Introduction

The Grid Oriented Hydraulic Fracture Extension Replicator is a planar 3-D geometry and fluid/solid transport simulator. The model relies heavily upon log-derived input data. Rock elastic properties, porosity and lithology can be input directly from standard log ASCII (LAS) files. This data along with pore fluid gradients, regions of overpressure or depletion, and tectonic offsets are used to constrain in-situ stress profiles. Effects of poroelasticity are considered in the calculation of closure stress.

The model has been used extensively in many regions including soft-sediment frac-packing, hard rock/tight gas environments, naturally fractured reservoirs, moderate perm oil sands, and for acid-frac design in carbonates. The formulation can be applied in all these areas without special "tuning" or incorporation of different assumptions. Major features of the model are described in the following brief sections. An example is presented to illustrate the model's utility.

Model Features

Data Input: The Grid Oriented Hydraulic Fracture Extension Replicator is based on a grid structure. The grid is mapped onto the surface of the created fracture and is therefore not necessarily planar in real-space. State variables, such as pressure, width, shear rate, fluid age, leakoff rate, prop concentration, velocity, fluid composition, proppant composition, etc., are defined at each grid cell. Similarly, input data such as rock elastic properties, pore fluid pressures, stresses, and reservoir properties, must be assigned at each cell location. This allows for both vertical and lateral variations in properties. Vertical variations can be described by well logs but lateral variations require additional knowledge of local geology and structure that can be obtained from 2-D and 3-D seismic or crosswell imaging. To take full advantage of the capability to model lateral variations, the model can be run in either "symmetric" or "asymmetric" modes. In the asymmetric mode both wings of the fracture are modeled so the effects of lateral pressure gradients, bed dip, and changes in rock properties can be modeled. This mode is also useful for modeling simultaneous multiple fractures initiated from a horizontal wellbore.

The program is structured to utilise all available well log data. Standard log ASCII format (LAS) files can be input to the Grid Oriented Hydraulic Fracture Extension Replicator and any data contained in the log tracks can be directly imported to the model an a foot-by-foot basis. Data is averaged over the specified node size.

For history matching, job data recorded on site can be used to re-create the actual pumping schedule used in the job. An input module automatically filters the actual job data and records any changes in pump rate or proppant concentration. The data is then converted to a rate schedule that can be directly imported into Grid Oriented Hydraulic Fracture Extension Replicator.

Fluid Rheology: Fluid rheology is described using a time-dependent Carreau rheology model that describes the low-shear Newtonian plateau, power-law, and high-shear plateau regions. Fluid break and thermal degradation is modeled using a complex model that has been validated against extensive laboratory measurements. Effects of solids addition on fluid rheology are also modeled.

Inputs to the model describe the initial (zero-time) characteristics of the fluid. These include n', k', the zero-shear viscosity (Newtonian plateau) and the solvent viscosity. The break profile, whether caused by spontaneous degradation at temperature or chemical breakers, is described by an additional eight parameters. These control the induction time for the onset of changes in n', k', and zero-shear viscosity, the exponential rate of break of each parameter after the induction time, and the maximum value of n' achieved and the minimum value of zero-shear viscosity reached (in case of an incomplete break).

Within the model, the age of the fluid in each cell is computed, along with the local shear conditions, and the volume fraction of each fluid present in the cell. At each cell and at each time-step the effective viscosity is determined from the time/shear dependent rheology, composition and solids loading of the fluid. It has been observed that local variations in fluid mobility have a significant effect on pressure and velocity distribution in the fracture, hence on ultimate frac geometry and proppant placement. For this reason the direct coupling between rheology and geometry is critical.

Fluid Loss: Fluid loss is described by an extension of the classical Howard-Fast or Carter leakoff model. Effects of the compressible region, viscous-invaded zone and filter cake are modeled. The formulation also handles the invasion of non-Newtonian whole gel prior to filter-cake deposition. Permeability, local shear, fluid age, filtration pressure, and other factors automatically scale spurt volume and leakoff coefficients. Filter cake erosion and equilibrium leakoff are also modeled.

Inputs to the model include spurt volume, wall building coefficients, equilibrium leakoff coefficient, filter cake compressibility, rel-perm and fluid damage factors, permeability, porosity, pore pressure, and local frac fluid pressure, viscosity and shear rate. The leakoff parameters are input under standard conditions of permeability, porosity, and filtration pressure. They are automatically scaled to the input permeability, porosity, and local conditions at each grid cell at each time. In this kind of model the idea of a single constant leakoff coefficient is in-appropriate since leakoff varies continuously, both spatially and temporally.

The model handles the effect of changing injected fluid since each fluid has its own array or leakoff parameters. As the bulk fluid composition in each node changes with time, the leakoff parameters are varied with the composition. The model clearly shows that changing from water or linear pad to crosslinked fluid (or vice-versa) can significantly change leakoff behavior and fracture geometry.

Pressure Dependent Leakoff: Pres-sure dependent leakoff caused by dilation of natural fissures is modeled. Field diagnostics have been developed to determine the critical fissure opening pressure and pressure dependent leakoff coefficient for input to the model.

Each cell in the model has an assigned minimum in-situ stress. The critical fissure opening pressure is handled by inputting a stress anisotropy offset for each cell. This offset defines the net pressure (above local closure) required to begin dilating existing natural fractures or pre-existing planes of weakness.

Each cell is also assigned a pressure dependent leakoff factor that controls the rate of increase of leakoff as a function of pressure above the critical fissure opening pressure. In many cases the observed leakoff can be many times higher than the matrix-dominated leakoff observed during pressure falloff at the end of a job.

Changes in apparent modulus or stiffness are also handled by an input pressure dependent modulus coefficient. Changes in modulus, either stiffening or softening are assumed to occur at the same critical pressure as pressure dependent leakoff.

Fluid/Solid Transport: Fluid and solid transport are computed using an implicitly coupled finite-difference model. The local fluid age, solids loading, shear rate, and fluid composition determine fluid mobility at each grid cell and are updated at each time-step (typically 1-5 seconds) throughout the simulation. The total potential field, including both the induced pressure gradient and density-driven flow drives flow. Fluid velocity controls local shear rate which in-turn controls fluid mobility. Hence local variations in fluid properties or flow conditions feedback to control future fluid and solid movement. Significant control of fracture geometry and proppant placement can be achieved by varying fluid properties.

Direct modeling of slurry transport in a 4 foot by 16 foot long slot has validated the solids transport model by 16-foot long slot model. These physical model studies have shown the importance of the Carreau rheology model instead of the traditional power-law approximation. The studies have also shown that the formulation used, which is equivalent to that used in most reservoir simulators, adequately describes most slurry flow phenomena. One factor, which is still being investigated, is the re-suspension transport of sand by low viscosity fluids.

In-Situ Stress Calculation: Rather than entering an assumed stress profile, the Grid Oriented Hydraulic Fracture Extension Replicator uses data from identifiable sources and applies consistent boundary conditions to determine stress profiles. Poisson's ratio and Young's Modulus are input manually or from log (LAS) files, and assigned to each grid cell. All properties can vary both vertically and laterally. Asymmetric fracture growth can easily be modeled.

Pore fluid gradients must be entered for each cell, along with pore pressure offsets to account for local regions of overpressure of depletion. Total pore pressure in each cell is calculated from the fluid gradients, grid cell datum elevations and pressure offsets. The resulting pore pressure is used, along with vertical and horizontal Biot's poroelastic constants, to determine stress resulting from uni-axial strain loading of the overburden sediment. The effective overburden gradient can be specified. It has been found, both through field observations and in laboratory measurements, that Biot's constant is not 1.0 (as is commonly assumed) for many formations. In moderately to well consolidated formations, values as low as 0.45 have been measured. This is an input parameter that can significantly affect computed stress that is commonly ignored but should be rigorously determined.

Since stress in most areas does not conform to the quiescent assumptions of the uni-axial strain model, regional tectonic strains are input. Experience with the model has shown that a constant strain boundary condition is more representative of "real-world" conditions than a constant stress offset. When considering a vertical interval of a few hundred feet it is more reasonable to assume that the vertical bed stack has been subjected to approximately constant strain, rather than constant lateral stress. In that case each bed will develop a different lateral stress related to its stiffness and the applied strain. High modulus layers will develop appreciable more stress than soft sediment layers. Total closure pressure is therefore a function of pore pressure, poroelasticity, Poisson's Ratio, and Young's Modulus.

Width Solution: Originally the Grid Oriented Hydraulic Fracture Extension Replicator used as an exact numerical solution of the complete surface integral formulation for the displacement of a semi-infinite half space proposed by Boussinesq. The solution assumed that the entire rock mass behaved as a linear-elastic closely coupled medium. This formulation resulted in large fracture widths and large stress concentrations at the fracture tips, leading to lack of height containment.

Observations of real jobs, based on treating pressures, production performance, tracer surveys, tiltmeter and micro-seismic mapping, all suggest that fractures grow very rapidly in length and are better contained than the elastically coupled model predicts. The assumption of complete elastic coupling throughout the entire rock mass is not representative of most formations. The lack of coupling is illustrated buy the existence of micro-seismic events caused shear failures in the deformed rock mass around the fracture. To approximate a shear de-coupled displacement solution the area of applied load integrated in the width solution was truncated to an input "shear dampening radius". This formulation results in improved height containment and a better representation of field observations.

Data Output: Because the Grid Oriented Hydraulic Fracture Extension Replicator provides state variables at each node at each time-step, there is a huge amount of output data to be processed. Along with typical time-vector plots of surface and bottom-hole treating pressure, injected rate, fluid efficiency, surface and bottom-hole prop concentration, Grid Oriented Hydraulic Fracture Extension Replicator also produces a time- lapse map of the fracture surface showing many of the tracked state variables.

By "re-playing" the time-lapse image the user can display any chosen variable over the surface of the fracture and watch its variation through time. Output variables include fluid pressure, net pressure, width, proppant volume fraction of up to ten prop types, proppant concentration, leak off rate, fracture and porous medium shear rates, fluid age, viscosity, injection rate and pressure drop through each set of perforations, and velocity in X and Z directions. For acid fracturing the model also outputs maps of surface and bulk concentration of all reactive species and products, along with the etched fracture width.

When history matching observed pressures, the model results can be plotted over the observed pressures while the model is running. This capability also provides real-time plotting for use at the job site. The program has real-time serial communication protocols to acquire data from the frac van and can plot it with the model results during the job.

The author is with Barree and Associates