Diagnosis and Evaluation of Fracturing Treatments

Abstract

Summary This paper introduces the pressure derivative in fracturing-pressureanalysis. The derivative is shown to enhance the analysis capabilitiessignificantly. The interpretation methodology is presented, and several fielddata sets and simulations are discussed to presented, and several field datasets and simulations are discussed to illustrate the technique. Introduction Numerous techniques have historically been used to evaluate fracturingtreatments. Temperature and radioactive survey logs are common for inferringfracture height at the wellbore. Production data help assess the success of afracturing treatment. Some seismic aplications have also been reported toindicate fracture geometry and azimuth. Perhaps the most prominent techniquefor evaluating fractured wells is transient-pressure testing. Several reservoirfractured-well models are used to infer fracture characteristics from test dataand to predict the expected well performance. However, the models used areidealizations of actual, more complicated physical phenomena. The resultsinferred may be representative physical phenomena. The results inferred may berepresentative of an "equivalent fractures which could partially explainsome of the discrepancies often noted from cursory use of fracture placementsimulators a"" test data analysis. The nonuniqueness of the placementsimulators a"" test data analysis. The nonuniqueness of the solution isnot a problem specific to transient-pressure analysis. It is inherent in anyinterpretive process e.g. log interpretation and seismic evaluation. In theearly 1950's, the relationship between pressure and fluid volume was recognizedas an essential element in understanding hydraulic fracturing. In 1979, Nolteand Smith proposed to analyze pressures recorded during the fracturingtreatment. We believe that interpreting treatment data is the foundation ofevaluating hydraulic fractures. It provides a direct means of checking andrefining the assumptions and input data used to design the fracture. This willhave a direct effect on achieving the maximum economic impact by ensuring thatthe opimum fracture designed is actually what is placed. Interpreting thefracturing-treatment data will simultaneously enhance and benefit from theanalyses of other evaluation tools (logs, well tests, and production data) byintegrating essential additional information that will help address thenonuniqueness problem. problem. Interpretation-Basic Concepts In 1979, Gringarten et al. introduced the concept of the inverse vs. directproblem for well-test interpretation. This concept has since spurred manysignificant developments in transient-pressure analysis, including the pressurederivative. The same concept is applicable to any other interpretationtechnique, such as log interpreation and fracturing-treatment data analysis. Schematically, the environment evaluated is considered as System S (Fig. 1), which, when excited with a given input signal, delivers a corresponding outputsignal. For a known model, an appropriate mathematical formulation can predictthe output signal for any given input signal. This is known as the "directproblem." Interpretation is the "inverse problem;" i.e., theappropriate model is unknown and the input and output data are used to identifyit. This model identification is essential to obtain parameters representativeof the physical system considered. In well testing, the system is the reservoirand the wellbore. The input signal could be a production rate change, and theoutput signal the corresponding variations in downhole pressure. In logging, the system is the wellbore and surrounding formations. The input signal couldbe the emission of neutrons or sonic waves, and the output signal neutrons, gamma rays, or sonic waves altered by the formation under evaluation. Similarly, in fracturing treatments, the system includes the wellbore andaffected formations. The input signal could be the injection of the fracturingfluid, and the output signal the pressures associated with the treatment. Inany of the above situations, the interpretation problem is the same. Theexternal input and output signals are used to infer the physical properties ofthe system and to build a representative physical properties of the system andto build a representative mathmatical model. No apriori model is or should beassumed. Solving the "inverse problem" (i.e., the interpretation)becomes a special pattern-recognition problem. The output data display certainpattern-recognition problem. The output data display certain patterns thatallow the interpreter to identify' an appropriate model. patterns that allowthe interpreter to identify' an appropriate model. The model is then used toestimate the parameters characterizing the system and to provide meaningfulinput to the design and/or the economic model. Inherent in the inverse problemis the nonuniqueness of the solution; i.e., two different models may exhibitthe same output for a given input. To reduce the number of alternativesolutions requires better pattern-recognition capabilities and/or additionalinormation. Integration of additional data to create a coherent solution thenbecomes necessary. Interpretation of Fracturing Treatments As with any interpretation problem, pattern recognition is initiallydeveloped on the basis of existing or known models, even simple ones asstarters (e.g., semilog straight line in well testing). Interpretation ofactual field data will inevitably suggest the development of more realisticmodels that enhance analysis capabilities. This is a major benefit of analyzingreal data as opposed to working exclusively in the orderly world of designmodels. Direct Problem: Basic Fracturing Models. Models representing fracturingtreatments can be built with three basic equations: the conservation of mass, afluid-flow equation, and the fracture compliance, which relates the fracturewidth to the net pressure. Net pressure is the difference between the pressureapplied within the pressure is the difference between the pressure appliedwithin the fracture and the in-situ stress, also known as the closure pressure, that acts to close the fracture. Simplified-geometry models commonly used todescribe fracture-width development and compliance consider Sneddon's and Sneddon and Elliott's classic solutions for pressurized cracks in an elasticmaterial of infinite extent. These solutions are for a 2D plane-strain crack, assumed to be elliptical in cross section and plane-strain crack, assumed to beelliptical in cross section and infinite in length, or for a radial crack. The2D solutions have been used in two different ways to model the fracture. The Khriovich-Geertsma-de KIerk (KGD) model considers the fracture height to be theinfinite dimension compared with fracture length. The Perkins-Kern-Nordgren(PKN) model considers the fracture Perkins-Kern-Nordgren (PKN) model considersthe fracture length to be the infinite dimension compared with fracture height. These geometry models and their ranges of applicability are discussed in detailin the literature. More complex 3D models and pseudo-3D models have also beenpresented. Nolte and Smith derived a relationship between the fracturingpressures measured at the wellbore and the pumping time by combining the basicrelations of material balance, fluid flow, and fracture compliance. P. 39