EM FIELDS DUE TO THE ADVANCING CRACK INTRODUCTION. J s = ˆxL p

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1 Electrokinetic Coupling in Hydraulic Fracture Propagation Nestor H Cuevas, University of California, Berkeley, James W. Rector, III, Lawrence Berkeley Laboratory, J. R. Moore, University of California, Berkeley and S. D. Glaser, University of California, Berkeley SUMMARY Electrokinetic coupling is the most popular mechanism proposed to explain observed electromagnetic (EM) signals associated with the hydraulic fracturing of rocks. Measurements in both, laboratory and in situ conditions show evidence of the phenomenon, however, as far as the authors know, there have been no reports on the description of the source mechanism, its relationship to a propagating crack, nor the electromagnetic field distribution due to such a source advancing through a conductive medium. In this paper it is shown that an electric streaming current density arising on the walls of a fluid driven crack gives rise to the EM fields observed in measurements of streaming potential in hydraulic fracturing experiments. A source function for the current density is established from the fluid pressure profile inside the propagating. Expressions for the EM fields due to such a source are derived for a crack propagating with a constant velocity, in a homogeneous isotropic conducting medium. The spatial and temporal behavior of the fields reasonably agree with measurements performed in laboratory experiments. In situ measurements are qualitatively described only, however it is shown that the magnitude of the fields and their temporal behavior can be well reproduced in a realistic hydraulic fracturing setting. degrees of success, and the results obtained are often of great help in the characterization of the fractured system. However most often the final fracture geometry inferred from the measurements seems to overlap with some of the microseismic events, but most of them appear to occur far beyond the edge of the fracture. As noted before changes in the stress field are expected to trigger microseismic events anywhere within (but not restricted to) the leakoff region. This paper explores the feasibility of interpreting electromagnetic (EM) data recorded during hydraulic fracturing experiments as a tool to aid in the characterization of tensile cracks propagating through the formation. An electrokinetic coupling mechanism is proposed to give rise to a current density source forming at the walls of the crack. More specifically, the fluid moving relative to the fracture s wall carries mobile electric charges that produce a streaming electric current, which in turn induces an electromagnetic field in the hosting medium. The EM fields due to such a source could be measured in order to provide direct diagnostic information of the hydraulic fracture propagation. The properties of the measured fields will depend on the properties of the fracture (geometry, location of the source, etc), the properties of the injected fluid as well as the properties of the surrounding formation (conductivity, etc). EM FIELDS DUE TO THE ADVANCING CRACK INTRODUCTION Hydraulic fracturing is used in oil reservoirs to stimulate oil production by means of creating conduction channels in the formation to improve oil and gas flow from the reservoir towards the wellbore. As the fracturing fluid is injected into the formation the increasing pore fluid pressure effectively decreases the confining stress holding together the planes of weakness in the rock. At some point the natural tectonic stress dominates, shear failure occurs (Pearson (1981), Cipolla and Wright (2000)) and micro-seismic events originate in the vicinity of the growing crack (the leakoff region), as well as near by the advancing tip. Evidently a rather popular technique to monitor hydraulic fracture growth is the so called Microseismic Fracture Mapping, in which the location of the microseismic events is recovered by detecting the p-wave and s-wave arriving at a distant seismic recording array (triaxial geophones, or accelerometers). The actual tensile crack growing in the medium can not be detected with this technique. Deformation measurement of the formation around the growing crack is the preferred methodology to monitor fracture s geometry (length, height and dip) and orientation. Together with microseismic recordings, downhole and surface tilt-meter measurements are nowadays the only techniques available to track hydraulic fracture growth from a direct far field perspective. Many are the reports of fracture diagnostics experiments using these methodologies Cipolla and Wright (2000) with various At the onset it is assumed that the EM source arising upon fracture propagation is electrokinetic in nature. As the fracture propagates, the driving fluid advances dragging along the electrical charges of the diffuse portion of the electrical double layer (Matijevic (1974)) on the walls of the fracture. From a macroscopic stand point the pressure gradient in the propagating direction of the crack causes the fluid flow and in turn a streaming current density, which couples to the pressure gradient through a coefficient L (Am/N), i.e., J s = ˆxL p x (x,t) J s A m 2, L Am N Notice that the same expression (1) describes the streaming current density arising upon fluid flow in a homogeneous porous medium (Nourbehecht (1963)). This is not surprising as the underlaying mechanism is the same, i.e., the transport of the charges in the double layer forming at the surface of the grains. However Cuevas and Rector (2008) showed that at the interface between a fluid continuum and a porous medium, the coupling gives rise to a current density component arising from the tangential pressure gradient, as well as another due to the fluid flow driven by forward momentum transfer across the permeable boundary. In this work it is assumed that the fluid front advances fast enough so that a non-slip condition for the fluid velocity holds at the walls of the fracture, and thereby the later component of the current density can be neglected. The domi- (1) 1721

2 nant surface component of the current density is that associated with the pressure gradient driving the flow along the outer most surfaces, and thereby the coupling is defined by the superimposition of the contributions appearing at each wall. In order to establish the pressure profile driving the flow, it is assumed that the opening fracture of half length l and thickness 2δ, is driven by the injection (constant rate Q) of a viscous, newtonian and incompressible fluid (viscosity η), and the fluid front lags behind the fracture s length at x = x 0. To a very good approximation all quantities can be regarded as one-dimensional if δ l, i.e., the fracture is regarded as a slot rather. In this case the pressure gradient is defined by, p x = 3Qη 2hδ 3 (x) and thereby the magnitude of the streaming current density is described by, (2) J s = 3QL sη 2hδ(x) 3 [A/m 2 ] (3) The function f (x,t) = δ(x,t) 3 defines the source-time function for the streaming current density, as the fracture propagates through the medium. The solution to this problem continues to be a active area of research, however for the sake of simplicity the approximate analytical derivation presented by Khirstianovic and Zheltov (1955) will be used in this work. Thus the source time function of the current density, at a given point, say x = x i (fig.1), determines the time dependence of f (x,t) as the fluid front reaches and passes by x i. The source is the strongest at t = 0 (at x = x i ) and it quickly decays to a constant amplitude (= 0) as the fluid is advances passed x = x i. The amplitude of the source depends on the length of the lag region, while the decay time depends on both the lag s size and the speed of propagation (v = v ˆx ;l(t)=l 0 +vt). Although an exact expression for f (x i,t) is easily determined (Khirstianovic and Zheltov, 1955), the expression f (x i,t)= f 0 exp( βt) δ(t τ) f 0 = δ(x i ) 3 (4) approximates the source time function by an exponential decay, the decay constant β linearly increasing with the propagation speed of the fracture. This last expression (4) will be used in the future in order to simplify the representation of the source time function. The convolution operation δ(t τ), reflects the delay character of the source, where τ is the time that takes the fracture to reach x = x i. The spatial distribution of the fields is obtained by superimposing the contributions due to the infinitesimal elements of current density forming at the position of the fluid front, assuming that the source arising at a given position (say r s ) does not perturb the current density triggered at an earlier time. Following standard practice, the fields due to an arbitrary excitation J(r s,t) are computed by applying an inverse Laplace transform to the (appropriately weighted) frequency spectra of the fields due to an impulsive source at r s, i.e., J = I 0 dsδ(r r s )δ(t). The effect of displacement currents in the total current density is neglected, and therefore the solution in the time domain is restricted (Wait (1960)) to times t such that (t 2 k 2 ) 1/2 2ε/σ, where k =(εµ) 1/2 r. The source function (eqn.4) transforms to the frequency domain Figure 1: Source time function near the tip of the fracture computed for two positions of fluid lag (s = iω) as, J s (t)= ˆxI 0 dsdz e βt J s (s)= ˆx I 0dsdz s + β where the element of current density I 0 dsdz represents the main contribution of the source, at the vicinity of the advancing fluid front. The time domain fields are then determined by, e r (t)=l E 1 r (s) J s (s) e θ (t)=l E 1 θ (s) J s (s) h φ (t)=l H 1 φ (s) J s (s) (6) where L 1 represents the inverse Laplace transform of the fields due to an ideal impulse (E r,e θ,h φ ). The time domain expression of the fields due to the element of exponential transient current density are then given by, e r (t)= I 0dsdz 2πσr 3 A(β,t)cosθ e θ (t)= I 0dsdz (5) 4πσr 3 B(β,t)sinθ h φ (t)= I 0dsdz A(β,t)sinθ (7) 4πr2 where the time functions A(β,t) and B(β,t) are described by, A(β,t)= er f c i(βt)1/2 2t1/2 B(β,t)= Re Re e βt iβ1/2 (1 + iβ 1/2 ) π 1/2 2t 1/2 e 2 /4t u(t) e βt iβ1/2 (1 + iβ 1/2 β 2 ) er f c i(βt)1/2 + 2t1/2 1722

$z is included to account for the superposition of fields due to the source density along the height of the fracture.$

3 π 1/2 2t 1/2 3 2t 1/2 e 2 /4t u(t) (8) 2 = σµr 2 and r = (x x i ) 2 + y 2 + (z z ) 2 is the distance from the source at the tip to the observation position. z is included to account for the superposition of fields due to the source density along the height of the fracture. As the fracture progresses, the fields observed at a distant point result from a superposition of delayed sources arising at the position of the advancing fluid front. Thereby the fields due to a delayed source are obtained by convolving eqn.6 with δ(t τ), which simply results in a translation of the time axis to t = t τ. Furthermore since the source appears at the position of the fluid front, the distance r depends on τ, and thereby the parameter and the angle θ as well. The superposition of the fields is accomplished by assuming that the position of the source x i (τ) advances with a constant velocity, such that x i (τ) =l 0 + vτ and thereby ds dx i = vdτ. The integration is then carried out over dτ, which in the fixed laboratory coordinate system yields, e x (t) = I 0dz v 4πσ e y (t) = I 0dz v 4πσ 0 0 2A (x x i) 2 r 5 B (z z ) 2 + y 2 r 5 dτ (x x i )y r 5 2A + B dτ (9) where A,B,x i,r, all depend on the delayed time t τ. The integration limit is defined according to the propagation time of the fracture T sec, such that = t for t T, and = T for t > T. A final integration in dz (not shown) must be done to account for the distribution of sources along the height h of the fracture, i.e., for z =[ h/2,h/2]. The distribution of the fields described by eqn.9 is demonstrated in fig.2 where the medium is assumed to be resistive (ρ 10 4 Ω m), the geometry is that of small laboratory sample (radius 5 cm) and the time of observation is that of active fracture growth. In this configuration the energy is focused in the vicinity of the advancing fracture s tip, however a tail can also be distinguished as the source behind the tip does not vanish instantaneously. The arrows represent the direction of the field lines which evidently resemble that of a dipole source with the charge distribution biased towards the tip. Figure 2: Radiation pattern of the EM fields due to the source at the tip of an advancing fracture (V/m) observed by dipoles E 6,E 3 and E 1 is shown in fig. 3, for the 1 sec long time interval of fracture propagation and failure. The data shows a spike in dipole E 6, associated with the crack EXPERIMENTAL EVIDENCE In the work of Moore and Glaser (2007) laboratory measurements were performed during hydraulic fracturing of intact Sierra Granite cores, of porosity and a m 2 permeability. After saturated with a 0.001M NaCl (σ f = S/m) solution, the bulk conductivity of the samples was σ b S/m, and the streaming coupling coefficient (Jouniaux and Pozzi, 1995) was Cc 175 mv/mpa. The measurements consisted of time sampling (50 KHz) of the electric potential drop between a common ground, and 6 evenly distributed positions on the surface of the cylindrical cores (E j = V /R, j = 1,...,6). The fracture was induced by fluid injection at the center, at a rate of 0.15 ml/s. The amplitude Figure 3: Electric field amplitude recorded upon failure propagation (circled fig. 3). Post mortem inspection showed that the fracture propagated asymmetrically along E 6. Therefore, in light of the theory described earlier, the dipole E 3 detects a signal 1000 times smaller. Thereby the field recorded by E 3 represents a base line for the SP signals alone, which appropriately scaled is used to describe the data recorded by the other dipoles before fracture initiation. Subtracting the SP component yields the residual spike due to the propagation of the crack, of interest in this work. Using the theory presented in the previous section a system of two fractures is needed to fit the data, one directed towards E 6 and another towards E

$As shown in fig. 4 the field strength and the transient behavior are well recovered, after adjusting the direction of fracture propagation, the initial fracture volume and the propagation velocity.$

4 As shown in fig. 4 the field strength and the transient behavior are well recovered, after adjusting the direction of fracture propagation, the initial fracture volume and the propagation velocity. Time series recording of electric field (500 Hz) performed (a) Figure 4: Isolated spike and predicted fields due to two propagating cracks. during an in situ hydraulic fracturing experiment are shown in fig. 5(a). Typically a main transient (highlighted event of rise time 0.1 sec) was followed by trains of pulses of varying width ( 50 ms) and amplitude. This pattern was observed intermittently during hydraulic fracturing, as well as long after the procedure was finished. The signals are clearly correlated and simultaneously occurring in all of the measurements, which suggest a source, electromagnetic in nature rather than a mechanically propagating perturbation. Unfortunately the experiment was poorly constrained as all of the fluid flow parameters (injection rate, pressure, etc) were sampled at very low frequencies (1 Hz), and there was no verification for the time stamp of the actual fracturing process. Therefore further analysis can be done from a qualitative stand point only. In light of these considerations the fields observed above a propagating fracture (at X=100 m, Z=400 m) in a 3 Ω m homogeneous medium are shown in fig.5(b), for a suitable injection rate of Q = m 3 /sec, fracture height 1 m, and final length of 5 m. Clearly the magnitude of the fields as well as the slow decay of the signals should be reproducible with the correct parameters of the experiment. CONCLUSIONS Electromagnetic fields observed as a consequence of hydraulic fracture propagation are described in this work as due to a surface electric current density arising in the vicinity of the fracture s tip. The source is electrokinetic in nature, i.e., it originates when mobile ions at the fracture s wall are dragged in the direction of the advancing fluid front of the opening crack. The source space-time function is derived from the pressure profile of the opening crack which is partially filled by the injected (b) Figure 5: (a) In situ recordings of surface electric field. (b) Predicted fields fluid. The electromagnetic fields arising from such a propagating source have been compared to those observed in laboratory hydraulic fracturing experiments, as well as those observed in situ during a hydraulic fracturing of an oil reservoir. In the former case the data can be explained by a superposition of two fractures propagating in different directions. However the existence of only one of them was verified: that which caused the laboratory sample to fail. The data recorded in situ can be qualitatively explained by the mechanism described in this work. However it is shown that the magnitude of the fields as well as the decay time of the signals should be reproducible from the mechanism described in this paper. Based on the theory described in this work a method can be developed to obtain direct diagnostic information of the hydraulic fracture propagation by measuring the EM fields observed at a distance. The spatial and temporal behavior of the fields depend on the properties of the medium but most importantly on the properties of the source, i.e., the propagating crack. The advantage of this method lies on the fact that the proposed source arises only as the fracturing fluid flows in the advancing crack, and therefore in principle the method should not be sensitive to those events occurring in the leakoff region. ACKNOWLEDGMENTS The authors wish to thank Schlumberger for the support during this work. 1724

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2009 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Cipolla, C. L., and C. A. Wright, 2000, Diagnostic techniques to understand hydraulic fracturing: SPE/CERI Gas Technology Symposium. Cuevas, N. and J. Rector, 2008, Electrokinetic coupling at the fluid continuum-porous medium interface: Journal of Geophysical Research. Jouniaux, L. and J. Pozzi, 1995, Streaming potential and permeability of saturated sandstones under triaxial stress: Consequences for electro-telluric anomalies prior to earthquakes: Journal of Geophysical Research, 100, Khirstianovic, S. and Y. Zheltov, 1955, Formation of vertical fractures by means of highly viscous fluids: Proceedings of the 4th World Petroleum Congress, II, Matijevic, E., 1974, Surface and colloid science. electrokinetic phenomena, vol. 7: John Wiley & Sons. Moore, J. R. and S. D. Glaser, 2007, Self-potential observations during hydraulic fracturing: Journal of Geophysical Research, 112, B2. Nourbehecht, B., 1963, Irreversible thermodynamic effects in homogeneous media and their applications in certain geoelectric problems: PhD thesis, Massachusetts Institute of Technology. Pearson, C., 1981, The relationship between microseismicity and high pore pressures during hydraulic stimulation experiments in low permeability granitic rocks: Journal of Geophysical Research, 86, Wait, J., 1960, Propagation of electromagnetic pulses in a homogeneous conducting earth: Applied Science Research, 8,

Modeling seismic wave propagation during fluid injection in a fractured network: Effects of pore fluid pressure on time-lapse seismic signatures

$Modeling seismic wave propagation during fluid injection in a fractured network: Effects of pore fluid pressure on time-lapse seismic signatures$ Modeling seismic wave propagation during fluid injection in a fractured network: Effects of pore fluid pressure on time-lapse seismic signatures ENRU LIU, SERAFEIM VLASTOS, and XIANG-YANG LI, Edinburgh