A Streamline Approach for Integrating Transient Pressure Data Into High-Resolution Reservoir Models
- Kari Nordaas Kulkarni (Texas A&M U.) | Akhil Datta-Gupta (Texas A&M U.) | D.W. Vasco (Berkeley Natl. Laboratory)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 2001
- Document Type
- Journal Paper
- 273 - 282
- 2001. Society of Petroleum Engineers
- 5.4 Enhanced Recovery, 7.6.2 Data Integration, 5.1.5 Geologic Modeling, 5.3.1 Flow in Porous Media, 4.3.4 Scale, 5.1.8 Seismic Modelling, 5.1 Reservoir Characterisation, 5.1.1 Exploration, Development, Structural Geology, 5.4.1 Waterflooding, 5.6.6 Cross-well Tomography, 5.6.4 Drillstem/Well Testing, 5.5.7 Streamline Simulation, 5.6.5 Tracers, 5.6.3 Pressure Transient Testing, 5.5 Reservoir Simulation
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We generalize the streamline approach to transient pressure applications by introducing a "diffusive" time of flight along streamlines. This allows us to define drainage areas or volumes associated with primary recovery and compressible flow under the most general conditions. We then employ developments in seismic tomography and waveform imaging to formulate an efficient approach for integrating transient pressure data into high-resolution reservoir models. The proposed approach exploits an analogy between a propagating wave and a propagating "pressure front" via high-frequency asymptotic solutions to the transient pressure equation. A key advantage of the asymptotic approach is that parameter sensitivities required for solving inverse problems related to production data integration can be obtained analytically using a single streamline simulation. Thus, the approach can be orders of magnitude faster than current techniques that require multiple flow simulations. We demonstrate the power and utility of the approach by applications to both synthetic and field examples. In the field example, the predominant fracture patterns emerging from the inversion are shown to be consistent with outcrop mapping and crosswell seismic imaging.
Proper characterization of reservoir heterogeneity is a crucial aspect of any optimal reservoir development and management strategy. In this respect, it is important to reconcile geologic models with the dynamic response of the reservoir. One important source of dynamic data is pressure interference tests that are performed by injecting or producing fluid from one well while observing the pressure response in one or more surrounding wells located several hundred feet away. The observed transient pressure response can be used to estimate the permeability distribution in the reservoir. Transient pressure information is an important dynamic data type because of its wide prevalence as well as the rapid response. Pressure response in observing wells can be obtained hours or days after starting injection or production, while it might take months or years to obtain sufficient fluid or tracer production for reliable reservoir characterization.
Previous efforts toward integrating transient pressure data into reservoir models have mostly utilized inverse modeling techniques in conjunction with finite-difference- or finite-element-based flow simulators.1-4 Such inverse modeling is computationally intensive, often requiring orders of magnitude more computational efforts compared to forward modeling or flow simulation. Streamline models have shown significant potential in this respect because of their computational efficiency compared to finite-difference models. Furthermore, sensitivities of the production response with respect to reservoir parameters such as porosity and permeability can be computed analytically using a single streamline simulation.5,6 These sensitivities quantify the change in production response because of a small perturbation in reservoir parameters and constitute an integral part of most inverse modeling algorithms.7
Until now, dynamic data integration with streamline models have been limited to tracer concentration history and multiphase production response, such as water cut (WCT) data at the wells. This is because current streamline models are particularly well-suited for modeling tracer transport and waterflooding as the velocity field remains relatively static and the streamlines need to be updated only infrequently. Under such conditions, streamline models can be orders of magnitude faster than conventional finite-difference simulators.8-11 A key development in streamline modeling has been the introduction of the concept of "time of flight" that has trivialized generalization to 3D flows.8 The time-of-flight formulation effectively decouples pressure from saturation and concentration calculations during flow simulations. Furthermore, mapping of solutions to 1D transport equations onto streamlines is also considerably simplified because we do not need to keep track of the geometry of streamtubes.
In this paper we first generalize the concept of streamline time of flight to compressible flow by introducing a diffusive or "pressure" time of flight. We then use developments in seismic tomography and waveform imaging to formulate an efficient methodology for integrating transient pressure data into high-resolution reservoir models. Our proposed method is based on an analogy between streamlines and seismic ray tracing. In particular, we adopt an asymptotic approach to develop the streamline time-of-flight equations for compressible flows using concepts from geometric optics and seismology.12,13 In subsurface flow and transport, there are several investigations of asymptotic solutions describing solute transport in the limit of long times.14,15 There are also relevant applications to the heat-conduction equation.16 However, only recently attempts have been made to develop an asymptotic series representation of the solution for flow and transport and to relate the orders of the expansion to attributes such as breakthrough or arrival times.5,6 In this paper we use asymptotic solutions to develop a general framework for production data integration into high-resolution reservoir models. A key advantage of the asymptotic approach is that parameter sensitivities required for solving inverse problems related to production data integration can be obtained using a single forward simulation. Thus, such algorithms can be orders of magnitude faster than current techniques that can require multiple flow simulations.
The Asymptotic Approach
The asymptotic approach (also known as Debye series method or ray series method) forms the mathematical basis for geometrical ray theory and has been extensively used in electromagnetic and seismic wave propagation.12,13 The method has also proved valuable in the analysis of front propagation in general.17 Many of its concepts, such as rays and propagating interfaces or discontinuities, have direct counterparts in hydrology and petroleum engineering.18,19 The method involves properties of the wave front and ray paths of the wave equation that have been studied for more than a century.
Our goal here is to find a solution to the diffusive pressure equation that mimics the one found in wave-propagation phenomena. The underlying idea is to look for a solution in terms of an asymptotic series. For transient-pressure response, we can utilize concepts from diffusive electromagnetic imaging to examine frequency domain solutions in inverse powers of as follows.20
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