models. This requires coupling a 3D heat flow model to a 3D elasticity model. A reason for ignoring the three-dimensional nature of heat conduction in the reservoir is the difficulty in treating the infinite geothermal reservoir geometry by numerical discretization. However, it has been demonstrated [4] that by using 3D Green's function for heat conduction and the integral equation formulation the need for discretizing the 3D reservoir is completely eliminated. In this paper we present a 3D integral equation formulation for calculating thermally induced stresses associated with cooling of a planar fracture in an infinite reservoir. A brief presentation of the fluid flow/heat transfer model is also provided for the sake of completeness. Additional details regarding the heat transfer modeling can be found in [4]. 2. FLUID FLOW & HEAT TRANSFER A schematic view of heat extraction from a fracture or a fracture zone in rock is illustrated in Figure 1. With only a few exceptions such as the finite element solution by [5-7] and the boundary element model in [8], the heat conduction in the reservoir is typically modeled as one-dimensional heat flow perpendicular to the fracture surface [9-11].

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-490

... ABSTRACT: A simple approximation of the rheological constitutive

**equations**for isotropic poro-viscoelastic material is proposed. The model is developed by modifying Biot's theory in the transient viscoelsatic regime. This is achieved by considering a liquid-filled spherical shell consisting...
Abstract

ABSTRACT: A simple approximation of the rheological constitutive equations for isotropic poro-viscoelastic material is proposed. The model is developed by modifying Biot's theory in the transient viscoelsatic regime. This is achieved by considering a liquid-filled spherical shell consisting of Maxwell viscoelastic material as the simplest representation of the poro-viscoelastic materials subjected to large volumetric strains. It is shown that in transition to viscous asymptotic, the stress-strain relations deviates from the canonic poroelastic form and Biot's constants become time-dependent. The proposed model is tested using the published experimental data on the deformation of partially melted rocks at elevated PT conditions. Experimental results are usually interpreted in terms of the power law viscous materials. However, in this work we consider the effect of strain damage on viscosity by treating the latter as a dynamic time-dependent parameter; with the variation rate proportional to the second invariant of strain rate. By taking healing into account, the dynamic power law viscosity has constant asymptotic at a given strain rate. The proposed rheological model is implemented in a 2D FEM code and used to study the formation of partial melt in biaxial tests. It is found that the numerically calculated stress-strain curves demonstrate maxima similar to those found in experiments. Also, the computed pattern of melt redistribution and strain localization at the contact with a stiff spacer is qualitatively similar to the experimental observations. The results also indicate that the matrix sensitivity to damage affects the scale of strain localization and liquid re-distribution. Additionally, the problem of the liquid migration in a folding of poro-viscoelastic layer was considered. 1. INTRODUCTION There is increasing interest in using damage theory in the formulation of geomechanics problems [1-3]. This is in response to the need to model the onset and accumulation of micro-cracks in the process of mechanical rapture of elastic materials. Healing of micro-scale damage is also possible. This occurs in the form of recrystallization when a fluid phase is present or due to thermally activated processes of grain boundary migration and dislocation motion. This approach can potentially provide an alternative to the traditional method of tracking crack propagation using the stress intensity analysis or classic theory of the plastic deformations. The concept of evolution of distributed damage has been used by Lyakhovsky et al. (2001) [2] to describe the non-stationary nature of the effective elastic moduli of porous and nonporous elastic geomaterials undergoing deformation. Such models produce a very realistic portrait of spatial strain localization in the elastic crust underlain by a flowing viscoelastic mantle. Spatial dynamics of the rheological parameters of viscous and viscoelastic materials lead to strain localization and formation of localized rapture zones similar to the cracks in elastic solids. Description of the compliance (viscosity) tensor as a dynamic parameter exposed to strain-weakening and thermally activated healing was suggested by Sleep (2001) [4]. At a constant strain rate, the conventional power law strain-rate dependent viscosity becomes the asymptote of the dynamic power law (for isotropic deformation). For geomechanical applications of the dynamic power law formalism, we use a simplified variant of the poro-viscoelasticity theory.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-497

... diffusion and heat diffusion to mechanical behavior. Constitutive

**equations**for this theory were first introduced by Palciauskas and Domenico [3] by extending the classic Biot's poroelastic theory [4] for the non-isothermal case. This theory was later established by other investigators, e.g., McTigue [5...
Abstract

ABSTRACT: Drilling of a borehole in fluid saturated rocks exerts significant influence on its surrounding reservoir. As have been observed extensively, stresses, pressure and/or thermal loadings on the borehole wall will lead to great change of stress field in the rock. As a result, stabilities of the borehole wall and pre-existing fractures will be at risks. In this paper, a mixed boundary element model (BEM) is developed to solve these problems within the framework of thermo-poroelasticity. The model is applied to borehole stability and fracture problems in high temperature subsurface environments. Instability of borehole wall in naturally fractured rocks can be simulated and predicted by studying the potential of crack initiation and growth. This model provides a flexible tool to deal with two-dimensional transient thermal-poroelastic problems. 1. INTRODUCTION Coupled thermal and poromechanical processes play an important role in many geomechanics problems such as borehole stability analysis and studies of initiation and propagation of hydraulic fractures. Thermal effects, as well as poromechanical effects, can greatly change the stresses and pore pressure fields around an underground opening. This is due to the fact that thermal loading induces volumetric deformation because of thermal expansion/contraction of both the pore fluid and the rock solid. A volumetric expansion can result in significant pressurization of the pore fluid. In order to take into account the influence of temperature gradients on pore pressure and stresses, it is necessary to use a non-isothermal poroelastic theory, or thermo-poroelasticity. However, many problems formulated within the framework of thermo-poroelasticity are not amenable to analytical treatment and need to be solved numerically. The boundary element method (BEM) has proven suitable for the poroelastic and thermoelastic problems [e.g., 1, 2]. The advantage of the method is that it reduces the problem dimensionality by one thereby reducing the computational efforts significantly. In this paper, a two-dimensional transient indirect BEM is developed to solve coupled thermo-poroelastic problems. The indirect BEM has two sub-formulations, namely, displacement discontinuity (DD) method and fictitious stress (FS) method. The DD method is particularly suitable for crack-shaped problem. A mixed FS-DD model is developed to take advantage of the strengths of both FS and DD methods. The model is applied to some thermo-poroelastic problems in borehole stability and hydraulic fracturing. 2. THERMO-POROELASTICITY Thermo-poroelasticity is developed on the basis of poroelasticity and thermoelasticity. It couples the time-dependent processes of fluid diffusion and heat diffusion to mechanical behavior. Constitutive equations for this theory were first introduced by Palciauskas and Domenico [3] by extending the classic Biot's poroelastic theory [4] for the non-isothermal case. This theory was later established by other investigators, e.g., McTigue [5], and Coussy [6]. The governing equations for thermo-poroelasticity can be found in the works of McTigue [5] which consist of constitutive equations, transport laws and balance laws. From the governing equations, the field equations can be derived for temperature, displacement, and pore pressure: Navier Equation: (available in full paper) Diffusion equation for pore pressure p: (available in full paper) Diffusion equation for temperature T: (available in full paper)

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-467

... ABSTRACT: In the fully-coupled thermoporoelastic wellbore stress modeling, pore pressure and temperature can be decoupled for a low-permeability shale and the decoupled

**equations**can be solved analytically in the Laplace domain. For a high-permeability rock, such as sandstone or carbonate...
Abstract

ABSTRACT: In the fully-coupled thermoporoelastic wellbore stress modeling, pore pressure and temperature can be decoupled for a low-permeability shale and the decoupled equations can be solved analytically in the Laplace domain. For a high-permeability rock, such as sandstone or carbonate, the fully-coupled pore pressure and temperature can also be decoupled by assuming the pore fluid flow reaches steady-state. This assumption of steady-state fluid flow in a high-permeability rock is validated by solving the fully-coupled pore pressure equations. It is found that the assumption is valid as long as the dimensionless time exceeds a certain time. Under this assumption, the temperature equation can be decoupled and can be solved analytically in the Laplace domain for both injection and production conditions. Closed-form solutions for thermally-induced stresses are also presented in the Laplace domain. The undrained loading effect, which usually occurs at short time and small distances in a low-permeability formation, may be negligible for a high-permeability non-shale formation. Results show that the undrained loading effect can be ignored for a high-permeability non-shale formation when the pore fluid flow reaches steady state. Modeling results of near-wellbore temperature and stresses can be applied to injection well design (pressure and temperature of the injection fluid), wellbore stability analysis, and sanding prediction analysis. 1. INTRODUCTION It has been demonstrated that thermal effects can be very important to both wellbore stability and injection design (Guenot and Santarelli, 1989 [1]; Paige and Murray, 1994 [2]; Charlez, 1997 [3]). Formation temperature and formation pore pressure are fully-coupled for fluid flow in porous media, such as in oil and gas drilling, injection, and production operations. The fully coupled thermoporoelastic equations and solutions for some initial and boundary conditions, and their applications in the petroleum industry, can be found in the literature (Kurashige, 1989 [4]; Wang and Papamichos, 1994 [5]). Theoretically, for any initial and boundary conditions, the fully coupled thermoporoelastic equations can be solved using the finite-difference method, but sometimes at the expense of computation time (Chen, 2001) [6]. In the meantime, the fully-coupled temperature and pore pressure can be partially or completely decoupled for various ranges of rock permeabilities. The decoupled equations might be solved analytically in the Laplace domain as well as in the real time domain under certain initial and boundary conditions (Kurashige, 1989) [4]. For low-permeability (~nanodarcy, or ~1e-21 m 2 ) shale formations, pore pressure, temperature and thermally-induced stresses can be determined analytically for a permeable wellbore boundary (Wang and Papamichos, 1994 [5]; Li et al., 1998 [7]; Chen et al., 2003 [8]) as well as for an impermeable wellbore boundary condition (Chen and Ewy, 2003) [9]. For intermediate and high permeability rocks, Wang and Dussault (2003) [10] presented temperature and pore pressure solutions for a permeable wellbore. Once pore pressure and temperature solutions are determined, the pressure-induced and thermally-induced stresses around the wellbore can then be solved. Rock failure can also be determined by comparing stresses with rock strength using various failure criteria.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-473

... components of these tensors, and the sensitivity of such measurements to changes in the vertical and horizontal stress field is discussed. reservoir geomechanics Upstream Oil & Gas wave velocity elastic wave velocity measurement anisotropic change

**equation**andÎ± 33 model 1 Reservoir...
Abstract

ABSTRACT: Elastic wave velocities in rocks vary with effective stress due to the presence of stress-sensitive discontinuities within the rock such as grain boundaries, microcracks, fractures, etc. Because of the low ratio of the thickness to lateral extent of hydrocarbon reservoirs, production of hydrocarbons leads to anisotropic changes in the stress field in the reservoir and surrounding rocks. Since the response of any discontinuities depends on their orientation relative to the principal stress axes, the anisotropic changes in the total stress field resulting from production can lead to significant elastic wave anisotropy that may be used to monitor the stress changes which occur. The elastic anisotropy resulting from anisotropic changes in the stress field can be written in terms of a second-rank and fourth-rank fabric tensor which quantify the effect of the stress-sensitive discontinuities on the elastic wave velocities in the rock. This allows elastic wave velocity measurements, amplitude versus offset, etc., to be inverted to obtain the components of these tensors. The theory allows the ratio of the normal to shear compliance of the discontinuities to be determined from seismic measurements. This ratio is of importance in determining the failure mechanisms involved in sanding, fault reactivation, etc. 1. INTRODUCTION Production of hydrocarbons usually leads to changes in pore pressure which give rise to changes in stress acting on the reservoir and surrounding rocks [1]. A decrease in pore pressure due to depletion leads to an increase in the effective stress acting on the reservoir which may be accompanied by reservoir compaction, surface subsidence, casing deformation, reactivation of faults and bedding parallel slip. Strong evidence for stress changes in and around reservoirs undergoing depletion is provided by seismic events resulting from production [2, 3]. The purpose of this paper is to examine the possibility of using time-lapse seismic measurements to monitor changes in stress in the reservoir and surrounding rocks. Elastic wave velocities in rocks vary with changes in effective stress due to the presence of stress-sensitive discontinuities within the rock such as grain boundaries, microcracks, fractures, etc. Since the lateral dimensions of hydrocarbon reservoirs are usually larger than their thickness, these changes in pore pressure often result in changes in the total horizontal stress which are significantly larger than the accompanying changes in total vertical stress. Since the response of any discontinuities depends on their orientation relative to the principal stress axes, the anisotropic changes in the total stress field resulting from production can lead to significant changes in elastic wave anisotropy that may be used to monitor the stress changes which occur. In this paper it is shown that the elastic anisotropy that results from anisotropic changes in the stress field acting on the rock can be written in terms of a second-rank and fourth-rank fabric tensor which quantify the effect of the stress-sensitive discontinuities on the elastic wave velocities in the rock. This allows elastic wave velocity measurements, AVO (Amplitude Versus Offset), etc., obtained using time-lapse seismic measurements to be inverted to obtain the components of these tensors, and the sensitivity of such measurements to changes in the vertical and horizontal stress field is discussed.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-491

... stress shear displacement shear fracture propagation hydraulic fracture

**equation**ABSTRACT: This paper deals with the shear-induced dilatation mechanism when a plane-strain fluid-driven fracture propagates on a plane of weakness, subjected to shear and compressive stresses. The fracture...
Abstract

ABSTRACT: This paper deals with the shear-induced dilatation mechanism when a plane-strain fluid-driven fracture propagates on a plane of weakness, subjected to shear and compressive stresses. The fracture surfaces are in contact and their sliding over each other gives rise to normal opening because shear across rough surface generates opening dilation. The analysis assumes an incompressible Newtonian fluid with zero viscosity which is injected into the fracture in an impermeable elastic medium. On the basis of the plane strain elasticity, the resulting slip, crack length and net shear stress, which is defined as the difference between the applied shear stress and the friction stress based on the Coulomb law, are calculated. A slip-weakening friction law is implemented to account for crack surface roughness changes with shear displacement. Using a slip-weakening law means no stress singularity exists at the shear crack tip for propagation along the plane of weakness. The governing equations are derived for equilibrium cracks and a scaling is proposed to simplify those equations. Numerical results based on a Chebyshev polynomial expansion show that the size of slipping region can grow under negative net shear stresses as a result of the slip-weakening mechanism which helps explain the success of hydraulic fracturing in promoting shear fracturing along planes of weakness. A critical length for shear fracture initiation exists as a result of use of a slip-weakening friction law and the removal of stress singularity. Similar to dislocation cores, expenditure of energy is required for crack nucleation from this critical size. For stable shear crack growth, the net shear stress should follow a critical curve obtained numerically. If the net shear stress is larger than its critical value, the shear crack will propagate unstably, otherwise it will be arrested as lack of driving forces. In addition, since the net shear stress is bounded, there is another critical length for the onset of unstable crack growth. 1. INTRODUCTION Shear dilatation is recognized as a mechanism that can enhance permeability of fluid-driven shear fractures. In contrast to the conventional tensile or opening mode fractures in which the fracture is kept open by an internal pressure that exceeds the minimum stress, the coupling between fluid pressure and conductivity in a shear fracture is via shear displacement or slippage. The fluid pressure in the fracture acts to reduce the effective normal stress acting across it which promotes shear of the fracture. The shear displacement changes the conductivity by causing shear-induced dilation. The shear dilation mechanism has been exploited to stimulate hot dry rock reservoirs and gas reservoirs (Pine and Batchelor[1], Vychytil and Horii[2], Mayerhofer et al.[3]). In addition, microseismicity generated by shearing is commonly observed during conventional hydraulic fracture treatments[4]. Recently, hydraulic fracturing has been applied to caving inducement and preconditioning rock masses for mining by block caving methods (van As and Jeffrey[5]). The preconditioning is aimed at modifying the rock mass strength and the size of rock fragments formed during later caving.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-481

...

**equations**in terms of effective stress ratios, resulting in a fracture criterion, which is independent of the borehole fracture angle. The data are leak-off data from oil wells. They are recorded in wells with different inclinations and azimuths, a requirement for a robust inversion. However, there is a...
Abstract

ABSTRACT ABSTRACT: The paper presents a theory for determining the in-situ stress state from multiple fracturing data and induced fractures from image logs. A solution can be obtained with a minimum of three data sets. However, using an inversion technique, a solution can be o btained with any number of data sets, as the solution is over determined. The magnitude of the stresses is mainly determined from the fracturing data. Fracture information from image logs is mainly used to determine the geographic direction of the principal in-situ stress. In the paper, plots of the Effective Fracture Pressure Ratio, the Fracture Angle and the Fracture Trace Angle gives a good overview how these three quantities behave, as a function of the borehole inclination and azimuth. This knowledge has advantages in planning new oil wells. The mathematical technique is to describe the general fracture equations in terms of effective stress ratios, resulting in a fracture criterion, which is independent of the borehole fracture angle. The data are leak-off data from oil wells. They are recorded in wells with different inclinations and azimuths, a requirement for a robust inversion. However, there is a non -uniqueness problem in fracturing modelling as the position on the wall where the fracture initiate, is usually not known. By using image logs, this uniqueness can be removed. The fracture trace on the image log is also helpful in finding the directions of the in-situ stresses, whether horizontal/vertical or inclined. The model presented in this paper gives opportunity to use the information of the fracture position and direction directly in the in-situ stress calculations. INTRODUCTION The application of rock mechanics in the petroleum industry has increased in later years. Due to the increasing complexity of petroleum wells, borehole stability issues have become challenges that have to be handled. Borehole collapse is one class of problems, whereas circulation losses due to unexpected fracturing accounts for significant additional expenditures. In general, drilling cannot proceed before mud losses are healed. It has become clear that assessment of the in-situ stress state is very fundamental for all modelling work of borehole stability. Data used for stress modelling includes: Leak-Off Tests at each casing shoe (usually 3 in each well) pore pressure and overburden pre ssure, and lithology. In vertical exploration wells we may also deduce the minimum horizontal stress direction from borehole breakouts. Aadnøy [1] developed an inversion technique in 1990. Since directional wells have different orientations (inclinations and azimuths), independent fracturing equations were derived. These were organized as an over determined system of equations and the in-situ stress state were solved in an inversion routine. In addition to determining the magnitude of the two horizontal stresses, it determines the direction of the stress field. Okabe et. al. [2] developed an inversion technique for data taken in the same borehole. Djurhuus and Aadnøy [3] presented a general solution to the problem and showed that the linearized version produces good solutions to the in-situ stresses and their directions.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-492

... chemistry drilling fluid formulation concentration shale diffusion

**equation**invasion capillary pressure permeability fluid filtrate ABSTRACT: We have proposed and studied a well-bore Stability model (a modified Failure Criterion), taking into account shale problems, effect of temperature...
Abstract

ABSTRACT: We have proposed and studied a well-bore Stability model (a modified Failure Criterion), taking into account shale problems, effect of temperature, vibration of drill-string, drilling fluid jet impact, osmotic pressure, dynamic pore pressure ahead of the bit, and capillary pressure. In our laboratory, we conducted a series of tests on the effect of caustic soda on the surface tension, resistivity, and capillary pressure of the pore fluid. In addition, we generated a computer program in order to solve two dimensional diffusivity and dynamic pore pressure equations using Darcy's laws simultaneously. Moreover, in this model we included the effects of vibration of drill-string and drilling fluid et impact on diffusion processes of filtrate into the shale membrane. Results show that the amount of invasion of drilling fluid into the drilled formation immediately ahead of drill bit is significant. Our study of the model shows that the higher than usual filtrate diffusion into the shale membrane occurs when there is not enough time for the mud cake to be formed at the rock-bit interface and the formation immediately ahead of the drill bit is pressurized by the sum of several pressures. The sources of these pressures are circulating mud, bit jet, weight-on-bit, and bit torque. While diffusion of drilling fluid occurs, caustic soda (NaOH) in the drilling fluid filtrate mixes with the pore fluid in the pore space. Depending on the concentration of caustic soda in the drilling fluid filtrate and permeability of the formation, the above mentioned mixture causes a tremendous reduction of surface tension and capillary pressure of the pore fluid in the pore space. This phenomenon enhances rapid diffusion of fluid filtrate into the shale membrane. Results also show that diffusion of the drilling fluid filtrate in the formation during dynamic and static periods is mainly controlled by osmotic pressure and adsorption of the drilling fluid's components on the rock grains.1. INTRODUCTIONLiterature Review. Before presenting the proposed model, the authors would like to focus on the previous models which have been studied mainly in recent years, with concentration on well-bore stability in shaly formations. Our proposed model will also be explained in more detail and the results will be discussed in the later sections. After studying the nature of shale and its properties, Apande (1980) 1 presented his model. Theoretically, he had worked on a model proposed by Hayatdavoudi which would connect some of the shale properties such as capillary and swelling pressures to well-bore stability considerations. Although his model did not include most of the other parameters like capillary pressure, the effect of temperature change, the effect of vibration of drill-string and pulsation of bit jet hydraulic, his model is still more realistic as compared with other works. His approach to the problem had a solid foundation. Huang et al. (1998) 2 presented a model based on considerations that the pore pressure change and the onset of swelling pressure are two main factors in well-bore instability issue.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-539

... proportionality loading perforation tunnel penetration theory jet velocity rock response

**equation**1. INTRODUCTION In cased completions of oil and gas wells the wellbore is connected to the reservoir via perforation tunnels created by explosive shaped charges. The shaped charges are usually loaded into...
Abstract

ABSTRACT: In most instances of the application of rock mechanics to oilfield problems the loading on the rock can be regarded as static or quasi-static. Furthermore the rock mechanical parameters that are used to quantify the rock response are obtained from static or quasi-static laboratory tests. Here we discuss an application in which the loading on the rock is large in amplitude and short in duration. We examine the creation of a perforation tunnel by a shaped charge jet. It is well known that a shaped charge jet travels at velocities in the range 1-7 km/s and that the jet tip exerts a pressure on the rock of the order of one million psi. Penetration of the rock and creation of the perforation tunnel last about 0.1 to 0.5 ms. The depth of penetration is one of the important perforating parameter for well productivity, particularly in hard formations, and its prediction and maximization are fundamental concerns. Recent mathematical models of the penetration process now include the effect of the rock strength through an ill-defined yield stress . We show that the results of these models provide good agreement with laboratory data when the yield stress is set proportional to the UCS with a constant of proportionality of about 20. Experimental data for jets shot into hard and soft concrete targets supports the predicted value of the yield strength, though we have severe reservations about the application of static parameters such as UCS in the context of this intense dynamic loading. Results from the models suggest that the rock strength severely limits the depth of penetration, which has implications for the design of shaped charges for use in hard rocks. 1. INTRODUCTION In cased completions of oil and gas wells the wellbore is connected to the reservoir via perforation tunnels created by explosive shaped charges. The shaped charges are usually loaded into the wellbore inside a steel tube known as a "gun". When a shaped charge is detonated the metal liner collapses under the intense pressure created by the detonation of the explosive and forms a jet that is projected towards the target at a high velocity, see Fig 1. (In the oilfield the target consists of gun, water clearance, casing, cement sheath and rock.) Typically the speed of the jet at its tip is of the order of 7 km/s, while towards the tail of the jet speeds of the order of 1 km/s are more typical. The whole penetration process is completed in a few hundred microseconds. Usually the front part of the jet travels at a speed in excess of the sound speed in the rock. The associated bow shock wave can be discerned in the numerical simulations shown in Fig 2. The pressure on the target at the jet impact site is of the order of 1 million psi. At least in the early stages of penetration the impact pressure is sufficient to overcome the strength of the target. The target material in the neighborhood of the jet tip then flows, almost as if it were a fluid being penetrated by another fluid, displacing the target material radially and creating a cavity or tunnel in the target.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-577

... circulation borehole minimum in-situ stress propagation

**equation**wellbore pressure critical pressure ABSTRACT: Existing borehole stability analysis tools are based on an assumed, fixed, circular borehole geometry. In a strict sense, these tools are valid only as long as the borehole remains...
Abstract

ABSTRACT: Existing borehole stability analysis tools are based on an assumed, fixed, circular borehole geometry. In a strict sense, these tools are valid only as long as the borehole remains circular. They cease to be valid and their applicability becomes limited when borehole breakouts or fractures are present at the borehole wall. Reliance on these methods can limit the ability to evaluate borehole stability and determine the safe mud weight window to the geometry or geometries on which the model being used is based-a circular borehole. When breakouts or fractures have formed, a different model is required to analyze the conditions around the borehole, to determine required mud weights, and to evaluate methods that can be employed to restabilize the borehole. The authors demonstrate this effect by analyzing and evaluating two specific borehole geometries-the circular borehole and a borehole intersected by an induced or natural fracture. Using existing borehole stability theories and basic hydraulic fracturing analysis, the authors review and evaluate various means available to alter the size of the mud weight window. Of particular interest are the two limiting cases: a circular borehole with no defects and a drilling-induced or natural fracture intersecting the borehole. Mechanisms reviewed include chemically altering the rock's mechanical properties or altering the borehole's surface characteristics, creating impermeable bridges within existing fractures, plugging existing or induced fractures with high-viscosity deformable or undeformable materials, and permanently sealing fractures with "rigid" cement. Calculations presented demonstrate the "borehole strengthening" that can be achieved by strengthening the rock matrix or by plugging an existing natural or induced fracture using materials that are deformable and do not rigidly adhere to the fracture walls (such as extremely viscous gel-like materials), materials that are not deformable (such as cement) and require time to cure, and materials that glue the fracture walls together. 1. INTRODUCTION Lost circulation during drilling operations continues to be a significant problem in the oil and gas industry. Control and remediation of lost circulation can require large expenditures and, if impossible, can result in the loss of the well. Lost-circulation problems will continue to plague the industry as it drills for oil and gas in reservoirs or fields with partially depleted sands that have to be penetrated to reach deeper productive intervals. Effective means to assess lost-circulation technologies are required. Lost circulation can occur in a number of ways. It can occur gradually through leakoff; abruptly as we penetrate into subsurface voids, rubbelized zones, and high-permeability intervals; or through the sudden initiation and continued propagation of a fracture. The remainder of this discussion focuses on the last of these lost-circulation issues-lost circulation through fracturing. Using existing borehole stability theories and basic hydraulic fracturing analysis, we review and evaluate various means available to help eliminate or remediate fracturing-related lost circulation. Of particular interest are two limiting cases: Avoidance of fracture initiation in a circular borehole with no defects ?Arresting or mitigating drilling-induced or natural fractures intersecting the borehole Mechanisms reviewed include chemically altering the rock's mechanical properties, altering the borehole's surface characteristics, creation of impermeable bridges within existing fractures, plugging existing or induced fractures with highviscosity deformable or undeformable materials, and permanently sealing fractures

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-526

... Fluid Dynamics thin plate mass evolution hollow cylinder Reservoir Characterization equilibrium gallery Upstream Oil & Gas transfer process moisture transfer evolution

**equation**flow in porous media thin plate diffusivity experimental data relative humidity saline solution...
Abstract

ABSTRACT: Hydromechanical and mass transfer phenomena in argillaceous rocks mass are currently studied in order to predict the perturbations around ventilated galleries of a nuclear waste storage. This paper presents the results of drying experiments performed on argillite samples bored at 500 m depth at Bure (France) where an underground research laboratory will be built. Relative humidity and mass evolution of thin plates and hollow cylinders samples are continuously measured in tight box in which relative humidity is imposed by saline solutions. The mass transfer phenomenon is characterized by the moisture diffusivity coefficient computed by comparing the measured mass evolution and the linearized analytical solutions for each drying step. The linearized water moisture diffusivity increases exponentially according to the relative humidity from about 0.5x10 -10 m 2/ s -1/ for the 44 % to 32% relative humidity step to 1.2x10 -10/ m 2/ s -1/ for the 97% to 90% relative humidity step. A theoretical moisture transfer model, accounting for the vapour water transfer and liquid water transfer is then proposed. The confrontation with the experimental data leads to the identification of permeability of the unsaturated rock. The model assumes an intrinsic permeability value of 10 -22/ m 2/ , which is a lower bound of measured permeability in the saturated state for this argillite. 1. INTRODUCTION The French Radioactive Waste Management Agency (ANDRA) has selected an argillaceous site in the east of France (Bure, Haute-Marne) as potential nuclear waste storage host. An underground research laboratory at a 500 m depth is currently under construction. The digging of the galleries is expected to create a damaged zone around the galleries, increasing the permeability by several orders of magnitude. Additional damage could be induced by the desaturation due to the ventilation of the galleries. Such hydric induced cracks have already been observed in the argillaceous Tournemire tunnel [1]. A complete experimental investigation is currently conducted at the LMS (Laboratoire de Mécanique des Solides) in order to characterize, on one hand, the unsaturated hydromechanical behaviour of the rock at moisture equilibrium and on the other hand, the mass transfer process during desaturation steps. The argillite under investigation belongs to the Callovo-Oxfordian layer at a depth of 400-500 m. The tested samples are bored from cores at a depth of about 480 m. The mineral content is: 40% of clay materials, 25 to 30% of quartz and 20 to 30% of carbonates. Microscopic observations have shown that the argillaceous phase is continuous whereas the quartz and carbonates phases are discontinuous [2]. The observed quasi-linear relationship between the water content and the logarithm of suction is typical of clay materials. Water permeability in saturated state has been measured by transient method and ranges from 10 -22 m 2 to 10 -20 m 2 [3], [4] for the undamaged material. The hydromechanical behaviour of Bure argillite has been extensively studied in the saturated state; however in the unsaturated state, few data are available, especially when the mass transfer process is concerned. We present in this paper the results of drying experiments leading to the experimental characterisation of the mass transfer process in term of moisture diffusivity coefficient as a function of air relative humidity.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-579

...

**equation**ABSTRACT: A key to the success of long-term storage of CO 2 in depleted oil or gas reservoirs is the hydraulic integrity of both the geological formations that bound it, and the wellbores that penetrate it. The integrity of this "bounding seal" system is affected by various mechanical...
Abstract

ABSTRACT: A key to the success of long-term storage of CO 2 in depleted oil or gas reservoirs is the hydraulic integrity of both the geological formations that bound it, and the wellbores that penetrate it. The integrity of this "bounding seal" system is affected by various mechanical, chemical and thermal forces that act during initial exploration, development and oil production operations, during CO 2 injection operations, and during the subsequent CO 2 storage phase. This paper provides a review of the geomechanical factors affecting the hydraulic integrity of the bounding seals for a depleted oil or gas reservoir slated for use as a CO 2 injection zone. Equations are given which are helpful for identifying the key parameters that govern these geomechanical factors, and further enable first-order estimates of the risks that they pose to bounding seal integrity. The results of this review are compiled into a table that summarizes key geomechanics-related risks, the mechanisms associated with these risks, and approaches to assess and mitigate them. Where possible, examples are given where these mechanisms have affected oil and gas field operations. 1. INTRODUCTION In order to achieve significant reductions in the atmospheric release of anthropogenic greenhouse gases, the implementation of technologies to capture carbon dioxide (CO 2 ) and store it in geological formations will be necessary. Deep saline aquifers have the largest potential for CO 2 sequestration in geological media in terms of volume, duration and minimum or null environmental impact [1]. The first commercial scheme for CO 2 sequestration in an aquifer is already in place in the Norwegian sector of the North Sea, where 10 6 tonnes of CO 2 are extracted annually from the Sleipner Gas Field and injected into the 250 m thick Utsira aquifer at a depth of 1000 m below the sea bed [2]. In light of the economic benefits of enhanced oil recovery (EOR) derived from CO 2 injection in oil reservoirs [3], these types of reservoirs will be attractive CO 2 injection targets and, most likely, CO 2 storage in depleted oil and gas reservoirs (or in conjunction with EOR) will be implemented before CO 2 storage in aquifers. An advantage of CO 2 storage in depleted oil or gas fields is the fact that much of the infrastructure for fluid injection (e.g., wellbores, compressors, pipelines) is already in place. The Weyburn CO2 Monitoring and Storage Project in Saskatchewan, Canada [4] is an example of a large-scale application of EOR operations using anthropogenic CO 2 , in which the oil reservoir is being evaluated for subsequent use as a long-term storage zone. A key to the success of long-term storage in depleted oil and gas reservoirs is the hydraulic integrity of both the geological formations that bound it, and the wellbores that penetrate it. Although the initial integrity of this "bounding seal" system is governed by geological factors, it is ultimately affected by various mechanical, chemical and thermal forces that act during exploration, development, oil production and secondary recovery (waterflooding) operations, during CO 2 injection, and during the subsequent CO 2 storage phase. This paper contains a review of the geomechanical factors that pose risks to the integrity of the bounding seal system.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-546

... strain rate compression deformation octahedral profile geomodel prediction

**equation**ABSTRACT: Sandia's GeoModel is a generalized plasticity model that was developed primarily for geological materials, but is also applicable to a much broader class of materials such as concretes, ceramics...
Abstract

ABSTRACT: Sandia's GeoModel is a generalized plasticity model that was developed primarily for geological materials, but is also applicable to a much broader class of materials such as concretes, ceramics, and even some metals. Nonlinear elasticity has been incorporated through empirically fitted functions found to be well suited for a wide variety of materials. The yield surface has been generalized to include any form of inelastic material response including pore collapse and growth. Deformation-induced anisotropy is supported in a limited sense through kinematic hardening. Applications involving high strain rates are supported through an overstress model. Inelastic deformation can be associated or non-associated, and the GeoModel can employ up to 40 material input parameters in the rare case when all features are needed, but simpler idealizations (such as linear elasticity, or Von Mises yield, or Mohr-Coulomb failure) can be replicated by simply using fewer parameters. 1. INTRODUCTION Simulating deformation and failure of natural geological materials (such as limestone, granite, and frozen soil) as well as rock-like engineered materials (such as concrete [1] and ceramics [2]) is at the core of a broad range of applications, including exploration and production activities for the petroleum industry, structural integrity assessment for civil engineering problems, and penetration resistance and debris field predictions for the defense community. For these materials, the common feature is the presence of microscale flaws such as porosity (which permits inelasticity even in purely hydrostatic loading) and networks of microcracks (leading to low strength in the absence of confining pressure and to noticeable nonlinear elasticity, rate-sensitivity, and differences in material behavior under triaxial extension compared with triaxial compression). For computational tractability and to allow relatively straightforward model parameterization using standard laboratory tests, the Sandia GeoModel strikes a balance between first-principals micro-mechanics and phenomenological, homo-genized, and semi-empirical modeling strategies. The over-arching goal is to provide a unified general-purpose constitutive model that can be used for any geological or rock-like material that is predictive over a wide range of porosities and strain rates. Being a unified theory, the GeoModel can simultaneously model multiple failure mechanisms, or (by using only a small subset of the available parameters) it can duplicate simpler idealized yield models such as classic Von Mises plasticity and Mohr-Coulomb failure. Thus, running this model can require as many as 40 parameters for extremely complicated materials to only two or three parameters for idealized simplistic materials. 2. GEOMODEL OVERVIEW The GeoModel shares some features with earlier work by Schwer and Murry [3] in that a Pelessone function [4] permits dilatation and compaction strains to occur simultaneously. For stress paths that result in brittle deformation, failure is associated ultimately with the attainment of a peak stress and subsequently work-softening deformation. Tensile or extensile microcrack growth dominates the micromechanical processes that result in macroscopically dilatant (volume increasing) strains even when all principal stresses are compressive. At higher pressures, these processes can undergo strain-hardening deformation associated with macroscopically compactive volumetric strain (i.e., pore collapse).

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-541

... cylinder non-stationary boundary pressure

**equation**coefficient stress path Boundary Pressure blowout ABSTRACT: The fundamental poroelastic solutions provide the framework for modeling of flow-induced stresses and deformations in saturated porous rocks, which is of significant interest in...
Abstract

ABSTRACT: The fundamental poroelastic solutions provide the framework for modeling of flow-induced stresses and deformations in saturated porous rocks, which is of significant interest in petroleum and mining geomechanics. We have developed an extended solution for the mechanical response of the poroelastic hollow cylinder under non-stationary stress and pressure boundary conditions. The solution was obtained in Laplace space and it was verified with published results for the special case of boundary conditions using numerical Laplace inversion. The proposed solution was successfully used to model and interpret laboratory tests. It was found that the solutions with simplified assumption of instantaneously applied pressure might overestimate the flow induced tensile radial stresses and under some conditions the results differ even qualitatively. The change in average axial stresses versus hole pressure obtained from the laboratory test was in good agreement with the model prediction. The developed solution can be used for laboratory test interpretation, optimization of the openhole completion and well control operations, prediction of wellbore collapse and bridging during oil or gas blowouts, and the subsequent estimation of probability of blowouts "self-killing". 1. INTRODUCTION One of the dominating methods of oil and gas blowout control is bridging, or wellbore sealing with rock fragments from the collapsing formation [1, 2]. The majority of bridged blowouts have a short duration [3]. It is generally assumed that at early time intervals the formation stability is controlled by a tensile failure and fluid flow induced stresses [4, 5], therefore formation stability can be predicted using fundamental poroelastic solutions of problems involving axisymmetric deformation of a saturated poroelastic media [6-10]. However, the prediction of pore pressure, stresses and deformations is usually obtained assuming some highly idealized pressure histories, either constant or changes in pressure following a step function (see e.g. Wang [11]). The actual boundary pressures and stresses encountered in laboratory experiments and industrial processes exhibit more complex behavior with time [12-14]. In this paper we have developed an extended solution for the mechanical response of the poroelastic hollow cylinder under non-stationary stress and pressure boundary conditions. The solution was obtained in Laplace space and it was verified with published results for the special case of boundary conditions using numerical Laplace inversion. The proposed solution was successfully used to model and interpret laboratory tests. The results of laboratory experiments on hollow cylinders of Berea sandstone indicate that the determination of realistic boundary conditions is crucial to the correct prediction of mechanical responses of the rock. 2. POROELASIC SOLUTION FOR THE AXISYMMETRIC PLANE STRAIN PROBLEM 2.1. Basic equations The purpose of the analysis is to describe the transient pore pressure, stresses and deformations, occurring during and immediately after the change in pressure at the walls of a saturated permeable thick-walled hollow cylinder of inner radius, R i , and outer radius, R o . We use conventional cylindrical coordinates (r; ¿; z) with the origin in the center of the lateral cross-section of the cylinder and the z axis along the axis of symmetry. The principal stresses are denoted as follows: radial, s rr ; tangential, s ¿¿ ; and axial, s zz (Figure 1). The analysis is linear poroelastic with full coupling. Possible pressure and stress dependencies of mater

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-520

... discontinuity

**equation**plasticity model localized deformation bifurcation geomaterial ARMA/NARMS 04-520 Bifurcation analysis of a three-invariant, isotropic/kinematic hardening cap plasticity model for geomaterials Richard A. Regueiro Materials and Engineering Sciences Center, Sandia National...
Abstract

ABSTRACT: Localized deformation such as shear bands, compaction bands, dilation bands, combined shear/compaction or shear/dilation bands, fractures, and joint slippage are commonly found in rocks. Thus, modeling their inception, development and propagation, and effect on stress response is important. This paper will focus on modeling the inception of these localized deformations-the onset of bifurcation to a localized material deformation response-for a three-invariant, isotropic/kinematic hardening cap plasticity model. Bifurcation analysis is the first step in developing a constitutive model for representing the transition of continuous rock-like material to fragmented rock material. Developing a post-bifurcation constitutive model and numerical implementation, whether via the finite element method or a meshfree method, is the next step and will not be discussed in this paper (but is part of our ongoing research). Applications of a constitutive model for modeling localized deformation in geomaterials include assessing the long term performance of nuclear waste repositories, designing tunneling construction, oil and natural gas production, and depleted reservoirs used for subsurface sequestration of greenhouse gases. 1. INTRODUCTION Localized deformation such as shear bands, compaction bands, dilation bands, combined shearcompaction or shear-dilation bands, fractures, and joint slippage are commonly found in rocks. These localized deformations can be triggered by either material inhomogeneities such as joint sets in rocks, inhomogeneous stress resulting from boundary conditions such as friction at end platens in a con- fined compression test, or by some microstructurally driven material instability. We can account for material inhomogeneities by constitutivemodeling in conjunction with a numerical simulation method such as the finite element method. Significant material inhomogeneities such as strata and joint sets can be meshed discretely, assigning different material properties for each spatial region of the finite element mesh, or they can be incorporated in an average sense into a continuum constitutive model via directional structure/anisotropy tensors or the like. Either way, depending on boundary and loading conditions, the material deformation response predicted by the constitutive model could become mathematically unstable. This mathematical instability could be made to coincide with the natural material instability observed in the field or laboratory. The most straightforward way to do this is to endow the constitutive model with as much material characterization and representative deformation response that is deemed significant for the problem of interest. For example, if joint sets are plentiful and dominate the material deformation response, they must be represented in the constitutive model. Depending on the boundary and loading conditions, the model must predict the onset of gross localized deformation resulting from activity of certain critical joint sets. In essence, the ability of a continuum constitutive model to predict material instability in the form of localized deformation is only as good as the model's sophistication in terms of representing material behavior. Some questions we should ask when choosing and developing constitutive models for geomaterials are: Is thematerial isotropic or anisotropic elastically and/or plastically? Is the material temperature and ratesensitive?Are joint sets or other in-situ material inhomogeneities prominent? Given a relatively sophisticated continuum constitutive model for geomaterials, this paper focuses on determining stress states at which the constitutive model predicts mathematical instabilities.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-613

... formulation of the governing

**equations**, we need to make a few assumptions with respect to the FEM model [6, 7]: Infinitesimal deformation theory holds. Domains of interest are fully saturated. The fluid and porous medium are everywhere in local thermodynamic equilibrium, i.e., the temperature of the...
Abstract

ABSTRACT: The paper presents a discussion on the strain-induced anisotropy in permeability for deformable granular media. A coupled deformation-flow-heat transfer simulator, which has incorporated the strain-induced permeability model, was developed using finite element method (FEM). It was used to conduct a coupling analysis of two-dimensional non-isothermal single-phase fluid flow in elastic porous media. As well, two fluid injection tests were carried out to investigate the effects of the permeability anisotropy. It is shown that the strain-induced permeability anisotropy does have significant impacts on the pressure response. It is also found that the proposed permeability model can accurately reflect the directional increase in permeability during fluid injection. 1. INTRODUCTION The dependence of permeability on direction or permeability anisotropy in porous media has been confirmed by many field studies. The existing reservoir simulators usually deal with permeability anisotropy in such a way that the permeability in the principal directions may vary in magnitude at different locations; however, the orientations of the principal permeability remain the same throughout the reservoir. In reality, both the magnitude and the orientation of the principal permeability may vary from region to region in the reservoir due to the variation of effective stress in the reservoir formation. The permeability anisotropy mentioned in the following context is referred to the latter case. Deformations in a reservoir are induced by the changes of pore pressure and temperature due to fluid injection and production in thermal recovery processes, thereby affecting permeability. However, in the geomechanics and petroleum literature, the permeability change of reservoir formation subjected to deformation changes is usually assumed as a function of porosity or volumetric strain, which is a scalar variable. Thus, the changes in permeability are equal in all directions even though the changes in strains are different in each direction. Wong [1] analyzed the grain fabric of intact and sheared oil sand specimens using the thin section imaging method. He observed that even in intact natural oil sand specimens, the hydraulic radius and tortuosity factors vary in vertical and horizontal directions resulting in an intrinsic anisotropy in permeability. Based on theoretical and laboratory works, he developed a new permeability model for deformable porous media. This model assumes the tensor permeability is governed by induced principal strains. It can quantify the changes in permeability when the material experiences shear deformation and the changes in permeability can be anisotropic. Coupled geomechanics-reservoir simulation is necessary in order to account for deformations due to pore pressure and temperature changes resulting from production and fluid injection. Conventional reservoir simulators usually use finite difference method (FDM) and assume permeability either isotropic or diagonal tensor. It is impractical to develop coupled geomechanics-reservoir simulators based on FDM numerical schemes due to its complexity. A coupled deformation-fluid flow-heat transfer simulator using finite element method (FEM) was developed. The full tensor permeability and the strain-induced permeability model were implemented into the simulator. It was then used to conduct a coupling analysis of two-dimensional non-isothermal single-phase fluid flow in elastic porous media. As well, two fluid injection tests were carried out to investigate the effects of the permeability anisotropy.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-560

... rheology fracture

**equation**strain rate fluid rheology ABSTRACT: Viscoelastic surfactant (VES) based fracturing fluids have shown distinct rheological characteristics from conventional polymer based fluids. A nearly constant shear stress plateau is commonly observed in the shear stress versus...
Abstract

ABSTRACT: Viscoelastic surfactant (VES) based fracturing fluids have shown distinct rheological characteristics from conventional polymer based fluids. A nearly constant shear stress plateau is commonly observed in the shear stress versus shear strain rate curve of the VES fracturing fluids. In this paper, a piecewise power law model is first proposed to describe the rheology of these fluids. A solution of Poiseuille flow with the piecewise rheology is then derived and implemented into the PKN hydraulic fracturing model. The effect of rheology on the fracturing process is evaluated numerically by using an explicit moving mesh algorithm. The numerical solutions with the piecewise rheology are then compared with those from power law models with parameters deduced from different shear strain rate ranges. The results indicate that the conventional "averaged" power law model may be inadequate to yield an accurate prediction for the fracture geometry due to the limited shear strain rate range sampled. 1. INTRODUCTION Hydraulic fracturing is a strongly coupled process involving fracture initiation/propagation, rock mass deformation, fluid flow and solid particle transport in a fracture, fluid flow in a porous medium and heat transfer. While fluid rheology is one of the most critical pieces of information for the successful design of hydraulic fractures, in classical hydraulic fracturing models, the rheology of fluids has traditionally been kept simple in order to accommodate the difficulties arising from the strong coupling between fluid flow and rock deformation. A Newtonian or a power law rheology model is generally assumed for conventional fracturing fluids such as polymer based gels. In a Newtonian model, the shear stress t behaves linearly proportional to the shear strain rate, i.e., t=¿ ¿ , where ¿ is the fluid viscosity. In a power law model, t=K¿ n , where K is the consistency parameter and n is the power index. As fracturing fluids evolve towards what the petroleum industry calls "lean fracturing fluids", these models may often fail to describe the complex behavior of the fluids. For viscoelastic surfactant (VES) based fluids, which form the base of many lean fracturing fluids, a nearly constant stress plateau in shear stress versus shear strain rate curve is commonly observed (see Figure 1 showing the rheology of a model fracturing fluid - fluid XE). One may wonder if the presence of this nearly constant stress plateau in the fluid rheology has any (available in full paper) implications for hydraulic fracturing design, in particular, for the geometry of hydraulic fractures. It is common practice in the petroleum industry to sample only limited data, typically within the shear strain rate range from 25 s -1 to 100 s -1 , in order to then fit a power law model to approximate the rheology of the fracturing fluids. Such an approximation has been validated for typical guar based polymer fluids. As for VES fluids, the presence of the stress plateau may, however, result in the corresponding power index n to become nearly zero. Not only will this nearly zero index n cause numerical difficulties for most hydraulic fracturing simulators, but also it is doubtful that such an approximation will capture the behavior of VES fluids in a hydraulic fracturing context.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper Number: ARMA-04-634

... reduction induced by rock compaction for practical production operations. In the following sections, first, we present the details of the numerical model that include coupled field

**equations**, initial and boundary conditions, constitutive relations, and numerical procedures. Second, the model was verified...
Abstract

ABSTRACT: This paper quantifies the relationship between rock compaction and well flow by including an anisotropic, stressdependent permeability tensor. When producing oil from a weak reservoir, rock compaction induced by pressure drawdown may occur, often with adverse consequences. For weak reservoirs such as unconsolidated sands, rock compaction often causes permeability reduction that may significantly influence the well productivity to the extent that production becomes uneconomic. A 3D finite element model was developed in order to simulate fluid production through a well in a deforming reservoir. Constitutive models for weak reservoir rock and deformation-dependent permeability tensor are also supplemented in the finite element model. The developed model was used to evaluate the influence of rock compaction on well productivity for compaction-sensitive formations. Results show that the model provides an effective tool to identify possible mechanisms associated with rock compaction that cause permeability reduction. The improved understanding of the permeability reduction mechanisms achieved by the model provides a guide to mitigate well productivity impairment resulting from the compaction of these deformable reservoirs. Numerical simulations for different reservoir characteristics, operating conditions, and well configurations were performed in order to establish quantitative relationships between well productivity declines and key field operational control variables. Modeling results for representative cases and the description and formulation of the finite element model are also presented. 1.INTRODUCTION Fluid production of a hydrocarbon reservoir results in decreasing fluid pressure and increasing effective overburden load on reservoir rock. The increase in effective overburden load will in turn compact the reservoir rock and change the stress state in the reservoir. Significant permeability reduction associated with rock compaction around the wellbore region is a well-known phenomenon in oil and gas production in weak reservoirs. As a consequence, rock compaction has adverse effects on well productivity. To quantify well productivity reduction caused by rock compaction, a coupled analysis is required because the physical process involves both geomechanics and fluid flow. Also, in a geologically and geometrically complex setting, it is very difficult, if not possible, to analyze the coupled problem analytically or semi-analytically. On the other hand, advances in numerical computing technology have made the numerical modeling of the coupled geomechanics and fluid flow problem rigorous, robust and efficient in a general fashion. Thus, the objective of this paper is to develop a general 3D finite element model that couples geomechanics and fluid flow, to quantify well productivity reduction induced by rock compaction for practical production operations. In the following sections, first, we present the details of the numerical model that include coupled field equations, initial and boundary conditions, constitutive relations, and numerical procedures. Second, the model was verified for its accuracy and its ability of retaining primary physics by test problems. Third, to demonstrate the capabilities of the model for practical field applications, we conducted modeling studies to examine and discuss how well productivity loss is influenced by operating variables such as production rate, well configurations, and reservoir characteristics including permeability anisotropy and layered heterogeneity in a compacting reservoir. Lastly, conclusions are drawn from this work.