@Copyright 1977, Amefican lnslltute of Mining; MeIallurglMl, and Petrol, nEngineers, inc. This paper was preaenterl at Ihe 52nd Annual FM Techmcsl Conference and Extnblt!on of the Society of Petroleum Engineers of AlME. held m Denver, Colorado, Oct. 9.12.1977. The material is sub;ect to Correchonbyt heauthor. Permission locopyia restdcted toanabstracl olnotmore lhan300 words. Write 6200 N. Central Expy. Dallas, Texas 75206. - ABSTRACT Particle transport within the Berea, NoxieandCleve-Iandsandstones having mean pore sizes of 10, 15and30 micronst respectively, was studied by infection of aqueous suspensions of sand particles hovingmean particle sizes of 4, 6and7 microns. Aconstarlt flow rate w~sused, and the pressure drop across the core wascontinuously monitored.The effluents were collected ina fraction collector, and the number of particles per milliliter and distribution of sizes of the particles were determined using apatiicle counter interfaced withc computer. These data were used to postulatea theory of discrete particle transport within porous media from a statistical point of view. A statistical random walk model was developed usinSPoiseuille's capillary flow equation and the actual pore size distribution of thecore to calculate the pressure drop across the core. Pa,-ticles areselecte6usin9a rondom number generator and the actual particle size distribution, and are tested for passage through the most probable c-pillary witha pressure threshold function.If the particle passes, another particle Isgenerated; however, ifthe particle lodges, onewpressure iscalwlated, andeach capillary is tested for particle breakout with the newpresxsre. This process is continued until ssplugging pressure i:~ttained ora ._ maximum number of particles pass through the core. The calculated pressures anddistributians of effluent particles close! y approximate the experimental Iy obsewed data. The process of particle transport, where COIIoidal forces are negl iglble, is a random statistical process which can be represented by a statistical mathematical model. ~RLECa f)oNALDsoN

Models of three-dimensional (3D) fracture propagation are being developed to study the effect of variations of stress and rock properties on fracture height and bottomhole pressure (BHP). Initially a blanket sand bounded by zones of higher minimum in-situ stress is considered, with stresses symmetrical about both the pay-zone axis and the wellbore. An elliptical fracture perimeter is assumed. Fluid flows are one-dimensional (1D) Newtonian in the direction of the pay zone. Two models, FL1 and FL2, are developed. In FL1, a discontinuous stress variation is approximated by a y2 variation in the vertical coordinate, and the fracture criterion, Ki = Kc, is satisfied at both major and minor axes. The net pressure at the tip, Lf, of the long axis required by the boundary condition Ki = Kc does not seem crucial in determining fracture height or BHP (compare with one group of published models that assumes p = 0 at Lf). Model FL2 properly represents the discontinuous stresses, and satisfies Ki = Kc at the wellbore but not at the tip of the long axis. A parametric study is made, with both models, of the comparative effects of stress contrast, Kc, pay-zone height, h, and Young's modulus, E, on fracture height and BHP. Results indicate that Kc does not have as much effect as either E or, at least for large stress contrasts. Model FL2 suggests the possibility of a rapid growth in fracture height as is reduced. Such modeling may be able to give an upper or "safe" limit on the pumping parameters ( and ) to ensure good containment. When the stress contrast is high, 700 psi [4826 kPa], an analytic derivation of BHP appears to be a good approximation for the parameters we use, if everywhere the fracture height is assumed equal to the pay zone height. Although leakoff is neglected here, subsequent modeling results show that, for leak off coefficients 0.001 ft- min [3.9 × 10 -5 m.s ], the results herein are a good approximation to the case when leak off is included. Introduction In their essence, models of hydraulic fracture propagation involve elasticity theory and fluid mechanics. The first is concerned with the fracture opening or width, w(p), as a function of net pressure on the fracture faces, while the second is concerned with the pressure drop, p(w), caused by the flow of viscous fluids in the fracture. Simultaneous solution of these equations includes a boundary condition that often takes the form Ki = Kc, where Ki is the stress-intensity factor at a point on the fracture tip, and Kc is the fracture toughness. The final solution is very complex in 3D, when a vertical fracture can expand vertically as well as horizontally along the pay zone. Thus, the first solutions were essentially two-dimensional (2D), and they assumed that the fracture height, hf, was fixed at the pay zone height, h. The 2D solutions were clustered in two groups as summarized by Nordgren, Perkins, and Geertsma and Haafkens. The first grouping, based on a model by Christianovich and Zheltov, assumed that the sides of an elongated, vertical fracture were parallel (i.e., free slippage between the pay and bounding zones, or no vertical stiffness). Other papers in this grouping included Geertsma and de Klerk, Daneshy and Settari. SPEJ P. 870^

The extent of formation damage due to invasion of fracturing fluids during the hydraulic fracturing process was studied. An apparatus, which can simulate reservoir conditions, is described for measuring permeability before and after fluids are pumped across the surface of a core. Dynamic fluid loss was also measured of fluid which passes through the core plug.

Laboratory permeability measurements of low permeability (less than 1 md) reservoir rock is significantly affected by test confining pressure. Knowledge of the confining pressure is required to predict permeability at reservoir conditions. A permeability at reservoir conditions. A model is proposed which relates electrical conductivity, permeability, and pore dimensions to confining pressure. The model assumes that rock pores are interconnected by thin cracks and microfractures which can be modeled by rectangular slits. For this model, core permeability is related to confining pressure by a third order polynomial, and electrical conductivity Is polynomial, and electrical conductivity Is related to confining pressure by a first order polynomial, Electrical conductivity and permeability versus confining pressure measurements were made on test cores having laboratory Klinkenberg permeabilities from 20 to 200 microdarcys. The experimental measurements were found to be consistent with the slit model theory. Introduction Thomas and Ward have shown that the measured permeability of many western gas sandstone cores is significantly decreased by increased test confining pressure, whereas effective porosity is only slightly affected. Jones and Owens developed an empirical method for relating permeability to confining pressure which is valid for a variety of low-permeability western gas sandstone core. The results are consistent enough to suggest that a rock matrix model can be developed to relate pore geometry to laboratory permeability. Their studies suggest that low-permeability reservoir rock consists of a matrix material containing larger pores which contribute mostly to the rock porosity. The larger-pores are interconnected by smaller pore throats that restrict permeability and are easily deformed by changes in confining pressure. Wyllie and Gardner developed a capillary model in which pores and pore throats are assumed to be short sections of capillary tubes. Stacking sections of the capillary tubes together in a random manner forms a model which can relate porosity, electrical conductivity, and permeability. porosity, electrical conductivity, and permeability. Archie's equation is often inferred from the Wyllie-Gardner model. It is possible to extend the capillary model to include the effects of confining pressure. The extension is made by assuming that each capillary tube is a thick-walled cylinder to which the confining pressure is applied. As the confining pressure is increased, the inside diameter of the capillary is reduced which reduces the permeability and electrical conductivity. This model could be useful if the pore throats were nearly round capillary tubes. It is more likely that pore throats in low-permeability sands can be described better as slits, cracks or micro-fractures. For the same cross-section area, a slit will be a much weaker structure than a capillary tube and will undergo a larger change in area due to an applied external stress. Our paper proposes that a slit model can paper proposes that a slit model can describe the effect of confining pressure on permeability and electrical conductivity for permeability and electrical conductivity for low permeability reservoir rock. Slit Model Theory The following assumptions have been made for the model:Flow paths through the core are independent and consist of a series of pores connected by rectangular slits.The slits are uniform in size in each flow path.Permeability and electrical conductivity are limited primarily by slit dimensions and fluid properties.Viscous flow is assumed; slit entrance and surface roughness effects are neglected. p. 391

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