Publications
REFEREED PUBLICATIONS IN JOURNALS
3.  Birdsell, D., S. Karra, and H. Rajaram On the Representation of the Porosity‐Pressure Relationship in General Subsurface Flow Codes, Water Resources Research, 54/2, pp. 13821388, 2018. Abstract The governing equations for subsurface flow codes in a deformable porous media are derived from the balance of fluid mass and Darcy’s equation. One class of these codes, which we call general subsurface flow codes (GSFs), allow for more general constitutive relations for material properties such as porosity, permeability and density. Examples of GSFs include PFLOTRAN, FEHM, TOUGH2, STOMP, and some reservoir simulators such as BOAST. Depending on the constitutive relations used in GSFs, an inconsistency arises between the standard groundwater flow equation and the governing equation of GSFs, and we clarify that the reason for this inconsistency is because the Darcy’s equation used in the GSFs should account for the velocity of fluid with respect to solid. Due to lack of awareness of this inconsistency, users of the GSFs tend to use a porosity‐pressure relationship that comes from the standard groundwater flow equation and assumes that the relative velocity is already accounted for. For the Theis problem, we show that using this traditional relationship in the GSFs leads to significantly large errors. We propose an alternate porosity‐pressure relationship that is consistent with the derivation of the governing equations in the GSFs where the solid velocity is not tracked, and show that, with this relationship, the results are more accurate for the Theis problem. The purpose of this note is to make the users and developers of these GSFs aware of this inconsistency and to advocate that the alternate porosity model derived here should be incorporated in GSFs. / Download 
2.  Birdsell, D., H. Rajaram, D. Dempsey, and H. Viswanathan Hydraulic fracturing fluid migration in the subsurface: A review and expanded modeling results, Water Resources Research, 51/9, pp. 71597188, 2015. Abstract Understanding the transport of hydraulic fracturing (HF) fluid that is injected into the deep subsurface for shale gas extraction is important to ensure that shallow drinking water aquifers are not contaminated. Topographically driven flow, overpressured shale reservoirs, permeable pathways such as faults or leaky wellbores, the increased formation pressure due to HF fluid injection, and the density contrast of the HF fluid to the surrounding brine can encourage upward HF fluid migration. In contrast, the very low shale permeability and capillary imbibition of water into partially saturated shale may sequester much of the HF fluid, and well production will remove HF fluid from the subsurface. We review the literature on important aspects of HF fluid migration. Single‐phase flow and transport simulations are performed to quantify how much HF fluid is removed via the wellbore with flowback and produced water, how much reaches overlying aquifers, and how much is permanently sequestered by capillary imbibition, which is treated as a sink term based on a semianalytical, one‐dimensional solution for two‐phase flow. These simulations include all of the important aspects of HF fluid migration identified in the literature review and are performed in five stages to faithfully represent the typical operation of a hydraulically fractured well. No fracturing fluid reaches the aquifer without a permeable pathway. In the presence of a permeable pathway, 10 times more fracturing fluid reaches the aquifer if well production and capillary imbibition are not included in the model. / Download 
1.  Birdsell, D., H. Rajaram, and G. Lackey Imbibition of hydraulic fracturing fluids into partially saturated shale, Water Resources Research, 51/8, pp. 67876796, 2015. Abstract Recent studies suggest that imbibition of hydraulic fracturing fluids into partially saturated shale is an important mechanism that restricts their migration, thus reducing the risk of groundwater contamination. We present computations of imbibition based on an exact semianalytical solution for spontaneous imbibition. These computations lead to quantitative estimates of an imbibition rate parameter A with units of LT−1/2 for shale, which is related to porous medium and fluid properties, and the initial water saturation. Our calculations suggest that significant fractions of injected fluid volumes (15–95%) can be imbibed in shale gas systems, whereas imbibition volumes in shale oil systems is much lower (3–27%). We present a nondimensionalization of A, which provides insights into the critical factors controlling imbibition, and facilitates the estimation of A based on readily measured porous medium and fluid properties. For a given set of medium and fluid properties, A varies by less than factors of ~1.8 (gas nonwetting phase) and ~3.4 (oil nonwetting phase) over the range of initial water saturations reported for the Marcellus shale (0.05–0.6). However, for higher initial water saturations, A decreases significantly. The intrinsic permeability of the shale and the viscosity of the fluids are the most important properties controlling the imbibition rate. / Download 
PROCEEDINGS REFEREED
2.  Birdsell, D., H. Rajaram, and S. Karra Code development for modeling induced seismicity with flow and mechanics using a discrete fracture network and matrix formulation with evolving hydraulic diffusivity, 52nd US Rock Mechanics/Geomechanics Symposium and Discrete Fracture Network Engineering Conference, ARMA 18565, 2018. Abstract Injectioninduced seismicity (IIS) depends on pore pressure, insitu stress state, and fault orientation; generally occurs in basement rock that contains fractures and faults; and moves away from the injection well as a nonlinear diffusion process. Therefore, to numerically model IIS a code should incorporate flow and geomechanics, the presence of fractures and faults, and the capability for hydraulic diffusivity to evolve with effective stress and failure history. In this work, we introduce and verify a modeling framework that allows hydraulic diffusivity to evolve as fractures open and close. Details and challenges in code development are discussed, including how the Bandis model for normal fracture deformation can be used to calculate hydraulic diffusivity as a function of effective normal stress. The discrete fracture network and matrix (DFNM) model is implemented in PFLOTRAN such that hydraulic diffusivity has different constitutive relationships for fracture and matrix grid cells. This model is applied to understand the recent IIS near Greeley, Colorado, and its results are compared to: (a) a traditional DFNM model where hydraulic diffusivity cannot evolve and (b) an equivalent porous media (EPM) model where the effect of the fractures are averaged over a large region of rock. The new DFNM model predicts critical pressure will propagate farther from an injection well. This modeling framework shows promise for applications where fracture and matrix flow are important and hydraulic diffusivity is a function of pressure, stress, and/or shear failure history. 
1.  Birdsell, D., H. Rajaram, D. Dempsey, and H. Viswanathan Numerical Model of Hydraulic Fracturing Fluid Transport in the Subsurface with Pressure Transient and Density Effects, 49th US Rock Mechanics/Geomechanics Symposium, 2015. Abstract Understanding the transport of hydraulic fracturing (HF) fluid that is injected into the deep subsurface for shale gas extraction is important to ensure that shallow drinking water aquifers are not contaminated. Pressure gradients, permeable pathways such as faults or improperly abandoned wellbores, and the density contrast of the HF fluid to the surrounding brine could encourage upward HF fluid migration. In contrast, very low shale permeability and well production may work to keep HF fluid at depth and remove it from the subsurface. Singlephase flow and transport simulations are performed to quantify how much HF fluid is removed via the wellbore and how much reaches overlying aquifers. If a permeable pathway connects the shale reservoir to the overlying drinking water aquifer, the pressure transient due to injection and the density contrast allows rapid upward plume migration at early times, but well production reverses the direction of flow and removes a large amount of HF fluid from the subsurface. We present estimates of HF fluid migration to shallow aquifers during the first 1,000 years and show that the pressure transient from well operations should be included in subsequent numerical models while buoyancy may be neglected depending on depth and permeability.

THESES
1.  Birdsell, D. An Investigation of Subsurface Fluid Injections Related to Oil and Gas Development: Modeling Hydraulic Fracturing Fluid Migration and Induced Seismicity, Dissertation, 161 pp., 2018. Abstract Unconventional oil and gas development is made economically feasible by horizontal drilling and hydraulic fracturing, and it can produce large volumes of wastewater that are injected into disposal wells. In the first part of this dissertation, we investigate the potential for fracturing fluid to migrate from the target formation to contaminate shallow drinking water aquifers. We present semianalytical calculations for capillary imbibition into shale, which can sequester up to 95% of the fracturing fluid, thus preventing migration to aquifers. Next, we present a numerical model of fracturing fluid migration, which is the first to combine injection and production, imbibition, and buoyancy. In the absence of a permeable pathway between the injection formation and the aquifer, no contamination occurs. In the presence of a permeable pathway, well suction and capillary imbibition significantly reduce the amount of fracturing fluid reaching the aquifer compared to scenarios that do not account for suction and imbibition. In part two, we present a new framework for modeling basinscale injectioninduced seismicity (IIS). The framework incorporates flow and geomechanics, the presence of fractures and faults, and the capability for hydraulic diffusivity to evolve with effective stress and earthquake history, which is modeled by the MohrCoulomb failure criteria. The model is implemented in the massivelyparallel code PFLOTRAN, which is important to capture the large length scales (\( \sim \)10 km) and many fractures and faults (1001000s). Applications of this model: (a) put constrains on the hydraulic diffusivity in basement rock, which may have been too large in previous modeling studies; (b) explain the heterogeneity of earthquake locations; and (c) capture the variations in critical pressure that cause earthquakes, based on stress state and fault orientation. In part three, we show work that verifies and increases the accuracy of subsurface simulators, which is important for the continued investigation of fluid migration, IIS, and other subsurface phenomena. First, we derive a porositypressure relationship for general subsurface flow codes (GSFs) which accounts for the relative velocity between pore fluid and rock matrix. Simulations using this relationship show excellent agreement between GSFs and the groundwater flow equation. Next we verify the fullycoupled flow and geomechanics implementation in PFLOTRAN by comparing to an analytical solution. 
show/hide list of publications
REFEREED PUBLICATIONS IN JOURNALS
3.  Birdsell, D., S. Karra, and H. Rajaram On the Representation of the Porosity‐Pressure Relationship in General Subsurface Flow Codes, Water Resources Research, 54/2, pp. 13821388, 2018. Abstract The governing equations for subsurface flow codes in a deformable porous media are derived from the balance of fluid mass and Darcy’s equation. One class of these codes, which we call general subsurface flow codes (GSFs), allow for more general constitutive relations for material properties such as porosity, permeability and density. Examples of GSFs include PFLOTRAN, FEHM, TOUGH2, STOMP, and some reservoir simulators such as BOAST. Depending on the constitutive relations used in GSFs, an inconsistency arises between the standard groundwater flow equation and the governing equation of GSFs, and we clarify that the reason for this inconsistency is because the Darcy’s equation used in the GSFs should account for the velocity of fluid with respect to solid. Due to lack of awareness of this inconsistency, users of the GSFs tend to use a porosity‐pressure relationship that comes from the standard groundwater flow equation and assumes that the relative velocity is already accounted for. For the Theis problem, we show that using this traditional relationship in the GSFs leads to significantly large errors. We propose an alternate porosity‐pressure relationship that is consistent with the derivation of the governing equations in the GSFs where the solid velocity is not tracked, and show that, with this relationship, the results are more accurate for the Theis problem. The purpose of this note is to make the users and developers of these GSFs aware of this inconsistency and to advocate that the alternate porosity model derived here should be incorporated in GSFs. / Download 
2.  Birdsell, D., H. Rajaram, D. Dempsey, and H. Viswanathan Hydraulic fracturing fluid migration in the subsurface: A review and expanded modeling results, Water Resources Research, 51/9, pp. 71597188, 2015. Abstract Understanding the transport of hydraulic fracturing (HF) fluid that is injected into the deep subsurface for shale gas extraction is important to ensure that shallow drinking water aquifers are not contaminated. Topographically driven flow, overpressured shale reservoirs, permeable pathways such as faults or leaky wellbores, the increased formation pressure due to HF fluid injection, and the density contrast of the HF fluid to the surrounding brine can encourage upward HF fluid migration. In contrast, the very low shale permeability and capillary imbibition of water into partially saturated shale may sequester much of the HF fluid, and well production will remove HF fluid from the subsurface. We review the literature on important aspects of HF fluid migration. Single‐phase flow and transport simulations are performed to quantify how much HF fluid is removed via the wellbore with flowback and produced water, how much reaches overlying aquifers, and how much is permanently sequestered by capillary imbibition, which is treated as a sink term based on a semianalytical, one‐dimensional solution for two‐phase flow. These simulations include all of the important aspects of HF fluid migration identified in the literature review and are performed in five stages to faithfully represent the typical operation of a hydraulically fractured well. No fracturing fluid reaches the aquifer without a permeable pathway. In the presence of a permeable pathway, 10 times more fracturing fluid reaches the aquifer if well production and capillary imbibition are not included in the model. / Download 
1.  Birdsell, D., H. Rajaram, and G. Lackey Imbibition of hydraulic fracturing fluids into partially saturated shale, Water Resources Research, 51/8, pp. 67876796, 2015. Abstract Recent studies suggest that imbibition of hydraulic fracturing fluids into partially saturated shale is an important mechanism that restricts their migration, thus reducing the risk of groundwater contamination. We present computations of imbibition based on an exact semianalytical solution for spontaneous imbibition. These computations lead to quantitative estimates of an imbibition rate parameter A with units of LT−1/2 for shale, which is related to porous medium and fluid properties, and the initial water saturation. Our calculations suggest that significant fractions of injected fluid volumes (15–95%) can be imbibed in shale gas systems, whereas imbibition volumes in shale oil systems is much lower (3–27%). We present a nondimensionalization of A, which provides insights into the critical factors controlling imbibition, and facilitates the estimation of A based on readily measured porous medium and fluid properties. For a given set of medium and fluid properties, A varies by less than factors of ~1.8 (gas nonwetting phase) and ~3.4 (oil nonwetting phase) over the range of initial water saturations reported for the Marcellus shale (0.05–0.6). However, for higher initial water saturations, A decreases significantly. The intrinsic permeability of the shale and the viscosity of the fluids are the most important properties controlling the imbibition rate. / Download 
PROCEEDINGS REFEREED
2.  Birdsell, D., H. Rajaram, and S. Karra Code development for modeling induced seismicity with flow and mechanics using a discrete fracture network and matrix formulation with evolving hydraulic diffusivity, 52nd US Rock Mechanics/Geomechanics Symposium and Discrete Fracture Network Engineering Conference, ARMA 18565, 2018. Abstract Injectioninduced seismicity (IIS) depends on pore pressure, insitu stress state, and fault orientation; generally occurs in basement rock that contains fractures and faults; and moves away from the injection well as a nonlinear diffusion process. Therefore, to numerically model IIS a code should incorporate flow and geomechanics, the presence of fractures and faults, and the capability for hydraulic diffusivity to evolve with effective stress and failure history. In this work, we introduce and verify a modeling framework that allows hydraulic diffusivity to evolve as fractures open and close. Details and challenges in code development are discussed, including how the Bandis model for normal fracture deformation can be used to calculate hydraulic diffusivity as a function of effective normal stress. The discrete fracture network and matrix (DFNM) model is implemented in PFLOTRAN such that hydraulic diffusivity has different constitutive relationships for fracture and matrix grid cells. This model is applied to understand the recent IIS near Greeley, Colorado, and its results are compared to: (a) a traditional DFNM model where hydraulic diffusivity cannot evolve and (b) an equivalent porous media (EPM) model where the effect of the fractures are averaged over a large region of rock. The new DFNM model predicts critical pressure will propagate farther from an injection well. This modeling framework shows promise for applications where fracture and matrix flow are important and hydraulic diffusivity is a function of pressure, stress, and/or shear failure history. 
1.  Birdsell, D., H. Rajaram, D. Dempsey, and H. Viswanathan Numerical Model of Hydraulic Fracturing Fluid Transport in the Subsurface with Pressure Transient and Density Effects, 49th US Rock Mechanics/Geomechanics Symposium, 2015. Abstract Understanding the transport of hydraulic fracturing (HF) fluid that is injected into the deep subsurface for shale gas extraction is important to ensure that shallow drinking water aquifers are not contaminated. Pressure gradients, permeable pathways such as faults or improperly abandoned wellbores, and the density contrast of the HF fluid to the surrounding brine could encourage upward HF fluid migration. In contrast, very low shale permeability and well production may work to keep HF fluid at depth and remove it from the subsurface. Singlephase flow and transport simulations are performed to quantify how much HF fluid is removed via the wellbore and how much reaches overlying aquifers. If a permeable pathway connects the shale reservoir to the overlying drinking water aquifer, the pressure transient due to injection and the density contrast allows rapid upward plume migration at early times, but well production reverses the direction of flow and removes a large amount of HF fluid from the subsurface. We present estimates of HF fluid migration to shallow aquifers during the first 1,000 years and show that the pressure transient from well operations should be included in subsequent numerical models while buoyancy may be neglected depending on depth and permeability.

THESES
1.  Birdsell, D. An Investigation of Subsurface Fluid Injections Related to Oil and Gas Development: Modeling Hydraulic Fracturing Fluid Migration and Induced Seismicity, Dissertation, 161 pp., 2018. Abstract Unconventional oil and gas development is made economically feasible by horizontal drilling and hydraulic fracturing, and it can produce large volumes of wastewater that are injected into disposal wells. In the first part of this dissertation, we investigate the potential for fracturing fluid to migrate from the target formation to contaminate shallow drinking water aquifers. We present semianalytical calculations for capillary imbibition into shale, which can sequester up to 95% of the fracturing fluid, thus preventing migration to aquifers. Next, we present a numerical model of fracturing fluid migration, which is the first to combine injection and production, imbibition, and buoyancy. In the absence of a permeable pathway between the injection formation and the aquifer, no contamination occurs. In the presence of a permeable pathway, well suction and capillary imbibition significantly reduce the amount of fracturing fluid reaching the aquifer compared to scenarios that do not account for suction and imbibition. In part two, we present a new framework for modeling basinscale injectioninduced seismicity (IIS). The framework incorporates flow and geomechanics, the presence of fractures and faults, and the capability for hydraulic diffusivity to evolve with effective stress and earthquake history, which is modeled by the MohrCoulomb failure criteria. The model is implemented in the massivelyparallel code PFLOTRAN, which is important to capture the large length scales (\( \sim \)10 km) and many fractures and faults (1001000s). Applications of this model: (a) put constrains on the hydraulic diffusivity in basement rock, which may have been too large in previous modeling studies; (b) explain the heterogeneity of earthquake locations; and (c) capture the variations in critical pressure that cause earthquakes, based on stress state and fault orientation. In part three, we show work that verifies and increases the accuracy of subsurface simulators, which is important for the continued investigation of fluid migration, IIS, and other subsurface phenomena. First, we derive a porositypressure relationship for general subsurface flow codes (GSFs) which accounts for the relative velocity between pore fluid and rock matrix. Simulations using this relationship show excellent agreement between GSFs and the groundwater flow equation. Next we verify the fullycoupled flow and geomechanics implementation in PFLOTRAN by comparing to an analytical solution. 