Abstract
Abstract
When a continuous sand is bounded by zones of higher, but unequal, minimum in-situ stress, a vertically asymmetric hydraulic fracture results. The modeling is much more difficult than in the symmetric case mainly because the width equation is harder to formulate and solve. In this paper we present the principal components of the modeling, which includes principal components of the modeling, which includes non-Newtonian flow, leakoff with spurt loss, and "storage" of fluid due to volume expansion. The assumption is that the fracture is highly elongated, i.e., stress contrasts between pay and bounding zones are relatively large (>few hundred psi). Vertical gradients of minimum in-situ stress and fluid pressure can be included in the modeling. To illustrate. the results, we present design calculations for a 30,000 gallon fracture, which was the first stimulation in the Multi-Well Experiment. The 80 ft fracture interval in the Paludal zone has at its upper edge a 520 psi stress contrast, and at its lower edge a 1195 psi contrast. Computed fracture height growth above and below the perforated interval, bottomhole pressure, and width perforated interval, bottomhole pressure, and width profiles in vertical sections are displayed. profiles in vertical sections are displayed. Comparison is made with diagnostic measurements of fracture length, height, and bottomhole pressure.
Introduction
At depths of a few thousand feet or more, induced hydraulic fractures will normally be vertical. Height growth containment is important so that the fracture will reach farther along the payzone, and so that the chance of vertical penetration into, for example, a water-bearing zone will be reduced. Although many factors influence height growth, the most important one appears to be the stress contrast between pay and bounding zones, where by stress we mean minimum in-situ stress. Here we study fracture height growth by developing a model for an expanding hydraulic fracture applicable when the fracture is highly elongated, with length along the payzone much greater than height. However, vertical variations in elastic parameters are not considered. The fracture shape in this paper is self-determined, in contrast to that in which an elliptical shape is chosens and the corresponding height or semi-minor axis determined. A variable-height fracture model has been intensively studied by Cleary and co-workers. The so-called "pseudo-3D" model treats the fluid flow as a dominant ID flow along the payzone, plus an auxiliary ID flow in the vertical direction. Although the models of Nolte and Palmer and Carroll take the vertical flow to be Palmer and Carroll take the vertical flow to be zero, thus simplifying the problem considerably, the general formulations are similar enough to Cleary's to be included under the rubric "pseudo-3D." In all these models, the fracture width is approximated by dividing the fracture into a number of vertical sections, and applying 2D elasticity theory to each vertical "line" crack. Thus the fracture is assumed to be highly elongated with length/height ratio >5. Finally, 3D modeling, with proper 2D fluid flow, is under development, but the problem is formidable and the computer run time enormous. In the interim we can learn much from pseudo-3D models. In general, the bounding layer stresses will not be equal, leading to a fracture which is vertically asymmetric, and furthermore both the minimum in-situ stress and the fluid pressure will vary with depth. This is the principal modification we make to the symmetric model, described previously. Other additions are:spurt loss has been included in the leakoff,non-Newtonian flow is included.
An extended model for the symmetric case, which has essentially the same components as herein, is described elsewhere. In that paper, a comparison is made between published results in three pseudo-3D models, some discrepancies are pointed out, and suggestions for reconciling the models are made. In the asymmetric model of this paper, calculation of fracture width is the most difficult task. We give most of the details here. Theoretical calculations of asymmetric fracture shapes have been reported by Settari and Cleary, but they appear to emphasize low stress contrasts (< couple hundred psi). Nolte gives one asymmetric width profile in a vertical section, but no method of calculation, nor any resultant fracture shapes, were given. Finally, to illustrate the results of the asymmetric model, we use the model to predict fracture height, pressure, and width for the first stimulation of the Multi-Well Experiment (MWX) carried out in December 1983. This prediction is compared with available fracture diagnostic measurements.
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