This is a study of the formulation, some basic solutions, and applications of the Biot linearized quasistatic elasticity theory of fluid‐infiltrated porous materials. Whereas most previously solved problems are based on idealizing the fluid and solid constituents as separately incompressible, full account is taken here of constituent compressibility. Previous studies are reviewed and the Biot constitutive equations relating strain and fluid mass content to stress and pore pressure are recast in terms of new material parameters, more directly open to physical interpretation as the Poisson ratio and induced pore pressure coefficient in undrained deformation. Different formulations of the coupled deformation/diffusion field equations and their analogues in coupled thermoelasticity are discussed, and a new formulation with stress and pore pressure as basic variables is presented that leads, for plane problems, to a convenient complex variable representation of solutions. The problems solved include those of the suddenly introduced edge dislocation and concentrated line force and of the suddenly pressurized cylindrical and spherical cavity. The dislocation solution is employed to represent that for quasi‐static motions along a shear fault, and a discussion is given, based on fracture mechanics models for fault propagation, of phenomena involving coupled behavior between the rupturing solid and its pore fluid, which could serve to stabilize a fault against rapid spreading. Also, the solution for a pressurized cylindrical cavity leads to a time‐dependent stress field near the cavity wall, and its relevance to time effects in the inception of hydraulic fractures from boreholes, or from drilled holes in laboratory specimens, is discussed. Various limiting cases are identified, and numerical values of the controlling porous media elastic parameters are given for several rocks.

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Summary A pseudo-three-dimensional (P3DH) model has been developed to describe realistically the evolution of geometry of a 3D hydraulic fracture produced by fluid injection into a reservoir. The present model bypasses produced by fluid injection into a reservoir. The present model bypasses the task of a fully 3D crack geometry calculation, which is at present prohibitively complex in the context of a complete practical fracturing prohibitively complex in the context of a complete practical fracturing simulator. The P3DH model presented formulates the problem in terms of equations for lateral fluid flow and crack opening for the main body of the fracture, coupled with a very efficient scheme for describing vertical fracture growth at each cross section. The equations for lateral flow are solved by finite differences, and the vertical propagation problem is solved by numerical implementation of a singular integral equation on a suitable set of Chebyshev points. The testing of P3DH components shows both an excellent agreement of the lateral propagation model with various analytical solutions (Ref. 1) and a strong sensitivity of vertical propagation to confining stress and stiffness contrast of adjacent strata and to fluid rheology. The sample simulations show that the model produces realistic fracture growth under a wide range of conditions, is extremely sensitive to the dominant containment parameters, and therefore can be used to study the effect of relevant design parameters on fracture shapes and pressures in stimulation treatments. The P3DH model is highly efficient and suitable for incorporation in a general 3D fracturing simulator. The computational effort for calculation of fracture geometry with P3DH is generally comparable to that required for the simulation of the fluid flow in the surrounding reservoir. Introduction Prediction of fracture geometry is one of the central Prediction of fracture geometry is one of the central issues in the engineering design of stimulation treatments as well as other oilfield processes involving fracturing of the reservoir rock. The complexity of the treatments has increased, but data-gathering techniques and the understanding of the basic phenomena have also advanced, so traditional design methods are now being replaced by more detailed simulations. In particular, a need exists for a more detailed and realistic modeling tool for prediction of 3D fracture growth. Such a model is needed for theoretical studies of containment and simulation of laboratory experiments as well as for optimization of fracturing-treatment design in the field. A model that describes fracture geometry more realistically allows much more information to be extracted from pressure data, such as those from minifracture tests and pressure measurements during actual treatments. On the other hand, a fracture-geometry design must be restricted to the essential features and dominant parameters because the completely general solution of the parameters because the completely general solution of the problem is still too complex and may be unnecessary from an problem is still too complex and may be unnecessary from an engineering viewpoint. This paper describes the development and testing of the P3DH designed for the range of conditions of interest in P3DH designed for the range of conditions of interest in fracturing operations. After an explanation and justification of the proposed concept, the paper treats the two types of conventional two-dimensional (2D) models that are used as components of P3DH. Results of these models, which are based on previous work (Refs. 1 and 2), are of great interest because they give correct, comprehensive solutions of the special 2D cases. Testing of the numerical techniques used to solve the components of P3DH is described in later sections. Some sample calculations demonstrate the role played by the principal containment mechanisms. The paper focuses on the prediction of fracture geometry. The details of the coupling of this problem with the fluid flow, heat transfer in the surrounding problem with the fluid flow, heat transfer in the surrounding reservoir, proppant transport, etc., are treated in a companion paper.

Summary We have derived model laws that relate experimental parameters of a physical model of hydraulic fracture propagation to the prototype parameters. Correct representation of elastic deformation, fluid friction, crack propagation, and fluid leakoff forms the basis of the scaling laws. For tests at in-situ stress, high fluid viscosity and low fracture toughness are required. Tests on cement blocks agreed with the scale laws based on elastic behavior. Introduction In hydraulic fracture treatment design, numerical simulation is used to relate measured pressure to fracture geometry. As yet, there is no way to observe fracture geometry in field treatments, except in special tests with extensive monitoring (e.g., Ref. 1). Even then, much room is left for data interpretation. Laboratory tests should therefore serve as benchmarks for numerical simulations. Although there is an enormous difference in the scale of fractures in laboratory tests and in field applications, a numerical model should at least be capable of describing model tests with the appropriate boundary conditions. Many researchers have attempted to study fracture growth in physical model tests. Still, we must critically review previous experimental work in this paper because we think that such efforts can be greatly improved, at least in regard to two important (related) issues: correct scaling of the physical phenomena and stability of fracture propagation. Correct scaling implies that the physics of fluid-driven fracture propagation at field scale must be represented in the test. For instance, if tests are set up at in-situ stress and water is used for fracturing in the laboratory, the fracture pressures required to produce reasonable experimental times (and stable crack propagation) become so low that fracture toughness dominates the process, which is contrary to field observations (e.g., equal pressures during initial propagation and fracture reopening). In addition, the nonpenetrated zone at the fracture tip will disappear and the fracture will grow dynamically. Such experiments can bear no relation to the quasistatic process implied by field conditions nor to any credible numerical simulation of field fracturing.

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