Patent Application: US-87111504-A

Abstract:
this invention is a method to measure fluid flow properties of a porous medium , including , but not limited to , the fluid flow permeability . in a preferred embodiment , the measurements are made down hole in drill wells exploring for hydrocarbons or acquifers . the measurement involves two types of instruments . one instrument creates a pressure wave in the porous medium , which generates motion of the fluid in the pore space . the second instrument measures the fluid motion in the pore space using nuclear magnetic resonance methods . any type of instrument that can generate a pressure gradient is suitable , including instruments that are remote from the nmr instrument . magnetic field gradients can be used to localize the nmr signal to a specific region within the porous medium . the magnetic field gradient also provides the method by which the fluid motion is encoded on to the nmr signal . the permeability is calculated from the known pressure gradient present in the porous medium by virtue of the applied pressure gradient or pressure wave and the velocity of the fluid in the rock pore space as measured by nmr .

Description:
the invention disclosed here describes methods that determine formation permeability by measuring nmr signals that respond to pressure gradients created by acoustic stimulation in the bore hole . these pressure gradients create motion of fluids in the rock pore space of the rocks in the formation surrounding the bore hole . in the case of hydraulic contact between the fluid in the borehole and the fluid in the formation rock , the displacement of the fluid in the borehole creates the displacement of the fluid in the formation rock . this is frequently referred to as the squirt boundary condition . motion of the fluid in the formation rock can also be created when there is no direct hydraulic contact between the fluid in the borehole and the fluid in the formation rock . hydraulic contact between these fluids can be broken by the presence of an impermeable barrier at the bore hole wall . this can for example be created by the deposition of drilling mud on to the bore hole wall or damage to the rock structure near the bore hole wall as a result of the drilling process . the presence of an impermeable barrier between the two fluids creates what is sometimes known as a no - squirt boundary condition . in this case , fluid motion of the fluid in the formation rock can be created by a propagating compressional wave generated in the bore hole . the permeability is then determined from the displacement of the fluid in the formation rock relative to the rock matrix . this fluid displacement is determined using nmr measurements as described by the procedures of the present invention . the description of the invention includes three components : theory , procedures of down - hole implementation , and procedures of data processing . current nmr tool measure nmr signals such as spin echoes which are used to determine nmr relaxation times that carry the information on the rock and fluid properties . the nmr relaxation rates are primarily controlled by the thermal motion of the fluid molecules and the collision of the fluid molecules with the internal surface of the pores in the rock . it is well known that the relaxation rate of carr - purcell - meiborn - gill ( cpmg ) echo train includes three terms 1 , 2 , 3 1 / t 2 = 1 / t 2 ⁢ b + 1 / t 2 ⁢ s + 1 3 ⁢ γ 2 ⁢ g 2 ⁢ τ 2 ⁢ d eff ( 2 ) where first term is the contribution from the fluid bulk relaxation mechanism , the second term represents the surface relaxation at the fluid - matrix interface and the last terms reflects the relaxation due to diffusion in the presence of a magnetic field gradient , g , where time τ refers to the time delay between the radio frequency pulse and the refocusing rf pulse in the spin echo pulse sequence and γis the nuclear gyromagnetic ratio . when fluid moves relative to the formation matrix , our theoretical analysis shows that there is an additional relaxation due to the coupling between fluid flow and the applied field gradient g , which we denote as the acoustic nmr relaxation rate 1 / t 2a . the theoretical analysis ( see appendix for details ) is comprised of the following three steps . first , the nmr calculation shows that the phase change of the transverse magnetization ( which is proportional to the nmr signal ) in response to an oscillatory flow motion during one echo time is proportional to the mean velocity of the absolute motion , 〈 φ ⁡ ( 2 ⁢ τ ) 〉 = γ ⁢ ⁢ g ω 2 ⁢ f ⁡ ( ωτ , ϕ 0 ) ⁢ 〈 v ⁡ ( ω ) 〉 ( 3 ) while the normalized amplitude of the transverse magnetization is proportional to the fluctuation of the pore flow velocity ,  m ⁡ ( 2 ⁢ τ ) m ⁡ ( 0 )  = 1 - [ γ ⁢ ⁢ g ⁢ ⁢ f ⁡ ( ωτ , ϕ 0 ) ] 2 2 ⁢ ω 4 ⁢ 〈 ( δ ⁢ ⁢ v ) 2 〉 pore . ( 4 ) where φ 0 is the initial phase of acoustic oscillation at the start of a cpmg sequence . both the phase change and the amplitude change decrease rapidly with increasing frequency . in the low frequency limit , such that ωτ & lt ;& lt ; 1 , eq . 5 reduces to the more simple expression , using eq . 4 , we obtain an analytical form for the acoustic nmr relaxation rate , 1 t 2 ⁢ a = 1 4 ⁢ γ 2 ⁢ g 2 ⁢ τ 3 ⁢ ⁢ cos 2 ⁢ ⁢ ϕ 0 ⁢ 〈 ( δ ⁢ ⁢ v ) 2 〉 pore . ( 7 ) in the second component of the theory , we use a pore network created from the x - ray computed tomography images of real rocks to calculate the fluctuation of velocities in the pores in response to an applied pressure gradient . from these simulations , we find that the velocity fluctuation is proportional to the applied pressure gradient , consistent with darcy &# 39 ; s law , 〈 δ ⁢ ⁢ v 2 〉 pore = β ⁡ ( k ) ϕ ⁢  〈 v 〉  = β ⁡ ( k ) ϕ ⁢ k η ⁢  ⅆ p ⅆ x  ( 8 ) where β ( k ) is a number between 1 and 2 with a weak permeability dependence ( see fig2 and fig3 ), and φ is the porosity of the rock . this expression relates the fluctuation of the velocities for the fluid in the pore space , which is measured by nmr , to the local applied pressure gradient . the third component of the theory relates the local pressure gradient to the pressure , p o , applied in the bore hole . this relation is a function of the contact boundary conditions , described above , between fluid in the borehole and the fluid in the formation . if the formation fluid is in hydraulic contact with bore hole fluid ( i . e ., the squirt boundary condition ), the local pressure gradient for a small offset x , relative to the wall of the borehole , can be estimated from , ⅆ p ⁡ ( x , t ) ⅆ x ≈ - p o λ ⁡ [ sin ⁡ ( ω ⁢ ⁢ t + x / λ ) + cos ⁡ ( ω ⁢ ⁢ t + x / λ ) ] ⁢ exp ⁢ { - x / λ } ( 9 ) where λ is the pressure excitation penetration length which depends on the excitation frequency , k / η is the formation fluid mobility , and m is the effective fluid modulus . biot theory 4 defines the relationship between these parameters , using the relationships in eq &# 39 ; s . 7 , 8 and 9 , we obtain the acoustic nmr relaxation rate which applies for the squirt boundary conditions and for low frequency acoustic excitation , 1 t 2 ⁢ a ⁡ ( x ) = 1 4 ⁢ γ 2 ⁢ g 2 ⁢ τ 3 ⁡ ( β ⁡ ( k ) ϕ ⁢ k η ⁢ p o λ ) 2 · [ 1 + sin ⁡ ( 2 ⁢ x / λ ) ] ⁢ exp ⁡ ( - 2 ⁢ x / λ ) . ( 11 ) the acoustic nmr relaxation rate under such squirt boundary conditions is a function of the offset x from the borehole wall with just two unknown parameters , the formation fluid mobility k / η and the penetration length λ . a nmr tool with multi - sensitive - volumes , receiving spin echoes from multi offsets , would recover the penetration length in the target formation , and the amplitude of acoustic nmr relaxation rate would directly give an estimate of formation fluid mobility . for the non - squirt boundary condition , the borehole pressure excitation generates a fast p wave in the formation , which creates an oscillatory displacement of the formation matrix together with pore fluid in the matrix . the mean velocity of the pore fluid is a sum of the matrix velocity and the mean velocity of fluid motion relative to the matrix , which can be observed by calculating the phase change of nmr . in the low frequency limit , by setting φ 0 ( x )= ωx / ν p , we have phase ( m ( 2τ )/ m ( 0 ))≈ γ gτ 2 cos ( ω x / ν p )& lt ; v & gt ;≈ γgτ 2 & lt ; v & gt ; ( 13 ) where ν p is the p wave velocity and the last step reflects the fact that the offset is much smaller than the wave length of the fast p wave . in eq . 12 , the relative fluid velocity is orders of magnitude smaller than the matrix velocity so that the phase shift of the nmr signal is dominated by the motion of the fluid with the matrix . however , based on the biot theory 4 , the relative flow velocity is proportional to the matrix velocity and the proportionality coefficient contains the information for the formation mobility , 〈 v r 〉 = ⅈ ⁢ k ⁢ ⁢ ωρ f η ⁢ v m ( 14 ) the acoustic nmr relaxation rate , eq . 7 , is a function of the velocity fluctuation for the fluid in the pore space . the pore network simulation establishes the simple relationship , eq . 8 , between the mean relative flow velocity and the velocity fluctuation for fluid in the pore space . using these relationships , we can estimate the formation fluid mobility by the acoustic nmr measurement alone with additional information on fluid density , which could be obtained from other independent measurements , such as neutron or resistivity logging . one possible implementation would be to use an nmr tool with multi - sensitive - volumes [ 5 ] in which two echo or cpmg measurements can be made at each downhole location . one nmr measurement is made under normal conditions and one nmr measurement is made in the presence of an applied pressure gradient generated by a pressure source . this pressure source could be located at the well bore surface or it could be located down hole near the location of the nmr instrument . the frequency of pressure excitation should be chosen to make the penetration length not too small compared to the offset of the nmr sensitive volume . in the case of nmr spin echoes , the echo spacing should be large enough so that the gradient related nmr relaxation dominates over the surface relaxation . an example of a timing sequence for the pressure wave , radio frequency pulses , and nmr spin echo signals detected is shown in fig4 . one procedure for determining the formation fluid mobility which is applicable under the squirt boundary condition includes the follow four steps : 1 . calculate the mean cpmg relaxation rate for each measurement ; 2 . obtain the acoustic nmr relaxation rate by subtracting the cpmg relaxation rate with no pressure excitation from one with pressure excitation ; 3 . fit the acoustic nmr relaxation rate as a function of offset x with a fitting form y ( x )= c ( 1 + sin ( αx )) exp (− αx ), where c and α are two fitting parameters . best fitting gives right number for both c and α , which are denoted as c ′ and α ′. 4 . estimate the penetration length and formation flow mobility in terms of c ′ = τ 3 4 ⁢ ( γ ⁢ ⁢ g ⁢ β ϕ ⁢ k η ⁢ p 0 λ ) 2 ⁢ ⁢ and ⁢ ⁢ α ′ = 2 λ one procedure for determining the formation fluid mobility which is applicable under the no - squirt boundary condition includes the following alternate steps after step 2 above : 3 . determine the velocity fluctuation & lt ;( δv ) 2 & gt ; from the acoustic nmr relaxation rate , eq . 7 . 4 . calculate the phase difference between the two measurements with and without pressure excitation . the phase difference will accumulate linearly with increasing number of echoes and the slope gives the mean velocity of matrix motion v m . 5 . calculate the formation flow mobility using eq . 8 and 14 , where v r = v : k η = ϕ βωρ f ⁢ 〈 ( δ ⁢ ⁢ v ) 2 〉 v m the effect of flow is to introduce an additional relaxation mechanism . in the anticipation of considering the effect of oscillatory flow , we consider the phase change for one nuclear spin undergoing a displacement in the time interval from t = 0 to t = 2τ : φ ⁡ ( 2 ⁢ τ ) = - γ ⁢ ⁢ g ⁡ [ ∫ o τ ⁢ x ⁡ ( t ) ⁢ ⁢ ⅆ t - ∫ τ 2 ⁢ τ ⁢ x ⁡ ( t ) ⁢ ⁢ ⅆ t ] ( 15 ) x ⁡ ( t ) = ∫ o t ⁢ v ⁡ ( t ′ ) ⁢ ⁢ ⅆ t ′ . ( 16 ) here , the displacement and velocity of the nuclear spin corresponds to those components along the direction of the applied magnetic field gradient . the nuclear spin velocity is a superposition of medium velocity v m plus a relative velocity v r of the fluid motion relative to the matrix : the corresponding phase shift of the transverse nmr magnetization can then be divided into two terms : m ⁡ ( 2 ⁢ τ ) m ⁡ ( 0 ) = 〈 exp ⁡ [ - ⅈφ ⁡ ( 2 ⁢ τ ) ] 〉 = 〈 cos ⁡ [ φ ⁡ ( 2 ⁢ τ ) ] 〉 + ⅈ ⁢ 〈 sin ⁡ [ φ ⁡ ( 2 ⁢ τ ) ] 〉 ( 19 ) where the average is taken over all spins . assuming that the spin phase change is much less than 1 , eq . 19 can be expanded in terms of φ . keeping terms up to second order , we have : m ⁡ ( 2 ⁢ τ ) m ⁡ ( 0 ) ≅ 1 - 1 2 ⁢ 〈 φ 2 ⁡ ( 2 ⁢ τ ) 〉 + i ⁢ 〈 φ ⁡ ( 2 ⁢ τ ) 〉 . ( 20 ) we are interested in the magnitude of the nmr response which is given by :  m ⁡ ( 2 ⁢ τ ) m ⁡ ( 0 )  ≅ ⁢ { 1 - 1 2 ⁢ 〈 φ 2 ⁡ ( 2 ⁢ τ ) 〉 } 2 + 〈 φ ⁡ ( 2 ⁢ τ ) 〉 2 ≈ ⁢ 1 - 1 2 ⁢ { 〈 φ 2 ⁡ ( 2 ⁢ τ ) 〉 - 〈 φ ⁡ ( 2 ⁢ τ ) 〉 2 } = ⁢ 1 - 1 2 ⁢ { 〈 φ r 2 ⁡ ( 2 ⁢ τ ) 〉 - 〈 φ r ⁡ ( 2 ⁢ τ ) 〉 2 } ( 21 ) the second step in eq . 21 indicates that the magnitude of normalized nmr signal is determined by the spin phase fluctuation . the motion of the medium induces a uniform spin velocity and therefore results in a uniform phase change that does not contribute to the relaxation of nmr signal . therefore , only the relative flow motion contributes in the third step of the eq . 21 . the velocity of a spin subject to a pressure gradient at a single frequency can be described as : φ ⁡ ( 2 ⁢ τ ) = γ ⁢ ⁢ g ω 2 ⁢ f ⁡ ( ω , τ , ϕ 0 ) ⁢ v ⁡ ( ω ) , ( 23 ) because the spin phase shift is proportional to the spin velocity , the differential nmr response to flow , { m ( 0 )− m ( 2t )}/ m ( 0 ), is proportional to the fluctuation of spin velocity , & lt ; δv 2 & gt ;=& lt ; v 2 −& lt ; v & gt ; 2 & gt ;, that is , 1 -  m ⁡ ( 2 ⁢ τ ) m ⁡ ( 0 )  = [ γ ⁢ ⁢ g ⁢ ⁢ f ⁡ ( ω , τ , ϕ 0 ) ] 2 2 ⁢ ω 4 ⁢ 〈 δ ⁢ ⁢ v 2 〉 pore . ( 26 ) the contribution of the relative fluid motion to the nmr relaxation is then given by : 1 t 2 ⁢ a = 1 4 ⁢ ω 4 ⁢ τ ⁢ γ 2 ⁢ g 2 ⁢ f 2 ⁡ ( ω , τ , ϕ o ) ⁢ 〈 δ ⁢ ⁢ v 2 〉 pore . ( 27 ) in the low frequency limit , the nmr relaxation due to flow is given by 1 t 2 ⁢ a = 1 4 ⁢ γ 2 ⁢ g 2 ⁢ τ 3 ⁡ ( cos ⁢ ⁢ ϕ o ) 2 ⁢ 〈 δ ⁢ ⁢ v 2 〉 pore . ( 28 ) relation of pore level fluid flow velocity measured by nmr to pressure gradient the flow velocity appearing in darcy &# 39 ; s law is the mean velocity of the fluid , υ b , which is an average of the flow over the whole volume while the nmr signal is a response only to the velocity of the fluid in the pore . the relationship between the bulk velocity and the pore level velocity has been determined from numerical pore network simulations . the pore network is defined from the digital images of real rocks where the images are generated using x - ray micro - tomography . typical digital images are volumes of 300 × 300 × 300 pixels where the resolution per pixel is between 5 to 7 microns . a pressure gradient is applied across this cube to calculate the average velocity for flow within a pore . the results show that the flow velocity varies by orders of magnitude from pore to pore . velocity distributions calculated for a number of sandstone rocks with permeabilities spanning a range of over three orders of magnitude are shown fig2 . in order to calculate the permeability , the mean flow velocity fluctuation , & lt ; δv 2 & gt ; pore , determined from nmr must be related to the velocity , & lt ; v & gt ; bulk , averaged over the whole medium volume . the porosity , φ , relates the flow velocity average in bulk to the average over the pore volume : & lt ; v & gt ; bulk = φ & lt ; v & gt ; pore . the pore network simulations were used to calculate these velocities for a variety of sandstones that had a range of permeabilities spanning over three orders of magnitude . the normalized velocity fluctuation , defined by the ratio of [& lt ; δv 2 & gt ; pore ] 1 / 2 to & lt ; v & gt ; pore , as a function of permeability for several sandstones is shown in fig3 . the ratio is of order unity and is only weakly dependent on the permeability . darcy &# 39 ; s law can then be written in terms of the average pore velocity fluctuation as : 〈 δ ⁢ ⁢ v 2 〉 pore = β ⁡ ( k ) ϕ ⁢ 〈 v 〉 = β ⁡ ( k ) ϕ ⁢ k η ⁢  ∇ p  ( 29 ) where β ( k ) is a function of permeability and varies between 1 and 2 . this relation provides the recipe for determining the permeability from the effect of the flow motion on the nmr relaxation rate . in the following , we assume that the formation fluid and borehole fluid are in hydraulic contact . under such squirt boundary conditions , the formation fluid pressure at the surface of borehole wall is the same as the borehole pressure p o . the pressure for an applied compressional wave is attenuated with the increasing offset away from the borehole wall due to geometrical spreading and to intrinsic attenuation mechanisms in the formation . for small offset , the geometrical spreading factor can be ignored so that the amplitude of fluid pressure in the formation is given : p ( x , t )≈ p o cos ( ω t + x / λ )· exp {− x / λ } ( 30 ) where k / η is the fluid mobility ( i . e ., the ratio of the permeability to the viscosity ), m is the effective fluid modulus , and λ is the pressure penetration length , which is defined using biot theory 4 by : 1 / λ = ω ⁢ ⁢ η 2 ⁢ mk . ( 31 ) the local pressure gradient can be calculated from the derivative of eq . 30 : ⅆ p ⁡ ( x ) ⅆ x ≈ - p o λ ⁡ [ sin ⁡ ( ω ⁢ ⁢ t + x / λ ) + cos ⁡ ( ω ⁢ ⁢ t + x / λ ) ] ⁢ exp ⁢ { - x / λ } , ( 32 ) which induces two oscillatory modes : one with initial phase φ 0 ( 1 ) = x / λ − π / 2 and another with initial phase φ 0 ( 2 ) = x / λ . the acoustic nmr relaxation rate under the squirt boundary conditions can then be calculated using eq . 28 , eq . 29 and eq . 32 : 1 t 2 ⁢ a ⁡ ( x ) = 1 4 ⁢ γ 2 ⁢ g 2 ⁡ [ 1 + sin ⁡ ( 2 ⁢ x / λ ) ] ⁢ τ 3 ⁡ ( β ⁡ ( k ) ϕ ⁢ k η ⁢ p o λ ) 2 ⁢ exp ⁡ ( - 2 ⁢ x / λ ) . 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