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Joint Flow–Seismic Inversion for Characterizing Fractured Reservoirs

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Joint Flow–Seismic Inversion for Characterizing Fractured Reservoirs Joint flow–seismic inversion for characterizing fractured reservoirs: theoretical approach and numer- ical modeling Peter K. Kang⇤, Yingcai Zheng, Xinding Fang, Rafal Wojcik, Dennis McLaughlin, Stephen Brown, Michael C. Fehler, Daniel R. Burns and Ruben Juanes, Massachusetts Institute of Technology SUMMARY ments into the seismic interpretation; on the other, improve the predictability of reservoir models by making joint use of seis- Traditionally, seismic interpretation is performed without any mic and flow data. account of the flow behavior. Here, we present a methodol- ogy to characterize fractured geologic media by integrating The basic tenet of our proposed framework is that there is a flow and seismic data. The key element of the proposed ap- strong dependence between fracture permeability (which drives proach is the identification of the intimate relation between the flow response) and fracture compliance (which drives the acoustic and flow responses of a fractured reservoir through seismic response). This connection has long been recognized the fracture compliance. By means of synthetic models, we (Pyrak-Nolte and Morris, 2000; Brown and Fang, 2012), and show that: (1) owing to the strong (but highly uncertain) de- recent works have pointed to the potential of exploiting that pendence of fracture permeability on fracture compliance, the connection (Vlastos et al., 2006; Zhang et al., 2009). Here, modeled flow response in a fractured reservoir is highly sensi- we propose a formal approach to improved characterization of tive to the geophysical interpretation; and (2) by incorporating fractured reservoirs, and improved reservoir flow predictions, flow data (well pressures and production curves) into the inver- by making joint use of the seismic and flow response. sion workflow, we can simultaneously reduce the error in the seismic interpretation and improve predictions of the reservoir flow dynamics. OVERALL FRAMEWORK Our approach seeks to combine seismic scattered wavefield data that provides spatial estimates of fracture orientation, spac- INTRODUCTION ing, and compliance (Fang et al., 2013; Zheng et al., 2013) (Fang et al, 2013; Zheng et al., 2013) with flow data (pressure Characterizing fractured geologic formations is essential in ex- and saturation values) at a number of well locations. The frac- ploration geophysics and petroleum engineering, as much of ture compliance values obtained from seismic data analysis are the oil and gas reserves worldwide are from reservoirs that are related to permeability through a rock physics model (Pyrak- naturally fractured. The relevance of fracture characterization Nolte and Morris, 2000; Brown and Fang, 2012). Both the has only increased in recent years with the growth of uncon- seismic data and the rock physics model contain potentially ventional resources like oil and gas shale. Determining the significant uncertainty. By combining the flow and seismic effectiveness and sustainability of hydrocarbon production in data in a single inversion we hope to obtain an optimal sub- those environments depends critically on our ability to charac- surface permeability field that can be used to predict reservoir terize natural and induced fractures. flow. Traditionally, seismic interpretation and flow modeling have The general workflow is shown in Figure 1. Our proposed been performed independently. Reservoir modeling typically framework is rather general and can be applied to field data, follows a unidirectional workflow. From an interpretation of but here we restrict our exposition and validation to synthetic seismic surveys and other geophysical and geological data, a computer models. The starting point for the synthetic mod- structural reservoir model—with reservoir geometry and faults— els is a ‘true’ compliance field, which entails generating: (1) a is built. Facies data and inference are then used to populate fracture network, which can disordered but have certain geo- reservoir properties (like porosity and permeability) on a fine metric statistics (fracture density, length and orientation); (2) elas- grid known as a static model (or geomodel). The number of tic compliance of the individual fractures, which also exhibits a cells in the geomodel is typically too large to perform reservoir predefined geospatial distribution (mean, variance, and corre- flow studies, so a dynamic model is built from either upscaling lation length). This model of interconnected discrete fractures, procedures or multiscale techniques, which solves the reser- embedded in a reservoir matrix located at depth, is the com- voir flow equations on a coarser grid. The rock physics prop- mon physical model from which seismic and flow response is erties (like porosity and permeability) and reservoir dynamics determined (Figure 2a). properties (like relative permeability and capillary pressure) are then modified to history-match production data. By then, The true compliance field (CT ) and true permeability field (KT ) all feedback to the originating seismic data, and often all geo- are related via a predefined rock-physics model, KT = f (CT ,a), logic realism, is lost. where a denotes a set of parameters governing the functional relation between CT and KT (Figure 2b). The objective is then In parallel, seismic interpretation in challenging geologic en- to infer the true compliance field and compliance–permeability vironments like naturally-fractured reservoirs is plagued with relationship by a procedure that unifies seismic and flow mod- uncertainty. The goal of our work is twofold: on one hand, re- eling. duce that uncertainty by incorporating dynamic flow measure- Kang et al.: Joint flow–seismic inversion Seismic modeling. We first run a forward seismic model on True compliance ( C T ) the true compliance field (CT ) to generate the detailed wave- field. We then treat this wavefield blindly, without knowl- KT Seismic forward Functional relation edge of the underlying structure, to estimate the seismic com- modeling bet. compliance ( C T ) & f(CT , ↵) pliance (CM) by means of the double-beam method (Zheng permeability ( K T ) C et al., 2013) (Figure 3b). The error in the estimated compliance T Uncertainty field, ec = CT CM, often exhibits a strong spatial correlation Inversion: double − beam method KT with the actual compliance field CT (Figure 3c); something that points to the need to model (de-trend) this error to reduce Flow response this dependence. Methodologically, this implies a transforma- = C + e (C ) tion C C such that the error in the transformed variable, CM T c T M ! M0 e = C C , is only weakly dependent on the underlying PT ,ST c0 T − M0 (and unknown) true compliance field. This error-modeling of Error model ( β ) : Uncertainty the compliance introduces a set of parameters, b, that need to CM = CT + ec Flow measurement be estimated. process Flow modeling. The flow response relies on the compliance- P ,S to-permeability relation, from which we generate the fracture- CM M M permeability field KT . We simulate flow on this permeability field, from which we extract a dynamic record of pressure (PT ) and production curves (ST ) at a discrete set of locations that Least Square Process represent well measurements. These records are subject to measurement errors, and therefore we denote the accessible, ˆ measured quantities as PM and SM, respectively. The param- ↵ˆ, β eters a generating this response are of course unknown. We run the flow model (GP) and transport model (GS) on estimates Figure 1: Overall framework for joint flow-seismic inversion. ˆ ˆ ˆ ˆ KT (aˆ ,b) to obtain simulated responses PT and ST . The sets of The above framework shows how seismic and flow models are ˆ parameters aˆ and b are then estimated by minimizing the er- integrated to better characterize fractured reservoirs. ror between the measured (PM,SM) and modeled (PˆT ,SˆT ) flow response. While sophisticated estimation and inversion pro- cedures exist, our work on simple least-squares minimization (a) True compliance !eld ( C T ) (b) Compliance-permeability relation 2 10 procedure: data 1 ˆ ˆ 3 10 Polyfit PM GP(KˆT (aˆ ,b)) SM GS(KˆT (aˆ ,b)) min − 2 + − 2 . (1) ˆ 0 CˆT ,aˆ ,b " svP svS # 10 (Normalized Aperture) 10 SEISMIC INVERSION ON ORTHOGONAL DISCRETE 10 1 2 3 4 10 10 10 10 FRACTURE NETWORKS Normalized Compliance We test our approach on discrete fracture networks consist- Figure 2: (a) Compliance field of the orthogonal discrete ing of two sets of parallel, equidistant, connected fractures fracture networks that we study. (b) Functional relation be- oriented at an angle of 0 and 90 degrees with respect to the tween fracture compliance and permeability from rock-physics x-axis. The fracture spacing is uniform and equal to 80m. On model; the parameter a determines the curve fitting (red line) average, the mean value of x-directional compliance values are to the data (blue circles). two times larger than the y-directional compliance values. Our 10 9 compliance values vary between 10− and 10− m/Pa. We construct a spatially-correlated compliance field that follows is highly correlated with CT itself (Figure 3). From the point a lognormal distribution with an exponential autocorrelation of view of estimation, this is of course undesirable because function in space (Figure 2a). We simulate seismic shot gath- it would require a priori knowledge of the true compliance ers using a 3D staggered grid finite-difference method (Fang field. Thus, one must introduce an error model that effec- et al., 2013) (Coates and Schoenberg, 1995; Willis et al., 2006). tively de-trends the modeled response and weakens its depen- For seismic inversion, we apply the double-beam method (Zheng dence on the true compliance field. Our error correction model et al., 2013) to estimate the modeled seismic compliance field CM. is motivated by the scatter plot between C C and C , T − M T which shows a linear trend (Figure 3d). From Figure 3d, we observe that e C C = g(C C )+e where e is c ⌘ T − M T h Mi ERROR MODEL FOR THE COMPLIANCE FIELD a random spatial variable that exhibits a much lower corre- lation with CT .
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